experimental physics: methods and apparatus

EXPERIMENT AL PHYSICS: METHODS AND APP ARA TUS

METODIKA FIZICHESKOGO EKSPERIMENTA

METOAHKA WH3HQECKOrO 8KcnEPHMEHTA

The Lebedev· Physics Institute Series Editor: Academician D. V. Skobel'tsyn

Director, P. N. Lebedev Physics Institute, Academy of Sciences of the USSR

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Proceedings (Trudy) of the P. N. Lebedev Physics Institute

Volume 40

EXPERIMENT AL PHYSICS ~

Methods and Apparatus

Edited by Academician D. V. Skobel'tsyn

Director, P. N. Lebedev Physics Institute A cademy of Sciences of the USSR, Moscow

Translated from Russian

CONSULTANTS BUREAU NEW YORK

1969

The Russian text was published by Nauka Press in Moscow in 1968 for the Academy of Sciences of the USSR as Volume 40 of the Proceedings (Trudy)

of the P. N. Lebedev Physics Institute

MeTO~llKa «{lu3H'IeCKOrO 8KCDepnMeHTa

TPYALI op~eHa JIeHHHa CDH3nQeCKOrO HHcTHTYTa HMeHH II. H. JIe6eAeBa TOM 40

ISBN 978-1-4684-0675-7 ISBN 978-1-4684-0673-3 (eBook) DOI 10.1007/978-1-4684-0673-3

Library of Congress Catalog Card Number 69-12522

© 1969 Consultants Bureau, New York A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y. 10011

United Kingdom edition published by Consultants Bureau, London A Division of Plenum Publishing Company, Ltd.

Donington House, 30 Norfolk Street, London W.e. 2, England

All rights resen'ed

No part of this publication may be reproduced in any form without written permission from the publisher

CONTENTS

Apparatus for Recording Neutral Particles by Reference to Decay Gamma Quanta . Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya

Magnetic Spectrometer for Charged Particles ...................... . V. N. Maikov, V. A. Murashova, T. 1. Syreishchikova, Yu. Ya. Tel'nov, and M. N. Yakimenko

Experimental Method of Determining the Efficiency Function of an Apparatus Containing a Magnetic Spectrometer ........................ .

V. F. Grushin and E. M. Leikin

Positive Pion Stopping Detector .................... . Yu. M. Aleksandrov, V. F. Grushin, and E. M. Leikin

Absolute Sensitivity of a Thick-Walled Graphite Ionization Chamber for I-GeV Photons ............................................ .

1. N. Usova

Statistics of Time Measurements Made by the Scintillation Method. V. V. Yakushin

A Cathode Stage with Amplified Feedback and Its Applications V. V. Yakushin

Wilson Chamber for Studying Photomeson Processes .......... . V. P. Andreev, Yu. S. Ivanov, R. N. Makarov, V. T. Zhukov, V. E. Okhotin, and I. N. Usova

Relative Monitor for a Wilson Chamber. V. p. Andreev, T. 1. Kovaleva, and I. N. Usova

1

32

57

70

75

84

137

164

185

APPARATUS FOR RECORDING NEUTRAL PARTICLES

BY REFERENCE TO DECAY GAMMA QUANTA

Yu. A. Aleksandrov, A. V. Kutsenko,

V. N. Maikov, and V. V. Pavlovskaya

An apparatus for the recording of correlated 'YY coincidences, designed in the high-energy electron laboratory of the Lebedev Physics Institute of the Academy of Sciences (FIAN), records and measures the energy of two 100 to 650 MeV 'Y quanta simultaneously incident on its detectors; this apparatus may thus be employed in order to study any processes in which such 'Y quanta are formed.

The detectors are formed by Cerenkov total-absorption spectrometers with an energy resolution of between ±19 and ±9.59'c over the range indicated. The energy of each of the correlated'Y quanta is measured in the presence of coincidences with a resolving time of ""5'10-9

sec.

The resultant energy values are recorded in the form of a numerical code in an intermediate memory based on ferrite cores. The form of the recording is such that the appearance of a correlated event is strictly related to the energy of each of the 'Y quanta. The resultant information is then printed and converted to perforated cards suitable for subsequent analysis of the results on an M-20 computer.

CHAPTER I

RANGE OF NUCLEAR REACTIONS STUDIED AND PRINCIPAL REQUIREMENTS OF THE APPARATUS

When studying the interaction of elementary particles it is frequently essential to record short-lived neutral mesons decaying into two 'Y quanta (1T o meson and 1'/ meson). These particles can only be recorded by reference to their decay products, i.e., by detecting one or two 'Y quanta.

It may be easily shown that the simultaneous detection of two decay 'Y quanta has a number of advantages over other methods, as it offers the possibility of studying a large number of processes under conditions in which other methods are practically useless. By way of example, let us consider the advantages of the method in question for studying the photogeneration of 1To mesons in hydrogen.

1

2 Yu. A. ALEKSANDROV ET AL.

For recording the process

r + p -'>-1tO + P (1)

it is sufficient to determine any two parameters, for example, the escape angle and the energy of the recoil proton. It was in this way that the first precision information regarding the angular and energy distributions of 'lr 0 mesons in this reaction was obtained. However, the regions of very small and very large meson escape angles e:r s 30° and e;. 2: 150° in the system of the center of inertia (CIS) have hardly been studied at all. This is because the corresponding recoil protons in the laboratory system of coordinates (LS) escape at angles close to 0 and 180°. In addition to this, the case e; s 30° corresponds to very low-energy protons (Ep ::s 10 MeV for Ey i:::i 500 MeV), and it is almost impossible to record these. Yet it is the photogeneration of 'lr 0 mesons at large and small angles which is of the greatest interest, since the more complex intermediate states in the meson-proton system should make their greatest contribution in this case.

The recording of a 'lr 0 meson by reference to its decay I' quanta offers the possibility of carrying out investigations at any 'lr°-meson escape angles. The kinematics of the decay of a 'lr 0 meson are such that the most likely event is that two I' quanta flying off symmetrically with respect to the 'lr 0 meson escape direction will be recorded. The angle between the quanta in the LS will be close to the minimum (critical) angle O!cr at which I' quanta may be emitted:

C1 m c2 • cr ~

sm 2 =--e-. ~

(2)

where Err is the total energy of the 'lr 0 meson and mrr its mass.

The energy range Err = 300-650 MeV corresponds to critical angles O!cr = 53.5-24°, i.e., I' quanta arising from the decay of 'lr 0 mesons generated at 0° in the LS (which corresponds to err ° = 0°) will make fairly large angles with the axis of the beam of primary I' quanta, and will thus be easy to record.

Thus, the simultaneous detection of two I' quanta arising from the decay of 'lr 0 mesons enables us to study the photogeneration of 'Ir ° mesons in an angular range very close to 0 and 180°, for which other methods prove extremely difficult.

A second very important advantage of the method under consideration is the possibility, in principle, of distinguishing the desired process from a background of other competing processes ultimately leading to the appearance of a particle analogous or similar in properties to the one being detected. For example, on studying reaction (1) at energies of E y > 322 MeV, the single generation of ~ mesons is accompanied by the paired generation of charged and neutral mesons, which also leads to the appearance of recoil protons. The separation of the recoil protons in reaction (1) from this background is quite a difficult problem, especially if one considers that reaction (1) is studied in bremsstrahlung beams having a continuous 'Y-quantum energy spectrum. The corrections introduced into a number of experiments of this type are very indefinite, and this reduces the accuracy of the results obtained.

When using the method based on recording two correlated I' quanta for studying process (1), a background process leading to the appearance of 'lr 0 mesons in the final state is, for example, the paired photoexcitation of 'lr 0 mesons (for E y > 322 MeV):

(3)

However, the maximum energies of the 'Ir ° mesons so formed will be smaller than the energies of the 'lr 0 mesons generated in reaction (1) at the same angle, and hence the critical angles between the decay I' quanta will also differ. For example, for Ey = 500 MeV, the critical angles

APPARATUS FOR RECORDING NEUTRAL PARTICLES 3

for 'ITo mesons generated in reactions (1) and (3), respectively, equal

By using this fact it is in practice quite easy to set up a geometry such that the recording of any of the 'ITo mesons generated in reaction (3) is in principle impossible (the angular aperture between the decay 'Y-quantum detectors should not exceed 45°).

Nevertheless, even after ensuring the fulfillment of this condition, reaction (3) may still be recorded on account of the incidence of'Y quantum from the decay of different 'ITo mesons in the two detectors. Cases (1) and (3) are then distinguished by means of 'Y-quantum spectrometry. The hardest to distinguish are the improbable cases in which photogeneration of the two 'ITo mesons occurs in the direction of the 'Y detectors, and this is followed by asymmetric decay. Quantitative consideration of the kinematics of these processes shows that they may be clearly distinguished by using 'Y spectrometers with good energy resolution as detectors and by varying the upper limit of the bremsstrahlung spectrum. In describing our experimental results in the following sections, we shall show that the recording of TJ mesons by this method has advantages analogous to those considered in the case of 7r ° mesons.

CHAPTER II

A TOTAL-ABSORPTION CERENKOV GAMMA SPECTROMETER

§ 1. Characteristics of the Existing Form

of the Spectrometer

The total-absorption Cerenkov spectrometer is intended for recording and measuring the energy of 80-650 Me V 'Y-quanta. The use of a spectrometer in experiments involving the recording of 'Y quanta arising from the decay of neutral particles ('IT 0, TJ) also presupposes its inclusion in fast-acting coincidence circuits (with a resolving time of the order of a few nanoseconds) with other analogous spectrometers or other particle counters. All this necessitates ensuring a high resolving time of the system as a whole, as well as good spectrometric characteristics.

In the present form of the spectrometer [I, 21, good spectrometric and time characteristics were achieved by simultaneously employing two types of photomultiplier ("spectrometer" and "time" types) and thus forming two independent channels for the electrical pulses arising from the same source, namely, the Cerenkov radiation in the radiator. In addition to this, the separation of the "spectrometric" and "time" functions of the apparatus is very convenient under the conditions of a physical experiment involving the recording of "fast" coincidences and, a "slow" amplitude analysis of the events.

The "spectrometric" channel used one Soviet-made spectrometric photomultiplier (FEU-49). The use of one multiplier instead of the usual several eliminated the necessity of having special electronic circuits for summing the pulses and obviated problems associated with choosing photomultipliers of the same sensitivity, as well as simplifying the construction and adjustment of the apparatus and increasing the reliability of its operation. The possibility of using a single photomultiplier arises from the good intrinsic resolution of "'6% and the comparatively large area of the photocathode (diameter 150 mm).

The "time" channel uses FEU-36 photomultipliers with a maximum spread of "'2 nsec in the time of flight of the photoelectrons and a high amplification factor. The number of multip-


Hers was chosen in such a way as to give 100% radiation-recording efficiency in the working 'Yquantum energy range.

Experiment has shown that the present version of the Cerenkov spectrometer (total-absorption type) has the best spectrometric characteristics of all those found in the literature (the energy resolution varies from ±19 to ±10% in the range 100-600 MeV) and fairly good time characteristics (resolving time 4-5 • 10- 9 sec). An extremely important property of the apparatus is, moreover, the separation of the recording function from the spectrometric analysis of the radiation.

§ 2. Spectrometric Characteristics

R a d i at 0 r • The radiator employed was TF-1 lead glass containing 53 % of lead oxide PbO. The radiator was made in the form of a truncated cone 240 mm high (10.1 t units) and diameters 260 and 300 mm (10.9 and 12.6 t units, respectively).

In order to improve the conditions for collecting the Cerenkov light, the side and ends of the radiator were carefully polished and surrounded by a reflector of polished aluminum. In contrast to a cylinder, a radiator in the form of a truncated cone should give total internal reflection of the greater part of the light emitted. For a cylindrical radiator, in fact, the condition of total internal reflection (sinO! = n-1, where O! is the angle made with the normal to the reflecting surface) can only be satisfied over the whole lateral surface for relativistic particles ({3 = 1) passing along the axis of the cylinder, since the angle of Cerenkov emission is determined by the relation cos e = (n{3)-1. In the case of a conical radiator, however, the dimensions quoted ensure the maintenance of this condition for particles passing at angles of ±5° to the axis of the radiator.

Spectrometric Photomultiplier [3]. Tests were carried out on several samples of the FEU-49 with an NaI-TI crystal (diameter 30 mm, h = 20 mm) and a collimated Cs137 source. The pulses from the photomultiplier were amplified and analyzed in a 100-channel AI-100 amplitude analyzer. The Uniformity of the sensitivity of the photocathode was verified roughly by means of a crystal placed in turn at the center of the photocathode and on the periphery. The FEU-49 multipliers tested had a fairly uniform sensitivity of the photocathode, the amplitude difference between the two points never exceeding 10-15%. The time stability of the FEU-49 characteristics was specially checked. Over a period of 8-h operation the change in amplitude and resolution never exceeded 0.5% after a 20-min heating.

The FEU-49 photomultiplier was very insensitive to weak magnetic fields. Thus, the earth's magnetic field, which usually affects spectrometer photomultipliers, had little effect on the resolution of the FEU-49 for different orientations of the multiplier to the field. The amplitude of the pulses from the photomultiplier in the horizontal and vertical positions differed by 2.5%. Screening the multiplier with a Permalloy screen annealed at 800 0 at atmospheric pressure completely eliminated this difference.

Optical Contact of the Photomultiplier with the Radiator. Inorder to transfer the light from the radiator to the photocathode of the photomultiplier without loss of intensity, a lubricant with a good transmission in the range of spectral sensitivity of the photomultiplier and a refractive index close to that of the radiator and the entrance window of the photocathode must be used" between them. These requirements are excellently satisfied by the usually employed liquid and viscous intermediate media, which have good optical properties, but require constant fairly reliable mechanical pressure on the multiplier. The use of insoluble adhesives eliminates the necessity of mechanical fixing, but makes the apparatus more difficult to dismantle.


We tested several types of optical contacts with corresponding fixing systems. Experiment showed the undesirability of working with liquids and viscous substances, particularly in the case of prolonged operation. Despite fairly elastic fixing of the multipliers, the optical contact deteriorated with time.

In the final version of the Cerenkov spectrometer we used a grease best satisfying the conditions both of light collection and of long-term stability of the optical contact. This was a water-soluble adhesive [4] based on the epoxy resin glycerin diethyleneglycerol (DEG-1 type) and hardener ethylene diamine in a weight proportion of approximately 10: 1.

In order to prepare the adhesive, the resin and hardener were mixed. The mixture retained the consistency of a viscous liquid for 40-60 min. The joint between the parts to be cemented together was effected in the same way as in the case of ordinary optical greases. The adhesive hardened fully at room temperature after 20 h (approximately). This kind of adhesive ensures good light conduction (the same as vaseline oil), a steady, reliable optical contact, and at the same time good mechanical fixing of the photomultiplier. If necessary, the cemented parts may be separated by dissolving the adhesive in water without harming their efficiency. The time required for this operation diminishes with increasing water temperature and washing intensity, and increases with increasing area of the cemented surfaces and diminishing thickness of the cementing layer. Thus, an FEU-36, 40-mm in diameter, may be dismantled by immersing the join in water at room temperature for 6-7 h, while for an FEU-49 with a diameter of 150 mm this requires several days.

Calibration and Characteristics of the Apparatus. The spectrometric properties of the Cerenkov spectrometers were studied and calibration was carried out in a monochromatic electron beam obtained by the deflection and magnetic focusing of electrons formed in a thin target bombarded by synchrotron 'Y quanta [5]. Smooth variation of the magnetic field of the f3 spectrometer enabled electron beams of preassigned energy (80-650 MeV) to be obtained in this way. The dispersion of the beam in the collimated window of a spectrometer 80 mm in diameter and of a given geometry was (according to a preliminary survey) ±1.5%, which was much less than the expected energy resolution of the spectrometer.

The intensity of the electron beam was "'103 electrons per pulse in the spectrometer window. For calibration purposes the pulse of 'Y radiation, and hence the duration of the electron beam, were "drawn out" to 0.5 sec (working frequency of the accelerator one pulse in 6 sec).

Immediately in front of the entrance window of the spectrometer, a telescope comprising two scintillation counters and serving as monitor was placed. The conditions of calibration are indicated in Fig. 20 of Chapter VI. The calibration consisted mainly of determining the resolving power and the amplitude of the output pulse as functions of the electron energy Ee. The spectrometer was calibrated for electron energies between 100 and 600 MeV. The results of a study of the spectrometric characteristics of the apparatus are presented in Figs. 1 and 2.

We see from Fig. 1 that the amplitude of the output pulse is a linear function of the energy up to "'600 MeV. The linearity of the apparatus over a wide frequency range indicates that the radiator dimensions selected ensure the absorption of nearly all the energy of the electronphoton shower.

However, in view of the finite size of the radiator, it would be natural to expect a slight deviation from linearity on raising the energy, owing to the different conditions governing the absorption of the last generations of the cascade shower in the course of its development. The observed linearity of the apparatus is evidently due not so much to the strict constancy of the effects of total absorption and proportional light collection as to the fact that any deviations from these tend to compensate each other [6]. For slight deviations there is clearly a mutual


A, reI. units 100

90

100 zoo 300

•

/f00 500 Ee. MeV

Fig. 1. Output pulse amplitude as a function of the energy of the electrons recorded. 1) Spectrometer No.1; 2) spectrometer No.2.

~B"r. ____________________________ -.

30

zoo 300 1100 500 /l00 Ee, MeV

Fig. 2. Energy resolution of the spectrometer as a function of the electron energy. Notation as in Fig. 1.

compensation of these effects. The energy resolution of the spectrometer 6 sp tis shown as a function of the energy of the recorded electrons Ee in Fig. 2.

The transformation of the energy of the primary particle into an electrical pulse at the output of the multiplier is a complicated process, comprising a number of successive stages, all subject to fluctuations: the generation of the shower, the emission of Cerenkovradiation quanta by the shower particles, the collection of light at the photomultiplier photocathode, the conversion of the photons into photoelectrons, the collection of the electrons in the multiplying system of the photomultiplier, and, finally, the multiplication of the electrons in the multiplier and the formation of the electrical pulse. The parts played by these processes and their relative contribution to the energy resolution of the shower spectrometer are considered in detail in [6]. Each quantity is regarded as random, and the transformation of one into another as a statistical process. Assuming that the distributions of "secondary" particles of various generations created by different "primary" particles are identical, the theory of branching processes is applied. On this basis, the relative mean-square fluctuation 1]~p of the shower spectrometer and the energy resolution of the apparatus were calculated in terms of the corresponding characteristics of the individual stages.

The continuous curve in Fig. 2 was obtained in [6] by means of the foregoing calculations for a spectrometer with a radiator of TF-l glass (thickness 10 radiation units, lightcollecting area "'0.3 of the area of the end of the radiator) and FEU-49 (FEU-24) multipliers with an intrinsic resolution of "'6%. These conditions correspond to the version of Cerenkov spectrometer under consideration. We see from the figure that theory and experiment are in excellent agreement. Figure 3 presents a comparison of the characteristics of the Cerenkov total-absorption'Y spectrometer described with other systems of the same type [7-11].

Despite the fact that in our case the FEU-49 only ensures the collection of light from % of the area of the radiator, the spectrometer has the same energy resolution as spectr9meters~ with seven or eight FEU-24' s, where the area of overlapping is about 50%; it essentially is distinguished by the Simplicity of construction.

t This quantity is the ratio of the half-width (width at half height) of the energy distribution to the position of the maximum (same units).

t Data relating to these spectrometers (up to an energy of 250 MeV) are presented in [6, 7].


0,%

80

50

If 0

30

20

,2 \

\ \ '- 3

\ ,-----------------\

\ \

\ \

\ , olf

05

mL-__ ~ __ _L __ ~ __ ~~ __ ~ __ ~ __ ~

o 100 ZOO 300 '100 500 800 Ee, MeV

Fig.3. Comparison of the energy resolution of various Cerenkov spectrometers (totalabsorption type). Continuous line - present work; broken curves as follows: 1) [7]; 2) [8]; 3) [9]; points: 4) [10]; 5) [ll].

§3. Time Characteristics

The "time channel" of the total-absorption Cerenkov spectrometer consists of time photomultipliers in the spectrometer radiator and a cutoff shaping circuit with subsequent connection to a "fast" coincidence circuit. Photomultipliers of the FEU-36 type were used in the spectrometer. In view of the small area of the photocathode (diameter 40 mm), a low intensity of collected light was expected.

In order to determine the efficiency of radiation recording by one FEU-36, approximate estimates were made of the mean number of photoelectrons passing into the accelerating system of the FEU-36 (u36)'

According to [6], the mean-square fluctuations n2sp of the pulse amplitudes of a shower spectrometer is

",2 1 + ",2 2 _ 2 + "2 + "3

1'JSp -1'Jl N ~ (4)

and, correspondingly, the energy resolution of the apparatus is

(5)

Here, N is the mean number of charged shower particles due to an electron of energy Ee; Ii is the mean number of photoelectrons falling into the accelerating system of the multiplier; 01 =

5.54111 characterizes the fluctuations in the number of shower particles, allowing for the escape of some of the particles from the radiator; o~ = 5.54 77 ~/N is the spread introduced by the light-collecting process, the quantum yield of the photocathode, and the collection of electrons at the latter; 0 § = 5.54 (1 + 115)/ Ii are the fluctuations associated with the multiplication of the photoelectrons.

From (4) and (5) we obtain the following for the FEU-49:

_ 5.54 (1 + 1']~) n49 = -:r=:==~

V 6;p- 6~2 (6)

Here, 1 + r,§ = (u + 1)/u; u is the mean secondary-emission coefficient.

Assuming for simplicity a uniform distribution of the Cerenkov light at the end of the radiator and neglecting the difference in the spectral sensitivity of the photocathodes of the FEU-49 and FEU-36, we obtain for the FEU-36

(7)

Substituting from [6] the experimental value of the energy resolution osp (see Fig. 2) and the theoretical value of 612 = {01 + O~ into (6), we obtain (for the lower energy of the working range of the spectrometer) u36 ~ 2-4 (for u = 2- 00). This estimate indicates the possibility in principle of recording Cerenkov radiation scintillations produced by an electron (or 'Y quantum) with E e ::: 100 MeV in a single FEU-36. In addition to this, we see that it is essential to select the FEU-36 very carefully, with the greatest possible quantum yield of the photocathode and the maximum amplification factor.


100

50 -

O~~-L~--~~~~~

-10 -8 -$ -If 0 If G 8 IU r, nsec

Fig. 4. Curve of delayed coincidences of the fast channel of the spectrometer in the case of four FEU-36's connected for coincidence in pairs.

Table 1

Source IT. nsec\ Ee. MeV

[12] 7 125 [13] 20 600-300 [14] 30-40 3000-500

Present work 4-5 500-100

Preliminary inspection was carried out and the optimum supply conditions for the photomultipliers were chosen using a plastic scintillator and a C060

source, and the final selection was made using semi-conductor low-intensity light emitters, the intensity corresponding to the removal of a few electrons from the photocathode of the multiplier. Tests in an electron beam showed that FEU-36's chosen in this way were able to give pulses sufficient to trigger the cutoff shaping univibrators from a single photoelectron, without additional amplification.

The time characteristics of a spectrometer incorporating an FEU-36 were studied and the number of multipliers required to give 100% efficiency of radiation recording was determined more reliably in an electron beam, Le., under the conditions in which the spectrometer characteristics were determined.

The electron beam was monitored with a telescope comprising two thin scintillation counters placed in front of the spectrometer and connected to a double-coincidence circuit (the coincidence count of these counters is called N M). Pulses from the FEU-36 fell on a shaping univibrator with a sensitivity threshold of "'0.1 V. The shaped pulses were passed through a long cable to the input of a coincidence circuit also connected to one of the monitor electron counters (the count of these coincidences is called N).

The ratio N/N M was studied as a function of the amplification factor of the FEU-36, which was

varied by changing the voltage U supplying the photomultiplier. The measurements showed that the probability that a system of this kind would recofd an electron with Ee = 100 MeV was 86% in the case of a single FEU-36. According to Poisson's law, the "inefficiency" of recording, (1- P) = 0.14, corresponds to Ii ~ 2.

Thus, in order to obtain 100% recording efficiency for 100-MeV electrons, the number of FEU-36's must be increased. (A multiplier with a much larger photocathode area could not be employed in our case owing to the small free area of the radiator.) In the case of two FEU-36's the curve relating the coincidence count N/N M to the voltage on the two multipliers had a fairly wide plateau with an efficiency practically reaching 100%.

The recording efficiency of the whole system for 100-MeV electrons was determined by comparing the number of pulses in the "spectrometric" channel on monitoring this channel, either by reference to the monitor telescope, or by reference to coincidences between the FEU-36 and one of the telescope counters. Both the form of the spectrum representing the output pulses from the FEU-49 and the total count obtained from the individual channels under the spectrometric curve, which corresponds to 100% recording efficiency, were the same.

It should be noted that, since ii36 is proportional to E e, 100% recording efficiency will be achieved even with one FEU-36 on raising the energy of the recorded electron above 200 MeV.

APPARA TUS FOR RECORDING NEUTRAL PARTICLES 9

N/NH ,%

':~ o : 0 ~ I I I I 700 100 ZOO 300 'f00 500 !l00 Ee. M eV

Fig. 5. Recording efficiency of the fast channel of the spectrometer as a function of the energy of the electrons recorded.

Fig. 6. External form of the Cerenkov spectrometer without its outer cover. FEU-36

The relation between the recording efficiency of the system and the electrical resolving time of the coincidence circuit was also measured.

For electron energies Ee = 100 and 500 MeV, 100% recording efficiency is reached for coincidence-circuit resolving times of T = 4.7 • 10-9 sec and 4.0 . 10- 9 sec, respectively. For the majority of physical problems in which total-absorption spectrometers are used, such time characteristics are entirely satisfactory. Table 1 shows the resolving times of the coincidence counters used in [12-14], which employed total-absorption Cerenkov spectrometers.

In the final form of the total-absorption Cerenkov spectrometer, the "time channel" was formed by two pairs of FEU-36's set for coincidence. This was due to the necessity of reducing the loading of the "time channel" by photomultiplier noise pulses. Figure 4 shows a typical curve representing coincidences between the fast channel of the spectrometer (with four FEU-36's) and the monitor telescope focused on scintillation counters, obtained in an electron beam (Ee = 100 MeV). The recording efficiency of the fast channel is shown as a function of electron energy in Fig.5.

§4. Mechanical Construction

of the Spectrometer

Figures 6 and 7 show the external form and construction of the total-absorption Cerenkov'Y spectrometer. The radiator with its

screens removed. aluminum screen is fixed in a steel cylinder. A single FEU-49 is attached to the end of the radiator (in the center) with epoxy adhesive,

and four FEU-36's are placed around the periphery. No additional mechanical fixing is used for the photomultipliers. The FEU-49 is surrounded with a Permalloy screen and the FEU-36's are covered with black paper screens opaque to light and surrounded by metal screens composed of copper foil. The multipliers are fed and the signals taken off through the end cover. A preamplifier for the signal from the FEU-49 and shaping stages for the signals from the FEU-36's are also placed on this cover. There is an aperture, revealing the glass, in the steel cylinder near the exit end of the radiator. Two semiconductor light sources for regulating the system are also sited there. In addition to these, a radioactive Cs137 source of radiation and an NaI-TI scintillation crystal (attached to the glass with thick grease at the same end window) may be used to check the spectrometer tract.

After assembly, the body of the spectrometer is slid into a steel tube simultaneously serving as an opaque sheath and magnetic screen. At the front end of the tube are a lead collimator 80 mm in diameter and an electron counter set on the optical axis of the spectrometer. The apparatus includes two identical Cerenkov spectrometers of the kind described.


Fig. 7. Construction of the Cerenkov spectrometer. 1) TF-l radiator; 2) Al reflector; 3) FEU-49; 4) FEU-36's; 5) magnetic screen (Permalloy); 6) pulsed light source; 7) window for placing the NaI-TI crystal and Cs 137 source; 8) light and magnetic screen; 9) lead collimator; 10) electron counter; 11) FEU-36; 12) scintillator; 13 output stage from FEU-36; 14) preamplifier associated with FEU-49; 15-23) voltagesupply and signal plugs.

§ 5. Electron Counters

The electron noise is eliminated when the spectrometers are recording ')I quanta, and the ')I-quantum noise is eliminated when the spectrometers are recording electrons, by means of electron counters. These are also used for monitoring the electron beam when calibrating the spectrometers.

We used ordinary scintillation counters consisting of a plastic scintillator of P-terphenyl in polystyrene with ROROR additive and an FEU-36 photomultiplier. The plastic (cut in the shape of a disc 90 mm in diameter and 15 mm thick) was attached by its cut end to the photomultiplier, using epoxy adhesive, and surrounded with aluminum foil. Relativistic electrons retained an energy of over 4 MeV in the counter, so that when the photomultiplier operated fairly large-amplitude pulses passed. directly into a long, matched cable. Since these counters were attached to coincidence circuits recording coincidences with the spectrometers or with each other, pulses arising from them passed to the input of the shaping univibrators, which had a threshold of ...... 0.2 V. The amplification factor of the FEU-36's employed had therefore to be greatly reduced. For the working supply voltages chosen the noise from the photomultipliers reached 5-10 pulses/ sec.


/ /

Fig. 8. System for setting the spectrometers on the target. 1) Rotatory support frames; 2) vertical framework (girder); 3) circular rails; 4) platform for spectrometer; 5) spectrometer; 6) target.

§ 6. Construction for Setting the

Spectrometers on the Target

The arrangement for fixing the spectrometers and setting them on the target is shown in Fig. 8. The system is axially symmetric with respect to the target, which may be withdrawn from the beam and reinserted during the experiment by means of a remote-control drive.

The Cerenkov spectrometer together with the electron counter and lead collimator is fixed to a movable platform placed on a vertical girder. The latter, in turn, is capable of radial and azimuthal movement relative to the target. The system is designed for operation with four spectrometers or other detectors.

The following movements may be made in order to adjust the spectrometers and set them on the target:

1) Azimuthal motion of the spectrometer through an angle of o!o = 0-180° by rotating the frames around the central axis of the system;

2) radial motion of the spectrometer by moving the vertical girder within the frames;

3) vertical motion of the spectrometer by moving the platform on the vertical girder;

4) tilting the spectrometer by rotating the platform in a vertical plane through an angle {3o (f3 = 0, vertical axis);

5) rotatory adjustment of the spectrometer in the plane of the platform relative to the entrance aperture of the collimator.

All the movements of the detector in the system may be read on scales as distances or angles, respectively.

In general, the detector has to be set with a given direction e to the axis of the primary beam at a distance R from the center of the apparatus. This may be done by successively setting the azimuthal angle O! 0 and the vertical angle {30, since these are related to e by the threedimensional angular relationship

cos e = cos IXo cos ~o.

The setting of the geometry for recording a neutral particle at an angle of e by reference to two decay 'Y quanta with an angle of O!cr between them, in particular, may be reduced to this problem. It may easily be shown that, if the particle escapes at an angle of e in a vertical plane passing through the axis of the primary beam and decays into two 'Y quanta symmetrically with respect to its own direction of motion, the setting angles are determined from the following relations:


et cr tanT

tan "'0 = cos e ;

et cos ~o = cos ~r sin e.

(8)

(9)

The setting of the spectrometers in accordance with specified setting angles a 0, (30, and a specific distance R may be effected in the following ways: 1) by means of the angular and metric scales; 2) with the help of a theodolite situated at the target position; 3) by extending the end of a movable probe (constituting a geometrical extension of the spectrometer axis) to the center of the target.

CHAPTER III

ELECTRONIC APPARATUS

The form of electronic apparatus developed was only intended to collect the events at the site of the experiment and accumulate information for subsequent analysis in the M-20 computer. This made it possible to reduce the amount of apparatus needed at the site of the experiment, thus easing the experimental procedure, increasing reliability, and at the same time offering wider prospects for the analysis of the resultant data.

We did our best to use easily accessible industrial parts in constructing the main components of the apparatus. The newly developed components were based on semiconductor technology (tranSistorized); this gave excellent characteristics and a high reliability of operation, together with economy in size and power supplies.

§ 1. Block Diagram of the Apparatus

The block diagram of the apparatus is shown in Fig. 9. This includes two spectrometer tracts, starting with an FEU-49 photomultiplier and ending with an amplitude-to-pulse-train converter, a logical circuit determining the operating conditions, an intermediate ferrite memory with a system specifying various operating programs, a system for extracting the information, and circuits for controlling the whole apparatus. The pulses from the output of the FEU-49 of each tract pass through a matching stage [15] along a long cable to a linear amplifier [16]. The pulses to be measured pass from the amplifier output through a line selector (gate) to the amplifier-to-pulse-train converter. In front of the gate is an expander converting the sharp pulses into rectangular pulses 2 p.sec long. This is necessary in order to eliminate the dependence of the conversion factor on the pulse length. Both expander and gate operate if and only if the command pulse "measure" falls on the circuits controlling them. The expander is directly controlled by this pulse and the gate through an "and" circuit with blocking inputs, which transmits a: command pulse in accordance with the selected operating conditions. A delay line in front of the expander and "and" circuit creates the necessary delay of the control signals with respect to the pulses on which they act. The output pulses of the linear amplifier (within the range 1 to 100 V) are converted into trains of 1 to 100 pulses and fall on the corresponding inputs of the number register of the ferrite memory.

Two decades are taken off for each counter in the number register; in these each train is converted into a two-place decimal number expressed in the binary system. In this form the information regarding the event obtained from the two spectrometers is transferred from the

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number register into the ferrite memory at a single address, selected before the arrival of the train. The choice of address is effected by the logical circuit, by adding +1 to the address register, which is zeroed before the information starts being collected. In addition to the foregoing form of operation of the memory, called the accumulator mode or condition [17], the programming arrangement of the system provides for another seven programs: the transfer of the data to a card puncher, printed digital output, operation as a one-dimensional analyzer (in obtaining the characteristics of the spectrometers and calibrating the latter), spectral observations, output to an automatic recorder, output to a numerical register, and verification of the operation of the memory.

Three programs usually take part when collecting statistics in the coincidence mode: the accumulator mode, punched-card output, and printed digital recording. The punched cards constitute the principal documentation; this is subsequently duplicated and analyzed in accordance with various programs, according to the demands arising in the course of the experiment. Digital printed recording gives a clear check on the progress of the experiment and selective control of the information collected. The remaining five programs are effectively used in various measurements when preparing the experiment and also in checking the apparatus.

Depending on the character of the investigation in hand, the whole system may operate in the following modes: 'Y'Y coincidence, measurement of the electron energies being produced; ee coincidence, measurement of the energy of the 'Y quanta being precluded; or 'Ye coincidence with any kind of energy measurement. In addition to this, the apparatus allows us to use each of the spectrometers as an independent system recording 'Y quanta or electrons and passing the results to the memory, operating either in the accumulator condition or as a one-dimensional analyzer. The different operating conditions of the system are selected by the appropriate setting of the tumblers on the logical circuit.

The logical circuit reacts to the outputs of the anticoincidence (A), time (fast) (B), and spectrometric (C) channels, and gives out the corresponding command pulses in accordance with the state of these. In the block diagram of Fig. 9 the circuit is shown as connected for working in the 'Y 'Y-coincidence condition.

The experimental logical circuit operates in the follOwing way. The fast "and" circuit (inputs Bt B2) establishes the fact of a coincidence between the two spectrometers. The circuits with inputs At Bl and A2~ emit a signal if an electron is recorded. This signal is shaped by the blocking circuit and transferred in the form of a positive or negative pulse to the corresponding input of the "and" circuit controlling the line selector. In the first case (measuring the energy of the 'Y quanta) this prevents the energy of the electron from being measured, and in the second case (measuring the energy of the electron) it allows only the electron energy to be measured. All other measurements are prevented by the "and" circuit of the selector.

If we have coincidence between the two spectrometers, the circuit B1B2 generates a pulse giving the command "measure" to the spectrometer tracts. The same signal operates a univibrator with a delay of 120 p,sec, through a triple "and" circuit. As the univibrator operates, the number register of the memory is zeroed and a +1 is sent to the address register in order to select a free location in the memory. While the univibrator delay is taking place, trains from the converters are written into the number register. When the delay period is over, the command "read and record" is sent to the control circuit of the memory, the "free" address is read, and the contents of the number register are recorded in this address.

The triple "and" circuit in front of the OD-120 univibrator does not transmit signals to the memory control if the event is of no interest for the investigation in hand. The identification of such events is effected in the spectrometer tracts by considering the amplitudes of the pulses, which are selected by integral discriminators (D1D2)'


Fig. i O. Circuit of the spectrometric-tract preamplifier.

II/K

'-----"-----.... -------~-m Fig. 11. Circuit of the main amplifier of the spectrometric tract.

The introduction of this arrangement, with thresholds of 7 V for the discriminators Dl and D2, greatly reduced the "expenditure" of addresses in the memory and freed the material coming into the computer from unnecessary information.

15

The univibrator OD-130 at the output of the "and" circuit B1B2 creates a dead time of 130 p,sec in this tract, thus preventing the access of the command while the previous event is being measured and recorded.

The slow "and" circuits in all three tracts of the logical circuit, which are controlled from the accelerator, only open these tracts on receiving pulses of appreciable intensity from the accelerator. For the kind of duty factor characterizing the S-60 accelerator, this reduces the loading from cosmic background by almost an order, which is extremely important when studying effects with a small cross section.

The operation of the apparatus is checked, while the statistics are being gathered together, by means of the auxiliary counters a, b, c, and d. The whole system is checked through by means of semiconductor light-pulse emitters.

§ 2. Spectrometric Tracts

The spectrometric tracts are designed to convert the pulse at the output of the FEU-49 into a pulse train, the number of pulses in which is proportional to the amplitude of the original pulse. Each spectrometric tract includes a preamplifier, a main amplifier, and an amplitudeto-pulse-train converter.


.---~r-------------------T---0·MQ ,....,.----_-4 + 1M

Input (from P1638, of panel 7)

l/lOpF +-1-.!)t+t1---0

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Fig. 12. Pulse-expanding circuit.

11 50 8 8 '10 03 § 30 ~ () 20 o o .-I

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aJ 0.5 0. 7 0.9 1.1 1.3 1.5 T, jJsec

Fig. 13. Operation of the input unit of the AI -100 as a function of the length of the pulse being measured. 1) Ordinary form; 2) with the expander.

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Fig. 14. Circuit for delaying the command pulse ofthe AI-100 blocking control channel.

The preamplifier is built on the "pair" principle with negatlV~ feedback [15J. The general circuit of the preamplifier is shown in Fig. 10. When operating with a long matched RK-150 cable the preamplifier has an amplification factor of K'" 1. The nonlinearity in the amplitude range 1-10 V is no greater than 1%. The pulse growth time is -e0.1 p.sec, the input resistance 90 Kfl, and the input capacity 15-20 pF.

The m a ina m p Ii fie r is built in accordance with the circuit described in [16J, which satisfies the main requirements laid upon amplifiers intended for amplitude measurements [18J. The main features of the circuit (Fig. 11) include considerable negative feedback, galvanic interstage couplings, and special measures taken in order to widen the dynamic range of the output pulses.

The input of the amplifier is calculated for a working range of spectrometer output pulses with amplitudes of 0.5 to 3 V and'Y quantum energies of 100-600 MeV. The amplifier output ensures linearity for amplitudes of 1-100 V, which satisfies the requirements of the amplitude-topulse-train converter of the AI-100 analyzer on which it acts.


The amplifier has the following characteristics. The amplification factor may be taken in ten steps from 8.5-85. The maximum amplitude of the output pulses is over 100 V. The deviation from linearity within these limits of output amplitudes is no greater than 1%. The growth time is <0.1 p.sec. The highest permissible input pulses (i.e., those not causing overloading at maximum amplification) have values of 20-25 V. The output resistance is 70 Q. The spectrometric tract as a whole (preamplifier, amplifier, and AI-100) has a linear amplitude characteristic.

Amp lit u d e - t 0 - P u Is e - T r a inC 0 n v e r t e r . This circuit is the last stage of the measuring tract. Apart from the transformation of the measured pulse into a pulse train, the circuit also controls the measuring tract; being guided by external commands, the tract only opens for specific pulses and at specific moments, when measurement of the previous pulse has ceased. These functions are fulfilled by the input unit of the AI-100 after introducing a number of changes and making some additions to allow for the specific arrangement of the apparatus as a whole and the characteristics of the pulses emitted by the Cerenkov spectrometers.

The converter circuit (block diagram in Fig. 9) consists of two channels: measuring and control. The measuring channel starts with a delay line and includes an expander, a line selector, and the actual amplitude-to-pulse-train converter. The expander [19J (Fig. 12) ensures measurement of pulses between 0.1 and 2 p.sec long. Figure 13 shows the influence of this circuit on the characteristics of the input unit. The introduction of aD. 5 p. sec delay to the measured pulse allows for a slight lateness in the command "measure" applied from outside.

The control channel consists of a shaping circuit, converting all pulses with amplitudes greater than 0.8 V into a standard pulse, two univibrators, one of which controls the expander and the other the selector, and an "and" circuit with two blocking inputs (the blocking inputs in the "and" circuit are shown by circles). '!\he delay to the pulse of the control channel in front of the "and" circuit (external-control circuit of the AI-lOO) is effected in the manner of Fig. 14 (notation as in the technical description of the AI-100).

According to the block diagram, depending on the position of the switch at its input, the control channel may be triggered either by the actual pulse being measured or by a signal from outside. Each triggering of this channel leads to the operation of the expander. The line selector is only triggered by those pulses which pass through the" and" circuit. If the switch of this circuit is in position 'Y then all the pulses arriving from the shaper through the delay line pass through the "and" circuit, trigger the univibrator, and open the gate. If a positive pulse arrives at the input 'Y, this forbids the transmission. If the switch of the" and" circuit is in position e, the pulses only pass through it if a negative signal falls on e. A positive voltage is fed to the fourth input of the "and" circuit from the converter circuit at the instant of its operation; this blocks the "and" circuit, prevents the triggering of the univibrator and hence also the line selector, and thus blocks the input of the converter while the conversion of the previous pulse is being carried out.

§3. Experimental Logic Circuit

The experimental logic circuit incorporates the time-coupling circuits of the spectrometers, the fast coincidence circuit (AjBj, A2~' and Bj~) forming the commands for controlling the measuring tracts, and a slow coincidence circuit (C jC2B) creating the commands for controlling the ferrite memory in the "accumulator" mode.

Spectrometer Time-Coupling Circuit. In order to ensure fast time coupling of the Cerenkov spectrometers, we used FEU-36's (four to each spectrometer). The pulses from the anodes of the multipliers fall directly onto standard shaping circuits Mj> M2, M4, and Mfi (Fig. 15). Structurally each stage is made in the form of an individual module [20,21J carrying a D10B diode limiter and a sensitive univibrator based on a 3I301G tunnel diode with an operating threshold of about 0.1 V.


Fig. 15. Block diagram of the time coupling of the spectrometer.

Univibrators Ms and Ms, acting as adders, are distinguished from the input devices by a lower sensitivity, since they receive pulses which have already been shaped.

The coincidence take-off unit CC is made on the same principle. The only circuit difference is the presence of a potentiometer controlling the position of the working point on the characteristic of the tunnel diode, thus varying the sensitivity of the univibrator. This enables us to use the device in tuning-up operations under conditions in which single pulses are pas

sed, which is necessary in order to select the working conditions of the FEU-36's. The main circuit diagram of the time-coupling unit of each spectrometer is shown in Fig. 16. The pulses from the output of the time-coupling unit of each spectrometer pass along a long cable to the input of a fast coincidence circuit.

Fa s t Co inc ide n c e C i r cui t . The fast coincidence circuit (CC) is shown in Fig. 17. This consists of three double-coincidence circuits. The upper (A1Bl ) and lower (A2 B:!) circuits are completely identical, and the middle one (B1B?) only differs in the output.

All four inputs of the CC start with standard shaping circuits analogous to those described earlier. The sensitivity with respect to the input is around 0.2 V. The signals from Bl and B2 are split. The shapers of these channels each trigger two univibrators, also based on tunnel diodes. In order to ensure identity of all the channels, similar univibrators are also placed after the shapers of channels Al and A2• The double-coincidence circuits are also made of analogous univibrators in which provision is made for moving the working point by means of a variable resistance, which within certain limits enables one to vary the resolving time of the coincidence circuit and also if necessary to convert the circuit into the single-count condition. The

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Fig. 16. Spectrometer time-coupling circuit.

APPARA TUS FOR RECORDING NEUTRAL PARTICLES 19

pulse from the take-off unit of the coincidence circuit is amplified by means of a two-stage transistor amplifier. The output stage of the amplifier of the upper coincidence circuit ("gage") T2,

together with transistor T3, form a gating circuit (an "and" circuit). The pairs of transistors T6, T7 and T15, T16 in the middle and lower "gages" work similarly. These circuits enable us to control the operation of all three tracts: to switch the count on and off ("start" switch), or only to allow the count on a command from the accelerator during the radiation of the latter (" cutout").

After the gate in the upper and lower" gage" circuits are output stages controlling (allowing or forbidding) the AI-100 analyzers and outputs to the control scalers. Since these stages are intended for matching the operation of the transistor and tube circuits, they require relatively large output pulses. Each output stage constitutes a blocking generator of the waiting type [22J based on a P-60 1 transistor. The circuit employs a transformer of type 5 from the series of pulse transformers of the M-20 computer.

The output stages of the middle gage form the command "measure" and one of the pulses for the slow-coincidence circuit C1C2B. In addition to this, in this part of the circuit the univibrator based on transistors Ts and TiO creates a constant dead time which prevents a second occurrence of the commands in question for 130 IJ-sec, so as to provide the time required for measuring very large pulses (up to 100 V). The stage T13 forms a pulse which may be used as a "read and write" command if there is no need to use the C1C2B coincidence circuit, and also if necessary to check the latter. In this case the pulse from T 12 is used to create the command +lRU, moving the address of the accumulator by unity.

In order to check the number of missed counts due to the introduction of the dead time, a so-called fast output (transistor Ts) is provided; this ensures the counting of all pulses appearing at the input of the univibrator creating the dead time. The count check based on the "fast" and" slow" outputs enables us to establish the permissible loading of the system, which is particularly important when carrying out adjustments, in which the loads may reach unacceptable magnitudes. The dead time of the upper "gage" is 14 IJ-sec, that of the middle one 135 IJ-sec (''fast'' output 8 IJ-sec), and that of the lower one 17 IJ-sec.

S low Co inc ide n c e C i r cui t . In the first form of the system described, control of the ferrite memory (address shift and read-write) in the accumulator condition was effected directly from the middle "gage" of the fast coincidence circuit. In real working conditions this meant that the intermediate ferrite memory, containing 99 addresses, could be filled in 10-15 min, the amplitudes of the recorded pulses lying principally in the region of the low channels. Considering that in the experiments in question the region up to about the lOth or 20th channel contains no useful information, we proceeded to reduce the recording efficiency in this region almost to zero; this made it possible to increase the time required for filling the memory (working under the same conditions as before) to between 1.5 and 2 h. The possibility of recording a pair of pulses, one of which had an amplitude smaller than 7 V, was eliminated by means of the slow coincidence circuit (in the block diagram the triple "and" circuit), the output of which shifted the address and initiated the "read-write" cycle.

The main circuit of the unit carrying out these operations is shown in Fig. 18. This consists of a triple coincidence circuit the inputs of which receive signals from the spectrometers C1 and C2 and a signal from the middle "gage" of the fast coincidence circuit. At the inputs of the circuit are diode limiters cutting off that part of the pulse from C1 and C2 which exceeds 8 V. The upper-limited pulses then fall on a diode discriminator, which has a threshold of the order of 7 V. The discriminator consists of three diodes and is made in such a way as to compensate the temperature drift of the back resistance of the diode, which increases the stability of the threshold. Pulses pass from the output of the discriminator through a pulse amplifier to the waiting "blocking" generator, which standardizes the pulse with respect to length and amplitude.

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The take-off unit of the coincidence circuit (standard module) is made from three D9D diodes which, being grounded on the side of the inputs (through the blocking generator winding or the open P403 transistor) maintains the output P403 transistor of the coincidence circuit in an open state. On the arrival of all three imput pulses the diodes are cut off. The output transistor is opened and creates a coincidence pulse. By means of the transformer windings this pulse is transferred to input of the circuits forming the me mory -co ntro I co mmands. The co mmand + lR U, which shifts the address by unity and zeros the number register of the memory, is formed by a waiting blocking generator based on a P601. The shaper for the "read -write" command is analogous to the output circuit of the middle "gage" of the fast-coincidence circuit already described. In the "accumulator" mode, both command pulses are fed to the AI-100 No.1 analyzer.

§ 4. Ferrite Memory and Extraction of Data

Fer r i t e Me m 0 r y . The principle underlying the recording of the information coming from the two spectrometers into the memory of the AI-100 No.1 analyzer, operating in the "accumulator" condition, is set out in the description of the block diagram and also in [17}.

In the "accumulator" condition, the train from the AI-100 No.1 converter is fed to the input of the first decade of the arithmetic unit (AU), the output of the second decade of the AU is disconnected from the input of the third decade, and a train from the AI-100 No.2 converter is fed to the input of the third decade. Simultaneously a command from the slow coincidence circuit passes to the input of the blocking generator (L161) (see the circuit of the AI-100 control system), which on being triggered carries out the operation of adding unity to the RU (recording unit) and zeroing the AU. In analogy with this, a recording command passes from the slow coincidence circuit to the input of the blocking generator (L 78 ) giving the "read-write" cycle.

In addition to this, the input of the blocking generator (:48)' which throws the RU, receives a pulse from the switch TV2-1 (180); when this is changed from the operating position to the recording position and conversely, the result is to trigger the blocking generator and zero the RU. Each coincidence B1B:! will open the measuring tracts, and trains from C1 (first and second decades) and C2 (third and fourth decades) will be written into the arithmetic unit. These trains will be added until an event producing a triple coincidence in the slow coincidence circuit (which is to be recorded in the ferrite memory) appears. Such an event operates the slow coincidence


olZ all 010 09 08 67 ~6 65 01/ 03 oZ 9' a a a a a <5 a a a a 9 LU8 DU (pos. 146)

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Fig. 19. Program-switch circuit of the AI-IOO for realizing two supplementary programs: "accumulator" and "data ex-traction" . 1) Read-write, 2) to the VD(G11) data-extraction unit, 3) input of third decade of the AU, 4) output of second decade of the AU, 5) train from converter No.2.

circuit command system. The signal + lRU zeroes everything which had previously been written in the number registers (AU) and selects the first RU address. After this the train comes from the converters and is written into the AU.

After a certain period, when the train comes to an end, the signal "record" appears and the "read-write" cycle begins. During this cycle, information from the selected address is rewritten in the A U. Since the ferrite memory is "empty," this reading does no damage to the AU record which the trains have already supplied. After this, the "recording" process takes place and the contents of the A U are written out in accordance with the first address, which still remains on record in the AU. Then the trains produced by the small pulses continue passing into the AU, as in the beginning. This continues until an event producing a triple coincidence again occurs. This event repeats all the operations just described, but selects the second address, and so on. This process of measuring and recording will continue until the 99th address is selected. When this address has been selected, the RU emits a signal and a blocking relay operates, stopping the collection of information. The contents of the ferrite memory are t>rinted in digital form and on punched tape, and the next series of measurements begins.

In order to ensure the foregoing sequence of operations in the AI-IOO No.1 control system, the following changes and additions are made to the program switching system. The first version (described in [17]) only provided for the introduction of one additional program, the "accumulator" program. This was achieved by using the fifth position of the switch (not used in the AI-lOO) and supplying an additional "21 d" plate. In this way the additional program was easily introduced without having to alter the factory production of the circuits giving the other programs.

The second version now realized is more complicated to make but gives two additional programs in the AI-lOO: the accumulator condition (position 5) and the condition in which the results are put into printed digital form, punched tape, and an automatic recorder (position 7). In addition to this, this version enables five additional programs to be introduced quite simply. This is achieved by replacing the standard program switch with a switch consisting of


ten plates, each with eleven positions. The arrangement of the new switch showing connections for two extra programs, is indicated in Fig. 19.

Extraction of Data in the Form of Printed Digital Records, Punched Tape, and an Automatic Recording. The system for data extraction (DE) is made in accordance with earlier developments [231; it enables the operator to extract the data in the form of a printed digital record on an EUM-23 electric typewriter, which presents the material in the form of a table. The system allows extraction to start from any preassigned channel, which is selected by means of special channel-selecting switches based on a countingrate meter. Until the selected channel is reached the interrogation (of the memory) takes place at a faster rate. The whole memory is extracted in about 1.5 min. In addition to digital printing, the system allows the data to be brought out onto an E PP-09 automatic recorder (0.8 channels/sec) and onto a tape puncher in the form of a five-digit code (1.4 channels/sec). Provision is made for all possible combinations of simultaneous output on these devices (DP+AR, DP+TP, TP+AR, and DP+AR+TP).

In order to convert the data into 80 -column punched cards, we use the output punch (re-sui ts punch) of the M -20 computer. The informa tion is extracted in the same form as it was presented in the arithmetic register of the AI-100: in the form of a pair of two-place decimal numbers, each place of which is represented in the binary system. The information from 12 channels is placed on one punched card. The numbers from one channel, together with the intervals between them, occupy from the 34th to the 52nd digit in one line. The first two decimal numbers on the left carry information from C2 and the next two from C1• The version of material display is not economical from the point of view of punched-card consumption and machine input time, but it has the advantages of simplicity in the circuits linking the AI -100 to the card puncher.

The matching circuit consists of 16 current amplifiers, each of which is composed of P16 and P201 transistors. The coils of the electromagnets of the corresponding digits in the magnetic cabinet of the puncher are connected to the collector circuit of the P20 l' s. The input of each amplifier is connected through a 120 KQ resistance to the arithmetic unit of the AI-lOO and linked to the anode circuit of a closed trigger tube. Thus the activated electromagnets always reproduce the state of the AU register if a voltage is supplied to them. Then the puncher operates in the usual way, questioning all the ahannels from 1-99 in order, and then stops. All the information from one space of the ferrite memory is extracted in 10 -15 sec and recorded on nine punched cards.

CHAPTER IV

ANAL YSIS OF THE RESULTS ON AN ELECTRONIC COMPUTER

The extraction of the information from the analyzer accumulator and its presentation on punched cards offers wide prospects of subsequent analysis on the M-20 electronic computer, thus freeing the physicist from the necessity of dealing with a further complex analyzing technique at the site of the experiment.

The output information is presented in clear form (in the decimal system as digital printing and in the binary-decimal system on punched cards) so as to allow the operator to check the operation of the system and the functioning of the punch and printer during the course of the experiment. However, the production of output information in this form meant that the output code of the physical apparatus failed to match the input code of the M-20 computer, so that in a number of cases the ordinary form of programing had to be modified. The experimental data were analyzed on the M-20 in two ways.


§ 1. Two-Dimensional Analysis

A special program was developed in order to allow punched cards obtained from the measuring system to be introduced straight into the computer without any changes, in such a way as to give data in tabular form (two-dimensional analysis) as well as numerically at the output.

The data analysis was carried out as follows: On the punched card obtained from the apparatus each event (pair of numbers Ai> A2) was written in a single line (digits 34-52) in the binary-decimal system. The program was set up so that this pair of two-place numbers selected the corresponding address in the operative memory of the machine (computer) and wrote unity in this address without converting the numbers to normalized form (the numbers are introduced in normalized form in the ordinary operation of the machine). After placing all the material in the operative memory (this takes place at the same time as feeding in) the machine is put into the state of production. The production is effected by high-speed printing on narrow paper strip in blocks of 100 lines. Four numbers are printed to each line. Twenty-five such blocks are printed out. On printing out, the information is arranged in such a way that, after sticking the resultant 25 pieces of narrow printed paper strip together, a complete matrix is obtained, the X axis of this corresponding to the amplitude At of the pulse from spectrometer Cj and the Y axis to amplitude A2 of the pulse from spectrometer C2; on the intersection of every two coordinates is a printed number corresponding to the number of cases of such coincidences. In order to satisfy the condition of a convenient matrix size, appropriate intervals are taken between the individual numbers. The program also provides for printing figures indicating the pulse amplitudes in volts along the X arid Y axes (numerical scale of the axes).

In addition to the two-dimensional representation of correlated events on the pulse-amplitude plane, an analogous distribution on the energy scale may also be obtained.

Two-dimensional analysis gives a clear representation of the spectrometric distribution of the events recorded and enables the results to be compared qualitatively.

§ 2. One-Dimensional Analysis

The desired energy distribution of the events may be obtained by direct one-dimensional analysis of the experimental data. In this case the analysis of information is carried out as follows: The punched cards obtained from the apparatus are fed into the computer and all numbers are normalized (the appropriate order is assigned to each of them). After this operation the experimental material may be analyzed on the M-20 by ordinary programing procedures. Programs were developed for two types of problems: a) To obtain distributions of N(Aj) and N(A2) over the whole range of recording, namely, Aj = 0 to 100 and A2 = 0 to 100 (At and A2 are expressed in terms of the channels of the AI-IOO analyzer); this provides a statistical verification of the correct operation of the spectrometer tracts and the cut-off circuits with respect to amplitudes of the recorded pulses, and also enables thl!l operator to check the operation of the charged-particle eliminating system and other parts of the apparatus; b) to obtain the energy distributions N(En), N(En), N(EYt + En) of correlated events over the whole recording range En, En. En + En, or any specified range

E y , min < E\, < E y , max .•

Ey, min < Ey, < Ey, max.

Emin < E y, + Ey, < Emax.

determined by the conditions of the physical experiment.


CHAPTER V

CHECKING THE OPERATION OF THE APPARATUS

The study of processes involving a small effective cross section demands extremely protracted measurements. A necessary condition for the conduct of the experiment is the longterm stability of the recording apparatus, the reliable operation of all its components, and a reasonably complete checking system indicating the mode of operation of the apparatus.

Since the apparatus in question is quite complicated and is intended for prolonged roundthe-clock operation, special attention is paid to the creation of simple, reliable, and operative means of checking.

The checking system employed may be classified in the following way: current checking in the course of the measurements, periodic thorough checking of the whole system, and unitby-unit checking.

§ 1. Current Checking

Current checking is effected during the measurements and at individual stages. During the experiment auxiliary counts are recorded continuously (see Fig. 9). These include: AjBio A2~' the numbers of charged particles passing through the electron counters and spectrometers and recorded by the corresponding fast coincidence circuit; BjB2, the numbers of fast coincidences in the two spectrometers due to both electrons and 'Y quanta; N(RU), the numbers of filled addresses in the AI-lOO No.1, i.e., the numbers of coinciding pulses from the two spectrometers, the amplitudes of which are measured. The relations between these counts (for a specified experimental geometry) constitute check figures (within the limits of statistical accuracy) and serve as a criterion for the stable operation of the apparatus. Any deviation from these check figures indicates a disruption in the operation of the apparatus and calls for a special check throughout the system or else a unit-by-unit check.

§ 2. Unit-by-Unit Check

The unit-by-unit checking of the system consists of checking the characteristics of individual units obtained when setting up the apparatus before calibrating the spectrometers.

1. Stability of the setting of the AI-lOO. Check pulses applied to the inputs of the analyzers are measured.

2. Stability of the amplification factors of the spectrometer tracts. Check pulses applied to the inputs of the spectrometer tracts (preamplifiers) are measured on the AI-lOO's.

3. Experimental logical circuit. Checks are made by simulating any combinations of pulses on the inputs of the fast coincidence circuits and the inputs of the principal amplifiers. (When working with this circuit careful matching of the applied pulses with respect to time is essential).

If there are no faults in the electronic apparatus, but the check figures still drift, this indicates either that optical contact between the photomultipliers and the radiator has been broken or that the operation of the photomultipliers in the spectrometers has been disrupted; this demands a thorough check of the whole system after the unit-by-unit check.

§ 3. Thorough Check of the Whole System

A thorough check of the whole system indicates whether the calibration characteristics of the entire apparatus have been preserved or disrupted.


1. Checking by Pulsed Light Sources. Semiconductinglightsourcesbased on gallium phosphide are kept on the radiator of each spectrometer the whole time. These are supplied from a non-second pulse generator along a long cable through a matched branching system (Fig. 9). The amplitudes of the light pulses are determined by the amplitude and duration of the generator pulses. Operation in the mode of single or periodic pulses may be used to imitate cases in which surges of Cerenkov radiation arise simultaneously in both spectrometers, thus enabling us to check the operation of the system as a whole. The counting of fast coincidences between the two spectrometers constitutes a check for the stability of operation of the time tracts, while the position of the maximum of the pulse amplitude distribution (on the AI-100) and the resolution serve as checks on the characteristics of the spectrometer channels. The experimental logical circuit is checked at the same time.

In addition to verifying the operation of the system in the mode of pulse-amplitude analysis in each of the tracts, with external control from a fast-coincidence circuit, the check system here described also enables us to verify the operation of the apparatus in the accumulator mode, Le., in the principal mode of operation.

Checking with the aid of semiconducting light sources is simple and easy to operate; it enables us to judge the correctness and stability of operation of the whole system under conditions close to those encountered in practice.

2. Checking With an NaI-Tl Crystal and a CS 137 Source. The NaI-Tl crystal is attached with thick grease to the end of the spectJrometer radiator and is irradiated by the collimated Cs 137 radioactive source. The position of the photo-peak of the Cs 137 is determined on the AI-lOO, together with its resolution. The spectrum may be recorded either when the analyzer is controlled by the pulses falling on its input or else when external control is supplied from a logical circuit, the threshold of the fast-coincidence circuit being changed so that it passes single pulses formed by the time channel of the spectrometer under the influence of the scintillations in the NaI crystal.

This method of separate checking of the spectrometer tracts is more reliable, but less convenient to operate.

3. Checking in an Electron Beam. Thebestmethodofcheckingthespectrometer tracts and verifying the efficiency of the fast photomultipliers is naturally that of returning to the conditions under which the spectrometers were calibrated, Le., by recording the Cerenkov radiation of electrons of different energies. However, this takes a long time and is only done in exceptional cases.

If unit-by-unit checking and thorough checking of the whole system give positive results, this eliminates any doubts regarding the operation of the apparatus itself, even if the experimental check counts involve deviations rather outside statistical accuracy. Such deviations may be due simply to incorrect reproduction of the geometrical conditions of the experiment, or to a variation in the parameters of the beam of'Y quanta coming from the accelerator.

Thus the successive applications of such test measurements enables us to discover any inaccuracies in the operation of the apparatus very quickly and also to locate these. In the practical use of the apparatus, in addition to current checks, the more sensitive thorough checks based on semiconducting light sources are applied periodically after each stage of measurements, so as to be quite sure of the correct operation of the apparatus.


CHAPTER VI

STUDY OF THE PHOTOGENERATION OF ETA MESONS IN CARBON NEAR THE THRESHOLD

27

The apparatus described in the foregoing chapters may be used for studying any processes involving the formation of two correlated,), quanta or electrons. At the present time the system is being used to study the photogene ration of T/ mesons in carbon nuclei in a 650 bremsstrahlung beam obtained from the Physical Institute synchrotron. We here present some preliminary results and also the experimental conditions of these investigations, in so far as they illustrate the operation and capabilities of the apparatus.

In order to record the,), + X - T/ + X process we used the fact that an T/ meson decays with a probability of about 35% [24J in accordance with the reaction T/- ')' +')' and may thus be detected by means of the equipment here described.

The main source of background in our case is made up of processes leading to the development of one or several1To mesons, which like the T/ mesons decay into two ')' quanta. However, the difference in the masses of the 1T and T/ mesons leads to a considerable difference in the critical angles even if the total energies of these particles are the same, and this greatly eases the conditions of the experiment. In order to eliminate the background from processes involving the formation of charged particles (the formation of pairs or threes of charged 7T~ mesons or electron-positron pairs) anticoincidence counters are employed.

The general way in which the apparatus is arranged in experiments on the photogeneration of mesons is shown in Fig. 20 (direction I). The same figure shows the electron beam (direction II) used for the calibration of the Cerenkov spectrometers. We see from the figure that the synchrotron y-ray beam is collimated and scavenged by magnets and scavenging collimators before falling on the target. The lead and concrete walls shield the apparatus from scattered radiation on the accelerator side, so that the background in the experimental room is quite low. The target is a carbon plate 70 x 70 mm in area and 80 mm thick, measured along the beam (0.32 t units).

The two Cerenkov spectrometers were arranged in a plane inclined at an angle of e = 30 0 to the axis of the beam of bremsstrahlung. The angle between the axes of the detectors in this plane was taken as £ref = 140 0

• The angular aperture of the entrance window of the spectrometer equalled ± 80 • With this geometry T/ mesons with energies between the threshold (E 1) min = 564 Me V) and the maximum possible at the upper limit of the brems spectrum, Ey max =650 MeV, could be recorded.

The efficiency e of recording the T/ mesons by reference to the two ')' quanta arising from their decay was calculated by the method of statistic::J tests (Monte Carlo) on the M-20 computer [25J for the geometry in question at an energy of Ey max=' 650 MeV. The results of these calculations are presented in Figs. 21-23. The total recording efficiency for T/ mesons formed by ')' quanta (with a mean energy of 608 MeV at an average angle of 40°30') equals (4.71 ± 0.16) '10- 5•

For the relative monitoring of the bremsstrahlung intensity we used an ionization chamber placed in front of the target. The absolute energy flow through the target was determined by means of a quantometer. Measurements of the count rate from the carbon target alternated with measurements of the background count without the target and also with measurements of the random -coincidence background.

The results of the measurements containing information regarding the energies of the two simultaneously-recorded,), quanta in the form of pulse amplitudes, expressed in numerical code, were written into individual addresses in the ferrite memory of the accumulator and then extracted in the form of digital printing and punched cards. A two-dimensional analysis of the experi-


:0.: . . :,C!.

!ODem >----' H f:IfO

IO~Il ,~ 8

II II

Fig. 20. Experimental geometry. I) Direction of 'Y quantum beam, II) direction of electron beam, 1) accelerator target, 2) collimators, 3) scavenging magnets, 4) electron-beam target, 5) electron spectrometer [5], 6) vacuum systems of the other experimental apparatus, 7) monitor, 8) mechanical system for setting up the spectrometers, 9) Cerenkov spectrometer, 10) electron counters, 11) lead shielding, 12) concrete shielding, 13) paraffin, 14) target, 15) 'Y counters, 16) positron counter of the 'Y spectrometer, 17) electron counters of the 'Y spectrometer, 18) main electron counter, 19) ionization chambers for measuring the intensity of the 'Y beam. The significance of points 15-19 is given in [5]. ao, {3o, are setting parameters.

(cos 8), reI. units 3

z

o~~ __ ~ __ ~ __ ~~ __ ~ __ ~ 0.3 0. If 0.5 as 0.7 0.8 0.9 1O

cos e Fig. 21. Recording efficiency of an 1/ meson as a function of the angle fJ.

t(£yJ, reI. units 50

Ey,MeV

Fig. 22. Energy spectrum of the decay 'Y quanta of a 1/ meson. Broken line obtained without considering the energy resolution of the spectrometer.


1(£1])] reI. units

15

10

5

r' I I I I I I

r" I I I I L.

500 550 800 fJJO 700 750 800

Fig. 23. Energy spectrum of the recorded TJ mesons. Notation as in Fig. 22.

mental material was carried out on the M-20 computer. Of all the events recorded, we selected just those which satisfied certain conditions, derived from recording-efficiency calculations (allowing for the finite energy resolution of the two spectrometers). The conditions were that the energies of the two simul-taneously-recorded'Y quanta, and also their sum, lay within a kinematically-possible energy range of the spectra presented in Figs. 22 and 23. The selection of events satisfying these conditions was carried out on the M-20 in accordance with the values of EYI + En' The results obtained without the target and the random coincidences were analyzed analogously.

This way of selecting the experimental data completely eliminated the possibility of recording the photogeneration of 1To mesons in the nuclei. An angle of 140° between the axes of the detectors was in fact critical for 1TO

mesons with an energy of E1fO = 144 Mev. The gamma quanta from the decay of such mesons will in fact have energies approximately equal to E1fo /2, and hence such cases will not satisfy either the first or the second condition for the selection of cases involving TJ mesons. For 1TO mesons with higher energies there is' only a low probability of scattering at angles of'" 140° and then only for very asymmetric decays. Thus, for example, for E1fo = 600 MeV there may be a decay into two 'Y quanta with a scattering angle of 140° between them if En = 591 MeV and EY2 = 9 MeV. It is clear that such cases fail to satisfy the first selection condition; hence the recording of the photogeneration of single 71'0 mesons or 71'071'+ pairs is completely excluded.

The only real process which could in principle give cases satisfying both selection criteria is that of the photogeneration of pairs of 71'0 mesons in nuclei. When two 71'0 mesons are generated at angles corresponding to the directions of the detectors, a very asymmetrical decay of each one of them may take place, and as a result of this two 'Y quanta with energies equal to the maximum allowed by the kinematics of the decay will move in the same direction toward the detectors, while the two other 'Y quanta with the minimum energy will move in the opposite direction. If both 71'0 mesons have a fairly high energy, the result may be that two 'Y quanta with energies such as to satisfy both selection criteria will fall into the two detectors at the same time. An estimate of the recording efficiency of this process based on the method explained earlier gives a value at least three orders lower than that of the TJ meson. Exact calculation of the contribution from this process in the range of present interest is difficult owing to the absence of any data regarding the differential cross sections of the photogeneration of pairs of 71'0 mesons in carbon. Some auxiliary experiments were therefore made in order to determine this quantity. The geometry of the experiment was altered in such a way as to exclude the recording of TJ mesons completely. For this purpose the detectors were placed so that no'Y quanta arising from the decay of an TJ meson with the maximum possible energy could fall into them. The angles between the axes of the detectors was here 101°; the angles between the axes of each detector and the axis of the beam of 'Y quanta remained unaltered. The results of measurements in the altered geometry showed that, within the limits of the statistical accuracy obtained, the count obtained in the range defined by the selection criteria was entirely determined by random coincidences and cosmic background.


The count of the random coincidences in the original geometry (in which the 11 mesons were recorded) was also quite high in the selected range of energies. This worsened the statistical accuracy of the results obtained and considerably increased the time required for the experiment.

The results of our fi:!;,st investigations into the photogeneration of 11 mesons in carbon [26J are now being refined.

* * * The apparatus here described may be successfully used for the study of processes invol

ving the formation of neutral particles decaying into two 'Y quanta. Long experience with the apparatus has shown it to be simple in operation and reliable in use. The existence of the simple method of coupling with the M-20 computer is particularly convenient, greatly easing the analysis of the experimental results.

In conclusion, the authors consider it their pleasant duty to thank the whole staff of the High-Energy Electron Laboratory and its Director, V. A. Petukhov, for help in the work, both in constructing the apparatus and discussing the results, their colleagues in the Computer Department, V. A. Sokolovskii, A. T. Matachun, and L. P. Konstantinova, for creating and developing the programs, and engineers in the Design Office and members of the Technical-Production Department of the Physical Institute for the mechanical construction of the apparatus.

The authors are also very grateful to S. G. Solovtev, I. D. Poddubnyi, T. 1. Kovaleva, N. 1. Moiseev, and P. 1. Malakhov, who took part in the work at various stages of the construction and application of the apparatus.

LITERA TURE CITED

1. A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, Pribory i Tekn. Eksperim., No.4, p. 38 (1964).

2. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, Pribory i Tekhn. Eksperim., No.5, P. 45 (1965).

3. G. S. Vil'degrube, N. K. Danilenko, and A. 1. Razumovskaya, Pribory i Tekhn. Eksperim., No.4, p. 74 (1961).

4. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, P. N. Lebedev PhYSics Institute ofthe Academy of Sciences ofthe USSR, A-73, Moscow (1965), Pribory i Tekhn. Eksperim., No.3, p. 221 (1966).

5. V. N. Maikov, V. A. Murashova, T. 1. Syreishchikova, Yu. Ya. Tel'nov, and M. N. Yakimenko, this volume, p. 32.

6. V. F. Grushin and E. M. Leikin, Tr. Fiz. Inst. Akad. Nauk, 34:187 (1966). 7. V. F. Grushin, V. A. Zapevalov, and E. M. Leikin, Pribory i Tekhn. Eksperim., No.2,

p. 27 (1960). 8. T. Yomagata, Doctoral diesertation, University of illinois (1956). 9. J. M. Brabant, B. Y. Moyer, and R. Wallace, Rev. Sci. Instr., 28:421 (1957).

10. A. F. Dunaitsev, V. 1. Petrukhina, Yu. D. Prokoshkin, and V. N. Rykalin, Summaries of the Scientific-Technical Conference on Nuclear Radio-Electronics, Gosatomizdat (1961).

11. C. E. Swartz, Nucleonics, 14:4 (1956). 12. G. Davidson, Doctoral diesertation, MIT (1959). 13. M. Feldman, V. Highland, I. W. Dewire, and R. R. Zittauer, Phys. Rev. Letters, 5:435

(1960). 14. M. N. Khachaturyan and V. S. Patuev, Pribory i Tekhn. Eksperim., No.6, p. 29 (1963). 15. 1. S. Allen, Rev. Sci. Instr., 18(10):739 (1947).


16. V. O. Vyazemskii, L. V. Drapchinskii, A. N. Pisarevskii, V. V. Trifonov, and E. 1. Firsov, Pribory i Tekhn. Eksperim., No.5, p. 40 (1958).

17. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, V. V. Pavlovskaya, and S. G. Solov'ev, Pribory i Tekhn. Eksperim., No.6, p. 84 (1965).

18. V. O. Vyazemskii, 1. 1. Lomonosov, A. N. Pisarevskii, Kh. V. Protopopov, V. A. Ruzin, and E. D. Teterin, The Scintillation Method in Radiometry, Gosatomizdat (1961).

19. A. V. Kutsenko and S. G. Solov'ev, Pribory i Tekhn. Eksperim., No.2, p. 228 (1966). 20. A. F. Dunaitsev, Pribory i Tekhn. Eksperim., No.6, p. 77 (1964). 21. A. Whetstone and S. Kounosu, Rev. Sci. Instr., 33:423 (1962). 22. S. Schwartz, Semiconductor Circuits [Russian translation], IL (1962). 23. M. P. Sokolov, Transactions of the Sixth Scientific-Technical Conference on Nuclear

Electronics, Gosatomizdat (1965). 24. A. H. Rosenfeld, A. Barbaro-Galtieri, W. Barkas, R. L. Bastien, 1. Kirz, and M. Roos,

Rev. Mod. Phy., 36:97 (1964); Usp. Fiz. Nauk, 86(2):335 (1965). 25. S. P. Denisov, Dissertation, Moscow University (1963). 26. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, Twelfth

International Conference on High-Energy PhYSics. Dubna (1964). Vol. 1, Atomizdat (1966), p.849.

MAGNETIC SPECTROMETER FOR

CHARGED PARTICLES

V. N. Maikov, V. A. Murashova, T. I. Syreishchikova,

Yu. Ya. Tel'nov, and M. N. Yakimenko

INTRODUCTION

The Compton effect associated with high-energy electrons (~650 MeV) is being studied to an accuracy of better than 1% on the synchrotron in the High-Energy Electron Laboratory of the Lebedev Physics Institute of the Academy of Sciences of the USSR. This high accuracy imposes a number of complex demands on the experimental conditions and individual parts of the apparatus. At least two methodical problems have to be solved. On one hand, the recording of lowenergy 'Y quanta (<e 2 MeV) with a reasonably high efficiency under synchrotron working conditions, and on the other hand the spectrometric recording of high-energy Compton electrons (~600 MeV) with an angular aperture relative to the primary beam of 'Y quanta of <2 .10-2• The first problem is partly solved in another paper of this collection [1] and the second in the present paper.

CHAPTER I

EXPERIMENTAL CONDITIONS AND REQUIREMENTS

IMPOSED ON THE SPECTROMETER

§ 1. Compton Effect with Electrons at High Energies

A study of the Compton scattering of 'Y quanta by electrons at high energies offers the possibility of verifying the radiation corrections to the Klein-Nishina-Tamm formula, and in the case of very accurate experiments may indicate deviations from the ordinary electrodynamic consideration of the scattering process. The majority of investigations into the Compton effect have been carried out with 'Y quantum energies of up to 350 MeV, the accuracy of the experiments never exceeding 3%.

The possibility of obtaining information relating to deviations from electrodynamics in this process by making relative measurements of the Compton effect at different angles and energies was considered in [21.

32

MAGNETIC SPECTROMETER FOR CHARGED PARTICLES 33

In the case of fairly large 'Y quantum energies k> 100 MeV, and for scattering angles of e>30°, i.e., for

k - (1 - cos 8) > 1 , m

the energy of the scattered l' quanta is

k' m -k = 1 e ~ , -cos

so that the energy of the recoil electrons is E '" k, while the angle made with the direction of the primary 'Y quanta is < 2 '10-2• In this case the effective cross section of the process may be expressed in the form

r~ dQ dcr =:2 kim (1- cose) (1 + a),

where a = a (e, k, k') «1, and the count rate of the Compton scattering acts is

r~ mdQ dn(k,8) =:2k(1-cose) (1 + a) NZ£6J(k)dk, (1)

where N is the number of atoms per cm2 of target, Z is the atomic number of the target substance, f(k) is the energy density of the intensity of the accelerator, t = t (e) is the efficiency of recording the scattered l' quanta, and ~ is the electron-recording efficiency.

Since the energy of the scattered 'Y quanta is practically independent of the energy of the primary 'Y quanta, their recording efficiency is constant for a variety of incident 'Y quantum energies. By providing a fairly large exit aperture of the spectrometer, it is easy to make the electron-recording efficiency constant over a wide range of quantum energies and scattering angles as well. Then by measuring dn(k, (J) for two angles and two energies we may set up the double ratio

where A is to a first approximation the algebraic sum of the values of a corresponding to the two angles and energies.

If this equation is disobeyed, this should indicate that the Compton scattering deviates from the Klein-Nishina-Tamm formula in an anisotropic, energy-dependent way.

(2)

Since A « 1, the expression on the right depends only slightly on the original parameters, while the accuracy of the expression on the left is mainly determined by the statistical accuracy of the quantities entering into it. Detailed analysis showed that both parts of expression (2) could be measured to an accuracy of better than 1%.

The accuracy of the quantity A depends on the accuracy of measuring such parameters as the energy, the angles, the form of the bremsstrahlung spectrum, and so on. It may be shown, for example, that the determination of the boundaries of the energy range of electrons recorded (k1 < E :s k2) to an accuracy of better than 1%, with a 100% recording efficiency within the range, introduces an error of the order of 0.1% into the quantity 1 + A. An analogous error appears if there is a 2 to 3% indeterminacy in the form of the spectrum within the range kl :s k :s k2• In order to reduce this indeterminacy (the form of the spectrum may change if there is any instability in the accelerator energy), the spectral density of the accelerator intensity is measured at the same time as the main experiment at the boundaries (k1> k2) of the working range.

All of the other parameters entering into A may also be determined to the required accuracy.

34 V. N. MAIKOV ET AL.

§ 2. Conditions for Recording Compton Electrons

The apparatus associated with the recording of high-energy electrons must provide for: 1) The extraction of 200 -600 MeV electrons with an aperture of:( ± 0.02 from the beam of

'Y quanta and their transfer to the recording apparatus; 2) the recording 'of these electrons in the energy range ± 5%; 3) the delimitation of the range in question to an accuracy of no less than 1%; 4) the maximum possible electron-recording efficiency, independent of electron energy.

These requirements in practice mean that electron beam has to be deflected through a certain angle, the spectral resolution has to reach a certain degree of dispersion, and that there has to be focusing of a high order in the horizontal plane with an energy resolution of "'1%, and finally focusing in the vertical plane such as to ensure the complete collection of the electrons.

Preliminary analysis of all these requirements showed that the most suitable apparatus in the case in question was a magnetic spectrometer with a constant, uniform magnetic field of the sector type in which end effects (fringing fields) were chiefly responsible for the focusing. In the following sections we shall consider various forms of sector fields and determine the optimum characteristics of the magnetic focusing system, such as to satisfy the conditions mentioned.

§ 3. Focusing Properties of Sector-Type Magnetic Fields with Sharply

Limited Curved Field Boundaries

In order to make a strict calculation of the properties of a sector magnet we must know the analytical form of the real field, including the fringing field at the ends of the pole tips, as well as the uniform component. However, a good approximation for selecting the parameters of the system may be secured by basing our calculations on a certain "effective" uniform field with sharp magnetic "faces." The validity of this is based on the fact that the space occupied by the fringing field for large sector angles and small gaps between the pole tips is much smaller than that occupied by the uniform field.

It was shown in [3] that the ion-optical properties (dispersion, aberrations) of a real sector field are very like those of a certain effective field with sharp boundaries (Fig. 1)

H (x) = {HOD for x <;; dx • for x>d:n

if the position of the effective boundary is defined in terms of the fringing field

in the following way:

H (x) = Hoh (x)

x~o h (x) dx = dx + xo. xt

(3)

(4)

(5)

The deflection angle, the turning radius, and the entrance and exit angles of the central ray are taken the same as in the case of the true field. Here Ho is the uniform magnetic field, hex) the form of the fringing field 0 < h (x) < 1; Xo, Xoo are the coordinates inside and outside of the uniform field [h (Xo) = 1] and [h (Xoo) = 0] respectively, x = 0 is the boundary of the pole tips, dx is the effective field boundary in the direction x, counted from the mechanical boundary of the pole tips (x = 0).

In the case in which the extent of the ends of the pole tips L is much greater than the gap K between them (L» K), h(xiK) depends little on K, and hence dx c< ax K, or in particular dn = anK,

MAGNETIC SPECTROMETER FOR CHARGED PARTICLES

y

x

Fig. 1. Method of introducing an effective field.

-x Ii

Fig. 2. Focusing in a uniform sector-type magnetic field with sharp boundaries. The points Fl and F2 give the positions of the source and image respectively.

35

where a n = a x cos e, e is the angle between the x axis and the normal n to the boundary of the pole tips, and an and ax are constants [3J.

Experience shows [3,4J that the ion-optical parameters calculated in this kind of effective fi~ld agree closely with those found experimentally, provided that an '" 1, i.e., dn '" K. We note also that h(x) varies little with Ho for Ho> 500 [3,5J. Hence the definition of the effective boundary (5) remains valid over a wide range of magnetic fields.

All calculations necessary for selecting the parameters of a magnetic focusing system were carried out on the assumption of a steady, sharply-bounded, uniform sector-type magnetic: field, subsequently regarded as an effective analog of the real field. Figure 2 shows a focusing scheme in this kind of field together with the shape of the latter.

Let us introduce the notation for the parameters of the magnetic system: ro, CPo are the radius of the curved trajectory and the turning angle of the "average" particle in the uniform magnetic field.

Rb, RH are the radii of curvature of the field boundaries at the points of entry and exit of the average particle. These are regarded as positive if the field boundary at the point of entry or exit is convex, and negative if this is concave.

eb, eH are the angles between the traJectory of the average particle outside the field and the normal to the field boundary at the point of entrance and the point of exit of the particle. The angles are regarded as positive if the trajectory of the "average" particle outside the field and the center of the circle lie on the same side of the normal to the field boundary and negative if they lie on different sides.


±o!"; ±w are the apertures of the particle beam passing out of the system in the horizontal plane perpendicular to Ho and in the vertical plane.

± by; ± bz are the horizontal and vertical linear dimensions of the particle source.

lb; lB are the distance from the particle source to the boundary of entry and the distance from the exit boundary to the image (in the direction of the trajectory of the "average" particle).

± z'; ± y' are the linear dimensions of the image. These are reckoned along the perpendicular to the trajectory of the "average" particle at the first-order focus in the Ho plane (vertical) and in the plane perpendicular to Ho (horizontal) respectively.

±[3 = ±(Eo - E)/E o is the relative range of energies, Eo the energy of the average particle, and E the energy of the particle with r = ro (1 + [3).

The trajectory parameters of the "average" particle with energy Eo, i.e., [3 = 0 are: a =

w = 0; by = b z = 0; ro; CPo; eb; e8; Rb; RB; lb; lB·

In order to calculate the image size of an extended source in the horizontal plane in the case of a sector-type uniform magnetic field with sharply-bounded curved boundaries, we used the expression given by Hintenberger and Konig [6] as far as terms of the second order:

(6)

The size of the image in the vertical plane, calculated to the first order, is given by Cross [7], and may be represented in analogous form:

(7)

Here B and A are known functions of the parameters of the magnetic focusing system (CPo, ro, eb, e B, lb, lB, Rb, Rb~·

On this basis we studied the conditions for focusing particles in a field with CPo = 60° and ro =200 cm (CPo and ro are determined by the geometry and conditions of the experiment) for a wide range of parameters eb, eS, lb, lB, Rb, RS, in order to be able to choose the optimum characteristics of the magnet for an electron spectrometer subsequently.

Preliminary analysis showed that, under the conditions of the experimental geometry proposed, it was impossible to effect double focusing [7] of the first order with respect to a and w (B1 = 0, A1 = 0), i.e., simultaneously to achieve minimum aberrations in both the horizontal and vertical planes. Attention was therefore concentrated mainly on reducing the aberrations in the horizontal plane, since these were associated with the energy resolution of the fOCUSing beam.

In the calculations of [8] we confined ourselves to positive entrance and negative exit angles eb and eB, since this gave the most favorable conditions for focusing in the horizontal plane.

We considered two classes of cases· 1) e' =- e" = e and 2) e' = e· . - e" = e for e = • 0 0 0 I' 0 ek = 30, 40, 47, or 60° (denoted by 1, 2, 3, 4 respectively). For CPo = 60° and ro = 200 cm, this embraces all practical entrance and exit angles eb and eB.

The reduction of aberrations demands that we should satisfy the condition of first-order focusing with respect to a, i.e., Bl (CPo, ro, eb, eB, lb, lB) = 0, which for the CPo and ro selected and


~--Z

3 -------'1-

o 1.0 Z.O l~/T'o

Fig. 3. Relation between lS/roand tb/ro for the case Bl = ° with 'Po = 60 0 and eb = -eS = ei; 1) ei =30; 2) 40,3) 47, 4) 60 0

•

-500 -'1-00 -300 -ZOO-fOO 0 fOO ZOO 300 'f00 50e -fOO R;, em

3

z

f----- --\-=i-=;;-----~~--____j

Fig. 4. Relationship RS = / (Rb) for the case Bl = 0, B11 = 0, 'Po = 60 0

, ro = 200 cm, eb= -eS = 47 0

; l)lb/ro = 0.5, ls/ro = 1.16; 2) lb/ro = 1.0, 18/ro = 1; 3) lb/ro = 2.0, lS/ro :: 0.89.

specified values of eb and eH imposes a relationship upon lb and lS (Fig. 3). All the calculations were carried out for the cases

(i). = 0.5, 1.0, 2.0 '0 }=1, 2, 3

and

I (~) \ B1=O = t (i) . ro j=l, 2, 3 ro i=l, 2, 3

For the parameters chosen we calculated the coefficients of {3 and by/ro, i.e., B:! ('Po, ro, eb, eS, lS) and B3 ('Po, ro, eb, c:Z, LS)·

In the second order with respect to O!, {3, by/ro, the coefficients depend on the curvature of the field boundaries Rb and RS. We therefore initially studied the condition of second-order focusing with respect to O!: B11 ('Po. ro, eb, eS, lb, lS, Rb, RS) = 0, which under the conditions indicated in the foregoing led to a relation of the hyperbolic type between Rb and RS. By way of example, Fig. 4 shows a typical RS =/ (Rb) relationship (case of eb = -eS = 47 0 ). For all the parameters considered, with Bl = 0, Bl1 = 0, in any practicable case IRb I, IRs I > 100 cm, and the condition RS > ° leads to greater aberrations than RS < 0. For this reason the aberrations for RS > ° were not considered.

All the coefficients depending on Rb, RS, namely, B11 ('Po, ro, eb, eS, lb, l8, Rb, RS), B12 ('Po, ro, eb, eH, lb, 18, RS), B13 ('Po, ro, eb, eS, lb, lS, Rb, RB), B:!2 ('Po, ro, eB, lS, RB), B23 ('Po, r 0, eb, eb, lB, RB), B33 ('Po, r 0, eb, eB, lB, Rb, RS), were calculated for RS < -100 cm with Rb = -100 cm, 00, and + 100 cm.

The dispersion and aberration in the horizontal plane and also the vertical dimensions of the image were considered for a practical case, corresponding to the conditions of recording Compton electrons:

~1 = ± 0.05, ell = ± 0.02, b!lJro = ± 0.002,

~2 = ± 0.005, Wl = ± 0.02, btl/ro = ± 0.002,

(8)

where 2{31 is the relative width of the energy range of the electrons recorded in the experiment and 2{32 is the required energy resolution of the spectrometer.

38 V. N. MAIKOV ET AL

!I(Jl b ,em 'Y~------mwr-------------------------~ 1.8

I.Z

1.0

--- - -- - --n"i\-0.8 '-'-'-'-'0"

\ ~\ \~'. 0.8 .. ~ \'

\ \~,/ .

",,/

/

/100

,/ /00

/ """" / """" / ./ ,.....--100

/./ .T'

0.1{- • \\ ·/:',,1 . • \ • \ '\' ,I... / L' ~2:;:=~~". . ,/,Y-,-r--., -InO 100 00

o.Z \J'·v .... _ ..... _-_.... UI

O~-----~----~----~~----~----~~~

-100 -ZOO -300 -'t-Oo -500 !fa, em

Fig. 5. Aberrations at the boundary of the detectable energy range as a function of the radius of curvature R8 for the case Rb = -100 cm, 00, + 100 cm and Zb/ro = 0.5, 1.0, 2.0. by == 0.4 cm; a == 0.02; [3 = 0.05; ro = 200 cm; eb = -e8 == 47°.

Bz 7..0

Z 1.5

J 82f3z r ol 8zJ3, r oJ em em

1.0 Z 20

0.5 I 10

0 0 ()

ZO 30 I{-O 50 fiO &iJ deg

Fig. 6. Size of the image of a point source with [31 = 0.05 and [32 = 0.005 at the site of the firstorder focus with respect to a (B1 = 0) as a function of eb == -e8 == e; CPo == 60°, ro = 200 cm. 1) Zb/ro = 0.5; 2) Zb/ro = 1; 3) Zb/ro = 2.

Figure 5 gives the aberrations y CI. b at the boundary of the recordable energy range <A) 1 Yl

b2

21 Ya b I = 1 Bnl <xiro + 21 B 12<XIB1 / ro + 21 B13<Xlbllll + 2/ BabYl + B23~lbYl/ + B33 -2!!. (9) I~ ~

at the site of the first-order focus with respect to a1 (B1 = 0) as a function of the radius of curvature R8 for Rb = -100 cm, 00, and + 100 cm and (lb/ro) == 0.5; 1.0; 2.0 for the case eb =-e8 = ei == 47° (similar curves were obtained for eb = 30, 40, 60°). Here the horizontal lines show the permissible level of aberrations corresponding to the specified energy resolution [32 at the edge of the energy range [31:


A,UJt rQ em At

20 5

18 If Z

12 J J

8

Fig. 7. Vertical size of the image ZWI bZ1 = A1wt r o + A3bZ1 ,at the site of the first-order focus with respect to ex as a function of £b=-£~ = £i; CPo = 60°; ro = 200 cm; w= 0.02; bz = 0, and also at the entrance (CPo = 0) and exit (cpo"" 0) of the magnet. 1) lb/ro = 0.5; 2) lJ/ro = 1; 3) lo/ro = 2.

Figure 6 shows the size of the image of a point source with i31 and i32 at the site of the first-order focus with respect to a (B1 = 0) as a function of eb = eH = ei and (lb1ro)i'

The vertical size of the image

39

(10)

(11)

at the site of the first-order focus with respect to a is shown as a function of eb = - e8 = e i for various (lb/r 0) and bZ1 = 1 in Fig. 7. Here the vertical dimensions of the beam on entering (lb sin W1) and leaving (zlzo/ro = 0) the magnet are also shown; these determine the required value of the gap between the pole tips.

Analogous relationships were obtained for y and z on studying conditions [21.

§ 4. Choice of Magnetic-Field Parameters

By considering the geometry of the proposed experiment, the angle of deviation of the particles by the magnetic field was taken as <Po"'" 60°. The radius of the circular trajectory of the particles in the steady field was taken as ro = 200 cm. The choice of this value is determined by the condition that there should be a reasonable value of the maximum field strength Ho .... 104

Oe for the maximum electron energy Emax "'" 600 MeV. A number of technical conditions associated with the manufacture of the spectrometer magnet imposed limitations on the possible width and height of the ma.gnetic path ~ro:S ± 20 cm, K:S 12 cm.

The choice of the remaining parameters of the magnetic focusing system was made by conSidering the energy and geometrical dimensions of the image [9-11) in conjunction with the requirements laid down in § 1. This resulted in the following conditions to be imposed on the characteristics of the focusing system and detecting apparatus.


a) In order to ensure a spectrometer resolution of better than 1% in an energy range of 10%, the distance between two isoenergy lines differing in energy by 1 % (f3 2 = ± 0.005) must be greater than the width of the lines at the ends of this range (f3 1 = ± 0.05), i.e., we must satisfy the condition

I Yalblfll < I Y~.I·

b) In order to focus a particle beam of energy Ei = const with a :5: ± 0.02 and (bylro) :5: ± 0.002 in a horizontal plane, the width of the beam must everywhere by smaller than the width of the uniform part of the magnetic path:

Il~sin all < I Mol. l(yClb )" 1<larol·

1 Y1 1,=0

(12)

(13)

c) In order to transmit a particle beam with w:5: ± 0.02 and (bz/ro) :5: ± 0.002 in a vertical plane without any loss, the vertical dimensions of the beam must be smaller than the height of the interpole gap of the magnet:

\lo sin 001 1< \ ~ I . I(Z",b)" 1<1~1·

I Zl lo= 0

(14)

d) At the geometrical site of the first-order focus with respect to O! the dimensions of the particle beam with E i = const, O! = W :5: 0.02 by/ro = bz /ro:5: ± 0.002, /3 ::s /31 + /32 = ± 0.055 should have a value reasonable from the point of view of detecting effiCiency. We took the following:

21 Z""b Z1 \ ~ 21 Y~I+fJ, I ~ 30 em.

The problem of choosing the magnetic system thus reduced to the choice of parameters eb, eS, lb, IH, Rb, RH, satisfying all the foregoing conditions.

Analysis of the results of calculations carried out over a wide range of variation of these parameters showed that, for the specified values CPo = 60°, ro ::::; 200 cm, ~rl < 20 cm, IKI < 6 cm, these requirements were best satisfied by a steady, uniform, sharply-bounded sector magnetic field with the following parameters:

8~ = 40° - 50°;

l~!ro = 0.5 -1,0;

R~< -150cm;

-8~=400-500;

l~!ro = [f (l~!rO)lBl=O; I R~ I > 100 em.

§ 5. Technical and Geometrical Parameters of the Magnetic Spectrometer

The spectrometer is constructed on the principle of an SP-97 sector magnet of current industrial production. The pole tips of the magnet are made in the form of an annular section with mean radius ro = 200 cm and width Ar = ± 20 cm. The sector angle between the mechanical faces of the pole path (at a radius of ro = 200 cm) is 50°15', but may be increased to 60° by providing additional end pole tips. In studying the fOCUSing properties of the magnet and also in


subsequent physical experiments we took an angle of 50°15'. As we shall show later, this gives an angle of rotation of the particles in the effective uniform magnetic field with ro = 200 cm equal to CPo '" 60°.

The construction of the pole tips provides for a positive entrance angle e' between the tangent to the central,arc and the normal to the mechanical face of the pole tip, and correspondingly a negative exit angle e ff . The angles el and ell may be varied from 35-55° (for cp = 60°) by using interchangeable end pieces to the pole tips. In the version described here we took angles of e' =_eff = 45°.

The end faces of the pole tips may be shimmed in order to create the required shape of the magnetic faces (R' and Rff) so as to improve the focusing properties of the magnet (in the second order with respect to a). The pole tips of the magnet may be shimmed near the side faces with a set of additional plates in order to obtain the desired radial homogeneity for the specified width of the magnetic path.

The size of the gap between the pole tips is K = 120 mm. The magnetic field in the gap may be varied from H = 1500 to 10,000 Oe by varying the current in the magnet windings between 50 and 350 A. Stabilization of the current in the magnet windings is effected by means of a circuit based on a BT-4 stabilizer, which gives a field stability of ~H/H < 0.1%.

The magnet has all the necessary mechanical movements for setting it in the specified experimental geometry. This magnet was used as the base for creating an electron magnetic spectrometer with a constant uniform magnetic field and edge focusing. The high demands imposed upon the focusing properties of the spectrometer, and particularly the energy resolution of the electron beam, necessitated the detailed measurement and correction of the characteristics of the magnet and a full study of its focusing properties.

In particular we had to plot the topography of the magnetic field and determine its degree of inhomogeneity, find the shape of the magnetic end faces and correct them in order to obtain the desired shape (Rb and RS) corresponding to minimum aberration, study the focusing properties of the magnet, and determine the geometrical arrangement of the counters recording the electrons.

CHAPTER III

UNIFORMITY OF THE MAGNETIC FIELD

§ 1. Effect of the Nonuniformity of the Magnetic Field on the Focusing Properties of the Spectrometer

The magnetic-spectrometer parameters giving the desired focusing conditions and presented in the previous chapter were determined on the assumption of a strictly uniform field with sharply limited magnetic faces. The actual field of the sector magnet, however, is not absolutely uniform, first because of the effect of the fringing field associated with the finite size of the magnetic poles, and secondly because of the possibility of the pole tips being not quite parallel or of other structural errors occurring. The nonuniformity due to these effects is usually of a monotonic character.

Let us consider the qualitative influence of this kind of nonuniformity on the focusing properties of the magnet. For simpliCity we shall calculate the fOCUSing conditions (as before) for a system with a sharply-bounded uniform magnetic field. In the linear approximation, a nonuniformity of the magnetic field with respect to the radius, i.e., perpendicular to the direction of


motion of the particles, may be introduced in the following way:

(1 r - ro) H(r) = Ho -n-rJ-, (15)

where rand ro are the radii of concentric circles of the magnetic path, constituting the geometric loci of equal field strengths H(r) and Ho(ro) respectively, while n is the index giving the rate of fall of the magnetic field over the radius.

In this case the first-order aberrations with respect to 01 take the form [9]

Bl = sin ~ <Po + ~ (cos r 1 n CPo + sin ~ <Po + tan e') + l~ (cos -V 1 n CPo + sin ~ <po tan en) + 1 - n ro 1 - n ro 1 - n

+ l;;~ [cos ~ CPo Ctane' +tan en) - sin ~ <po (1- n -tan e'tan e")]. (16)

Clearly in the case of a field linearly rising or linearly falling over the radius [n(r) = no and no> 0 or no < 0] first-orderfocusing with respect to 01 (Bt = 0) is quite possible. As compared with the case of a strictly uniform field no = 0, only the relation between lb/ro and lS/ro changes, i.e., the distance between the positions of the source and image .. However, in the case of a field falling toward the edges, when n(r) changes sign on passing through the magnetic axis, the condition Bl = 0 is not exactly satisfied for a wide beam. In order to make an estimate let us take the following coarse approximation to the real field:

n (r) = no>O for r >ro;

nCr) =-no<O for r<ro;

then charged particles with energy Eo traveling at angles of +01 and -01 will move in fields with indices of different signs and hence focus in two different points. Thus for no = 5 -10-2 and 1 biro = 1 the position of the image (first-order focus with respect to a) will be at the points I biro = 1.59 and 1.49 respectively, which will lead to a broadening of the image by 0.8 cm. Analogously the broadening of the monoenergetic line resulting from the nonuniformity of a magnetic field with an index no of ± 1 .10-2 (which corresponds to a field nonuniformity of .6.H/H = ± 1 -10-3

at the edge of the magnetic path) will be "'0.2 cm.

From this approximate consideration we may make an estimate of the permissible nonuniformity of the magnetic deflecting field with respect to the radius. Theoretical calculation shows that, in order to establish an energy resolution of the spectrometer better than 1%, the width of the monoenergetic line (sum of all possible aberrations) should not exceed 1.6 cm (see Fig. 5). We thus have the following rigid restriction imposed on the nonuniformity of the magnetic field with respect to radius: .6.H/H::: 1.10-3•

In order to keep the energy resolution of the spectrometer of the order of 1% in an energy "window" of "'25%, we must ensure the required uniformity of the magnetic field with respect to radius over a path ± 12.5 cm wide, Any deviation of the magnetic field from uniformity with respect to azimuth associated with the possibility of the pole tips being other than parallel may change the magneto-optical axis of the system, but will have no great effect on the dispersion and energy resolution.

MAGNETIC SPECTROMETER FOR CIL<\RGED PARTICLES 43

§ 2. Measurement of Magnetic-Field Nonuniformity

The topography of the magnetic field was plotted for two field values, Ho'" 2000 and 8000 Oe. The measurements were made with a magnetic-induction meter of the IMI-2 type, the operation of which is based on nuclear magnetic resonance.

For fields of ..... 2000 Oe we used a standard detector 37 x 17 mm in area and 11 mm high attached to the apparatus. The working substance in this was water. The error in measuring the field depends on the degree of nonuniformity. In our case for the central part of the magnetic path the accuracy of measuring the magnetic field was ..... 0.03%. At the ends of the path, where the nonuniformity reaches 0.2% over 1 cm, the error rises to 0.1%.

Large fields ..... 10,000 Oe were measured with the same apparatus, using a hand-made detector. Structurally the detector was made in the form of a cylinder 8 cm in diameter and 5 cm long. The glass ampoule containing the working substance occupied a volume of 2 x 1 x 1 cm. As working subtance we used heavy water D20 with a trace of Fe(N03h. In this case the apparatus allows measurement of the field strength for a nonuniformity not exceeding 0.02% per 1 cm.

The magnetic field at the edges of the pole tips was measured with an IMI-3, based on the Hall effect. The inaccuracy committed in measuring the magnetic field with this apparatus has the form

(17)

The detectors were fixed rigidly on a platform enabling them to be moved with respect to magnet aximuth, radius, and height in the interpole gap. The main series of measurements was carried out for a field of H ..... 2000 Oe. These showed that the magnetic field was constant to the required accuracy (better than 1 ·10-~ along the radius in the median plane (z = 0) at various azimuths over a magnetic path of .6.ro = r- ro = ±9 cm.

In order to broaden the working range of the magnetic field, we shimmed the pole tips near the side faces. We obtained the topography of the magnetic field for several sets of shims. The variation in the strength of the uniform magnetic field along the radius for one particular azimuthal angle (J is shown in Fig. 8 for various kinds of shimming (analogous results were obtained for 12 values of azimuthal angle). The best version was that given by shims 2 mm thick at the outer and 3 mm at the inner arc of the magnetic path for a width of 40 mm. We also checked the quality of the shimming for H ..... 8000 Oe.

JH/Ho,%

0.5

D.J

-D.J

IJ -0.5

-20 -to o 10 Jr, em

20

Fig. 8. Nonuniformity of the magnetic field along the radius for an azimuthal angle of e = 24°24' with various shims. The figures on the curves show the thickness of the shims in mm. The broken lines give the boundaries of permissible values of .6.H/H.


tJH/Hu1 % ~--------------D.5'-----------------~

-D.J

-20 -10 o 19 29 dr,cm

Fig. 9. Relative variation in the magnetic field along the radius in various planes z/K for an azimuthal angle of e = 24°24'. Numbers of the curves give the values of z/K. The broken lines give the boundaries of the permissible values of,6, H/H.

As a result of the shimming indicated we were able to obtain a magnetic path,6,r = ± 13 mm wide in which the nonuniformity of the magnetic field never exceeded 1 . 10-3 • Within the range Ar = ± 16 cm" ,6, H/Hz 1 '10-2• We found that the magnetic field was not constant along the magneto-optical axis because the pole tips were not quite parallel. The maximum nonuniformity never exceeded "'0.3% on the actual axis and ..... 0.5% over the working part of the magnetic path. As already noted, the azimuthal nonuniformity of the magnetic field did no harm to its focusing properties. We therefore made no attempt to remove this nonuniformity.

We also studied the uniformity of the magnetic field in various planes with z ~ O. Figure 9 shows the relative variation in the magnetic field along the radius at a fixed azimuth for various z, expressed in units of the gap between the pole tips K. On approaching the plane of the pole, the field at the ends of the magnetic path changes considerably; however, in the region Ar ± 10 cm the uniformity of the field remains within acceptable limits.

CHAPTER III

MAGNETIC CHARACTERISTICS OF THE EDGE (FRINGING) FIELD

§ 1 . De t e r min at ion 0 f the E ff e c t i v e Mag net i c B 0 u nd a r y

In sector magnets the focusing effect is due to the edge or fringing field. It is therefore important to know the shape of this within the aperture of the entering (and leaving) beam of charged particles, the true entering (or leaving) angle of the central ray relative to the lines of equal field strength of the fringing field, and the shape and length of these lines. The fullest information regarding these characteristics may be obtained by plotting the whole topography of the fringing field, for example, by measuring the field strength with miniature induction coils or in an electrolytic tank.

However, this difficult measuring procedure may be avoided and the required fringingfield data may be obtained if we directly determine the effective magnetic face d x defined in the form of [51. Since the effective face is one of the lines of equal field strength, a knowledge of this face enables us to estimate both the sharpness with which the fringing field falls and the


E \ >, ! I 9 aDorn curvature of its boundaries. This method offers a more practical way of selecting the required curvature of the field boundaries when shimming the ends of the pole tips and regulating the sharpness of the fall in the fringing field by applying a diaphragm system. In addition to this, a knowledge of the effective

I _. ______ ____ __ ----__ __J

II face enables us to introduce an effective sharply-bounded magnetic field equivalent to the real one and calculate the focusing properties of the latter to a good approximation.

Fig. 10. Arrangement of the apparatus for determining the effective magnetic face. Rj and Rz are the internal resistances of the coils, R z ' Rsh , and R are resistance boxes, Rp is a potentiometer, and M-17 is a galvanometer.

In order to determine the effective magnetic pole face we constructed a special measuring apparatus consisting of two induction coils and an electrical circuit. The apparatus is shown schematically in Fig. 10. The length of the larger (first) coil Lj = 1100 mm ...., 10K is approximately ten times the interpole gap; the width a = 20 mm is constant to a high accuracy over the whole length. The area of the

coil is 2.2.104 mm2and the number of turns Wj = 10. The small coil II has a diameter of 4 mm and a number of turns W2 = 7200. The two coils are fixed on a common mechanical axis so that their own axes are parallel to one another and perpendicular to the axis of rotation 00'. The whole system can be rotated through 1800 by hand.

The extent of the fringing field [3) is determined by the height of the interpole gap of the magnet K and is roughly 5-6 times K. For measuring purposes the apparatus is therefore placed in the field of the magnet in such a way that one end of the large coil (Xj) is in a region of really uniform field H(xj) = Ho, and the other (x2) is practically outside the field H(x2) l::: O. The small coil is always in the uniform field Ho and serves to calibrate the large coil.

On rotating the whole system around the 00' axis, an emf is induced in coils I and II: Xz • Xz

El =j H(x)~awldx = Sl;) H(x)dx, (18)

EII = HO~S2' (19)

where cP is the angular velocity of the coil and Sj and ~ are the coil constants.

The ratio er leu determines the position of the effective magnetic face along the coil axis (x):

x,

2 = SSL1 ~ h (x) dx = SSL1 (dx + Xl), err 2 1 J 2 1

(20) X,

whence the effective boundary, reckoned from the edge of the pole tip, is

(21)

The voltage ratio is meas ured with a ballistic galvanometer of the M -17 type, using the compensation method (see Fig. 13). For a zero reading on the galvanometer, corresponding to the zero point,

nRp(Rl + R l) RZ(Rz '" Rpl

(22)


where nRp is the proportion of the resistance Rp from which the voltage on the galvanometer is taken. In other words

2. _ ~ = R l (R2 + R p) ElI - A ' where A Rp(R1+Rl) (23)

Thus the measurement of the effective magnetic face reduces to the determination of the value of n corresponding to the zero reading on the galvanometer.

The apparatus is fixed to a special platform, rigidly connected to the pole tip. The axis of rotation of the coils is set in a plane perpendicular to Ho. The axis may be moved parallel to itself in the horizontal and vertical planes and also rotated through an angle of ± 10° in the horizontal plane relative to a selected direction (the ion-optical axis of the magnet). The positions of the axes of rotation of the coils for measurements in the working system of coordinates x', y', ~ are indicated in Fig. 11. Experiments were carried out for pole-tip edges making angles of + 45° with ro at the entrance and -45° at the exit.

The small coil is always in a steady uniform magnetic field Ho for all different positions of the axis of the system (to an accuracy no worse than O.ltto). The large coil always embraces the whole range of the nonuniform field in the selected direction. It is clear that the position of the effective magnetic face ~f(Y', ~ = 0) cannot be determined to an accuracy better than ± 1 .10-3 L j ~ 0.1 cin, i.e., "'0.8% when the axis of the system is moved parallel to itself within

the limits of the magnetic path. The value of n may be measured to an accuracy of "'0.6%. However, when measurements are made for various angles ~ and a constant position of the small coil, the coverage of the fringing field by the' large coil varies, and for large + ~ the coverage is sufficient. The accuracy of determining thE: effective face dx (~, y') is in this case ~ 1.5%.

Measurements were made at intervals of oy' = 5 cm within the range b.y' = ± 25 cm and o~ = 1° within the range b.~ = ± 10°. The direction ~ = 0 (x' axis) made an angle of 45° with the normal n to the mechanical face of the pole tip. This choice was dictated by the desire to obtain the field characteristics in directions close to those at which the charged-particle beam entered and left.

Measurements were made for two values of magnetic field, Ho = 2000 and 9000 Oe. The effect of the shimming of the end faces on the shape of the field boundaries was studied. An effective face with a curvature corresponding to the minimum aberrations was selected by adding additional plates.

§ 2. Results of the Measurements. Shimming the End Faces

Figure 12 shows the position of the magnetic faces for Ho = 2000 Oe from the direction of the entrance and exit ends of the magnetic path of the spectrometer for the case of rectilinear pole-tip edges and additional trimming plates.

Figure 13 compares the position of the magnetic faces for two field values, Ho = 2000 and 9000 Oe. A difference in the shape of the faces only appears near the sharp ends of the pole tips, i.e., in a distinctly nonworking region.

The wedge-shaped pole trimmers, made of mild steel, were chosen in such a way as to obtain the geometrical shape of the effective face in the form of a circular arc (or a similar curve) with radii IR'1 > +100 cm and IR" I> +300 cm. In Fig. 12 the broken lines indicate the circles of limiting radii with a common tangent (parallel to the y' axis) at the point of their intersection with the x' axis. We see from the figures that in the range -20 cm < Y2 < 20 cm (b.y = ± 13 cm) the curvatures of the field boundaries lie between the limiting radii and hence satisfy the conditions of minimum aberrations.


Fig. 11. Position of the axis of rotation of the coils when measuring the position of the effective magnetic face. 1) Pole tip; 2) small coil; 3) large coil; 4) magnetic face.

!I

"? , , 7"=1670

/ ---:c' / II n

7"=ZOOO 7"=ZI30 Exit

~:EJ.'!-L.~~~~~~~~~~r=1600 11'

H=9000 H=ZOOO~~---r-----if-.~-~

/ r=Z130

/ /

:c' , n 7" "ZOOO 7"=1670

Fig. 13. Position of the magnetic faces for two field values Ho = 2000 and 9000 Oe at the entrance (I) and exit (II) ends of the magnetic path in the case of rectilinear pole-tip edges with trimming plates.

v'

~--~~-~-------n

Fig. 12. Position of the magnetic faces for Ho = 2000 Oe at the entrance and exit ends of the magnetic path for rectilinear pole-tip edges and trimming plates. The trimmers and the changes in face shape due to these are shown by the broken lines.

Figure 14 shows the magnetic faces at the exit side in various planes z. In all cases the shape of the curves remains relatively constant over the working region.

According to the measurements in the central part of the magnetic path, the effective magnetic face is parallel to the pole-tip edge and lies at dn = 13.5 cm = 1.125K from it at the entrance and dn = 12.5 cm = 1.04K at the exit. For a sector angle of 'Po = 50°15' between the mechanical boundaries at the points at which the central ray enters and leaves, the sector angles between the effective magnetic faces at the points at which the central rays enter and leave the effective uniform magnetic field is <Po = 61°30'. The angle at which the central ray enters the sharplydelineated effective field is eb = 46° and the corresponding angle at which it leaves is eH = -46°.

Z3~ __ a&==~===*==~ 1 / /

7"=2130 :c' 7"=ZOOO

/

n't 'r=1870

Fig. 14. Magnetic faces at the exit side for various planes z. 1) zl2K = 0; 2) z/2K = Ys; 3) z/2K = -Ys.


CHAPTER IV

FOCUSING PROPERTIES OF THE SPECTROMETER

§ 1. Method of Investigation

The creation of a magnetic field of reasonably high uniformity and the establishment of a fringing field of the desired shape led us to expect that the spectrometer would have excellent focusing properties.

The dispersion and aberrations of the magneto-optical system were determined directly, using a method based on an extended current-carrying conductor in a magnetic field [10].

The arrangement for plotting the trajectory of the electron beam in the spectrometer field is shown in Fig. 15. We used the following measuring apparatus. As current-carrying filament we took a copper wire 0.1 mm in diameter. One end of the filament was fixed in a movable stand enabling it to be moved in two orthogonal directions, thus imitating the geometry of the target (charged-particle source). The movement of the stand along the magneto-optical axis of the system also allowed it to be set at a specific distance from the entrance end of the magnet. Then the filament passed through the field region, and its other end was passed through a tensometer, which measured the tension. This apparatus was fixed to a special platform and had three mutually perpendicular movements: a vertical movement, in order to establish the position of the filament as regards height, and two other movements (as well as rotation around the vertical axis) in order to establish the trajectory of the filament in the desired direction. The tensometer was placed in such a position as to ensure the stable location of the filament in the steady magnetic field [10].

The ends of the filament were connected to a stabilized source of current in such a way as not to alter its tension. The position of the wire at the entrance and exit of the magnet was established by means of graduated scales of the mirror type, acting as markers. One marker (I) was placed on the side from which the beam entered the magnetic field (outside the latter) and served to establish the angle (± 0') between ~he trajectory of the filament and the original axial direction, which coincided with the direction of the primary beam of'Y quanta (imitating the aperture of the incident electron beam).

Three mirror marker scales (II, III, IV) established the direction of the wire on leaving the magnet and gave a clear indication of the linear parts of the trajectory outside the field, thus enabling the observer to find the focus of the system by subsequent linear extrapolation.

As indicated earlier [101, the method of the current-carrying filament is based on the following relation between the characteristics of the filament [current I (A), tension T (g)] and the energy of the particle imitated E (MeV) for a constant magnetic field Ho = const:

T E=2.939 j . (24)

The accuracy of determining the particle energy E in a given direction corresponding to the trajectory of the filament for Ho = const depends on the errors in the measurements of T and I and also on the stability of these quantities during the experiment. The agreement between the position of the filament and the trajectories of particles of energy E characterized by a current I and tension T depends on the accuracy of measuring the field Ho and the stability of this field during the experiment, as well as on the errors in reading the position of the filament in space and in the extrapolation of the data. Let us consider all the measured quantities individually.


V Fig. 15. Arrangement for tracing the electron beam in the spectrometer field. 1) Target; 2) balance beam; 3) Focal plane; I to IV) mirror reference scales; V) mirror with scale markings.

49

Magnetic Field Strength Ho. In the course of trajectory plotting the field was measured with an IMI-2 meter to an accuracy of 0.03%. The stability of the field was better than 0.1%.

F i I am e n t T raj e c tor y . When using mirror scales, the position of the filament is read by a noncontact method (by establishing visual coincidence of the filament, its image in the mirror, and the reading scale). The accuracy of reading the reference scale may be greater than 0.5 mm. The determination of the position of the focus and the corresponding errors are analyzed in detail later.

Cur r e n t I. The absolute value of the current is measured and its stability checked by means of a compensation voltmeter. The accuracy in determining I is better than 0.190 and its instability never exceeds 0.19'0.

Ten s ion T. The principle of measuring the tension of the current-carrying filament is simple: the filament is threaded through a pulley, and the tension T is balanced by a weight P. However, T = P +jT' wherejT is the frictional force. The accuracy of measuring the tension T = P may be raised by reducing the friction of the axle in its bearings. However, in addition to reducing the friction in the pulley the effect of the friction on the measurement of tension may be reduced by using a balance-beam system. One form of this system is proposed in [11]. We ourselves constructed and used two other forms of the same system, as shown in Fig. 16a, b. The principle of the apparatus may be seen from the figures. Here the small pulleys 5 and the large pulley 6 only serve to select the free length of wire and take no direct part in determining the tension. In both cases the pulleys are set in yokes (I, 2) which in turn rest by means of steel trigonal prisms (3) on supporting agate prisms (4) in the manner of the balance beam of an analytical balance. The yoke may be balanced so that the center of gravity of the system lies on the vertical passing through the center of rotation. The motion of this component along the vertical determines the restoring moment of the apparatus and hence its sensitivity. The development of a rotating moment in the yoke is due to the inequality between the tension and the weight of the load; it is associated with the difference between the rotating moments of the frictional forces in the bearings of the pulleys and the prisms. The deviation from the vertical is indicated by a pointer reading on a fixed scale [7].

The position of the filament at the exit from the field (corresponding to E = 2.9 T /1) is read and the tension T = P determined when the specified entrance aperture has been established by moving the tensometer in a direction perpendicular to the filament trajectory, and when at the same time the position of dynamic equilibrium of the balance beam has been restored by moving the tensometer in the direction of the trajectory.

T

50

o I Z cm L.....l.-J

V. N. MAIKOV ET AL.

T

9

b

o I Zcm ~

7

Fig. 16. Schematic view of the two forms of balance beams. 1, 2) Yokes holding the pulleys;

8

p

3) steel prisms; 4) agate prisms; 5), 6) pulleys; 7) stationary scale, fixed in a transparent casing not shown in the figure; 8) stand; 9) balance adjusters.

The accuracy of measuring the tension, i.e., the accuracy of the relation T = P, depends on the sag of the filament (specific gravity, diameter, and length of the filament and the value of T), on the sensitivity of the apparatus (frictional forces in the prisms), and on the degree of equality between the arms of the tensometer (in Fig. 16b, in contrast to Fig. 16a, equality of the arms means constancy of the radius). In order to smooth the results, T may be measured in two mutually opposed positions of the yoke relative to the vertical axis of the apparatus.

The absolute determination of the accuracy of the tensometer is a difficult matter, since we know of no other apparatus for measuring tension having a higher accuracy. Relative measurements with two tensometers under our present experimental conditions showed that the tension was being measured with a random error of DoT/T < 0.1%.

§ 2. Magneto-Optical Characteristics of the Spectrometer

The trajectory plotting was carried out under the following conditions. We chose an axial direction a = 0, by = 0, bz = 0, w = ° and ro = 200 cm, corresponding to Eo = 2.9To/Io for Ho• We determined the position of the filament for:

a) Ho = const, a = 0, by = 0, and various {3 by varying T with I = const.


b) Ho = const, (3 = const, by = 0, T = const, I = const, and various a ((3 being a parameter).

c) Ho = const, (3 = const, a = const, and various by (a, (3 being parameters).

This cycle of measurements was carried out for several values of the field Ho. Apart from measuring errors (AHo/Ho'" 1.5-10-2; AE/E '" 2'10-3) the particle energy E for the specified directions of the particles at the exit from the magnet was proportional to the field Ho.

In order to estimate the energy resolution of the spectrometer we made a special analysis of the trajectory-plotting results.

Let us introduce a coordinate system xy so that the x axis is directed along the median trajectory Yo (a = 0; (3 = 0, by = 0), and the origin lies on the effective pole face. In this system of coordinates, starting from a certain distance from the magnet at which the magnetic field may be neglected, the trajectory of some particle may be expressed in the form

(25)

while the scales of the markers delineating the trajectory are to a fair accuracy straight lines: xIII = 58.3 cm, xIV = 136.5 cm. As indicated earlier, at the first-order focus the y coordinate of this trajectory is determined by the following expression:

Let us transform this expression, using the fact that the coefficients Bt> B11 , B12 are linear functions of x, i.e.,

Then

BI = alx + bb

Bll ~ aux + bll ,

Bl2 = al2x + b12 • etc.

y = (alx + bl ) roa + (allx + bll) roa2 + (a12x + b12) roa~ + ... By comparing expressions [25] and [26] we obtain the following equations:

Ai = aur oat + allirOa~ + al2trOal~i + a2lrO~1 + a22IrO~21+ a3lbYi,

Bi = bur oai + bu1r oar + b12irOai~i + b2ir o~1 + b22ir,,~~ + b3ibyi '

(26)

(27)

(28)

The left-hand sides of equations [27] and [28] may be calculated from the experimental trajectory-plotting results. In introducing the coordinate system we used results obtained by measuring the effective magnetic pole face at the exit from the magnet: dn = 12.5 cm. Altogether we plotted more than a hundred trajectories for various values of the parameters 0', {3, by, i.e., there were more than 100 equations of the [27] and [28] type to be analyzed. On the assumption that random errors only occur on the left-hand side of the equations, while the values of the parameters a, {3, by are known exactly, the determination of the coefficients ali' alii' a12i' a2i' a22i' a3i and bH, bili , b12 i> b2i, b22 i, b3i is a problem which may easily be solved by the method of least squares. Strictly, mathematically speaking, the problem is formulated as follows: to determine the coefficients at> a2, ... , aN in an excess system of equations of the form

In our case n = 6.

n

Yi = ~ ajX2j (r = 1,2 ... N). j=l

52 V. N. MArKOV ET AL.

Since the correction to the magnetic field with respect to the radius and shape of the effective magnetic surface at the spectrometer entrance and exit was based on an energy range 1131 < 0.125, in order to estimate the energy resolution of the spectrometer it seemed reasonable to consider trajectories lying within this energy range. There were 60 such equations in all; all these contained a nonzero parameter 13, 38 equations contained a nonzero a, while only 12 equations contained a nonzero by. Since the accuracy of determining the coefficients depended on the value of the ratio Nln, the coefficients as and b3 of the parameter by were found to a lower accuracy (greater error) than all the other coefficients.

Knowing the coefficients a and bi, we may determine all the parameters characterizing the focusing properties of the spectrometer: the position of the focus, the dispersion, and the aberration.

The first-order focus with respect to the angle a is found from the equation

(29)

For our case, in which the source-target lies at 177 cm from the mechanical face (163.5 cm from the effective face), the image lies at (237 ± 15) cm from the effective face. At this point the dispersion of the apparatus is (1.8 ± 0.09) cm/%.

The width of the monoenergetic line of a point source

(30)

for a particle escape angle of la 1 :s 1 . 10-2 in the region of the focus and at a distance from the latter not exceeding the error in locating it (± 15 cm) is no greater than 0.6 cm.

For an extended source of diameter 8 mm (by = ± 4 mm) the total width of the monoenergetic line

in the same region is (1.2 ± 0.7) cm.

!I, cm---___________ --,

8

7

G

5

If

3

Z

'\. '\

'" '~,

o~~~~-a~~~~~~~~~~

'f0 80 120 180 200 2'f0 280 320 380 '100 'f"0 "80 Z; em

(31)

Fig. 17. Width of the monoenergetic line and dispersion of the spectrometer as functions of the distance from the effective exit magnetic face. a = ± 0.01; ro =200 cm; b = ± 0.04; CPo = 60°, 1) y = B1 aro + B11 a2ro + B3by; 2) Y = (B1a + B11a2) ro; 3) y = (~~ + B22~2) ro for ~ = 1 ,10-2; 4) Y = B3By.


On the assumption that the lines are regarded as resolved in energy if they are completely free from overlapping, we may estimate the energy resolution of the spectrometer for the case of a point source. This is in fact (0.3 ± 0.2)%, or for an extended source (by = ± 0.4 cm) it is (0.65 ± 0.69)%, i.e., better than 1.4%.

Figure 17 shows the width of the monoenergetic line of a point source at various distances from the effective magnetic face, and also the general form of the aberrations associated with the width of the source y = B3by and the total curve; the same figure shows the dispersion y =

~f3 + ~2f32 for f3 = 1 -10-2•

§ 3. Plotting the Trajectories of a Beam of Positrons

As indicated in the foregoing, provision was made for the construction of a pair 'Y spectrometer in order to check the constancy of the form of the synchrotron 'Y spectrum. From constructional considerations, and also from the point of view of the purpose of the 'Y spectrometer, it was decided that the 'Y spectrometer should have one positron and two electron channels connected to two coincidence circuits.

Since the energy "window" of the electron spectrometer is comparatively narrow ( ..... 30%), the simultaneous recording of recoil electrons with energy E I':; k and both components of pairs with the same total energy E+ + E_ = k imposes extremely rigorous conditions on the choice of positron recording energy. It is necessary that

Since a considerable proportion of the path of such positrons lies in the region of the fringing field, it is hard to calculate their trajectories, and hence even the preliminary estimate of the focusing properties of the positron counter was carried out by the method of the currentcarrying conductor described earlier.

The difficulty lay in the fact that, at the point at which it was proposed to place the positron counter, at the exit from the magnet, there was still quite a strong magnetic field, so that the trajectories were not rectilinear. Since the focus with respect to the angle a is usually determined by linear extrapolation, in order to avoid serious errors it is essential to place the point of suspension of the wire as close as possible to the focus, insofar as this satisfies the requirement of wire stability, and to determine its position at as large as possible a number of points. For this purpose the apparatus described in an earlier section was used. The front end of the filament was threaded through the balance-arm pulley simulating the target, and the rear end was fixed firmly to a stand capable of moving in three orthogonal directions.

In order to establish the required angle a between the trajectory of the filament and the axial direction, we used the same marker with a mirror scale as was used in tracing the trajectories of the electrons. In front of the rear support was a mirror with six scales (Fig. 15, V) lying at a distance of 4 cm from each other.

We made three series of measurements analogous to those described earlier for the electron beam, although it was clear that we could hardly expect good focusing properties in the present case, in which we were only using the arbitrarily-shaped edge of the magnetic path of the spectrometer. Considering the comparatively low energy of the positrons, the permissible angle a in the trajectory plotting was widened in comparison with the principal measurements to 3 '10-2

rad.

First we obtained a field of trajectories for a large range of energies ( ..... 30-90 MeV), and then, after selecting a site for the positron counter, we made a final evaluation of the trajectories falling into the counter window. This cycle of measurements was carried out with the

54 V N. MArKOV ET AL.

counter screen (which distorts the field near it and may affect the shape of the trajectories) in its final position.

According to the measurements, the dispersion was about 4 MeV fcm. The width of the monoenergetic line was very great, about 15 mm, so that the resolution of the counter could not be better than 10%. There was no vertical focusing at all at this point. Since the divergence of the positron beam was inversely proportional to the energy of the positrons, the efficiency of the positron counter fell with falling energy. These failings rather spoiled the characteristics of the 'Y spectrometer.

For a fixed direction of the positrons we verified the linear relationship between their energy and the magnetic field (AH/R = 1.5' 10-2; Llli/E = 2 . 10-~ and proved that this held up to a field of ,..., 10,000 Oe.

CHAPTER V

EXPERIMENTAL GEOMETRY AND RECORDING APPARATUS

A spectrometer with the properties described in the foregoing chapter is a component part of the apparatus used for measuring the Compton effect for an electron (see Fig. 20 on p. 28 of this book [12]).

A beam of 'Y quanta, passing through the collimator system 2 and scavenging magnets 3, falls on a thin target 4 composed of materials of low atomic number Z (CR, Be, C). The target lies in the vacuum chamber 6. Around the target, outside the chamber, is a series of counters 15 for recording the 'Y quanta scattered at angles of ,..., 45 and"'" 1350 • The primary flow of 'Y quanta is measured by the thin-walled ionization chambers 19. The components of the direct spectrometer are arranged in such a way as to make a maximum use of the "fan" of particles analyzed by the spectrometer. The electron counter 18 recording the Compton recoil electrons is set in the region of the best resolution of the spectrometer. The counter scintillator, 300 x 300 x 20 mm in size, is placed in the focal plane and covers 10% of the region of the beam. Lower down (as regards energy) are the electron counters of the 'Y spectrometer 17. The position of these is chosen so that the energy of the 'Y quanta which they record in coincidence with the positron counter 16 should be respectively equal to the upper and lower boundaries of the energy range of the electrons falling into the main counter. For a field of 9160 Oe the electron counters of the 'Y spectrometer record electrons over a range of 6 Me V for mean energies of 440 and 493 MeV.

The scintillators of these counters have dimensions of 60 x 15 x 2 mm. The positron counter set for coincidence with these counters is fixed to the front (with respect to the beam) wall of the magnet yoke. Since the magnetic field is large at the site of the counter, the latter is placed in a multilayer magnetic screen. For a field of 9160 Oe the scintillator of the counter, 50 x 15 x 6 mm in size, effectively records particles with energies between 62 and 68 MeV.

Thus the line width of the 'Y spectrometer is no greater than 12 MeV at a level of 505 and 558 MeV. As already noted, the efficiency of this spectrometer depends greatly on energy. All the electron counters are situated behind a thick (20 cm) lead wall in which there is an embrasure passing the whole fan of particles analyzed by the spectrometer.

A free beam is provided in the spectrometer for calibration measurements. The energy spread of this beam is no greater than 3%. The beam has a very small divergence, so that the apparatus being calibrated may be placed several meters from the spectrometer. The intensity


of this beam behind the 80 mm collimator set 5 m behind the spectrometer reaches thousands of particles per accelerator pulse.

CONCLUSION

The charged-particle spectrometer thus created enables us to record electrons simultaneously over the energy range ±16.5%, for a maximum average energy of 600 MeV. The minimum value of the stabilized magnetic field corresponds to an energy of about 90 MeV. The comparatively large range of particle values analyzed simultaneously facilitates extensive use of the apparatus for a wide variety of investigations.

We also studied the possibility of carrying out a magnetic analysis of particles of different sign with an accuracy of about 10% in the range 9-60 MeV. This should enable us to place a system constituting a two-channel pair 'Y spectrometer with an energy resolution of about 2% in the spectrometer complex, in addition to the main counter for recording recoil electrons produced by the scattering of 'Y quanta by an electron. This 'Y spectrometer forms an extremely sensitive system for studying the stability of the top accelerator energy.

We also provided a monochromatic electron beam for calibration measurements (for calibrating the total-absorption counters, telescopes, coincidence circuits, etc.). The intensity of this beam ( ..... 103 electrons/sec) was also sufficient to enable us to carry out certain other physical experiments [13].

In conclusion the authors wish to express their sincere thanks to V. A. Petukhov for interest and help in building the apparatus, V. V. Yakushin and M. 1. Blagov for constant assistance in the work, A. A. Lagar'kov and V. V. Gorshkov, who constructed the measuring apparatus, and also E. V. Pantyushkov for help in producing the illustrations.

LITERA TURE CITED

1. V. V. Yakushin, this volume, p. 84. 2. V. A. Petukhov, A. A. Komar, and M. N. Yakimenko, The Compton Effect and the Limits

of Applicability of Quantum Electrodynamics, Preprint of the Joint Institute of Nuclear Research, R-283, Dubna (1959).

3. L. A. Konig and H. Hintenberger, Z. Naturforsch., 10a:877 (1955). 4. A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, An Electron Beam for Calibrating

Cerenkov Spectrometers, report, P. N. Lebedev Physics Institute of the Academy of Sciences of the USSR, Moscow (1963).

5. W. P. Ploch and W. Walcher, Z. Phys., 127:274 (1950). 6. H. Hintenberger and L. A. Konig, Z. Naturforsch., lla:1039 (1956); 12a:140 (1957); 12a:377

(1957). 7. W. Cross, Rev. Sci. Instr., 22:717 (1951). 8. V. N. Maikov and M. N. Yakimenko, Focusing Charged Particles in Homogeneous Sector

Type Magnetic Fields. Focusing Properties of a /3-Spectrometer with cp = 60° and ro =

2000 mm, report, P. N. Lebedev Physics Institute of the Academy of Sciences of the USSR, Moscow (1960).

9. A. A. Kolomenskii, A. B. Kuznetsov, and N. B. Rudin, Rotational Focusing System for Introducing Particles into a Synchrotron, Preprint of the Joint Institute of Nuclear Research, R-250, Dubna (1958).

10. M. S. Kozodaev and A. A. Tyapkin, Pribory i Tekhn. Eksperim., No.1, p. 21 (1956). 11. A. Citron, F. J. M. Farley, E. G. Michaelis, and H. Veras, Floating Wire Measurements

on the SC Magnet, CERN 59-8, Synchrocyclotron Division (Feb. 20, 1959).


12. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlovskaya, this volume, p. 1.

13. F. R. Arutyunyan, K. A. Ispiryan, A. G. Oganesyan, and A. A. Frangyan, Zh. Eksperim. i Teor. Fiz., 4:277 (1966): Zh. Eksperim. i Teor. Fiz., 52:1121 (1967).

EXPERIMENTAL METHOD OF DETERMINING

THE EFFICIENCY FUNCTION OF AN APPARATUS CONTAINING A MAGNETIC SPECTROMETER

V. F. Grushin and E. M. Leikin

In this article we shall consider a physical system for recording charged particles formed as a result of the interaction of a beam of primary particles with target nuclei. Between the target and the detector is a magnetic spectrometer in which the secondary particles are analyzed with respect to momentum.

In order to analyze the results of such experiments quantitatively, we must know the efficiency function of the apparatus, i.e., the relation between its solid angle and the momentum of the particles being analyzed. The solid angle of the whole apparatus is determined by the entrance aperture of the magnetic spectrometer, which in turn is given by the dimensions of the entrance aperture of the particle detector, the mutual geometrical arrangement of the target, spectrometer, and detector, and also the focusing and analyzing properties of the magnet. In general, the finite dimensions of the target also have to be taken into account.

In this article we shall describe an experimental method of determining the efficiency function; this method was used in [1] for measuring the differential cross sections for the formation of n+ mesons in hydrogen by 'Y quanta from the bremsstrahlung of a synchrotron.

§ 1. General Presentation of the Problem

Figure 1 shows the main components of the apparatus schematically.

The spectrometer employed has a homogeneous magnetic field of the sector (wedge) type, the focusing properties of which have been fully studied theoretically [2-4]. Magnetic spectrometers of this type have found wide application in experiments with high-energy particles and have a number of important advantages. They are simple from the point of view of construction, have a good transmission, and focus a charged-particle beam in two planes, horizontal and vertical.

The secondary-particle detector is placed behind a protecting wall, a rectangular aperture (diaphragm) in which serves as entrance aperture for the detector. We used a positive pion stopping detector [1], which recorded those particles in a specific energy (momentum) range specified by the thickness of the filter and stopping counter used.

The geometrical dimensions of the entrance aperture of the detector are determined by a) the dimensions of the detector itself, or more exactly the transverse dimensions of its counters, b) the specified momentum range of the recorded particles, c) the dispersion of the magnetic spectrometer at the position of the diaphragm.

57

58 v. F. GRUSHIN AND E. M. LEIKIN

Fig. 1. Schematic arrangement of the elements of the apparatus. The continuous line with arrows gives the axis ofthe primary particle beam; the broken lines are the trajectories of the secondary charged particles. 1) Target; 2) entrance aperture of the spectrometer; 3) magnetic spectrometer; 4) entrance aperture of the detector; 5) particle detector; e) average escape angle of the recorded particles.

The horizontal dimension of the entrance aperture of the spectrometer is (generally speaking) chosen independently of the horizontal dimension of the detector diaphragm. We have to seek a compromise between the tendency to increase the solid angle of the apparatus, on the one hand, and to ensure a paraxial beam of analyzed particles on the other. The vertical dimension of the entrance aperture of the spectrometer is directly associated with the vertical dimension of the diaphragm and the focusing properties of the magnet in the vertical plane.

The problem of determining the efficiency function of the apparatus under consideration for a specified shape and size of the entrance aperture of the detector arises in connection with the necessity of knowing the exact form of the relationship between the number of particles falling into the detector and their energy or momentum. This characteristic is usally expressed in terms of the solid angle of the apparatus, i.e., ~(p).

The finite target dimensions are allowed for by summing the set of efficiency functions obtained for different points of the target (averaging over its working volume). These elementary or "point" efficiency functions Wi (p) may be calculated theoretically and determined experimentally by a variety of methods.

In order to calculate the "point" efficiency functions, we must carry out an exact calculation of the trajectories of the analyzed particles. This introduces serious complications,associated with the possible existence of nonuniformity in the magnetic field of the spectrometer and also with the presence of fringing fields. In our case, the importance of obtaining precise physical results led us to choose an experimental method of determining the functions Wi (p).

There are two very widespread experimental methods of solving this problem. The first of these (see, for example, [5]) is based on the use of a "point" radioactive charged-particle source, which is placed at different target points; the detector particle count rate is studied as a function of the coordinates of the source. This method has a number of failings, the chief of which are the following: First, the detector usually lacks the universal capacity to record particles of different natures and very different energies in the same way; secondly, the particles emitted by the source are monoenergetic, which eliminates the possibility of studying the dispersion of the magnet; also the checking conditions are not the same as those encountered in practical experiments.

EXPERIMENTAL METHOD OF DETERMINING THE EFFICIENCY FUNCTION 59

a

b

c s

Fig. 2. Schematic illustration of the focusing of a charged-particle beam by a sector magnetic field (horizontal projection); t represents the spread of the image of a point source in the case of a monoenergetic beam.

The second method, plotting the trajectories of the analyzed particles with a current-carrying wire [6], is free from these faults. The difficulty of the method is justified by the accuracy which it gives. One version of this method used in the present investigation is detailed below.

It should be noted that the experimental simulation of the trajectories of the particles passing through the magnetic field enables us to study the optical characteristics of the magnetic lens formed by the spectrometer in question. The focusing and analyzing properties of the spectrometer are illustrated in Fig. 2, which represents the projections (in a horizontal plane) of the trajectories of particles emitted by a point source situated outside the magnetic field. Figure 2a corresponds to the case of the ideal focusing of a monoenergetic beam, for which a point is imaged as a point (stigmatic image). Figure 2b explains the origin of astigmatism due to the nonparaxial nature of the beam of monoenergetic particles and also to the departure of the magnetic field from ideal conditions (uniform with sharp boundaries).

Figure 2c generalizes this version for the case of a beam of nonmonoenergetic particles. The dispersion of the system is determined from the image spread S resulting from the momentum spread of the incident particles ~p. If the magnetic lens is long-focusing and the value of ~p is reasonably small, the image spread takes place almost in the direction of the y axis of the coordinate system indicated in the figure. Then the ratio ~y/~p is usually called the dispersion. For convenience we shall in future call the reciprocal of this, W = ~p/~y, the disperSion, and express it in units of MeV Ic/cm.

Let the detector diaphragm be situated near the image of the source, oriented parallel to the y axis, as in Fig. 1. In this case it is not difficult to predict the form of the efficiency function wi (p). If the transverse dimension of the diaphragm in the horizontal plane Dh exceeds the spread t of the image of the point source (monoenergetic beam), then Wi (p) will have the form of a trapezium.t The smaller base of the trapezium has a size of 2(.0.P)1 =W(Dh - t) and the larger of

t We suppose that W = const within the limits of D .

60 V. F. GRUSHIN AND E. M. LEIKIN

2~ph = W(Dh + t). It is natural to associate the solid angle Qi cut off by the beam of particles subsequently passing into the diaphragm with the height of the trapezium. In the case of Dh ::::: t the function Wi (p) takes the form of a triangle. Thus the problem of determining the "point" efficiency functions wi (p) reduces in essence to one of finding the values of W and Qi corresponding to different points of the

a target.

b

Fig. 3. Focusing of particles in a) horizontal and b) vertical plane.

Before passing on to the direct description of the experimental method of determining these quantities, we must briefly consider some general questions regarding the theory of focusing particle beams with a sector magnetic field. The main theoretical formulas will later be used for making a priori estimates of the errors in Wand Q i' In addition to this, it is interesting to compare the theoretical calculations with experimental results derived from trajectory plotting.

§ 2. Focusing of a Beam of Particles with a Uniform Magnetic Field

of the Sector Type

A large number of papers have been devoted to this subject (for example, [2-4, 7]). Here we shall set out the main results concerning the nature of particle focusing in two different planes (vertical and horizontal) and the possibility of achieving double focusing. The solutions may be obtained in analytical form on the assumption that the magnetic field is absolutely uniform with sharp boundaries.

1. F 0 c us in gin a H 0 r i z 0 n t a I P I a n e. Let us consider the general case in which a particle source of finite size is placed at a distance Z' from the front boundary of the magnetic field (Fig. 3a). We consider the trajectory of particles with momentum p and initial conditions (e1' d1). After passing through the magnetic field, the particles will move in a direction specified by the conditions governing their emergence from the far boundary of the field (e2' d2). Then the trajectory of particles with momentum p + Ap emerging from a point at a distance q from the previous and moving at an angle of a to the latter will have a displacement S at a certain distance zg beyond the magnet.

It was shown in [2] that the value of S may be expressed in explicit form t in terms of the parameters just mentioned:

S = Cl {COS III COS B2 [1- (l' sin u _ cos (tD - Btl) • (it sin u _ cos (tD - B2) )]}_ Sill U COS2 £1 COS Bl COS2 B2 COS e2

_ {COS Bl (ihSin u _ COS (tD - B2))1. --L 2 Llp {COS~SintD/2 [cos (e _ <D/2) + cos (e _ rD/2) (itSin u _ cos (eI> - 82) )]} q cos 82 COS2 82 cos e2 J I P. Sill u 1 2 COS2 e2 COS e2 '

(1)

t In this expression we only consider terms of the first order in a (Gaussian optics).


where u is the angle of the magnetic-field sector and <I> = ej + f:2 + u is the angle through which the magnet rotates the particles. All distances in formula (1) are expressed in units of p, the radius of curvature of the trajectory of particles with a momentum p.

The appearance of astigmatism is associated with the fact that the coefficients of 0' and q cannot vanish simultaneously. If we equate the coefficient of 0' to zero, we obtain the condition for the focusing of the particle beam in the horizontal plane:

1/~ = tan(CD-'iJ)-tane2}

tan 'iJ= tan 81 + 1/1'

tan 8, + 'II' + tan E2 + 'Jlh or tan CD = ---------"i;-

1-(tane1 + '/l')(tan e2+ 111~ An analogous condition was also obtained in [3].

(2)

On allowing for second-order aberrations in 0' we obtain a transverse spread of the image of a point source [2];

1 2 {(' cos e1 co, E2 l2 \" ". 1 t = -2:Yo P ll" (<D) -cos <1)1(1 + lh tan82) + lhStnU~. \ Slil u - cos e, cos - e1 . ) J

( 3)

The third term in formula (1) enables us to find an expression for the dispersion of the magnet at a distance lh' Then

II" = ~P = L [(1- cos <1) (1 + lh" tane2) + lhsin (1)]1. LJ.!I P

( 4)

2. Focusing in a Vertical Plane. rhis case is shown schematically in Fig. 3b. We consider the trajectories of monoenergetic particles in the median plane of the magnet, not experiencing vertical focusing, and also the trajectories of particles with initial conditions a, /3, where a is the initial displacement and f3 is the escape angle relative to the median plane.

After passing through the magnetic field, the latter particles will be displaced relative to the median plane by an amount Y(I ~), where Z ~ is the distance from the front boundary of the magnet. This displacement may be expressed [2] in terms of the same parameters as in formula (1):

( 5)

The vanishing of the coefficient of f3 again gives the focusing condition in the plane under consideration:

" 1 11lv = tane 2 - i. 1 ) . (6)

<D-1NallB,- r The second-order aberrations and the dispersion of the magnet in the vertical plane may

be considered in the same way as before.

3. Do ubi e F 0 c us i n g . By solving equations (2) and (3) simultaneously we may obtain the conditions for double focusing, i.e., focusing the particle beam in the vertical and horizontal planes at the same distance zg from the rear boundary of the magnet. Then we have:

" 1 11ld =tan(Q) - 'iJ) - <D - I/ttan ec 1/1')'

(7)

Double focusing is possible for values of the parameters ej, I', and <I> such that the expression on the right-hand side of formula (7) is positive.

4. Characteristics of the Optical System. It follows from formula (2) for the horizontal focusing that lh is a function of ej for constant l'. We have already noted this


fact (Fig. 2b) as the phenomenon of astigmatism, which is due to the fact that the beam of incident particles is not paraxial. However, in analogy with the g'9ometric optics of light rays, the spread in the image of a point source (or Alb ) is in practice negligibly small if narrow beams are employed (small A£'j).

The situation is different for vertical focusing. The corresponding expression (6) for l~ also contains (£'j), but the relationship l~ (£j) may be a much stronger one than lI; (£'j). The choice of initial conditions (£'j, A£'j, l', etc.) for the specified magnet parameters (u, p) is therefore very important.

Before confirming this assertion with an example, we must note that at the very beginning of the present investigation we chose such initial conditions (£'10 d j , l'), as would correspond to our subsequent quantitative trajectory plotting aimed at determining Wand Qi' The initial conditions in the horizontal plane were chosen in accordance with subsequent requirements, associated with the particular features of the detector employed. First, owing the limited transverse and considerable longitudinal dimensions of the detector, the magnetic lens has to be fairly longfocUSing so as to avoid severe divergence of the beam near the image. Secondly, the required dispersion of the magnet near the image of the target is strictly related (§ 1) to the size of the detector diaphragm and the energy range of the particles recorded. Finally as the first stage in the work we took the following parameter values: l' = 1 m, d1 = 20 cm, £1 = 55°. t

Let us now compare the astigmatic effects in two different planes predicted by the foregoing formulas for the conditions chosen. The horizontal entrance aperture of the spectrometer was ~£1 = 4°. Figure 4 shows the resultsofacalculationof the relationships It (£1) and l~ (£1)

for the value of A£,j indicated. The negative values of l~ signify defocusing of the beam in the vertical plane. We see from Fig. 4 that for ~£1 = 4° the values of ~lR/lg and ~l~/l~ are respectively 0.11 and 4.0. This means that the particle beam has only been made sufficiently paraxial in the horizontal plane for horizontal stigmatic focusing. In order to reduce the astigmatism in the vertical plane, the value of A£'j must be greatly reduced.

It is natural to expect that, under certain different initial conditions, we might achieve stigmatic focusing in both planes without having to reduce A£'j' However, under such conditions we could in practice never satisfy the demands imposed upon the optical power and dispersion of the magnetic lens, all the more so because the non ideal nature of the spectrometer field would lead to a difference between the computed and experimental data. In § 6 we shall specially consider a comparison between the theoretically expected results (in particular those indicated in Fig. 4) and the experimental data. As regards the characteristics of the optical system, these should be compared with the aberrations associated with asymmetry of the system, which are already well known from geometric optics [8J.

§ 3. Determination of the Energy Dispersion

As already indicated, we determined the parameters of the point efficiency functions Wi (p) by plotting the trajectories of the particles under analysis, using a current-carrying conductor.

The thin, almost weightless conductor carrying the current I is placed under tension (tensile force T) between the poles of the magnet. When the magnetic field is switched on, it acts on the wire, forcing it to bend and follow the arc of a circle in the interpole gap. In the case of a uniform magnetic field H, the radius of curvature p may be obtained from the condition of equilibrium of the wire:

t The spectrometer had pole tips of the sector type with an angle u = 50°; the air gap of the magnet was 6 cm; the maximum field in the gap reached about 8000 Oe.


l;' em 300 ZOO

100 I~ r-- I

0

~'; -100 -ZOO

-300

-/f00 -500

-GOO

Fig. 4. Calculated astigmatism of the magnetic lens in two mutually perpendicular planes. 1) Horizontal plane; 2) vertical plane.

where c is the velocity of light in vacuum.

____ 8~: ~II 1)( I I 8 'Cf..~r 1i I I I / /

~p 1 OE'-/ 11

Fig. 5. Schematic arrangement of the pendulum -suspended load.

(8)

On the other hand, it is well known that a particle carrying a charge e and having a momentum p in a magnetic field H moves alpng a circular arc of radius

I pc P =eJj' (9)

It may be rigorously shown [9] that the trajectory of a charged particle coincides at every point with the line representing the stable position of the conductor if their initial conditions are the same at the front boundary of the magnetic field. Thus by equating the right-hand sides of expressions (8) and (9) we may determine the momentum of the particle the trajectory of which is given by the conductor in question. If for convenience we express the tensile force T in grams and the current I in amperes, the momentum of the particle in MeV Ic is

p = 10-7 cg ~ = 2.942 ~ , (10)

where g is the acceleration of gravity in cm/sec2• The numerical factor corresponds to the latitude of Moscow.

For tracing purposes we used a copper wire 0.1 mm in diameter carrying a current of 0.75 A. The wire was put under tension by the pendulum suspension of a load from one of its ends, the other end being fixed rigidly at one point of the target. The weight of the wire employed was under 0.2% of the tensile force and had hardly any effect on the results of the trajectory plotting. Figure 5 shows the arrangement used for determining the force on the wire in schematic form. An extended part of the wire was fixed at the point A on a vertical Plexiglas plate and a balance pan with variable weights was suspended from the point B.

The value of the tensile force T acting on the wire depends on the Q between the direction of the extension and the direction of the main part of the wire; for Q = 450 it coincides with the weight of the load P. If we keep the angle Q = 45 0 constant (by adjusting the whole attachment) we may create a tension of any prescribed value. This method has a great advantage over that of applying the tension to the wire through a light pulley, because the tension in such a pulley introduces an indeterminate error into the determination of T.


Trajectory plotting was carried out from various points of the target to a fairly large distance beyond the spectrometer, at which the stray magnetic field was so weak that the trajectories were practically rectilinear. The tracing procedure consisted of determining the projections of the trajectories of particles with a specified momentum on the horizontal and vertical planes.

The results of the horizontal tracing enabled us to determine the energy dispersion of the spectrometer Wat the site of the diaphragm of the particle detector. Since the horizontal focusing action of a sector-type field does not (generally speaking) depend on the vertical displacement of the particles, it was sufficient to plot the trajectory in the median plane of the magnetic field.

It was found after repeated horizontal trajectory plotting for every trajectory that the value of W was almost independent of the displacement of the point of the source and the incident beam, within the horizontal aperture of the spectrometer. The value of W for the case in which the momentum of the particles traced corresponded to p = 1 m was 1.4 Me V /c/cm.

§ 4. Determination of the Working Entrance Aperture of the

Spectrometer

In view of the characteristics of the optical system just mentioned, let us consider the mechanism of vertical focusing in more detail.

The focusing of a beam of particles in the vertical plane (Fig. 3b) is associated with the action of nonuniform fringing fields acting as two thin lenses [4]. Particles which move outside the median plane of the magnetic field of the spectrometer under the conditions £'1 ;e 0, £'2 ;e 0, experience a deflection at both ends of the magnet in a direction perpendicular to the median plane. This is associated with the "barrel" shape of the fringing field, which has a nonzero horizontal component. The edge or boundary lenses may be either converging or diverging, depending on the signs of the angles £'1 and £'2' In our case the choice of the initial conditions made the first lens focusing (£1 = 55 0 > 0) and the second defocusing (£'2 = -56 0 < 0). The total effect with respect to the vertical was weakly defocusing.

As mentioned in § 2, the relationship 1; (£1) characterized the vertical focusing. Of course, if when the particle beam passes out of the magnetic spectrometer, it is diverging with respect to the vertical (Fig. 6), then for a fixed l~ the vertical displacement of the trajectory Y will be a quantity depending on £'1 (on the assumption that h is constant within the limits of D.£'1)' This is also evident from formula (5) for Y, which corresponds to Fig. 6 if we put a = 0 and l~ < O.

In order to determine the entrance aperture of the spectrometer we shall be interested in the relationship h( £1) or 2f3( £1) for a fixed l~ and a specified detector diaphragm size Dv' The problem is thus solved as follows.

Using the system indicated in Fig. 5 (adapted for vertical displacement), we traced the required trajectories in the vertical plane. We determined the variation of Y with the vertical entrance angle f3 for a given l'~ experimentally, taking various values of £'1 and various points of the target. We note that, according to the coordinate system shown in Fig. 6, a change of D.£'1 in the angle £'1 corresponds to a change of x within the limits of ±Xo at the spectrometer entrance (y = 1', z = 0). The coordinates Xi' Yi' zi correspond to various starting points on the trajectory; the range of variation of these is limited by the size of the target.

We see from formula (5) that the relationship Y(f3) is linear. On varying the coordinate Zi of a point target, the straight line Y i (f3) moves parallel to itself. We found experimentally that to an accuracy better than ± 1% this displacement was equal to the displacement of the initial tracing point within the limits of the vertical dimensions of the target (± 1 cm). Thus the total


Fig. 6. Schematic form of a beam diverging vertically. Origin of coordinate.s (x = y = z = 0) coincides with the central point of the target.

V,cm--------------~

5 3

19.B j3, mrad

Fig. 7. Experimental results of vertical tracing obtained for one of the target pOints. 1) (£ 1) min; 2) (£ 1) av; 3) (£ 1) max'

Fig.8. Experimental entrance aperture of the spectrometer, corresponding to the median point of the target (front view).

vertical aperture angle of the beam 2f3 corresponding to the displacement ±Y = Dv /2 is independent of the target coordinate zi' This fact enables us subsequently to restrict vertical tracing to trajectories starting from points in the median plane of the magnet zi = O.

Here it should be noted that, according to experiment, the dispersion of the magnet in the horizontal plane (W) was two orders lower than the dispersion in the vertical plane (.6.p/.6.y). For this reason the value of 2f3 was almost independent of the momentum of the particles being traced, within the limits of the range of momenta separated out by the detector (± 5 MeV/c). Hence the vertical tracing had subsequently only to be carried out for a single (average) momentum Po.

Ultimately the vertical tracing procedure reduced to obtaining experimental data regarding the Y(f3) relationship by varying the target coordinates (Xi, yd and the entrance angle c.j over the desired range. By way of example, Fig. 7 shows results obtained for the middle point of the target (x i = 0, Y i = 0). The three lines shown correspond to the average and two extreme values of the entrance angle c.j.

By analyzing results of this kind we determined the values of 2f3(xiJ Yi, £1) corresponding to ± Y = Dv /2. We then allowed for the longitudinal and transverse dimensions of the target by averaging 2 f3 (Xi, Yi, C.1) over the range of variation of Xi and Yi' We used the averaged values of 2f3( £1) to calculate the area of the entrance aperture of the spectrometer

+x, s= ~ l'213(x)dx= ~h(x)dx. (11)

-Xli -XQ

The shape and size of this area are indicated in Fig. 8.


Finally, using the value of § obtained, we calculated the desired values of the solid angles of the apparatus from the formula

(12)

where l' means the distance to the middle point of the entrance aperture of the spectrometer.

§ 5. Accuracy of the Method

The accuracy with which the efficiency function of the apparatus may be found by the method indicated is conditioned by the existence of errors in determining the parameters of the "point" functions wi (p), namely, the dispersion Wand the solid angle Q.

Let us estimate the accuracy of these parameters.

The error in the quantity W = ~p/~y includes the experimental errors committed indetermining the range of momenta ~p and the corresponding displacement of the trajectories ~y (in the horizontal plane).

The mean square error in the quantity W may be found from the formula

2 _ ( 1 )2 2 '1/2 .. 2 Ow - (';.y 0Ll.P +" ULl.~, (13)

where the symbol (J means the absolute and the symbol {j the relative error in the corresponding quantities. The error (J6.p has a systematic character and is associated with the indeterminacy of the momentum of the particles being traced. According to formula (10), remembering that T = P /tan Cli, we obtain

(14)

The load P was known to an accuracy of ±0.1%. The error in the angle Cli was mainly de-· termined by the accuracy of laying the sighting lines on the Plexiglass slab and equalled :s 0.1%; the error introduced by visual inspection carried out in drder to check the coincidence of the extended part of the wire with the sighting line, using a mirror scale, was still smaller. The accuracy of the absolute value of the current I in the wire was limited by the class of ammeter used (in our case an M-104 of class 0.2). Thus the error ~ was no greater than ±0.25%.

As regards errors in the quantity ~y, these were related to the accuracy of determining the coordinates of the trajectories of the particles being traced. The coordinate y was measured to an accuracy of ± 0.1 mm. However, the main error in the coordinate was associated with the indeterminacies of the entrance parameters characterizing the motion of the particle, since the character of the trajectory beyond the magnet depended on these. In order to find this relationship we may use Fig. 3 and, in accordance with the notation indicated, write the equation of the trajectory beyond the magnet in the form

(15)

where the quantities e2 and d2 are functions of the original parameters ej, d j , and P.

In this case the absolute error of an individual measurement of the coordinate y may be estimated from the formula

(16)


Considering that the error in the quantity dj is only associated with the error in the entrance angle £.j (for a fixed position of the point particle source relative to the magnet), we may show that (ad/a£.j) == l'/cos £.j. Then the previous formula may be conveniently rewritten in the form

y[IOY)"( ('0828 1) ] (OY)" ay = ± \OE1 1 + (f52 cr~, + ap cr;. (17)

The explicit calculation of expressions oy la £.1> 0 y la d1• and a y lap usually is not very difficult; however, since the result is extremely cumbersome, there is no need to give it here.

Formula (17) relates the standard deviation of the average coordinate y to the inaccuracies committed in reproducing the quantities £.1 and p in repeated plottings of the same trajectory.

In the present method, the reproduction of the angle £.1 in successive measurements was effected to a high accuracy, thanks to the use of a mirror scale (OC1 ,.., ± 0.01%). Hence the spread relative to the average coordinate y was chiefly due to the spread in the value of the radius of curvature p of the particle trajectory in the magnetic field. This meant that formula (17) was practically converted into the relation

~ _ + oy " Vy - _ apvp • (18)

The spread in the quantity p in turn was due to inaccuracy in reproducing the particle momentum p and magnetic field H. The reproducibility of H was checked with a coil rotating in the field, connected to a ballistic galvanometer. These data enabled us to estimate the relative spread in the value ofp, which proved to be about ±0.3%. Then an estimate of u y from formula (18) gave about ±1.5 mm.

The foregoing estimate agreed closflly with experimental results. For a ten-times-repeated tracing of the same trajectory, the maximum scatter in the coordinate y at the site of the image was ± (2 to 2.5) mm. The value of the coordinate y averaged over the ten measurements thus contained an error of about ± 0.5 mm.

In order to estimate the error in the dispersion W, we now put the following values in the original formula (13): up == op • Po == 2.5.10-3 • 180 MeV Ic = 0.45 MeV Ic; 6.y = 10 cm; W = 1.4 MeV Ic/cm; ot::.y = 20 y = 0.01. Finally we obtain

Ow = ± 0.05 MeV/clem,

i.e., the accuracy of determining the value of the spectrometer dispersion equals ± 3.5%.

Let us now turn to estimating the accuracy with which the solid angles Qi were determined. We see from formula (12) that the error in the quantity Qi is associated with indeterminacies in the distance n and area So Since the distances E'i were known to a high accuracy (about 0.1% for an average l' = 100 cm), the error in Qi was entirely determined by the error in S.

It was indicated in the previous section that the value of S was calculated from averaged values of 2i3 (£'1). The main contribution to the error in S thus comes from the scatter in the values of 2f3 relative to 2i3, which equalled about ± 2.5% (the greatest scatter was associated with the variation of the coordinate xi). The error in calculating the actual values of 2{3(x i • Yi' (1)

was about half this, and was determined by the error in measuring the displacement Y and the scatter of the experimental points (see Fig. 8) relative to the straight lines Y(f3). An estimate of the error in the slope of these straight lines carried out by the method of least squares gave about ± 1%.

It should be noted that the procedure for vertical tracing with the system illustrated in Fig. 6 involved the introduction of an error into the value of the momentum being traced owing


TABLE 1

W.MeV/e/cm I t, em t;,.l"/l" h 'h t;,.l:V/ l V

2.5 0.95 0.11 4.0 1.4 1.20 0.20 1.5

to the infringement of the condition T = P (the current-carrying wire failed to coincide with the normal to the direction P). However, this did not lead to any serious indeterminacy in the value of Y (as compared with Dy), since the deviation was no greater than 0.02 rad, while the dispersion of the magnet with respect to the vertical (top/toy), as already mentioned, was extremely large, about 100 MeV le/cm.

Thus the method in question enabled us to determine the values of Q1 to an accuracy of ± 2.7%. The resultant solid angle of the apparatus, referred to the middle point of the target xi = Yi = zi = 0, was no = (1.15::1: 0.03)'10-3 sr.

Summing up the whole question of the attainable accuracies of the parameters of the point efficiency functions wdp), we note that the errors in Wand Di mentioned still fail to reflect their contribution to the error of the physical result obtained with the apparatus under consideration. In calculating the efficiency function of the apparatus Q(p), allow ing for the dimensions of the target, the error in W is in practice "concealed" by averaging the functions wt(p) , and only the error in the solid angle fully contributes to the systematic error in the differential cross section of the process being studied (see [10]).

§ 6. Comparison with Theoretical Calculations

In conclusion, it is interesting to compare some experimental results on the focusing and analyzing properties of a sector magnetic field with the data calculated from the formulas of § 2.

In order to be specific, let us make the comparison under the particular conditions e1 = 55°; d1 = 20 em, u = 50°; l' = 1 m; ~e1 = 4°. The results of the comparison are given in Table I (upper line - calculation, lower line - experiment). Here W is the energy dispersion of the magnet at the site of the image; t is the transverse dimension of the image of a point source, associated with second-order aberrations; tol~/t~ and tol~/l~ are parameters characterizing the effect of astigmatism in the horizontal and vertical planes.

The difference between the calculated values and those obtained experimentally from trajectory plotting serves as an indication of the way in which the real field differs from the ideal one assumed in the calculations, and also emphasizes the necessity of carrying out an experimental investigation.

LITERATURE CITED

1. Yu. M. Aleksandrov, V. F. Grushin, V. A. Zapevalov, and E. M. Leikin, Eksperim. i Teor, Fiz., 49(7):54 (1965).

2. M. Camac, Rev. Sci. Instr., 22:256 (1959). 3. W. Cross, Rev. Sci. Instr., 22:284 (1959). 4. E. Segre, Experimental Nuclear Physics, Vol. 1. Wiley, New York (1953). 5. C. Robinson and P. Baum, Report No. 40, PRL, illinois (1961). 6. M. C. Kozodaev and A. A. Tyapkin, Pribory i Tekhn. Eksperim., 1:21 (1965).


7. J. Septier, CERN Report (1958). 8. G. S. Landsberg, Optics, GITL, Moscow (1946). 9. J. Thomson, Phil. Mag., 6:561 (1907).

10. V. F. Grushin, Dissertation, P. N. Lebedev Physics Institute of the Academy of Sciences of the USSR (1 96 5) .

POSITIVE PION STOPPING DETECTOR

Yu. M. Aleksandrov, V. F. Grushin, and E. M. Leikin

The experimental study of interactions between elementary particles ultimately depends on the solution of various methodical problems regarding efficient particle recording. In a number of cases it is required to record positive pions carrying a specified quantity of energy. Over a certain range of pion energy (about 100 MeV) the solution of this problem may be based on the separation of these particles by slowing them to a complete halt and using the fact that the halted pion decays thus: 11" - Jl + v.

This approach enables us to create a fairly simple positive pion stopping detector capable of Identifying these particles quite reliably. The principle of operation of the detector described below (a series of three scintillation counters with a retarding filter between the first and second of them) is as follows.

As first criterion for selecting the positive pions with energies in a specified range of values, we separate out those particles the ionization path of which ends within a prescribed counter of the series. Then in the stopping counter we select the cases of 11" -Jl decay; these constitute delayed events, distributed in accordance with an exponential time law, with a time constant Trr = 2.55'10-8 sec (life of the quiescent pion).

The simplest version of a stopping detector is shown in Fig. 1, together with a block diagram of the recording apparatus. The passage through the detector of a charged particle above the range specified is accompanied by the appearance of a triple-coincidence pulse. This pulse is then shaped into a pulse of standard amplitude and length ("gate"), which falls on one of the channels of a gating circuit. A pulse from the stopping counter 3 falls on the other input Of this circuit. The apparatus provides for the recording of the number of gating pulses NG and the number of pulses Nil from the counter 3 lying within the width of the gate. If by means of a delay line we introduce an appropriate time displacement in the gating-pulse channel, we may thus record simply the delayed coincidences to which the events of the 11" - Jl decay belong. The length of the gating pulse in our case was about 1.2'10-7 sec. The choice of a gating pulse length T large in comparison with Trr enabled us to avoid additional losses in counting the decay events, while the indeterminacy in T introduced practically no error into the recording efficiency.

The number of recorded events of the 11" - Jl decay depends greatly on the time displacement ~t between the moment of stopping the pion and the beginning of the gate into which the pulses from a muon decay should fall. The value of ~t should be chosen in such a way that the recording of the delayed coincidences should only take place after the instantaneous coincidences between the gating pulse and the pulse from the stopping counter have ceased to be recorded. In practice this means that ~t must be approximately equal to the length of the unshaped pulse from the photomultiplier of counter 3.

70

POSITIVE PION STOPPING DETECTOR 71

! F Z J

Fig. 1. Schematic representation of a pionstopping detector and block diagram of the apparatus (version designed to operate with a moderate background). I), 2), 3) Scintillation counters; F copper filter; CC coincidence circuit; GSC gate-shaping circuit; DL delay line; G gate.

Of all Soviet-made photomultipliers the one with the shortest pulse is the FEU-36. However, owing to the inadequate band width of the output stage of these photomultipliers, there is usually a series of after-pulses immediately following the main pulse at the output, comparable with the latter in amplitude. The appearance of the after-pulses is due to the action of the resonance circuit formed by the inductance of the conductor attached to the final emitter and the parasitic capacitance of this emitter with respect to ground. Thus ordinary types of FEU-36 were unsuitable for use in counter 3, since the after-pulses simulated cases of a 11" - P. decay. Reliable operation of the apparatus associated with the recording of the delayed coincidences was ensured by using some experimental FEU-72 photomultipliers [1]. The introduction of decoupling condensers inside the bulb of these

photomultipliers in the circuit of the last two dynodes led to the suppression of the after-pulses, reducing their amplitude by an order of magnitude.

It was mentioned earlier that, if we record the delayed coincidences Nil for various delay periods of the gating pulse, we may expect that, after a certain critical delay time, only events of the 11" - P. decay will be recorded. However, since the resolving time of the delayed-coincidence circuit (gate) was quite long, it was possible that random delayed coincidences might also be recorded together with the decay events. This is particularly important when there are large loads on the telescope counters.

Np.--------------.

f,\ z ~ , , , ,

10Z " , , 7 ,

5

3

2

101

0 Z 3

~ If 5 G 7

tI 'IO-~sec

Fig. 2. Typical curve of delayed coincidences.

The background of random coincidences is measured by introducing long delays (7"d > 47"11")' The presence of random coincidences may also be reliably checked by observing the time distribution of the delayed coincidences. Figure 2 shows the curve of delayed coincidences, taken for light loads on the telescope counters, on a semilogarithmic scale. We see that, as 7"d increases, first the instantaneous coincidences cease being recorded and then only 11" - P. decay events begin to appear (the slope of the straight line corresponds to

7"11" = 2.55'10-8 sec). If there is a substantial contribution from random coincidences, this means that the results of measurements obtained for "working" delay periods will not fall on a straight line with a specified slope.

For working with considerable loads it was essential to carry out improvements to the detector, taking special measures to reduce the number of times the gate was triggered as a result of fast particles passing

72 Yu. M. ALEKSANDROV, V. F. GRUSHIN, AND E. M. LEIKIN

, F Z:3 'I-

tlG Nfl

Fig. 3. Schematic representation of a stopping detector and block diagram of the apparatus (version intended for working under considerable loads). Main notation as in Fig. 1; PC = prohibiting circuit; D3 = fast amplitude discriminator.

through the telescope. For this purpose an extra anticoincidence counter 4 was introduced into the telescope, based on the difference between the ranges of particles with different energies; fast particles were also discriminated by reference to the amount of energy which they released in counters 2 and 3.

Figure 3 shows a version of the stopping detector intended to operate under heavyloading conditions. A triple-coincidence pulse on circuit CC2, due to the passage of particles through all four counters of the telescope, prevents the pulse from CC l from passing to the gate-shaping circuit. The suppression of false gate-triggerings by this method was not quite perfect, partly because the anticoincidence tract was not 100% efficient and partly because of cases in which fast particles

suffered radiative retardation in counter 3 (thickness 2.1 g Icm2 of styrol).

The possibility of discriminating fast particles by reference to their energy evolution in counter 2 (thickness 1.05 g/cm2 of styrol) is associated with the fact that, in this counter, the minimum loss of energy by a pion stopped in counter 3 is roughly twice the probable energy loss of a relativistic particle. Figure 4 shows the energy released by pions in counters 2 and 3 as a function of their kinetic energy. The figure shows that the introduction of a corresponding energyevolution threshold into the channel of counter 2 reduces the recording efficiency of fast particles without losing any recording efficiency of the pions stopped in counter 3. It follows from an estimate based on the Landau distribution for ionization losses of relativistic particles that the recording efficiency of the latter may in this way be reduced to 3 or 4%.

6'£, MeV

za[ --/:Ti\-----~ I~~/ 1 i

1/ i "'" I 10 8 6 -=-' ---=-----k.;;;: --1

I ' I I I 11 I

/0 20 80 'f0

Fig. 4. Calculated curves giving the energy released by the pions (c5E) as a function of their kinetic energy Ek• 1) Counter 2; 2) counter 3; horizontal straight lines indicate the thresholds of counters 2 (lower) and 3 (upper).

In practice discrimination in this channel was effected by varying the amplification factor of the photomultiplier of counter 2. Figure 5 shows the count characteristic of the detector, illustrating the results obtained. By varying the voltage of the photomultiplier V2 within certain limits, the number of gate triggerings NG not associated with the stopping of pions could be reduced by a factor of several times. The working voltage V 0 chosen corresponds to the maximum of the ratio Nil 1NG

within the range of the Nil plateau.

The fast amplitude discriminator D3 introduced into the channel of counter 3 (see Fig. 3) further reduced the number of gate triggerings due to fast particles. The efficiency of recording 1f - JJ. decays was determined experimentally as a function of the threshold of the discriminator D3• Figure 6 (curve 1) shows ell(VD ), this efficiency relative to the voltage on the screen grid of the discriminator tube.

POSITIVE PION STOPPING DETECTOR 73

IG/

Fig. 5. Count characteristic of the detector.

Eq , MeV

51'

- 1.0

L I

/50 1GO VDJ V

Fig. 6. Characteristic of the discriminator D3. I) Experimental curve relating the efficiency f.J.l of recording 7r - P, decay events as a function of the threshold VD of the discriminator D3;

II) calibration curve Err (V D) of the detector relating to the evolution of energy in the counter 3, obtained from the experimental relationship C,1l (VD ) and the theoretical curve c, Il (E rr ).

sion factor for transforming N Il(Ll) to N'if

We see that an energy-evolution threshold of 6 MeV in cou'nter 3 (which is one and a half times the probable loss of energy by relativistic particles in the counter) corresponds to a reduction of no more than 5% in the efficiency of recording 7r - P, decay events, while reducing the number of gate triggerings due to relativistic particles by roughly one order (see Fig. 4). The relation between the level of energy evolution in counter 3 (E rr) above which pions are recorded by the apparatus and the value of VD appear as curve II. This curve is obtained from the experimental relationship C,1l (VD ) and the theoretical relationship C,1l (Err) calculated from the given experimental conditions. The calibration curve Err (VD ) is universal for the particular detector relative to the energy evolution in counter 3.

Let us briefly consider the procedure for converting the number (Nil) of recorded 7r - P,

decays (see Fig. 2) into the total number (N 7r) of positive pions stopped in counter 3. The relation between these quantities, allowing for the shape of the growth front on the multiple-coincidence curvet, has the following form:

(1)

where ~ is the value of the time delay for which the number of cases ~ is recorded, T is the length of the gating pulse, and j(t) is a function describing the multiple-coincidence curve, which is determined in a control experiment free from any incremental count due to delayed events.

The integral in formula (1) may be calculated graphically to a fair accuracy. Thus the reciprocal of the product of this integral arid the exponential factor e -M'r'if constitutes the conver-

for any given delay time Ll. Then

(2)

where a (Ll) is the conversion factor; a typical value of this factor for ~ = 4 (see Fig. 2) is 3.15.

In conclusion, we note that the method of recording positive pions described in the foregoing has been used for studying the reaction 'YP - n7r+ in the threshold region of y-quantumenergies [2]. The experiment also involved measuring the yield of pions at very small angles, for which about 10 5 positrons passed through the detector for each pion stopped in the latter.

t That is, allowing for the finite growth time of the efficiency of recording decay events.

74 Yu. M. ALEKSANDROV, V. F. GRUSH IN, AND E. M. LEIKIN

LITERA TURE CITED

1. V. G. Pol'skii, Dissertation, Moscow Power Institute (1961). 2. Yu. M. Aleksandrov, V. F. Grushin, V A. Zapevalov, and E. M. Leikin, Zh. Eksperim.

i Teor. Fiz., 49:7 (1965).

ABSOLUTE SENSITIVITY OF A THICK-WALLED GRAPHITE IONIZATION CHAMBER FOR I-GeV PHOTONS

I. N. Usova

A large number of different methods have now been developed for making absolute measurements on the flow of energy in the y-ray beam emitted by electron accelerators with energies of 100 MeV and upward [1-4J. These methods differ from one another in degree of accuracy, reliability, and convenience in operation. Recently R. Wilson's apparatus [3J, which has become known as a quantometer, has been the one most frequently used for making absolute measurements. The main advantage of this apparatus is the fact that its sensitivity depends very little on the energy of the photons [3,5J. However, this advantage is accompanied by a number of failings: The construction and prep~ration are quite complicated, and the assembly has to be carried out very carefully indeed, as the gaps between the electrodes are very small. In fact, before using the quantometer one has to calibrate it very carefully by one of the more complicated but more reliable methods of absolute intensity measurement, for example, the calorimetric method or the pair spectrometer. For this reason it may prove troublesome to employ the quantometer in a number of specific cases.

A much Simpler apparatus is the thick-walled graphite chamber. The construction and prepara~ion of this are extremely simple. This chamber constitutes a gas space about 1 cm thick, placed between two graphite blocks each 0.2 t units thick. The main difficulty in using this method lies in calculating the absolute sensitivity of the chamber, since this requires fairly accurate allowance to be made for processes taking place when high-energy photons pass through a fairly thick layer of material. However, if the proportion of photon energy converted into ionization is calculated, the calculation of the absolute sensitivity of the chamber for any photon spectrum is not particularly difficult.

A thick-walled graphite chamber for measuring the flow of energy in a high-energy 'Y-ray beam was proposed in 1947 by M. Lax [1J, who carried out an approximate calculation of the absolute sensitivity of the chamber for photons with energies up to 100 MeV. The sensitivity of the chamber for energies up to 250 MeV was calculated on the same assumptions by B. M. Bolotovskii [6]. The accuracy of the assumptions made in [1J was analyzed in [7, 81, in which it was shown that these assumptions led to a conSiderably greater error than Lax had supposed [1]. In addition to thiS, the importance of considering a number of effects not mentioned in [11 was pointed out.

A calculation of the absolute sensitivity of a thick-walled graphite chamber for 260 MeV photons based on a more accurate representation of the processes taking place when 'Y radiation passes through the graphite walls of the chamber was described in [8]. The improvements to the calculation led to a 1.23-times change in the absolute sensitivity of the graphite chamber.

75

76 I. N. USOVA

It was also indicated in [8] that, since, in calculating the absolute sensitivity, the second stage in the development of the cascade process in the chamber walls had to be considered, while the contribution of the third stage could be neglected with an error not exceeding 0.5%, the accuracy of the formulas obtained for the absolute sensitivity should be practically unaltered on passing to energies of 1-1.5 GeV. Since electron accelerators are being constructed for maximum energies of several GeV, it is of great practical interest to extend calculations of the sensitivity of the thick-walled graphite chamber to energies of this order. We therefore carried out a calculation of this kind, and also used a thick-walled graphite chamber for the first absolute measurements of intensity on the 680-MeV synchrotron of the Lebedev Physics Institute (FIAN).

In this paper we describe the calculation of the absolute sensitivity of a thick-walled graphite chamber for up to 1-GeV photons. The calculation was carried out fora chamber with an air gap 1 cm thick and graphite walls with a thickness of 4.5 cm.

We calculated the sensitivity on an M-20 computer, thus eliminating the necessity of making some of the Simplifications proposed in [8] and carrying the calculation out quite rigorously.

The accuracy of the readings of the chamber (and hence the accuracy of the calculation) was verified for an energy of 640 MeV by comparing with measurements made calorimetrically [9]. This verification showed that the error in determining the energy flux of such chambers was no greater than a few percent. Since the main physical principles underlying the calculation were set out quite fully in [8], we shall only mention them briefly here.

The use of thick-walled ionization chambers for measuring the energy flux in a high-energy 'Y-raybeam is based on a generalization of the theory governing the operation of small (thimble) ionization chambers to the case in which the ranges of the secondary electrons created by the photons in the walls of the chamber become much greater than the thickness of the front wall. In this case the equilibrium between the primary and secondary radiation around the working space is infringed. However, as indicated in [1], for secondary electrons with ranges R smaller than the thickness T of the front wall, and for the intensity of the 'Y radiation averaged over the thickness of the front wall, equilibium is preserved. Hence for this fraction of the secondary electrons the energy transformed into ionization in the layer ~t will be equal to the energy of the secondary electrons created in the same layer.

For that fraction of the secondary electrons which has a range greater than the thickness of the front wall T, only a proportion of the energy is transformed into ionization in the layer ~t; this proportion equals

(1)

where ~E(E) is the energy lost by electrons of energy E in passing through the graphite layer of thickness T.

Since in calculating the absolute sensitivity one must allow for the dependence of the retarding capacity (stopping power) of the material on the electron energy, and also for the proportion of energy lost from the working volume of the chamber by long-range () electrons (see [7,8]), the proportion of photon energy transformed into ionization must be calculated directly for the gas space of the chamber.

Then expression (1) is rewritten in the follOWing form:

1'] (E) = lL\~J")J p (E) [1- K (E)], (2)

ABSOLUTE SENSITIVITY OF A THICK-WALLED GRAPHITE IONIZATION CHAMBER 77

where p (E) is the ratio of the stopping powers of air and graphite and

K(E) = 11£'<;;0, + O.t7;3·106 (E·- 2eo) (1+~- e~)_1 Pi (E) Fi (E) (E - eo) 2£ 2E2

is the proportion of energy lost by the secondary electrons in the gas space, which is carried away as a result of the creation of long-range () electrons. For electrons with an energy of 1 GeV this correction equals about 9% of ~E(E). Here c:,o is the energy of the () electrons with a range of R(c:,) = T, c:, is the loss of energy in a single collision, and FdE) are the average losses associated with ionization in graphite. For T = 4.5 cm of graphite £0 Ri 20 MeV.

In calculating the energy ~E CE) converted into ionization in the layer ~ t for secondary electrons with ranges of R < T (or with energies up to 20 MeV), we must allow for the losses in bremsstrahlung, since the probability of such losses is quite high for these electrons. However, the probability that the secondary photons thus created might be able to produce electrons again on passing the chamber walls is small; the additional contribution to ionization from this process is under 0.5%.

Hence the value of ~E(E) for electrons with ranges R::::: T will be

( 3)

where ~Erad is the energy lost by the electrons in bremsstrahlung. For the case R> T, in calculating the value of ~E(E) we must allow for the fact that, when the electrons pass through the chamber walls, their energy is reduced as a result of ionization and radiation losses. Hence the value of ~E (E) is in this case given as follows:

E ,. Pi (8) ,

CiE (R) = \ F . + P () de +:xCiRrad (B.), J 'i (e) rad e

ET'

(4)

where Frad (£) is the average energy loss of the electrons in the graphite, while O!~Erad (£) is the proportion of the energy radiated by the secondary electrons converted into ionization as a result of secondary processes. The lower limit of the integral ET " was determined from the equation

E

\ 1 de = T' .1 Pi (e) + Fi:ad (e) , ET'

( 5)

where T' = T Icos f8'i; ij2 is the mean square angle for the multiple scattering of secondary electrons of energy E in the chamber wall.

Thus in calculating the proportion of energy lost by an electron in ionization by formula (4) we automatically allow for: 1) the energy dependence of the ionization losses of the electrons, 2) multiple scattering, and 3) the first two stages of the cascade process.

The values of A(E) = [b.E(E) + O! b.Erad(E)] [1 - K(E)] for electrons with energies up to 1 GeV and a graphite wall 4.5 cm thick are given in Table 1, which also shows the values of p (E) for air and graphite corresponding to similar electron energies.

In order to determine the mean proportion of photon energy converted into ionization in the working space of the chamber, we must average E/w·1) (E) over the secondary-electron spectrum.t

tHere W is the energy of the photon and E the energy of the secondary electron created by the latter.

78

E, MeV I

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 28 30

I. N. USOVA

TABLE 1

A(E), MeV l P (El, II air /graphite

E, MeV I A(E), MeV II P (E), air I graphite

0 -0.999 0.00591 1.994 0.00607 2.984 0.00618 3.962 0.00623 4.930 0.00635 5.886 0.00642 6.830 0.0064.7 7.770 0.00652 8.697 0.00657 9.611 0.00661

10.51 0.00665 11.40 0.00669 12.28 0.00672 13.15 0.00675 14.01 0.00678 14.87 0.00680 15.72 0.00683 16.56 0.00685 17.30 0.00687 18.19 0.0069G 18.19 0.00692 18.24 0.00694 18.29 0.00698 18.35 0.00702 18.43 0.00705 18.49 0.00707

35 40 50 60 70 80

100 120 140 160 20~

250 300 350 40~)

450 500 550 600 650 700 750 800 850 900 950

1000

18.61 18.73 18.91 19.03 19.11 19.17 19.21 19.27 19.31 19.34 19.37 19.43 19.41 19.44 19.45 19.45 19.46 19.46 19.46 19.46 19.48 19.23 19.23 19.22 19.24 19.49 19.47

0.00712 0.00716 0.00722 0.00727 0.0:)730 0.00733 0.00736 0.00738 0.00740 0.00741 0.00742 0.00743 0.00743 0.00743 0.00743 0.00743 0.00743 0.00743 0.00743 0.00742 0.00742 0.00742 0.00742 0.00742 0.00741 0.00741 0.00741

Here we have to allow for: a) the increase in the number of Compton electrons resulting from multiple scattering [the correction a(W) in formula (6) 1, b) the annihilation of positrons, which reduces the number of positrons contributing to ionization [the correction b(E) in formula (6)], c) the different ionizing powers of electrons and positrons.

Allowing for these corrections, the formula for the mean proportion of photon energy converted into ionization in the working volume, s(W), takes the form

Emax ~ (:) f!(El{[l +a(Wl) "c (E, WH [1.98-b(El) [1 +ct(W, Z))"p(W, E)}dE

s(H1)=~o ______ ,, ________________________________ ___ Emax

(6)

S {"dE, Wl+ [1 +ct(W, Zl)" (E, W)}dE o P

Here O"c(E, W) and op(E, W) are respectively the cross section associated with Compton scattering and with the formation of pairs at a nucleus; the factor [1 + a(W, Z)] allows for pair formation in the field of the atomic electrons.

The absolute sensitivity of the graphite chamber is obtained by averaging s(W) over the spectrum of 'Y radiation falling on the chamber, allowing for the attenuation of this in the walls. For secondary electrons with R ::::; T this averaging is carried out over the thickness of the layer from which secondary electrons pass into the working volume, while for electrons with R::: T it is carried out over the whole thickness of the front wall. The attenuation factor for these two cases respectively equals


(Pl (W) = exp[ -'t'o(W) T) {exp [To (W) Rw (E)] - 1} for the case R < T; 't'(W)Rw(t:)

IJldW) = T't';W) {1-exp [-'to(W) TJ} for the case R"> T. (7)

Here TO (W) is the absorption coefficient of photons of energy W in graphite, Rw(E) is the range of the secondary electrons with energy E, and E is the average energy of the secondary electrons created by a photon of energy W.

The values of qJ1 (W) and til.! (W) are given in Table 2, which also shows the absorption coefficients of 'Y radiation in graphite for two particular cases: 1) allowing for nuclear absorption (TO (W», and 2) not allowing for this factor tr (W»,

Knowing the values of qJt (W), til.! (W), s (W) and the phot on spectrum, it is easy to calculate the absolute sensitivity of the chamber, However, owing to the necessity of considering the weakening (attenuation) of the intensity, the quantity s (W) must be divided into two components (see Table 2) with respect to the limits of integration in the numerator of formula (6), For the first term St(W) -the upper limit of integration is the energy Eo of the secondary electrons with range R = T(Eo = 19.5 MeV), while for the second term S2 (W) we have the rest of the integral:

Eo ~ (:) 1] (E){[1 +a (W)] crc (E. lV) + [1.98-b(E)] [1 +a(lV.Z)]crp(W, E)}dE

81(W) = 1;; , max S {crc (W, E) + [1 + a (lV, Z)] crp(W, E)} dH o

Emax (8) S (.!i...) 1] (E){[1 + a(W))crc (E, W) + [1.98 - b (E)][t + a (W, Z)] crp(W, E)} dE E. W

8 2 (W) = Emax . S {crc(W, E) + [1 +a(W. Z)] crp(W, Z)} dE o

The formula for the absolute sensitivity is written as follows:

wmax S 't' (W) W f (W) [<PI (W) 81 (W) + <ps (W) 82 (W)] dW

S(Wmax ) = 0 Wmax (9)

S Wf(W)dW o

Here f (W) is the number of photons with energy W in the emission spectrum, r (W)Wf(W) is the energy of the 'Y radiation absorbed in a layer ~t at a depth T. In calculating this quantity the nuclear absorption of photons is not taken into account, since in view of the smallness of the cross section of photonuclear reactions and the short range of the reaction products the contribution of this process to the ionization is negligibly small. The values of B (W) = r (W) [(/Jt (W) st (W) + <P2(W)~(W)] are given in Table 2. If this quantity is known, the cauculation of S(Wmax ) for any spectrum creates no difficulty.

The flow of energy in the 'Y-ray be.am on measuring the ionization from the charge leaking into a capacity C will equal

Ow U = ges(W~ax) .!lV, (10)

where C is the capacity of the condenser, we is the energy for the formation of one pair of ions, g is the depth of the working volume of the chamber, e is the charge on the electron, and ~ V is the potential difference to which the condenser is charged during the measuring period.

80 I. N. USOVA

TABLE 2

E. Mevj To(W).em-' I T(W), em-' I 'P,(W) I 'P_(W) I s,(W) I s,(W) I B(W),em-1

I 0 - - - - - - -1.5 0.3934.10-3 0.572 0.1158 0.260.10-4

3 0.4224 U.682 U.0797 0.230 4.5 U.4458 0.749 0.0541 0.2141 6 0.4J24 U.781 0.0553 0.0568 0.1996 7.5 0.4i67 U.805 0.04J7 0.1908 9 0.4857 0.825 0.U4:'58 0.0456 0.18iO

10.5 0.494:'5 0.8~0 0.0430 0.1789 12 0.4994 0.8:'51 0.0409 0.04.15 0.1738 13.5 0.5u53 0.861 0.0392 0.0398 U.1705 15 0.5102 0.870 0.0379 0.0382 0.1683 16.5 0.5143 0.879 0.031:i9 0.0374 0.1653 18 0.5174 0.886 0.0360 0.0369 0.1649 19.5 0.5208 0.0000 0.890 0.922 0.0353 0.0365 0.1638 21 0.4212 0.0998.10-3 0,892 0.922 0.0347 0.0363 0.1624 22.:'5 0.3496 0.166 0.893 0.922 0.0342 0.0372 0.1590 24 0.3029 0.205 0.894 0.923 0.0338 0.0::\39 0.1555 25.5 0.2571 0.231 0.894 0.924 0.0334 0.0354 U.151O 27 0.2383 0.250 0.894 0.924 0.0332 0.0348 0.1474 28.5 0.2148 0.264 0.895 0.925 0.0329 0.0344 0.1434 30 0.1950 0.275 0.895 0.926 0.0327 0.0340 0.1403 40 0.1157 0.294 0.998 0.930 0.0319 0.0328 0.1199 50 0.779.10-4 0 . .283 0.902 0.931 0.0316 0.0318 0.1052 60 0.562 0.266 0.903 0.932 0.0316 0.0316 0.0945 70 0.425 0.248 0.903 0.932 0.0317 0.0317 0.0853 80 0.332 0.231 0.902 0.932 0.0319 0.0319 0.0782 90 0.286 0.215 0.902 0.931 0.0321 0.0321 0.0725

100 0.218 0.201 0.901 0.931 0.0323 0.0323 0.0669 110 0.182 0.189 0.901 0.931 0.0326 0.0326 0.0626 120 0.154 0.178 0.900 0.930 0.0328 0.0328 0.0590 130 0.132 0.168 0.900 0.929 0.0330 0.0330 0.0554 140 0.114 0.1586 0.899 0.929 0.0332 0.0332 0.0525 150 0.899.10-5 0.1515 0.899 0.929 0.0334 0.0334 0.0498 160 0.879 0.1432 0.898 0.928 0.0336 0.0474 170 0.780 0.1365 0.898 0.928 0.0337 0.0448 180 0.697 0.1303 0.898 0.927

I 0.0339 0.0431

190 0.627 0.1247 0.897 0.927 0.0340 0.0416 200 0.566 0.1196 0.897 0.927 0.0342 0.0397 210 0.514 0.1148 0.896 0.926 0.0344 0.037S 220 0.468 0.1104 0.S96 0.926 0.0345 0.0366 230 0.429 0.1065 0.895 0.926 0.0346 0.0353 240 0.394 0.1026 0.895 0.925 0.0348 0.0343 250 0.363 0.990·10""" 0.895 0.925 0.0349 0.0331 260 0.335 0.957 0.894 0.925 0.0350 0.0320 270 0.311 0.926 0.894 0.925 0.0351 0.0311 280 0.239 0.897 0.894 0.925 0.0352 ).03)1 290 0.270 0.870 0.894 0.924 0.0353

I 0.0292

300 0.232 0.844 0 .. 893 0.924 0.0354 0.0284

320 0.221 0.797 0.893 0.924 0.0355 I 0.0268 340 0.196 0.755 0.893 0.924 0.0357 I

0.0256 350 0.175 0.717 1J.89~ 0.923 0.0338 , 0.0243 380 0.156 0.683 0.892 0.923 0.0330 0.0232 400 0.141 0.631 0.891 0.923 0.03Gl 0.0222

420 0.128 0.623 0.891 0.92;) 0.0332 0.0212 440 0.116 0.597 0.891 0.923 0.0363 0.0203 460 0.106 0.573 0.891 0.922 0.0365 0.01960 480 0.987.10-6 0.551 0.890 0.922 0.0366 0.01892


TABLE 2. (continued)

I -1 S,(W) B(W), em

500 0.900 ·10-f> 0.530.10 4 U.890 0.922 0.0367 I

I 0.01824·1 520 0.831 0.511 0.890 0.922 0.0318 0.01759 540 0.770 0.493 0.890 U.922 0.0308 0.01700 560 0.715 0.477 0.839 11.9:21 0.0359 0.1)1642 580 0.667 0.461 0.8S9 0.921 0.0370 0.01595 600 0.622 0.447 0.889 0.921 0.1J371 0.01551 620 0.583 0.433 0.889 0.921 0.0371 0.01499 640 0.546 0.420 0.889 0.921 0.(1372 0.01454 660 0.513 0.408 0.888 0.921 0.0372 0.01421 680 0.483 0.397 0.888 0.920 0.0373 0.01380 700 0.455 0.386 0.888 0.920 0.0373 0.01346 720 0.431 0.375 0.888 0.920 0.0374 0.01305 740 0.401 0.366 0.888 0.920 0.0374 0.01272 760 0.386 0.357 0.888 0.920 0.0375 0.01244 780 0.366 0.348 0.888 0.920 0.U375 0.01213 800 0.348 0.339 0.887 0.920 U.0376 0.01184 820 0.331 0.331 0.887 0.920 0.0376 0.01156 840 0.315 0.324 0.887 0.920 0.0377 0.01130 860 0.301 0.317 0.8S7 0.919 0.0377 0.01071 880 0.287 0.310 0.887 0.919 0.0378 0.01083 900 0.274 0.303 0.887 0.919 0.0378 0.01062 920 0.262 0.297 0.887 0.919 0.0378 0.01040 940 0.251 0.291 0.886 0.919 0.0379 0.01023 960 0.241 0.285 0.885 0.919 0.0379 0.01001 980 0.231 0.279 0.885 0.919 0.0380 0.00982

1000 0.222 0.274 0.886 0.919 0.0380 0.00964

The absolute sensitivity of the graphite chamber was calculated for synchrotron bremsstrahlung with a maximum energy of 640 MeV, and also (in order to check the previous calculation) for bremsstrahlung with a maximum energy of 260 MeV. The accuracy of these calculations was in practice determined by the accuracy of formula (9), or, more exactly, by the accuracy with which the quantities entering into the formula were known. The main contribution to the error of this formula (about 3%) came from calculating the quantity 1](E) (see [2J and [8]).

On allowing for the errors introduced into the formula by the process of pair formation (± 2%), the attenuation of the 'Y radiation in the wall of the chamber (± 2:5%), the contribution of secondary processes (± 0.7%), and the absorption of photons in graphite (± 2%), the error in formula (9) becomes equal to ± 5%. In addition to this, in determining the energy flux from formula (10) an additional error arises from the inaccuracy in the experimental determination of we' According to recent published data we = 34 ± 1 eV for air (see, for example, [10]).

Thus the value of S(Wmax ) may be calculated to an accuracy of about ± 5%, and the intenSity measurement carried out to an accuracy of ±6%, the measuring error (usually no greater than 1 or 2%) being included in this.

The values of S(Wmax ) calculated from formula (9) for the bremsstrahlung from the accelerator targett for several values of Wmax are given in Table 3. For Wmax :: 260 MeV the result

t The calculation was carried out for the bremsstrahlung spectrum allowing for the finite thickness of the target: For Wmax = 260 MeV with a tungsten target 0.12 t units thick and for 640 and 700 MeV with a tantalum target 0.15 t units thick.

82

260 640 700

TABLE 3

8.48±0.43 4.8:aO.24 4.63±0.23

TABLE 4

Method of intensity measurement

Fr om thin-converter pair of differences

Pb-AI ........... . pb-eu ........... .

From induced activity in a graphite sample ... ' ..... .

Thick-walled chamber ..... . From the area under the cascade

curve (collimated beam) ... Thick-walled chamber (colli-

mated beam) .......... .

TABLE 5

Method of measurement

From the area under the cascade curve ................ .

With a calorimeter ........ . With a thick-walled chamber ..

1. N. USOVA

8.3±0.6

Energy flux '10-12 ,

referred to 100 counts of the comparison

monitor, MeV

3.68 ± 0.24 3.64 ± 0.25

3.98 ± 0.40

3.69 ± 0.22

1.93 ± 0.11

1.97 ± 0.12

Energy flux .10-12 ,

referred to 100 counts of the comparison

monitor, MeV

0.318 ± 0.020 0.326 ± 0.006 0.337 ± 0.024

of a calculation taken from [8] is also given. We see that the difference in the values of S (W max)is very slight, being equal to 2.2%.

The accuracy of the readings of the thick-walled graphite chamber for photons with energies up to 260 MeV was checked on the Physics Institute synchrotron with Wmax ::: 260 MeV by comparing with intensity measurements based on three other methods: thin -converter pair differences, the area under the cascade curve [11], and the inducedactivity method. The results of this comparison are shown in Table 4.

The accuracy of the readings of the graphite chamber for higher energies was checked on the Physics Institute synchrotron with Wmax = 680 MeV by comparing the results of intensity measurements in this chamber with measurements of the area under the cascade curve in lead, and also by comparing with calorimetric measurements carried out on the same synchrotron by a team of physicists from the Leningrad Physicotechnical Institute of the Academy of Sciences of the USSR. In both cases the energy flux in a collimated beam of 'Y radiation was determined. The diameter of the beam at the site of the apparatus was 40 mm and the diameter of the working volume of the graphite chamber 80 mm.

The number of pairs of ions formed in the working space of the chamber was determined from the potential difference to which the condenser C was charged during the measurement. The potential difference was

measured by the compensation method, using an electronic circuit based on a lElP tube connec-ted in the Bart manner. The experimental error in determining to V was ± 29'0 (the error of the comparison monitor was included in this). The results of this comparison are shown in Table 5.

We see from Tables 4 and 5 that all the intensity measurements agree closely with each other, which shows that the thick-walled graphite chamber may be successfully used for absolute measurements of intensity in electron accelerators.

In conclusion, the author wishes to thank B. K. Artem'ev for setting up the program and making the calculations of absolute sensitivity on the M-20 electronic computer, and also E. D. Kochetkov, who took part in constructing the measuring apparatus and carrying out some of the measurements, as well as S. P. Kruglov and I. V. Lopatin for making the comparison with the calorimetric measurements possible.


LITERATURE CITED

1. M. Lax, Phys. Rev., 72:61 (1947). 2. W. Blocker, R. Kenney, and W. Panofsky, Phys. Rev., 79:419 (1950). 3. R. Wilson, Nucl. Instr. Methods, 1:101 (1957). 4. E. Edwards and D. Kerst, Rev. Sci. Instr., 24:490 (1953). 5. S. P. Kruglov, Dokl. Akad. Nauk SSSR, 145:309 (1962). 6. B. M. Bolotovskii, P. N. Lebedev Physics Institute of the Academy of Sciences of the USSR,

Moscow (1952). 7. 1. N. Usova, Pribory i Tekhn. Eksperim., No.6, p. 17 (1959). 8. 1. N. Usova, Pribory i Tekhn. Eksperim., No.2, p. 36 (1962). 9. S. P. Kruglov, Zh. Eksperim. i Teor. Fiz., 33:21060 (1957); S. P. Kruglov, Z. Kovarzh,

and 1. V. Lopatin, Izv. Vuzov, Fiz., No.1 (1960). 10. 1. N. Usova, Dissertation, P. N. Lebedev Physics Institute oftheAcademyofSciencesofthe

USSR, Moscow (1961). 11. 1. N. Usova, Pribory i Tekhn. Eksperim., No.4, p. 27 (1961).

ST A TISTICS OF TIME MEASUREMENTS MADE

BY THE SCINTILLATION METHOD

V. V. Yakushin

§ 1. Scintillation Counters in the Coincidence Method

The present state of the technique of recording nuclear particles is characterized by the extensive use of scintillation counters. The distinguishing features of these are their high sensitivity and low inertia, which enables individual light quanta emitted in the scintillator to be recorded over a short period of time. In this article we shall consider questions relating to the use of scintillation and Cerenkov counters for studying nuclear events by the coincidence method. Other characteristics of the counters such as their energy resolution will not be discussed.

In future we shall understand "recording apparatus" or "recording system" to mean the whole aggregate of the elements of the scintillation or Cerenkov counters: the sensitive volume (scintillator with or without a light guide, or radiator), the photomultiplier, and the threshold (cut-off) electronic circuit.

The emission of photoelectrons from the photocathode of the photomultiplier is common to the scintillation and Cerenkov counters; in this connection we shall in general have to consider the counting of the photoelectrons reaching the first emitter of the photomultiplier. Thus the more general problem of the present article will be to consider the potentialities of photoelectron counters.

The recording of nuclear events by the coincidence method is used when setting up physical experiments in which information is obtained by studying the spatial and time correlations of interaction products. The use of the coincidence method when studying spatial correlation [1-4] is essential when effects of different kinds accompany the experiments, for example, subsidiary products of nuclear interactions, accelerator radiation background, cosmic background, induced activity, and so forth. Other experiments of this kind involve the e~ct measurement of time intervals in order to determine neutron energies from the time of flight of the neutrons [5] or finding the efficiency function of a Compton spectrometer [6, 7]. The study of time correlation forms the content of experiments aimed at establishing a genetic relation between the products of nuclear interactions, for example, when determining the lifetime of excited nuclear states [8], the half-decay period of short-lived radioactive isotopes [9], and so on. Thus the coincidence method enables us to establish simultaneity between the nuclear-interaction products emitted by an experimental target or determine the time sequence of these.

The use of photoelectric counters in the coincidence method, as in other detectors, is by no means free from failings of principle. These failings lead to a loss of experimental information and reduce the rate at which this information is secured. The particular presentation of the experiment and the energy and type of particles recorded largely determine how far these failings are likely to appear.

84

STA TIS TICS OF TIME MEASUREMENTS MADE BY THE SCINTILLATION METHOD 85

[;-1---1

________ 1

Fig. 1. Multichannel coincidence circuit. 1) Recording systems; 2) unit accepting the coincidences [111; 3) amplifier; 4) counting circuit. S) Fragments of the nuclear reaction from the experimental target.

"'(t max) --Z-

Fig. 2. Distribution of W(t) for periods before the instant of time corresponding to the record-ing of an event, and its width w. Here t d max

is the instant of time corresponding to the maximum of the distribution.

The total number of light quanta reaching the photocathode of the photomUltiplier is determined by the energy of the particle, the geometry of the sensitive volume of the recording apparatus, and so on. In experiments in which the particle energy directly converted into light quanta is small, there may also be losses of quanta in the sensitive volume of the counter chosen for the experiment (absorption, geometrical losses). In this case, even for an unrestricted resolving time of the coincidence circuit, the total number of quanta may be insufficient for reliable recording of the particle. However, the possible loss of counts from the recording apparatus in prinCiple involves with one further fact.

In the coincidence method, we consider a certain range of sensitivity c,j' the recording apparatus within which we expect the recording of a coincidence between different events. The extent of the range T

is determined by the conditions of the experiment, and its initial point coincides, for example, with the moment at which an event simultaneous with the one under consideration is recorded by an analogous apparatus. Coincidence between the events is regarded as recorded if, within the period T, the inputs of the element accep

ting coincidences from the cut-off electronic circuits of the channels (Fig. 1) receive pulses resulting from the execution of a whole series of successive processes in the recording systems. The act of recording may be identified with the completion of an incomplete series of processes ending, for example, when the Q-th photoelectron is emitted by the photocathode of the photomultiplier, or when the potential difference on the RC filter in the circuit of the photomultiplier collector exceeds the cut-off value of the electronic circuit.

In this case the efficiency £0 with which an event is recorded by one channel of the coincidence circuit equals the total probability that not less than Q photoelectrons will be emitted by the photocathode in time T; a similar definition may be applied to any other such process. The efficiency of the recording apparatus is defined as the proportion of events recorded within the period T relative to the total number of such events, i.e., it is a quantity averaged over a fairly large number of events. As the time T, called the electrical resolving power of the coincidence circuit, diminishes, the fall in recording efficiency means that the number of physical experiments which may be carried out with the aid of the coincidence method is restricted to experiments with" good" statistics [10]. The manner in which the efficiency £0 depends on the resolving time T is determined (among other factors) by the intensity of the final process, but independently of this aspect an unlimited reduction in T is accompanied objectively by an unlimited, monotonic fall in efficiency. A reduction in T may be made in experiments with "good" statistics in the interests of improving the resolving power of the recording apparatus or reducing the absolute level of the background.

86 V. V. YAKUSHIN

A resolving time T is ascribed to the coincidence circuit (see Fig. 1) if the electronic cutoff circuits of the channels form voltage pulses of a certain length, so that for equal time delays in the channels and a specified discrimination level the sensitivity of the coincidence-accepting element is constant for this period [11]. However, the mutual disposition of the pulses on the time axis is not strictly fixed, but experiences fluctuations owing to the statistical nature of the processes leading to the act in which the coincidence of the events is recorded. Let us now suppose that the position of the pulse corresponding to one event in a two-channel coincidence circuit (for example, the recording of a high-energy particle generated by nuclear interaction) experiences slight fluctuations, and that the time origin is associated with its initial point in every act of nuclear interaction.

Let us also suppose that both events are exactly simultaneous. Then the position of the pulse corresponding to another event (the generation of another particle in the same reaction) is "spread out" along the time axis for a set consisting of a fairly large number of events. This objective distribution of W(t) over the "waiting" period in which the coincidence of the events is to be recorded occurs if the value of T is comparable with the width w of the distribution at half its peak value. The value of the time fluctuations is characterized by the width w if, at the end of an interval of duration w (T «w), the expected number of recording acts is a factor of two below the maximum (Fig. 2). The determination of the width w for a real recording apparatus is prevented because the shape of the distribution W(t) is masked by the fact that it has been integrated over an interval of duration T. If T > wand is increased further then the form of the coincidence curve (which constitutes the result of the integration and may be determined by introducing a variable time delay into one of the channels of the coincidence circuit) becomes nearly rectangular, while its width at half the peak value approaches T. On reducing the resolving time without limit (T - 0, or at any rate T « w), the coincidence curve degenerates into the distribution of W(t) for the period of time preceding the instant of recording the coincidence of the events. It should be noted that the foregoing argument relates to the case in which the voltage pulses formed by the cut-off circuits of the channels are standardized wi th respect to length and amplitude, and the effect of their superimposition may be neglected. Any infringement of these conditions may be observed as a change in the expected shape or a broadening of the coincidence curve resulting from the appearance of additional time fluctuations [12].

The concept of the resolving power W(T) of the recording apparatus or the coincidence circuit as a whole arises in connection with experiments aimed at discovering the nature of the genetic relationship between various events in time. In general the resolving power W(T) is to be regarded as the width of the coincidence curve at half its peak value, obtained by integrating the distribution W (t) with respect to an interval of length T. As T is reduced without limit, the resolving power becomes equal to w, i.e., the accuracy of revealing the genetic relationship between the events is limited by the value of w. The character of the genetic relationship is manifested by the way in which the shape of the coincidence curve deviates from the measured value for simultaneous events [8]. Strictly speaking, the character of the genetic relationship cannot be revealed to this accuracy, since for T « w the efficiency of the recording apparatus is unacceptably low.

Thus there are two asymptotic types of physical experiments using the method of coincidences. The first type includes experiments with "good" statistics or with a rapid access of information, in which case the requirements regarding the accuracy and duration of the experiment are satisfied even for T < w; the resolving power in these experiments is W(T) Rl w. The second type includes experiments with "poor" statistics [10], in which rigorous demands are made on the efficiency of recording the events, i.e., T > w, and the resolving power w( T) Rl T. In both types of experiments, independently of the ratio of the quantities T and w, the level of back-

STATISTICS OF TIME MEASUREMENTS MADE BY THE SCINTILLATION METHOD 87

ground is determined by the random overlapping of pulses in time, i.e., the background is to a first approximation proportional to T m- I , where m is the number of channels of the coincidence circuit [13]. The use of photoelectron counters in a physical experiment presupposes a knowledge of the relationship between the efficiency of the coincidence circuit and its resolving power (we consider that the geometry of the experiment, the energy and type of particles recorded, and so on, are known). The duration of the projected physical experiment and the expected accuracy with which it is to be carried out are determined by precisely this relationship.

§ 2. Distribution W(t) for Intervals Prior to the Instant of Time

at Which the Event Is Recorded

With the act of recording a nuclear event we identify the appearance (within a period of time T) of a voltage or current pulse at the output of a cut-off electronic circuit, taking place as a result of the fulfilment of a series of successive processes in scintillation or Cerenkov counters, beginning with the release of energy ~E by a nuclear particle in the scintillator or radiator. However, the act of recording may in principle also be identified with the fulfillment of a certain proportion of these processes rather than the complete series. The choice of a process which in this sense executes the act of recording is determined by the nature of the physical experiment and the type of recording apparatus employed. A reasonable criterion for chOOSing a recording process is that its potentialities should not be seriously affected if all the successive component processes are taken into account instead of just some of them.

The accuracy of experiments on time selection and the identification of the genetic relationship between nuclear events is limited by the efficiency eo and the width w, quantities which may be calculated from the distribution W(t); the derivation of W(t) should, strictly speaking, constitute the result of considering the whole series of processes in the recording apparatus. Let us first find the solution of a limited problem, neglecting some of these processes and also the corresponding error in determining w. This approach is reasonable, for example, in considering the time characteristics of the photomultiplier, for which we may confine consideration to the entrance chamber and two or three stages of secondary-electron multiplication [14], and in certain cases neglect the time fluctuations introduced by the photomultiplier as a whole also. Analogous considerations are also valid for the cut-off electronic circuit (see § 5), which introduces no additional time fluctuations.

Numerical solution of the problem in hand is in principle restricted by the statistical character of the processes in the recording apparatus, since the occurrence of an event on the molecular or atomic level (for example, the emission of a photoelectron after the absorption of a quantum by the scintillation photocathode of a photomultiplier) is of a random nature. The probability of such an event occurring, determined from a fairly numerous set of individual events, is called the conversion efficiency of the process in question.

The time fluctuations introduced by the photomultiplier and the electronic cut-off circuit are negligibly small compared with the fluctuations in the recording apparatus as a whole if the distributions xW(t) and QW(t) for the intervals preceding the instant at which the voltage pulse exceeds the threshold (cut-off) value x of this circuit and the intervals preceding the emission of the Q-th photoelectron by the photocathode of the photomultiplier coincide. Clearly this condition can only be rigorously satisfied in practice if the photomultiplier introduces no additional time fluctuations, if Q = 1, and if there is an unlimited sensitivity (x = 0) of the cut-off circuit. The latter condition enables us to avoid transforming the amplitude spectrum of the voltage pulses on the RC circuit in the collector circuit of the photomultiplier into the distribution corresponding to intervals prior to the instant at which the cut-off circuit is tripped (Fig. 3). Nevertheless, the distribution Q W(t) (Q> 1) enables us to estimate the potentialities of the recording apparatus in the more general case of x > 0 also. The distribution Q W(t) also enables us to estimate the

88 V. V. YAKUSHIN

contribution of the time characteristics of the entrance chamber of the photomultiplier. Let us proceed to derive the distribution QW(t) for intervals prior to the instant at which the Q-th photoelectron is emitted by the photocathode of the photomultiplier.

In recording nuclear particles, we must distinguish two types of counting-statistics problems, leading to quite different time distributions. Let us picture a fairly large number of completely identical systems consisting of identical particle sources irradiating identical counters. Suppose that in the time interval 0 to t the first counter operates Ql times, the second Q2 times, and so on. How are the numbers of counts Ql' Q2, ... , etc. distributed? If the probability of recording a total number of counts R in one of the systems in an unlimited period of time is described by a Poisson distribution [15]

where Ro is the total number of counts averaged over the systems, then the probability of effecting Q counts in one of the systems in the time interval 0 to t is

t t

QPRo (t) = [~t* (i) dtr exp [-Vi (t) dt] [QW\ (2.1) o

where f* (t) is the intensity of the counts averaged over the systems at the moment of time t (the ratio of the numbers of counts Ro and Q may be arbitrary) [16]. Let us now consider just one counter and the corresponding source, and let us record numbers Q10 Q2' ... counts in a fairly large number of successive and equal nonoverlapping time intervals (the intervals are called nonoverlapping if the second starts no earlier than the first ends). If the intensity of the source is strictly constant and the total number of counts over the whole measuring period greatly exceeds the numbers of counts Qio Q2' ... in the intervals, then clearly the distribution of these numbers is also described by (2.1), since the counts in the nonoverlapping time intervals

u(t) ,------

1.0

0.2 0.1

0.05

b

O~----------~t~O~----------------~ Vmax

W(tmax i - - -- --

O~--~-~--~--------~

Fig. 3. Transformation of the amplitude spectrum of voltage pulses on the RC circuit in the collector circuit of the photomultiplier into the distribution corresponding to the period of time prior to the instant of triggering the cut-off circuit. a) Form of the voltage pulse vet); broken lines show discrimination thresholds; b) amplitude spectrum of the pulses n(v max); c) distributions xW(t) with thresholds x = 0.2, 0.1, 0.05. The maximum amplitude of the pulse from the spectrum is taken as 1.0. With increasing sensitivity x the width w of the distribution xW(t) diminishes: wO.05 < wO.l < wO,2'

STA TISTICS OF TIME MEASUREMENTS MADE BY THE SCINTILLATION METHOD 89

are statistically independent. However, for a variable intensity, the validity of the Poisson distribution for the number of counts in equal nonoverlapping intervals is infringed, except for the special case in which the intensity j* (t) is a periodic function of time [10].

In recording nuclear particles, statistical problems regarding the time intervals between counts often arise, for example, the problem of the distribution of the time intervals between two successive counts, between a fixed instant and the Q-th count, and so on.

Analogous problems arise in considering the emission of photoelectrons by the photocathode of a photomultiplier in a fairly large number of acts of particle recording.

Let us suppose that the act of recording an arbitrarily-chosen particle is effected as a result of the emission of a certain proportion AE of the energy E of the particle in the sensitive volume of the photoelectron counter. In effecting each of the successive processes in the recording system, a certain initial energy is expended. The ratio of the energy emitte~ as a result of the process to the total energy expended characterizes the conversion efficiency ~i of this process, if ~i is defined as a quantity averaged over a fairly large number of acts in which it is in-

volved. Thus the product of the energy yields II ~i of the processes (the conversion efficiency i

of the scintillating material, the geome~ric efficiency {17] of the sensitive volume fg, and the quantum efficiency of the photocathode ~) determines the total number Ro of photoelectrons expected (on average) in an unlimited period of time after the release of the energy AE in the sensitive volume. Owing to the statistical character of the processes in question, there is a possibility (in principle) that the photocathode will emit any total number of photoelectrons R, not exceeding R max' for which the energy AE is transformed without loss into overcoming the work function Of photoelectrons emitted by the photocathode. Then the probability of the emission of a total number R of photoelectrons is described by a binomial distribution [15] with a mean Ro and a maximum Rmax

(2.2)

where Ro / Rmax = II ~iis the product of the energy yields of successive processes, including the i

emission of photoelectrons; this is a quantity much smaller than unity [18, 19] (in a scintillation counter with the best scintillators and photomultipliers II ~i ~ 10-2 , in a Cerenkov counter II ~i < 10-7 ). However, for R!R max « 1 the binomial distribution differs very little from the i

Poisson distribution [15]

(2.3)

The distribution (2.3) objectively reflects the fact that on average the energy spent in overcoming the photoelectron work function is several orders smaller than the energy evolved in the sensitive volume of the photoelectron counter. Thus the probability of the emission of a total number R of photoelectrons for a particle chosen arbitrarily from the whole number of particles is (2.3), provided that the energy AE is constant.

90 V. V. YAKUSHIN

Let us consider the emission of photoelectrons by the photocathode as a function of time. For one arbitrarily-chosen particle with a total number of photoelectrons R, the probability of the emission of Q photoelectrons in a time 0 to t is described by a binomial distribution analogous to (2.2)

(2.4)

t

where the photoelectron emission function f (t) = ~ f' (t) dt is the mean number of photoelectrons a

expected at the moment of time t, 1(0) = 0,1(00) = R, 1(t)/R is the normalized mean number of photoelectrons, independent of the total number R. The distribution (2.4) only exists for R ;::: Q. The probability of the emission of Q photoelectrons for a particle with a total number of photoelectrons R, chosen from among the whole set of particles, allowing for (2.3) and (2.4) is R PRoQPR(t). Summing the analogous probabilities for the particles of the whole set, we have

(2.5)

The distribution (2.5) relates to a counting-statistics problem of the first type, in which we consider independent events in periods of time relating to the recording of different particles by the same apparatus. In fact the form of (2.5) coincides with (2.1):

(2.6)

t

where f (t) = \ t* (t) dt = Ro! (t) / R. Thus the Poisson distribution (2.6) describes the probability '6

of the emission of Q photoelectrons in a time 0 to t in one of the set of acts of particle recording. We note that, if the total number R undergoes no fluctuations and R '" Q, then the probability of the emission of a number of photoelectrons in one of the recording acts is described by the binomial distribution (2.4). Let us consider certain special features of this problem of counting statistics.

1. The probability of the emission of Q photoelectrons in the time interval 0 to t is described by the Poisson distribution (2.6), independently of the form of the functionJ* (t), but it is necessary that the intensity of the emission of photoelectrons J* (t) ;::: O. This condition is satisfied if the act of particle recording is identified with the emission of the Q-th photoelectron (in the count) by the photocathode of the photomultiplier. We can only speak of the sensitivity of the system to the Q-th electron in this connection if the time constant of the RC circuit attached to the collector of the photomultiplier is so large that, when the Q-th portion of charge reaches the collector, the potential difference on the resistance R differs negligibly from the potential difference existing on the capacity C when there are Q - 1 elementary portions of charge on the plate of the latter and the resistance R is infinite. In this sense one usually speaks of a threshold (cutoff) electronic circuit of the integrating type. Recording systems with threshold circuits of other types (in which the recording is based on a current pulse, the centroid of the pulse, and so on) are characterized by large time fluctuations and will not be considered in this article (see § 6).


2. The distribution (2.5) is valid for particles with any total number Ro of photoelectrons, including one comparable with Q, provided that in accordance with the definition of (2.4) R ~ Q. If the quantities Ro and Q are comparable, then the proportion of particles with R = Q relative to the whole set becomes considerable, and for Ro « Q overwhelming. In fact for Ro « Q the probabilities (2.3) OP Ro , 0 + IPRo ' ... of the emission of a total number Q, Q + 1, ... photoelectrons are

RQe-Ro

QPRo=T'

Q+1P R o = QP Ro Q ~ 1 '

and the distribution (2.5) degenerates into one term of the sum, with R = Q:

(2.7)

(2.8)

The distribution (2.8) describes the probability of the emission of Q photoelectrons for particles leaving a small energy ~E in the sensitive volume of the counter. In addition to this, if the number of scintillation quanta falling on the photocathode is for some reason or other small [a) absorption in the long light guide transmitting the scintillation quanta to the photocathode from a scintillator situated in an electric or magnetic field, b) a scintillator with a high self absorption coefficient relative to the quanta, c) a photomultiplier photocathode with an area small in comparison with the dimensions of the collecting face of the sensitive volume, d) the recording of the nuclear particles by means of scintillation films, and so on], then the probability of the emission of Q photoelectrons for the whole set of particles is also described by (2.8).

3. The distributions xW Ro (t), OWRo (t) are identical as, already mentioned, x = 0, Q = 1, and the time fluctuations introduced by the photomultiplier may be neglected. As a result of the statistical processes involved when the secondary electrons are multiplied in the dynode system of the photomultiplier, the total number (R) of photoelectrons described by the Poisson distribution (2.3) is transformed into an amplitude x of the voltage pulse in the RC circuit (Fig. 4) attached to the collector of the multiplier, represented by a distribution l/J(x) (Fig. 5) (RC exceeds the growth time of the leading edge of the pulse)

'Il (.T) = (:: t e-Roe-xlaIl [2 ( R,;; f'J, (2.9)

where a is the mean amplitude of a single-electron pulse [20]. The relative number of particles with a total number R 2: 1 of photoelectrons out of the whole set of particles converting at least one scintillation quantum, or the statistical limit of efficiency eO max of the recording apparatus (resolving time T infinitely long), is clearly determined by the sum of the probabilities (2.3) for all the particles with R 2: 1, i.e.,

00 RRe- Ro (BQ max )Q=l = ~ --1IT- = 1- e-Ro •

R=I

(2.10)

Thus, (eO max)O = 1 is proportional to the number of particles corresponding to the emission of at least one photoelectron, since particles with a total number of photoelectrons R < 1 are not recorded at all (R = 0). According to (2.9), particles with R 2: 1 correspond to voltage pulses with any amplitude, 0 ::: x ::: 00, and hence (eO max)o = 1 = (ex max) x = 0' if l/J (x) is normalized. In fact,

92

photomultiplier

V. V. YAKUSHIN

00

(ex max )x=o = ~ '\j)(x)dx = 1- e-R ••

o

However, if Q > 1, then only for Ro » Q can the sensitivity of the recording apparatus to the Q-th photoelectron be reasonably identified with the sensitivity of the cut-off electronic circuit to

Fig. 4. RC circuit in the photomultiplier collector (anode) lead.

a voltage pulse with amplitude x aQ (the mean amplitude of a pulse with a total number of photoelectrons Q) since the efficiency

If these conditions are satisfied, the distributions QWRo (t) and x WRg (t) will also be identical when determining the time characteristics of the recording apparatus.

(2.11)

4. The Poisson distribution (2.3) is not always valid for the total number of photoelectrons. Sometimes one has to record particles having an energy spectrum, or leaving a spectrum of energies ~E in the sensitive volume of the counter. In this case, instead of distribution (2.3) we should use the normalized spectrum of the total number of photoelectrons R. There is, however,

t/(:c) 'lOr-----------------------------------------,

1.65

7.8 13.9

ZIf.I IfZ.O

87.1

3ZZ 'f90

U,Of

u,om~ ______ ~ __ ~ ____ ~~~~~~~~-L~= 100 1000

x/a

Fig. 5. Spectrum of pulses in the photomultiplier collector circuit for different total numbers Ro of photoelectrons (numbers on curves). The ordinate axis represents the relative number of counts per unit amplitude.

STATISTICS OF TIME MEASUREMENT MADE BY THE SCINTILLATION METHOD 93

a condition under which the distribution of the number R, is described by (2.3), for any spectrum of ~E. Let us consider a cascade process consisting of two stages: the conversion of scintillation quanta and the emission of photoelectrons. The square of the relative fluctuation of the number of photoelectrons R is [21)

(2.12)

where 6~ and o~ are the squares of the relative fluctuations of the number of quanta N in the scintillation and ~ is the probability of the emission of a photoelectron on the conversion of one quantum, N being the number of quanta in a scintillation, averaged over the spectrum N. Since f « 1, the probability of a number ~ is, in analogy with (2.3), described by a Poisson distribution

where f is a quantity averaged over a fairly large numbers of acts of quantum conversion. According to (2.7), the probabilities of the emission of a number of photoelectrons greater than unity for one quantum are negligibly small compared with the probability of the emission of just one photoelectron, i.e., ~ = 0 or ~ = 1. We have

Thus

(2.13)

where 6& = DN IN2, DN is the dispersion of the spectrum N. If the spectrum N is described by a Poisson distribution, then DN = N- and 62 = 1/fN-. However, since f N = Ro. the spectrum of the number of photoelectrons R is also described by (2.3). In actual fact, the square of the relative fluctuation of the number R is 1 IRa, where Ro is the mean distribution of R; hence the probab~ity of the realization of R is described by the Poisson distribution. In this case, however, DN ;I! N, and distribution (2.3) will only be valid on condition that

(2.14)

Satisfaction of the inequality (2.14) can only be achieved if the spectrometric characteristics of the recording apparatus are considerably worsened, since as the number Ro falls, the relative fluctuation in the total number of photoelectrons R increases, and the spectrunl of the number R "masks" the original spectrum of the total number of quanta N, degenerating into the Poisson distribution. If we suppose that the energy ~E evolved in the sensitive volume of the counter is constant for all the recorded particles and is completely converted into scintillation quanta, then the distribution of the number N is an infinitely narrow spectral line with a dispersion equal to zero: DN = O. Then, according to (2.13), it is sufficient to put f « 1, in order to be able to describe the distribution of the number of photoelectrons R by (2.3).

The conversion efficiency of a scintillator, and still more of a Cerenkov radiator, as already mentioned, is less than unity, and the quantum efficiency of the photocathode of a photomultiplier is never greater than 30%; hence the distribution of the total number of photoelectrons R for a monochromatic line ~E is described by (2.3), even in the case of 100% collection of the scintillation quanta on the photocathode. Hence f « 1 is a quantitative condition for the applicability of the Poisson distribution in describing the recording statistics of nuclear particles giving up a

94 V. V. YAKUSHIN

IIPI/q from (Z.z) PRo from (Z,8)

2.0

Fig. 6. Probability ratio RPRo calculated from (2.2) and (2.3). Ro = 5; 1) Rmax = 6; 2) Rmax = 10; 3) Rmax = 20.

constant energy AE in the sensitive volume of the counter. For wide spectral distribution of the number N, the dispersion DN ,.... :N2 and f «lIN, i.e., fN = Ro «1. In this case, in accordance with (2.7), the photocathode in general emits only single photoelectrons for each particle. The satisfaction of inequality (2.14) constitutes the basis of our method of estimating the sensitivity of photomultipliers intended for experiments involving the time selection of nuclear particles [22J.

5. If we replace the distribution (2.2) for the total number of photoelectrons R by the distribution (2.3), we introduce a certain error into the final result; the value of this error is determined by the product ht = Ro/ Rmax . For Ro/Rmax «1 the

t

ratio

is indistinguishable from unity in the overwhelming majority of cases in which R photoelectrons are realized. If, however, Ro/Rmax = '" 1 (hypothetical case), then in the ranges 1 :::: R < Ro (Q = 1) and Ro < R :::: Rmax this ratio is smaller and in the range R ,.... Ro greater than unity (the probability (2.3) is finite for R::::: Rmax , where (2.2) is not defined). It follows from the asymmetry of the distribution (2.3) that, even for Q = 1, the number of cases of realizing R with R> Ro is not less than half the total number. The recording of particles with R> Ro involves smaller time fluctuations than in the case of R < Ro (see '§ 5); hence, after an averaging operation analogous to (2.5), the value of the fluctuations appears to be smaller than is really the case. However, for Ro « Rmax this error is negligibly small (Fig. 6).

Let us now derive the distribution for the number of counts in nonoverlapping time intervals relating to the recording of particles with a total number of photoelectrons R (or, in general, events of a particular class represented by the emission of a total number of photoelectrons R) by a photoelectron counter. As mentioned earlier, one of the decisive aspects in a countingstatistics problem of the second type is the question of the constancy of intensity. Strictly speaking, a constant intensity, i' (t) = const. is only hypothetical for the recording systems under consideration, since in the overwhelming majority of cases the number i (t) of photoelectrons emitted is not a linear function of time (see ~ 3). However, even constancy of the intensity cannot, generally speaking, constitute an exhaustive criterion for the choice of time distribution, since, for one arbitrarily-chosen particle, the probabilities that a number of photoelectrons Qt, Q2' ... ,

Qn' ••. will be emitted in nonoverlapping time intervals of finite length are determined not only by the expected (average) number in each such interval but also by the number of photoelectrons R which is expected to be emitted in the fugure. If the number R is limited, then, for one particle, the probability of the emission of a number of photoelectrons Qn in one of the nonoverlapping time intervals is described, independently of the form of the intensity function i' (t), by the binomial distribution (2.4)


[/(t) I(t )]Q n [ J(t) I(t 'In-Q-Qn P (t _ t ) = cQn n - n-l 1 _ n -, n-l) Q n R-Q n n-l R-Q H-Q fJ-Q ' (2.15)

where QnPR _ Q (t n - t n _ 1) is the probability of the emission of Q n photoelectrons in the interval tn - tn _ 1> Q is the number of photoelectrons already emitted before the start of this interval,

In

!(tn)-!(tn-l) = ~ /'(t)dt i n - 1

and 1(00) = R - Q. The probability (2.15) onhe realization of a number Qn depends statistically on the number of photoelectrons Q emitted before the beginning of the time interval tn - tn-I, The "statistical dependence" in this sense has the following meaning.

The emission of each individual photoelectron is a random or chance event, the probability of which is given by the quantity II ~i , i.e., it is related to the conversion of one scintillation

i

quantum and is independent of the emission of other photoelectrons. However, the probability of the simultaneous emission of a group of photoelectrons Qn is given (other conditions being equal, i.e., the duration of the interval tn - tn-I' the expected number of photoelectrons in the interval, and so on) by the total number R - Q, which by definition has yet to be emitted for the particle in question. The probability of the emission of a number of photoelectrons Qn (Q already having been emitted) for a set of equal nonoverlapping intervals is clearly given by the sum of the corresponding probabilities (2.15), allowing for the function 1 (t n) - 1 (tn-I) representing the emis sion of photoelectrons in each of the intervals. Let us now consider that R - Q exceeds Qn so much that, for the whole set of intervals, the intensity may be regarded as constant (for example, the intensity described by the exponential function RA exp (-At) is constant for At « 1), or else is constant by definition, l' (t) = const, Then the probability of the emission of a number of photoelectrons Qn in one of these intervals is described by the Poisson distribution. The same distribution is valid for the whole set of intervals. If, however, R - Q, » Q n, but the intensity is not constant (see § 6), then in each of the intervals the Poisson distribution is valid as before (the number of statistical tests is unlimited), but for a set of equal nonoverlapping intervals it is not satisfied, except in the case in which the intensity is a periodic function of time [10].

Thus a limited total number of photoelectrons is a sufficient criterion to justify use of the binomial distribution for the number of counts in one of the equal nonoverlapping time intervals, independently of the form of 1 (t). If, for example, in some apparatus, the intensity of the source is constant over a certain period of time and equal to zero outside this interval, while the total number of pulses is limited, R ~ Qn' then the probability of the recording of a number of counts Qn is described by (2.4). On the other hand, if the number of statistical tests is practically unlimited, R »Qn' then the numbers of counts in one of the intervals are described by a Poisson distribution, also valid for the whole set of these intervals, provided that the intensity is practically constant for the whole period of measurement.

The derivation of the distribution Q W Ro (t) for intervals prior to the moment of recording the event (emission of the Q-th photoelectron by the photocathode) involves a statistical problem relating to intervals separating a fixed instant of time (the beginning of the photoelectron emission process, t = 0) and the instant of recording the Q-th count in the apparatus (emission of the Q-th photoelectron). The distribution in question clearly describes the probability density for the instant corresponding to the emission of the Q-th photoelectron. In accordance with the presentation of the problem, we consider two nonoverlapping and mutually adjacent time intervale 0 to t

96 V. V. YAKUSHIN

and t to t + dt, the latter of which is infinitely small. Let us suppose that Q photoelectrons are emitted in the period 0 to t + dt, the Q-th being emitted in the period t to t + dt. Then Q - 1 will have been emitted in the interval 0 to t. For a fixed instant of time t and an arbitrarily-chosen particle, with a total number of photoelectrons R, the probability of the emission of one photoelectron in the interval t to t + dt is, after allowing for (2.4):

1 [q>(dt) J[ q>(dt) JR - Q - ' IPR-(Q-I) (dt) = CR-(Q-I) R _ (Q -1) 1 - R _ (Q -1) - f (t) dt,

since cp(t) = 1 (t) - (Q -1), cp(oo) = R -(Q -1), i.e., at the end of the interval t to t + dt

q; (dt) = q; (t + dt) - ep (t) = f (t + dt) - f (t) = f'(t) dt,

if 1 (t) is a continuous monotonically-increasing function.

(2.16)

The index R - (Q - 1) in 1PR _ (Q _ 1) (dt) reflect the fact that the number of photoelectrons yet to be emitted after the instant t will in principle not exceed R -(Q -1). If we now take (2.1) into account, we obtain

IP (dt) = [ep (dt)] e-'P(dt) = l' (t) dt, (2.17)

i.e., for any t, the probability of the emission of one photoelectron in the interval t to t + dt is independent of the total number of photoelectrons R. We note that the probabilities of the emission of two or more photoelectrons in this interval are according to (2.7) infinitely small, i.e., of higher orders with respect to (2.17). In this sense, the act of recording a nuclear particle is identified with the arrival of the next portion of charge in the anode circuit of the photomultiplier (corresponding to the emission of one photoelectron by the photocathode) in the time t to t + dt.

For a particle with a total number of photoelectrons R, the probability of the emission of the Q-th (counting from the beginning of the emission process) photoelectron in the time t to t + dt (adjacent to but not overlapping the interval 0 to t) is the product of the probabilities of the emission of Q - 1 photoelectrons (out of the total number R) in the interval 0 to t and one more in the interval t to t + dt:

(2.18)

since the emission of the photoelectrons in the latter interval is an event statistically independent of events in the interval 0 to t, and 1PR-(Q-1) is determined only by the intensity 1'(t) and the length of the interval dt. However, this photoelectron can only be the Q-th if Q - 1 have been emitted in the previous interval 0 to t. Expression (2.18) reflects the probability that these two events take place together.

In (2.18), the probability Q_ 1 PR (t) is in general (2.4), n being a normalizing factor determined from the condition

00

~ QWR (t)dt = 1. (2.19) o

The total probability of the emission of the Q-th photoelectron, defined for the unlimited (infinite) time interval 0 to 00, equals unity, since in accordance with the principle of causality a necessary condition for the emission of R photoelectrons in this interval (following the emission of an energy .6.E by the particle in the sensitive volume of the counter) is the emission of each of the preceding photoelectrons Q :s R.

We note that condition (2.19) is not dependent on the form of the intensity function fr(t). Considering (2.4), (2.18), and (2.19), we have n = (R + 1) fR, since'


1

C~-l ~ [I ~) r-1 [1 - f~t) r- Q+

1 d [f (t)] = R / (R + 1), o

where 1

("' t,,-l (1 _ t)1H dt = I' (x) f' (:;) = B()' 1) J r (.l:-r y) ., y , o

i.e.,

Q vVR (t) dt = n,~ 1 O-lPU (t) l' (t) dt. (2.20)

If f'(t) is constant, then for particles with a fixed total number of photoelectrons R the probability density (2.20) is described by the binomial distribution (R '" Q) or the Poisson distribution (R» Q) corresponding to the probability of the emission of Q - 1 photoelectrons in the interval 0 to t (apart from a constant factor). Summing the probabilities for a set comprising a fairly large number of nuclear particles with R 2:: Q, in analogy with (2.5), we have

W (t) = ~ R~e-Ro R + 1 CQ- 1 [!J!lJQ- 1 l1 _ !J!lJR- Q +1 I' (t)

Q Ro .LJ R! RI R R R . R=Q .

(2.21)

The distribution (2.21) may, for example, be interpreted thus. The quantity QWRo(t) is the probability density of recording the Q-th count at the instant t in one of the systems, consisting of identical counters and particle sources (the total number of counts for these systems is described by a Poisson distribution with a mean Ro), or the probability density of the first count (at the instant t) by a conversion circuit with a conversion factor Q in one of the systems with identical counters and particle sources and identical electronic apparatus (the dead time of the conversion unit is fairly small).

For Ro «Q the distribution (2.21) degenerates, in accordance with (2.7), into one term of the sum with R = Q:

(2.22)

Expression (2.22) is the probability density of the instant of emission of the Q-th photoelectron, for which the average total number of photoelectrons Ro is so small that the overwhelming majority of the recorded particles is made up of particles corresponding to the emission of a total number of Q photoelectrons.

Let us now put (2.21) in the form

[j* (t))Q-1" -R 00, 1 R~-Q+1 [f (t)lU-Q"l QWRo(t) = (Q-l)! I (t)e 0 ~ Ro (R_Q+1)!(R+1) 1-RJ

R=Q

{ [ f (t)l}R [( (t))Q-1e-Rot' (t) ~1 (1 + _Q_) Ro 1- R [1 _ f (t)J

(Q-1)! LJ R+1 R! R ' R=o

where f* (t) = Rof(t) fR. The distribution (2.21) reduces to the form

[1* (t)] Q-1e-t* (t». QWRo(t) = (Q-l)! I (f),

(2.23)

( 2.24)

98 V. V. YAKUSHIN

if Ro »Q, i.e., in (2.23) 00

R~O (1 + R ~-1)[ 1 - f ~t)J {Ro [l-lr (I) / R]}R ::::: exp {Ro [1- f (t) / RI}. (2.25)

On satisfying (2.25), the contribution of terms in the sum of (2.21) with R"" Q is negligibly small, i.e., the total number of photoelectrons R for the overwhelming majority of particles is so large that the emission of a number of photoelectrons Q is described by the Poisson distribution for each particle. However, since the intensity of photoelectron emission in the time interval considered may not be constant, within this interval the probability density Q W~(t) depends on the form of the function, i.e., it does not coincide with (2.6). In this case the intervalstatistics problem in hand is analogous to the counting-statistics problem of the second type treated earlier. If we now consider time intervals in which j'* (t) = const, i.e., j* (t) ,.,. t, then, apart from a constant coefficient, (2.24) is the Poisson distribution for the emission of Q -1 photoelectrons.

Thus the condition Ro »Q is sufficient for the distribution (2.21) to coincide with the asymptotic form (2.24), but the proportionality of (2.21) and (2.6) is only valid when at the same time Ro » Q and j'*(t) = const. On the other hand, consistancy of the intensity is by no means a sufficient criterion for the validity of any particular distribution in relation to the interval-statistics problem under consideration.

The asymptotic form of (2.24), which was obtained by Post and Schiff [23], enables us to estimate the resolving power and efficiency of a photoelectron counter in recording particles leaving a comparatively large amount of energy AE in the sensitive volume, so that the mean total number of photoelectrons Ro greatly exceeds the sensitivity Q of the electronic cut-off circuit. However, up to the present, there has been a tendency to use this distribution in the range Ro ,... Q as well [10, 14, 24, 25], for which its validity no longer holds. The errors in estimating the resolving power of the recording apparatus committed in the papers cited create a false impression of the possibility of setting up a wide range of physical experiments (see § 5).

A situation in which Ro F::J Q may, for example, arise when recording low-energy particles, recording particles of fairly high energy with scintillation films, using long light guides in order to carry the scintillation quantum out from a region subjected to electric and magnetic fields, using large sensitive volumes with considerable self absorption (with a photomultiplier photocathode area small in comparison with the area of the collecting face of the sensitive volume) for a recording apparatus with a cut-off electronic circuit of low sensitivity Q F::J Ro, and so on. The range of applicability of the distribution (2.24) is determined by the extent to which the time characteristics of the recording apparatus may be allowed to deviate from the true ones, in accordance with (2.25).

The distribution (2.21) for the probability density of the instant of time corresponding to the emission of the Q-th photoelectron enables us to find the statistical characteristics of the recording apparatus, the efficiency eQ and the resolving power w(1') if we know the resolving time T of the coi.ncidence circuit, or else to find l' from given values of eQ and w(7'). It may be found, however, that the relation between the values of eQ' W(T) , and T required for carrying out the experiment is not satisfied, owing to time fluctuations introduced by the existing recording apparatus. In this case, the experimental conditions may be satisfied by improving the parameters of this apparatus, by using a scintillation material with a smaller scintillation time, by choosing a faster photomultiplier, and so on.

The distribution (2.21), derived for the discrete process of the emission of photoelectrons by the photocathode of the photomultiplier, may also be used for determining the time characteristics of the photomultiplier itsel~ to an accuracy which is fully acceptable for the majority of uses; this will be considered in § 6.


§ 3. The Photoelectron Emission Function jet)

The efficiency eQ and resolving power WeT) of the recording apparatus and also the form of the function QWRo (t) are determined by the average total number of photoelectrons R, the sensitivity Q of the electronic cut-off circuit, and the time dependence of the photoelectron emission function j (t). The estimation of the time characteristics of the recording apparatus reduces to a standard operation (see § 5) if the intensities of the successive processes involved are described by a unified function. We shall show that for the recording systems under consideration the exponential constitutes such a function.

The conversion of scintillation quanta in inorganic scintillating substances is described by the function [26]

~ It (p (l) = 'ANe-' ,

i.e., the mean number of photoelectrons expected at the instant of time t in a counter with a "slow" scintillator of small volume and a "fast" photomultiplier is

j (l) = Uv (1- e-At ), (3.1)

where ~ is the quantum efficiency of the photocathode, averaged over the scintillator emission spectrum, the absorption spectrum of the sCintillating substance, and the spectral sensitivity of the photocathode, and :N is the average total number of scintillation quanta for the whole set of particles recorded, or the total number for one arbitrarily-chosen particle.

A study of the scintillation of organic scintillating substances carried out in recent years has shown that the conversion of scintillation quanta in these substances takes place as a result of the successive realization of two processes: the excitation of the centers of luminescence, as a result of the diffusion of the energy ~E evolved in the solvent, and the deactivation of these centers. The intensities of both processes are described by exponential functions, i.e., at the instant of time t

df (.1) _ u, t -V ( -At -A t) -----,,1 e -e 1 dt AI-I.

( 3.2)

and

For Ai = A

jet) = ~N[1-(1 + 'At)e-At ],

where the constant 71.1 refers to the statistical process involved in the transfer of the energy released in the solvent to the centers of emission [27, 28].

If the photocathode of the photomultiplier is irradiated with a source of light quanta of fairly short duration, then the function j (t) will only exist in this time interval, since the indeterminacy at the instant of emission of the photoelectrons is a negligibly small quantity [29]; the same applies to the delay in the time of emission of the photoelectrons. In such cases we must consider the inertia of the processes taking place in the photomultiplier. If in considering these we neglect the multiplication of secondary electrons in the dynode system, then the time characteristics of the photomultiplier will be determined by the time function of the number of photoelectrons reaching the first emitter of the dynode system [30],

f (t) = ~k~ N (1 -- e- xt ), (3.3)

100 V. V. YAKUSHIN

a

~ ! \ b

t Fig. 7. Spherical approximation for the surface of a conical volume. a) Light collected on the larger face; b) light collected on the smaller face.

Fig. 8. Diagram to illustrate the determination of the photoelectron-emission intensity function for the instantaneous scintillation of a source of light quanta.

where ~k is the efficiency and 'X is a coefficient representing the number of photoelectrons arising from the volume of the entrance c_hamber and striking the first emitter, while N is the average total number of light quanta falling on the photocathode in the course of one pulse. The secondary-electron collection function associated with the corresponding dynode is also exponential (see § 6).

The time characteristics of the recording apparatus are associated with the geometry of the sensitive volume (the scintillation volume, the radiator of the Cerenkov counter or the Cerenkov shower spectrometer, and the optical path), its shape and size, the absorption of light quanta by the substances compos

ihg it, and the state of its surface. Also important from this point of view are the charge, mass, and type of particles being recorded, which determine the range, the ionization density distribution along the track (for a charged particle), and so on. Let us consider an arbitrarily placed point source (Le., one very small compared with the main volume) of light quanta formed by interaction between a nuclear particle and the material of a conical sensitive volume. If the sides (walls) of the volume are ideally reflecting or transmitting, the upper face is absolutely absorbing, and the photocathode of the photomultiplier has ideal optical contact with the perfectly transparent lower face~ then, independently of the shape of the lower face, the photoelectron emission function f (t) will be to a fair approximation described by the function representing the emission of photoelectrons from the surface of a sphere of radius r + lo (Fig. 7) absolutely absorbing the light quanta. For r = 00 the surface of the sphere degenerates into a plane and the conical volume into a cylinder. The intensity of the emission of photoelectrons from the element dS of the sphere (Fig. 8) at the instant of time t is [17]

[dl (/)J = [N (~)2 cosye-l'ctlndS, dt I;;' to 4:t ct


where to is the time for the propagation of the light quanta by the shortest path to the cathode, N is the total number of quanta instantaneously emitted by the source for an arbitrarily-chosen particle, nand J.l are the refractive index and absorption coefficient of the light quanta in the material of the volume, averaged over the source emission spectrum, the absorption spectrum of the substance, and the spectral sensitivity of the photocathode, and

( ct )2 as = 2n -;- a (cos 8) / cos y,

since dS = 27T (r + lo) d (r + lo); d (r + lo) = (r + lo) sin f3 df3. From the triangle ONdS

a (cos 8) = - [~ (~)2 (1 + 210 ) + ~l at

to t I' 1r n

and

ld/ (t)] = ~N e-p.ctln [(~)'211 +~) + ~J ' dt t;?-to 2to 1 \ - 2r - 2r _ (3.4)

For convenience of calculation, the illumination of the lower face at a moment of time t 2:: tc is taken as uniform, independently of the diameter of the photocathode. For r = 00 and JL = 0 (3.4)

reduces to the expression obtained in [311:

rdi (/)J _ Hi (~)2 L rli t;?-to - 2/0 I • (3.5)

Integrating (3.4) from to to t, we obtain for the photoelectron -emission function

-- t tlto I

f (t)t>t = E,2N J1e-p.lo :L' 1 + -#- 11 + ~l )l_ ~ e -1'0/0 t; l-1 + ~ (1 +~, ~ \)l- ft(o (1 + 210 ) \ e- fL

oX ax} , ~ 0 - d' f.L 0 j 1 -- _r f.L 0 10 i - r J x (3.6)

1

The plus sign in (3.4) and (3.6) is obtained for the ymission of photoelectrons from the lar ger face and the minus sign (after analogous considerations) for the emission of photoelectrons from the smaller face of the conical volume (Fig. 7). From (3.5) we have

f1(t) = E"N (1 -~) , t;?-to 2 t (3.7)

For the exponential scintillation of the source of quanta (Fig. 9)

__ I __

ldi (t)J = ~ N r Il(~)2 (1 +~) + ~J 'Ae-A(t-t') e-!J.ct'lndt' = E,N 'Ae-At {e- a [1 + ~ (1 + ~)J-(it t;?-t. Lt~ J t - 2r - 2r 2 - 2r a

tn -

t tlto

- ~e-aToll + ~ (1 + ~~)J -a (1 +~) ~ e- ax ax}, t --- 2r a to _ - 2r .} x (3.8)

1

where a = (lJ.c/n - i\) to.

From Fig. 7, for a volume with reflecting walls, we find the upper limit t = t max corresponding to the emission of the total number of photoelectrons. For the larger face

Il( r )2 (r 1 )2J'/' [( r)2 (r 1 )21'/'} t = to ~ -+ 1 - -- -+ 1 + - - ----, 1 , max I 10 10 10 _ 10 10 10 _

102 V. V. YAKUSHIN

I I

I I'

I

"", I x I

'" /

I I

I

I

I I

/

I I

I

4 I

I

r--' I \ I \ IN' I t-O' I -, ,

\ \ , I , , ,

7l/Z \

Fig. 9. Diagram to illustrate the determination of (df(t)/dth ~to for an exponential source of light quanta.

f(tmax J --t~~-.I-----==-

o

I I I I I I I I

Fig. 10. Approximation of an exponential function.

Fig. 11. Collection of Cerenkov radiation quanta in a cylindrical radiator.

where the plus and minus signs correspond to e> 7r 12 and e < 7r12 (Fig. 8) and for the smaller

( r )';' lmax = to 2 t; + 1 .

For r = 00, tmax = 00. The value of t max is independent of the shape of the cross section of the volume cut-off by a plane perpendicular to its height. However, for transmitting walls we must determine the effective aperture angle of the volume, for example, by taking this as equal to that of a cone of revolution with the same height and area of the lower face. Then for the larger face

tmax = to (1 + i-) {[1 - C ~lJ sin2 8 r-__ r_ cos8}

r + 10

and for the smaller

h 8 Jt . 1 + ( F' 7 were = 2 -- arc sm n __ ct see IgS.

and 8).

For a cylindrical volume with trans-mitting walls t max = nto independently of the shape of the cross section of the volume cut-off by a plane perpendicular to its height. The function f (t) for a source of quanta in a volume with a reflecting upper face is determined by the sum of the functions corresponding to two identical sources, symmetrically placed with respect to the upper face, appropriately displaced in time. Light quanta falling on the larger face of a conical volume experience, on average, a smaller number of reflec-tions from the walls than in a cylindrical volume. The greatest number of reflec

tions is experienced by light falling on the smaller face. Hence, in the case of a real volume, expressions (3.6) and (3.8) for the emission of photoelectrons from the larger face constitute a better approximation to the true state of affairs. Subsequent considerations show, however, that the accuracy of the resultant expressions is not too important when determining the time characteristics of the recording apparatus if the tot a I number of photoelectrons emitted by the photocathode is measured.


We make use of the approximation (3.6) for the exponential

(3.9)

where, allowing for (3.4), the constant Ag and geometrical efficiency ~ are

A. - 1 (dl(t») g - I (tmaX) dt 1=1,

1 1+10 /"

to 2 ~g (3.10)

The set of functions (3.6) normalized with respect to j (t max) and j' (t)t = to lies within an area bounded by the linear function describing the emission of photoelectrons by the surface of a sphere with a source of light quanta near its center (see Fig. 7): rho« 1, lilo ~ 1 and t max :::::

toCr/lol + 1), i.e.,

/2) (t) = ~N(l + -"'--\ (~-1) t~to .2 'r ) \ to ' (3.11)

and the function (3.7) (Fig. 10).

On turning to (3.4), in fact, we note that f' (th ~ tv_ falls with increasing t:::: to, i.e., the intensity of photoelectron emission is no greater than '(~Nl2to) (1 + lo/r) in any case. Further, the function (3.7) for an absolutely transparent cylinorical volume with ideally-reflecting sides has the weakest time dependence of the intensity (3.5) normalized with respect to f(oo). It is not hard to convince oneself that, even for the emission of photoelectrons from the surface of a sphere of radius r -lo (Fig. 10), owing to the limited nature of t max the normalized intensity (3.4) for any t:::: to is no lower than for /1) (t). The function jet) corresponding to any deviation from the accepted idealization and the exponential (3.9) are situated within the area of Fig. 10. The applicability of approximation (3.9) is estimated in § 5 by reference to the accuracy with which the efficiency and resolving power of the recording apparatus are determined.

On considering the collection of Cerenkov radiation quanta in the radiator of a Cerenkov shower spectrometer, we obtain a nonexponential function of the photoelectron emission intensity; hence the determination of the resolving power WeT) of the spectrometer is not a standard operation (see § 5). If the energy Eo of the primary relativistic particle greatly exceeds the critical energy ecr of an electron for the substance composing the radiator, Eo» e cr ' then the overwhelming proportion of the Cerenkov radiation is emitted by the components of the shower at points on their path at which the average effective angle of their Coulomb scattering is quite small. If, furthermore,ecr > E T , where Er is the threshold energy of the Cerenkov radiation (its intensity and the angle e, shown in Fig. 11, rise rapidly from zero to the maximum value [32]) and the effects of diffraction, the disperSion of the medium, the recoil of the quanta, etc. may be neglected, then the overwhelming proportion of the Cerenkov radiation will propagate in an infinitely thin layer along the surface of a cone of revolution with a vertical angle of e, where cos e =

1/n{3, n being the refractive index. Thus, considering the radiating components of the shower as homogeneous and moving with relativistic velocity, we must identify the shape of the function N(x) giving the intensity of the Cerenkov radiation with the cascade curve (Fig. 12). Let us put N(x) in the form [33]

C 3.12)

104 V. V. YAKU8HIN

The quantities Nsp , ri' and r2 are determined from the system of equations (see Fig. 12)

(3.13)

where Nm = aim, a has the dimensions of quanta/cm, Xm = 8m X o, and Xo is the radiation length. Let us consider the cylindrical radiator of a counter without focusing, with reflecting sides, a plane collecting face, and an absolutely-absorbing upper face. If the radiation-source velocity vector is normal to the collecting face and the moment at which the particle intersects the upper face is taken as the time origin t = 0, then the time for the propagation of Cerenkov radiation quanta from the point x to the collecting face

The intensity of the quanta falling from the element dx onto the collecting face is

x -p.-

N(x)e cosudJ'/tan8.

For ideal optical contact between the photocathode (of one or several photomultipliers) and the perfectly transparent collecting face (the face is completely overlapped by the photocathode), the photoelectron emission intensity is

(dt (t)) _ ~ IV ~c [~n~2C dt 1;>10 - , (x) (n2~2 _ 1)'1, exp -!l n?~2-1 (t - to) 1· ( 3.14)

Allowing for (3.14), for the instant of time t we have

I

f (t) = ~N sp ~ (n 2~2 - 1 (I, ~. [e)·,(t-Io)-r,l_ Q4 ~-~ J

10

( 3.15)

where

k is a normalizing factor determined from the condition j(t max ) = Ro, and

For a TF-1 lead glass radiator (ecr = 13.8 MeV, ET = 100 keV) we take 1 = 24 cm, n = 1.7, J.I. = 0.01 cm-t, a = 174 quanta/cm [32]. Then, remembering the condition Eo» Ecr and using (3.14) and Fig. 12 (Eo = 500 MeV), we have

1'1/1'2=2.5; 1'1 = 1/6.18Xo; Xo=2.4 em; k=0.2em-1

and/...1 =2.41.109 see-1, /...2=0.8.109 sec-I.

Let US now consider a Cerenkov counter with the same simplifying assumptions, but l < Xo, i.e., N = const. Then in the geometry of Fig. 11 the photoelectron emission function is

f(t)f;pt, = ~JNl[1-e-)·g(t-t.)],


i 9.--------------4

im 8

7

f

5

If

3

2

o Z " !l 8 /0 IZ S, rad. length

Fig. 12. Cascade curves for lead; i == the number of electrons at various depths. 1) Eo == 500 MeV; 2) Eo == 300 MeV; 3) Eo == 200 MeV.

where

n~2c - (n2~2 _ 1)_If,

A.g = f.t n2132 -1' Sg= [1ln~

N has the dimensions of quanta/cm, provided thatl ~ 3/j.J-nj3 (from the condition A. g (tmax - to) ~ 3). In practice, however, l« 3/!J,nj3, i.e., using an approximation analogous to (3.9) (Fig. 10), we have

* ~ -,- -I, (t-to) f (t)r;?>lo = s~NI [1- e g ],

where

).g 1 - exp (- [1n~l)

and

Thus the determination of the time characteristics of a Cerenkov counter without focusing reduces to a standard operation which may be carried out to a fair accuracy (see

§5), but this only considers the contribution of the radiator geometry.

Allowing for m successive exponential processes in the recording apparatus, the photoelectron-emission function j(t) is determined by analogy with the function representing the voltage rise at the output of an m-stage electronic amplifier with RC circuits as plate loads in each stage (or in general a quadrupole conSisting of m series-connected RC filters of the integrating type).

Putting l/RC == A, we have for various constants Ai [34]:

(3.16)

In the denominator of the sum (3.16) the factor 1 - Ai IAk' which vanishes, is excluded. For identical constants Ai == Ai

m

f X) = 1 - e-At ~ i =1

().t)m-i

(m-i)! (3.17)

Let us consider the case in which t« 11A i . From (3.16) and (3.17) we have respectively

(3.18)

!.J!l ~ -4 (A.t)m = _1_ (Me)m. R m. V 2rrm m (3.19)

These relationships enable us to estimate the time characteristics of the recording apparatus to a fair accuracy for the whole number of successive processes taking place in the latter (see § 6).

106 V. V. YAKUSHIN

§ 4. Efficiency of Recording Nuclear Particles

The efficiency of the coincidence-circuit channel is the average number of nuclear events expected to be recorded within the resolving time T of the coincidence circuit, referred to the total number of events. The magnitude of the efficiency characterizes the probability of recording the event under consideration within the time T, since it is defined for a set comprising a fairly large number of events.

If the act of recording is identified with the emission of the Q-th photoelectron by the photocathode of the photomultiplier, then the efficiency is calculated by integrating the probability density QWRo(t) with respect to a period of T:

td+T 00 R td+T

EQ = C' W 'V Ro e- Ro R + 1 Q-l ~ [I (I)J Q- 1 [ 1 (1)]R-Q+1 , _, Q Ro(t)dt = .Li -R-' ---R- en .i R 1- R , (t)dt, td R=Q . I'd-

(4.1)

where td is the time displacement (delay) in the channel or the origin of the range of integration. Within the range T the sensitivity of the recording apparatus is constant, Q == const; the quantity td is reckoned from the start of the process in which the photoelectrons are emitted.

In experiments with "goodV statistics, in which we may have T < wand the quantity QWRo (t) may be regarded as constant over the range T,

( 4.2)

i.e., the efficiency with which the event is recorded by a coincidence counter with a short resolving time is greatest if the time displacement td corresponds to the maximum of the probability density QWRo (t). The value of eQ for Ro"'" Q, Ro» Q and Ro« Q follows from (4.2) on allowing for (2.21), (2.22), and (2.24). In experiments with "poor" statistics, in which we choose T > w, integration is effected over practically the whole period of time for which the probability density is finite, starting from td = O. If T is unlimitedly large, then the effiCiency reaches the statistical limit [351

oo~ 00 RR -R 1 00 R R E = Wit) dt = 'V 0 e 0 (R + 1) \' [!J!llQ -1[1 _ !J!l]R-Q Tl d [!J!l] _ 'V Ro e- 0

Q max . Q R, LJ (Q _ 1)1 (R - Q + 1)1 ~ R R R -.Li R! . o R=Q R=Q

(4.3)

An analogous result follows from the integration of the asymptotic form (2.24) of the probabilitydensity function over an unlimited range of T:

co

E _ 1 \' ,* ( ) Q -1 -f*(I) - 00 R~ e- Ro Q max - (Q -1)! .\ [ t 1 e '(t) dt = ~ ii! (4.4)

o R=Q

The agreement between (4.3) and (4.4) is not unexpected, since the total number of events expected over an infinite period of time T == 00 is independent of their distribuLlOn within this interval. The statistical limit of efficiency E.Q max for an unlimited sensitivity of the recording apparatus, Q == 1 or x == 0, in which all the events leading to the emission of even a single photoelectron are recorded, is (see § 2):

(4.5)

In real coincidence circuits, the value of the resolving time T is finite; exact expressions for the efficiency can only be obtained for Q = 1. Putting Q = 1 and td == 0 in (4.1), in ~xperiments


with "poor" statistics we have [17]

Further, for Ro » 1

(4.7)

On the other hand, allowing for (2.24), we obtain

1'(')

81= ~ e- f«t)drr(t)1=1-e-R"fl,)/R. (4.8)

Expressions (4.7) and (4.8) coincide to the extent to which the following condition is satisfied:

[1 - f ~) fo = e-R,f('VR.

It should be noted that expressions (4.7) and (4.8) are in principle approximations. For Ro «1 the sum (4.6) degenerates into one term with R = 1:

(4.9)

The expressions obtained enable us to find the efficiency of the recording apparatus with an arbitrary functionj(t) of photoelectron emission and for any energy of the recorded particles. However, the problem may be presented in another way: How should we choose the resolving time r of the coincidence circuit in order to ensure a given efficiency, for example, 0.95£1 max,

for an existing apparatus with which it is intended to record nuclear particles? The expressions for £1 also enables us to estimate the resolving time r necessary in order to keep the efficiency constant on changing the energy of the particles or the parameters of the recording apparatus (the geometry of the sensitive volume, the type of photomultiplier, and so on), provided that the sensitivity of the electronic cut-off circuit is Q = 1 as before.

§ 5. Resolving Power w (r) of the Recording Apparatus Without

Considering Processes in the Photomultiplier

The resolving power w (r) of the recording apparatus, as already indicated, is usually taken to be the width of the curve of instantaneous coincidences at half its height; the curve is the result of integrating the probability density QWRo(t) over a period T and is determined by introducing a variable time delay into one of the channels of the coincidence circuit. This choice of w (r) is associated with the fact that nuclear events separated by an interval not less than w (r) in duration may be easily resolved with respect to time over a limited number of repetitions of the experiment. The problem of the present Section is to obtain the resolving power as a function of the energy of the nuclear particles studied and the parameters of the recording apparatus (Le., as a function of the average total number of photoelectrons Ro, the sensitivity to the Q-th photoelectron, the form of the function j(t), and the resolving time of the coincidence circuit), without allowing for processes taking place in the photomultipliers. Consideration of the latter generally involves the deformation of the probability-density function (2.21) (see § 6).

108 V. V. YAKUSHIN

The efficiency of the recording apparatus is objectively associated with the value of the resolving power w ('1'). In experiments with "good" statistics, the acceptable efficiency may be so low as to enable the maximum resolving power of the particular recording apparatus, approximately equal to the width of the function Q WRO(t), W(T) ~ W, to be reached. In experiments with "poor" statistics it is, strictly speaking, necessary to have the maximum efficiency fQ = fQ max,

i.e., the statistical limit of efficiency resulting from the integration of the probability density Q WRo (t) over an unlimited period '1'; however, even for '1' »w, a very large number of experiments cannot be set up, owing to the unacceptably high level of chance-coincidence background. The reduction of the latter by shortening '1' ultimately leads to a considerable fall in the efficiency with which useful events are recorded. In e~eriments with "poor" statistics it is usual to consider the minimum efficiency as being 95% of the statistical limit [35].

Thus the time characteristics of the recording apparatus are to be regarded as the width w of the function Q W RO(t) , the resolving time '1' opt of the coincidence circuit, and the resolving power Wopt ('1') corresponding to 95% of the statistical limit of efficiency. These quantities enable us to make an objective comparison of the potentialities of recording systems.

In addition to the width w, a quantitative estimate of the characteristics of a recording apparatus may sometimes by given by the dispersion DtQ of the function Q W Ro (t)

D 2-2 tQ - tQ - tQ, (5.1)

where

(5.2)

and 00

t~ = ~ t~W Ro (t)dt eQ max 0

(5.3)

are the first moment (mathematical expectation) and second moment of the probability density QWRo(t);

00

eQ max = ~ QWR,(t) dt o

is the statistical limit of efficiency, effecting the normalization of expressions (5.2) and (5.3). The dispersion DtQ is not a universal parameter of the function QW Ro (t) and has a limited application. In the overwhelming majority of cases the value of DtQ can only be determined by numerical methods. It is also important to note the relation between DtQ and the form of the photoelectron-emission function f (t) [171. If in fact we use, for example, (3.7) and (5.1) to (5.3), we have for Q ='1:

00 R R 1 ~ Ro e- 0 1 J (1)

8 .LJ R' If' Dtl = 00, 1 max R=1 .

(5.4)

i.e., for the function (3.7) QWRo(t) relates to the class of Cauchy distributions [36], characterized by the absence of one of the moments. The intrinsic absorption of light quanta in the real sensitive volume corresponds to the function (3.6) and is sufficient, in principle, to ensure the convergence of expression (5.2); however, the relation between the dispersion DtQ and the width w of the distribution QWRo(t) is associated with the value of the absorption coeffiCient, i.e., with the form of f (t) .


The calculation of to and Dto for the distribution OW Ro (t) and the photoelectron-emission function f(t) = R(l - e- At) is nevertheless useful in order to compare with the results obtained by Post and Schiff [23J, which are usually employed in the literature relating to the time resolution of scintillation counters. The quantities to and DtO may be calculated by means of the generating function

c

G (,1) = ~ ;'It QTV R, (t)dt, o

i.e.

- r 1 ac (x) J tQ= -- ._-~ G(x) ax _X=l

and

D = [_1- [- a2c (J) , ac (x) J _ [_1_ ac (,x) J21 tQ C(x) ax" T aJ _ c (x) a,c JX=I'

Allowing for (2.21) and (5.5), we have [35j

~ RR e-R"(R __ 1) f(R-Q+2-1nx/A) G(x)= R~Q ~H-Q+1)! l'(R+2-lnx/A)

Analogous expressions follow from (5.8) for (}G (x)/ax and a2G (x)/ax2• Further,

[ G (x) ]xc"l = B Q max '

and using (5.6) and (5.7), we obtain

RR e-Ro

OR! [1jJ(R + 2)-1jJ(R - Q + 2)].

+ 1jJ' (R - Q + 2) -1jJ' (R -+ 2)} - lb.

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

(5.ll)

In expression (5.8) r (R, Q, x) is a gamma function and in (5.10) and (5.ll) l/J (R,Q) and l/J' (R,Q) are the logarithmic derivative of a 'Y function (a l/J function) and the derivative of the l/J function. The expressions for to and DtO cannot be put in any form convenient for calculation; only for Q = 1 does (5.10) simplify:

(5.12)

Let us determine the number of terms in the sums of (5.10) and (5.ll) to which consideration may be restricted. For W > Ro we have [37J:

1jJ(W+2)-1jJ(W- Q +2)<Q1jJ'(W-Q +2),

1jJ'(W -Q + 2) -1jJ'(R' + 2)< Q \1jJ"(R+- Q + 2)1.

- -:+ - -2 -+ 2 -2 + -+ -+ 2 + If to = to + ~to' to = (to) + ~to, Dto = DtO + ~D to' wher~ to, (to) , and DtO are the sums of the first (R+ -Q) terms in (5.10) and (5.ll), then~Dto = ~tb -~tO(2to + ~tO) < ~tb. Further we have

110 V. V. YAKUSHIN

A~< (~)2 8R+max {[1Jl'(R+ _ Q + 2)]2 + ~ /1Jl" (R+ - Q + 2) I} . r. 8Q max Q

The accuracies Dt and DD of the calculation of tQ and ~ follow from the expressions

~1<m/7Q' ~D<At~/DtQ'

The functions R~ e-Ro/(R!),eQ max' ljJ, ljJ', and ljJ" are tabulated [10, 37]. The calculation of1Q and DtQ from (5.10) and (5.11) for Ro> Q becomes cumbersome as the number of terms in the sums giving an appreciable contribution to the result of the calculation increases. Let us put

f (t) = R (1- e-)'t) = R [~t _ (~~)2 + (~:)3 - ... ] '

then [38]

1 [f 1 ( f)2 1 ( f)3 J t (I) = T If + "2 \ If +"3 R + ...

and, considering (2.21) and (5.2)

00

- 1 ~ tQ = -;;----'J..8Qmax

R=Q

R~e-RO(R+1) C[(f)Q 1(f)Q+1 (Q-1)1 (R-Q+1)1 J If +2" If +

o

1(I)Q+2 J( ,)R-Q+1 (') Q e(Q+l) max + '3 If + .. , 1 -If d If = Ro'J.. 8Q max [1 + Q -1 + (Q -1)(Q + 1) + ] 2Ro 3Ro ... •

We note that for Q = 1 (5.13) is identical with (5.12). Analogously

t 2 (I) = ~2 [( ~ r + ( ~ r + g (~ r + ... ].,

72 _ Q (Q + 1) 8(Q+2) max [1 +.!l.. + (Q + 2)(11Q- 3) + ... J Q - 'J..2R~ eQ max Ro 12R~

and

D Q 8(Q+2)maX [1 +~+6Q2+3Q-1 + ... J. tQ = 'J..2R~ eQ max Ro 2R~

(5.13)

(5.14)

Table 1 gives the values of Ro above which expressions (5.13) and (5.14) are valid to an accuracy of better than 10% for three terms of the expansion. Expressions (5.13) and (5.14) differ from those obtained by Post and Schiff, even if, in accordance with [23], we take e(Q + 1) max /eQ max = e(Q + 2) max leo max = 1:

- Q [Q+1 1 tQ = i..Ro 1 + 2Ro + '" J '

(5.15)


TABLE 1 TABLE 2

Q I

1 I

2

I 4

I 10 0

I 1

I 2 I

4 I

10

-I 0 3.2 4.0 7.6 -

tQ tQ 0.6 1.0 1.8 4.7

DIQ 5.4 30 v;;;- 0":"5 ------I 4.2 4.;', 1.0 2.1 7.0

D =_Q_[1+ 2 (Q+1)+ ... ] tQ A,2R~ Ro . (5.16)

The reason for the discrepancy lies in the approximate nature of the representation of the probability density Q W Ro(t) in terms of the asymptotic expression (2.24) used in the paper mentioned [23] .

As the average total number of photoelectrons Ro falls, the first moment and the di-spersion of the distribution (2.21) approach their asymptotic values. Considering (2.7), (5.10), and (5.11), we obtain

- 1 Q .tQmax=T['i'( +2)-'\1(2)], (5.17)

D'Qmax= ~2 ['IjJ'(2)-'i"(Q+2)]. (5.18)

The results (5.17) and (5.18) correspond to the emission of a total number of photoelectrons R = Q by the photocathode of the photomultiplier. The quantities tQ max and DtQ max are maximum values for this sensitivity of the recording apparatus, siEce any nuclear particle with a total number of photoelectrons R> Q corresponds to smaller tQ(R) and DtQ (R):

(Q(R)= ~ ['i'(R+2)-1P(R-Q+2)]<to max , (5.19)

DIQ (R) = ~2 ['Ijl' (R - Q + 2) - 'i" (R + 2)] < D10 max (5.20)

(see [37]), i.e., after averaging the expressions (5.10) and (5.11) with respect to R we obtain tQ < tQ max and Dro < DtQ max' Table 2 shows the limiting values of_Ro, up to which expressions (5.17) and (5.18) are valid to better than 10%. Thus the first moment to and the dispersion DtQ have asymptotic values in the regions Ro- 0 and Ro» Q (Fig. 13).

With increasing total number of photoelectrons Ro (or with an increasing amount of energy ~E released by the particle in the se\nsitive volume of the counter, with any improvement in the collection of scintillation quanta at the photocathode, and so on), the fluctuation v of the instant at which the event is recorded, usually identified with the quantity v = D~2, falls, and for Ro »Q

v z VQP,Ro. (5.21)

Independently of the value of Ro, the fluctuation v has a minimum for the highest sensitivity of the recording apparatus, Q = 1, when the recording efficiency is a maximum (4.6), eo = e1' i.e., the counter records all the events leading to the emission of at least one photoelectron by the photo-

112 V. V. YAKUSHIN

cathode. If particles leaving a small energy AE in the sensitive volume, Ro « Q, are now recorded, then for a constant sensitivity Q the value of the fluctuation v will, in accordance with (5.18), not exceed

v~ ~ [1\" (2) -1\" (Q + 2)]'/'

and will be a minimum for Q = 1 (Fig. 14):

v =0.5/'J... (5.22)

The time characteristics of the recording apparatus will from now on be determined for the maximum sensitivity, Q = 1, corresponding to an infinite sensitivity "Of the electronic cut-off circuit, x = O. The statistical limit of effiCiency el max is according to (2.10) proportional to the number of nuclear particles to which the emission of at any rate one photoelectronic corresponds, i.e., the number of missed counts of the recording apparatus is here minimal. However, the sensitivity Q = 1 is advantageous in another respect. The sensitivity of the system to a single photoelectron is equivalent to the sensitivity to the first photoelectron, quite generally, independently of the total number R. It is shown in papers on the time characteristics of scintillation counters [14, 25, 39] that, with increasing sensitivity, the width w of the function Q WRn(t) contracts, and in the majority of applications, independently of the form of the function j(t) governing the emission of the photoelectrons, it reaches a minimum value for x ~ O. An analogous assertion is clearly valid for the resolving power w (T) of the recording apparatus (see § 6).

Let us determine the relation between the fluctuation v and the width w of the distribution lWRo(t) for a functionj(t) = R(I-e_Xt) and maximum sensitivity Q = 1. Turning to (2.22), for Ro « 1 we have

(5.23)

Analogously for Ro» 1 from (2.21)

~ [ r (I) JR. lWR.(t)=t(t)1-~ , w = O.694j'J..Ro. (5.24)

An expression for lWR (t) coinciding with (5.24) was also obtained in [30] on the assumption o that the separation of a "slow" timing pulse from the amplitude spectrum of the pulses arising from the output of the photomultiplier woultl ensure that only events involving a total number of photoelectrons R would be recorded. However, for recording all the events involving the emission of at any rate one photoelectron, the results of [30] are, according to (5.24), only justified when Ro »1. On comparing expressions (5.21) to (5.24), we find that for Ro « 1 and Ro » 1,

.3!.- = 1.44. w (5.25)

Thus in experiments with "good" statistics the resolving power may only be identified with the width w or the fluctuation v of the distribution l WRo(t), if the photoelectron-emission function j (t) is exponential. In experiments with "poor" statistics, the resolving power Wapt (T) may in asymptotic cases be expressed in units of the width w. From (4.9) for Ro« 1 we have

(5.26)

then from (4.7) for Ro »1

(5.27)

STATISTICS OF TIME MEASUREME1\l"TS MADE BY THE SCINTILLATION METHOD 113

tQmax [II}..]

Dtqmax [Ij}..z]

2.S Z.O

1.5

1.0

O.S 0.5 0.'1

D.J o.ZS

./ ./

L V

taul)..} JJt/tlX]

d Z.OZ

-I.Z8 1. 0

~1~~ ~ O.'IS

0. .If ./ D.JG

0.25 0. Z

0. .!

'I 0.0

0.0. 'Z

, 0.0

-c

!l!.b d

:Fa b-

a

!'I

Z --0=1 If -Z-- ---- -0=1 ----- ---

"-"- ....

10 ----~ ~

\

~ \. ~ ~~ .... ~ '\

d~ .... \

'\~ ~ "" ~ .... ,

.... \ , ~ ~ " ,

\ , , \ , \ ~ "-\ \

d~ '" , \ \ alb \ c l '\.

\1 \ , , \ \- '" I

' I \ \\ \\' , , \

"'-0.1 0.2 D.1f /.0 Z.O '+.0 10 ZO If 0 75100 Ro,

Fig. 13. Mathematical expectation tQ (continuous lines) and dispersion DtQ (broken lines) of the distribution Q WRo (t) as functions of the average total number of photoelectrons Ro for an exponential function j (t); a), b), c), d) are asymptotes for Q = 1, 2, 4, 10.

IV V

z ..-v

/" V

f.----

whence in accordance with (5.17) and (5.13)

Topt = 3t1 max= 1.5/').,; Topt = 3f1 = 3/')., Ro.

Taking account of the foregoing, for an exponential function j(t), Q = 1, and

az, 2 3 'I 8 10 Q

Ro «1, Ro » 1, we have 1" opt /w = 4.33, i.e., the resolving time 1" of the coincidence circuit for an efficiency el of the recording apparatus equal to 0.95 of its statistical limit el max exceeds the width w of the distribution lWRo(t) by a factor of 4.33. The resolving power W (1"), or the width of the curve of instantaneous coincidences at half its height, is in principle greater than the resolving time 1".

Fig. 14. Asymptotic values of tQ max (1) and DtQmax (2) as functions of sensitivity Q (Ro « Q). The function eo (td, T) describing the

form of the coincidence curve for a circuit in which the time fluctuations are negligibly small for all channels except one is, on allowing for (4.1):

td+~

-t<td<O, eQ(td,t)= ~ QlfRo(t)dt=eQ(td+T), o

td'··~ td

O<td, eQ(td,t) = ~ QWllo(t)dt - ~ QWR.(t)dt =eQ(tdj-T)-eQ(fd), (5.28) o 0

if the sensitivity of the recording apparatus is constant (Q = const) within the resolving time T.

In (5.28) td is the variable delay time introduced into one of the channels of the coincidence circuit.

114 V. V. YAKUSHIN

W('lJ/W

3 0.75

z 0.5

0.Z5

~ __ ~ ____ ~ __ ~ ____ ~O o Z 3 l(. ~/w

Fig. 15. Resolving power w (T) and efficiency el (t d max' T) as functions of the relative resolving time T/W of the coincidence circuit; 1) w (T)/W; 2) w (T)Lw from formula (5.33); 3) £1

(td max, T).

For td = 0

(5.29)

i.e., the efficiency of the recording apparatus (or the coincidence circuit as a whole) becomes equal to eO (T) as the delays in the channels become equal. However, the maximum efficiency for a resolving time T may exceed eO (T). The corresponding delay td max is given by the equation

(5.30)

and for a limited value of T may differ from zero if the intensity f' (t) of photoelectron emission equals zero, or at any rate is not maximum, at the onset of the count. The latter condition is satisfied, for example, in the case of the functions (3.2), (3.8), and (3.15). In real coincidence circuits the broad-ening of the coincidence curve under consid

eration can only arise as a result of a fall in the sensitivity of the recording apparatus at the ends of the interval T.

Efficiencies of the coincidence curve equal to half the maximum efficiency eO (td max' T) correspond to delays td (1) and tct<2) , determined from the equations

- 't -< lct -< 0, eQ (tal) + 't) = eQ (td max -r)/2,

0< td, eQ (ta2) + -r) - eQ (td2 )) = eQ (td max T)/2,

whence the width of the instantaneous-coincidence curve at half height, or the resolving power of the recording apparatus,

(5.31)

The quantity w (T), determined for an exponential function f (t), with due allowance for (5.26) and (5.27), for a sensitivity Q = 1, and for Ro « 1 and Ro » 1 is according to the foregoing computing procedure

(5.32)

where w = 0.69 t1; tl follows from (5.13) or (5.17). Expression (5.32) agrees to a fair accuracy with the approximate expression (Fig. 15)

(5.33)

The relation between the efficiency el (td max, T) and the resolving time T of the coincidence circuit is also indicated in Fig. 15. For an exponential function f(t) , the photoelectron-emission intensity f' (t) is maximum at the beginning of the time count, i.e., td max = O. In this case for el (td max, T) 2: 0.95 el max the values of the resolving power w (T) and the resolving time T practically coincide. A reduction in T leads to a considerable improvement in the resolving power w (T) provided that the corresponding fall in efficiency is allowed by the experimental conditions.


Cfmax ------.. - -- 1.0

0.9

0.5

Fig. 16. Energy dependence of the resolving time T opt and the statistical limit of efficiency £.1 max (broken lines) (see § 4). 1) NaI-Tl; 2) naphthalene; 3) anthracene; 4) stilbene; 5) pterphenyl + a NPO in toluene; 6) 2% p-terphenyl + 0.02% POPOPinpolystyrene. Theabscissas of the circles on the T opt (AE) curves correspond to Ro = 1 for the scintillator in question. The dotted and dashed curves are' taken from [10, 24].

According to the foregoing, existing estimates ofthe resolving time Topt should be reconsidered, particularly for low values of the energy ~E left by the recorded particles in the sensitive volume of the counter (as before, we suppose that in all channels of the circuit except one, the time fluctuations are negligibly small). Figure 16 gives the dependence of T opt on AE for several scintillation materials (recording efficiency £.1 =

0.95 £.1 max). The data relating to the relative efficiencies of these are taken from [18]; the energy scale corresponds to a 20% collection of scintillation quanta at the cathode of the photomultiplier and a photocathode quantum efficiency of 10%. The conversion efficiencies of the scintillation materials are regarded as independent of the density of the ionization created by the particles being recorded. In order to obtain the curves shown in Fig. 16, we used the idealized model of the scintillation process, with an exponential photoelectron-emission function j(t) (3.1). The energy relationships of Fig. 16 are in qualitative agreement with the experimental results obtained over a wide energy range for inorganic scintillation materials [40-42] (see later in § 5 and § 6). For organic scintillation materials the function j (t) is des

cribed by (3.2), i.e., it is not an exponential, and this distorts the corresponding relationships of Fig. 16. The resolving power given, for example, in [43] for a coincidence circuit with organic scintillators in the channels was 4.10-10 sec as opposed to the value of 1 .10-10 sec expected for an exponential.

The dotted and dashed curves in Fig. 16 are taken from [10] and [24] and may be obtained from (5.15) in accordance with [23] if we ignore the consequences arising from the fact that the total number of photoelectrons Ro may be comparable with the sensitivity Q of the recording apparatus. The overestimate in the resolving time Topt indicated in the papers cited, which is particularly significant for small energies AE, creates a false idea of the possibilities of setting up a considerable class of experiments. The asymptotic behavior of the quantity Topt as the energy of the recorded particles falls, which is characteristic of the curves of Fig. 16, may be explained by the fact that, as the average total number of photoelectrons Ro falls, Ro < 1, the relative number of recorded nuclear particles involving the emission of one photoelectron, R = 1, increases until T opt coincides with the value (independent of the energy AE) calculated for R = 1 (the probability density (2.21) degenerates into a single term with R = 1).

We note one particular feature in the setting up of experiments for recording low-energy particles, arising from the asymptotic behavior of w in the range Ro < 1. In this range the scintillation constant A may be the only criterion for the choice of scintillation material, not depending on its conversion efficiency. In fact, if the effiCiency of recording the products of the nuclear reaction under consideration in the i-th channel of the coincidence circuit is not less than 95% of the statistical limit £.1 max, while the time fluctuations in the remaining m - 1 channels of the circuit and the loading of the i-th counter by side reaction products are negligibly small, then, independently of the energy of the particles recorded by this counter or the conversion efficiency

116 V. V. YAKUSHIN

of the scintillator, the ratio of the number of true coincidences to the number of random (chance) coincidences is proportional to

ntr -- "'-'--n m-l ' ran T

i.e., for Ro < 1

and for Ro » 1 , (' t )m-l n tr Inran ""'-' ""i'oi ,

where ~i is the conversion efficiency of the scintillator (see § 2). The expressions presented were obtained on the assumption that the number of true coincidences in the m -channel circuit was comparatively small [10].

The curves of Fig. 16 have a universal character, as they enable us to determine the time characteristics of the recording apparatus on an appropriate energy scale when the photoelectronemission function is an exponential with a parameter (constant >..) corresponding to the time constant of one of the successive processes in the counter (conversion of scintillation quanta, collection of quanta in the large sensitive volume, collection of photoelectrons in the large chamber of the photomultiplier, and so on).

The setting up of experiments on the time selection of long-range nuclear particles (neutrons [44], 'Y quanta of medium energy [2], and particularly neutrinos [45]), cosmic rays [46], and other nuclear particles [47] by means of scintillation or Cerenkov counters involves the use of large sensitive volumes. The time selection of counter particles in the presence of strong electric and magnetic fields is effected by means of light guides. Under these conditions the potentialities of the photoelectron method are limited by the statistical properties of the volumes employed. Estimates of the time resolution of such counters are of a qualitative nature [46, 48], and existing numerical results [31] only relate to a counter with an absolutely transparent, cylindrical light guide and a thin scintillator (the sides of the light guide are ideally reflecting). Let us estimate the time characteristics of the sensitive volumes of scintillation and Cerenkov counters and a Cerenkov shower spectrometer, neglecting the remaining statistical processes in these systems. The over-all consideration of the time characteristics determined by one of the successive processes in the counters culminates in the solution of this particular problem.

Let us determine the accuracy with which the function f(t) representing the collection of light quanta in the sensitive volume is approximated by an exponential (3.9). The statistical accuracy of the results of experiments on the time selection of nuclear particles is determined, inter alia, by the number of simultaneous events recorded within the resolving time 'T of the coincidence circuit. From this point of view the acceptability of the approximation (3.9) is determined by the accuracy of calculating the efficiency with which these events are recorded. The approximation of functions (3.7) and (3.11) by the exponential of (3.9) is clearly accurate enough for the whole class of functions (3.6). Turning to (3.7), (3.10), and (3.11), we have [17]

~1) = 1/to; A.~) = lo/rto.

Omitting the to for the sake of simplicity, we obtain from (4.9) the following for Q = 1 and Ro - 0 after allowing for (5.17):

for the function (3.9) e;. = Ro [1_e-'<ii'lmaXj, 11 max = 0.5/Ag,

for the function (3.7) ail) = Ro [1- (1 + T/21~1)maxr2J, Ifl)max = 0.5 to,

for the function (3.11) ai2) = Ro [1- (1-T/21~2~ax)2J, 7i2bax=0.5 rto/lo.


%.-------------------~----------~

15 w [!/A-gJ %

!f0

-zO

3 10 ZO 50 100 I?o

Fig. 17. Percentage error involved in approximating the functions j (1) (t) and j<2) (t) by the exponential jet) in experiments with "poor" statistics (el - ep) (continuous curves) and (el -ep) (broken curves); the numbers on the curves indicate the average total number of photoelectrons Ro.

Now, putting Ro » 1, we have

Fig. 18. Width of the distribution jWRo(t) and error in approximating the functions j(l) (t) and j(2) (t) by the exponential j(t) in experiments with "poor" statistics: 1) \w - w (1) /w(1); 2) (w - w(2)/W<2) .

for tpe function (3.9) ~ = 1-e-~/r,. 71 = 1/A.gRo•

for the function (3.7) Bll) = 1- (1 + r:/Ro711)rR" 711) = to/Ro,

for the function (3.11) B12) = 1-(1-r:/Ro7i2»Ro. 7iZ) = rto/loRo.

The ratios (el - e?) )/e)l) and (ej - e\2l)1e)2) are given in percentages in Fig. 17 as functions of Tit1 (Ro being a parameter); it follows from this that the error in the approximation (3.9) is no greater than + 13% or - 5% if T ~ 3t1•

In experiments with good statistics, the accuracy of the approximation is determined by the error in calculating the width w of the function jWR (t). Turning to (5.23) and (5.24) and allowing

a for (3.7), (3.9), and (3.11) we obtain

Ro-+O. wmax = O.347/1vg; willax'= O.26/~1); w~ax = O.5j1..!t.

Ro~1,w=O.694/IvgRo; W(l) = [21/(2+Ro) _1l1A.~1); W(2) = [1-2-1/Roj/Ai2).

The quantities W max, w; w li?ax, w(t); w(JIax, w(2) and also the ratios (w - w(1) /w(1) and (W - w(2)/w(2) are given in percentages in Fig. 18. The error in estimating the value of w, using the approximation (3.9), is in the worst possible case +35% and -29%. According to Figs. 17 and 18, the error in the approximation of (3.9) may be neglected even for comparatively low energies AE (Ro;::: 10).

The time characteristics of the sensitive volume of the photoelectron counter are determined, according to (3.9) and (3.10), by the geometrical constant Ag or the rate of collecting light quanta from a point source

1 1 ± lolr A.g = to 21;g exp (Illo) ,

118 V. V. YAKUSHIN

and the geometric efficiency ~ g' or the degree of absorption of the light quanta emitted by this source in the volume on their way to the collecting face

~g = t (tmax) I ~ N.

The constants Ag and ~g are equivalent in this sense to the scintillation constaEt A and the conversion efficiency of inorganic scintillation materials. The quantities Ag and ~g enable us to compare different geometries in order to choose the best shape of the sensitive volume. If the average total number of photoelectrons Ro is fairly large, then the criterion for the choice of geometry is the value of the product

~_ 1 + lo/r A.g-og = 2to exp (1110) • (5.34)

since according to (5.13)

Topt = 371 ~ 3/A.gRo•

where Ro ,..., ~g' However, in the range Ro »1 the numerator of the right-hand side of (5.34) is an approximate quantity, since the function (3.6) was obtained by replacing the plane surface of the collecting face with the surface of a sphere (see Fig. 7). Actually the recording of highenergy particles is effected, according to (5.13), in a period of time small compared with the value of 1IA g' so that the solid angle of propagation of the light quanta recorded in this time is no greater than the solid angle over which the collecting face of the volume is visible.

Thus, if the collecting face has a plane surface, then, independently of the geometry, j(t) is no different from the photoelectron-emission function for a cylindrical volume lo/r = O. For the same reason, the counter recording of particles with comparatively high energies (Ro » 1) is independent of the type of walls used for the sensitive volume or the way in which these have been treated. However, the average total number of photoelectrons Ro is in any case proportional to the quantity f (tmax)' as before; hence even a slight reduction in the light collection may worsen the energy resolution of the counter [19]. The foregoing considerations for Ro » 1 are also valid for any Ro if the solid angle of propagation of the quanta is limited (for example, in a long light guide with absorbing walls). With increasingdimensions [lo = (c/nHo] and absorption of light quanta by the material of the sensitive volume, the resolving power of the counter in the range Ro »1 becomes worse.

The situation is rather different in the range Ro « 1 if, in addition to the scintillation material, the value of Ag is taken as a criterion for the choice of geometry. Considering (5.12), for Ro « 1 we have

T - 37 ~ ~ _ 3to~g opt - 1 max~ A.g - (1 ± lo/r) exp (- filo) ,

i.e., a conical volume, with the coll~ction of light quanta on the large face, is preferable. An increase in the geometric efficiency ~g , although worsening the resolving power ~f the counter in· the range Ro «1 (since, for an arbitrarily chosen particle, R = 0 or R = 1, as ~g increases it takes longer to reach any specified value of the function j(t), is nevertheless essential for T/tmax > 1 in experiments with poor statistics, in which according to (3.9) and (4.9) the recording efficiency is proportional to the mean total number of photoelectrons Ro

B1 = Ro (1- e-T / iiJ max), Ro = ~ ~N. In the general case, the time characteristics of the volume are determined not only by the

geometry but also by the mass, energy, and charge of the particles being recorded (we note that the conversion efficiency of the scintillation material is associated with the density of ionization created by the charged particle, i.e., with its velocity and charge; in this sense it is appropriate


Fig. 19. Diagram to illustrate the determination of the function f (t) for a particle forming a light "track" ab in a conical volume; alb' gives the radial projection of the track.

:::.:::::

--0.003 0.01 0.03 0.1 0. 3 1.0 !.2f

t," [I/)'zl

Fig. 20. Efficiency ej (continuous curves) and shape (broken curves) of the distribution jWRo(t) for a Cerenkov shower spectrometer. Numbers on the curves give the values of Ro.

to draw an analogy between J\.g and J\.). The functionf(t) for articles creating a "light track" in the volume with a known intensity distribution is determined by integrating along the radial projection of the track (Fig. 19), allowing for the velocity of the particle. It is not hard to show that, if the length of the radial projection of the track is a small proportion of the height of the volume l or p. > 1llo, then f(t) will be satisfactorily described by an exponential. Long-range particles form point sources of light quanta (secondary ionizing particles usually have short ranges). In this case the characteristics of the volume for a set of a fairly large number of particles are determined by a probability density function constituting a superposition of the functions 1 WRo (t) corresponding to the point sources.

The efficiency of electron collection on the first emitter of the dynode system of the photomultiplier is nearly 100%; hence the contribution of the entrance chamber to the resolving power of the counter as a whole is determined by the spread in the transit time of the photoelectrons as they travel to the first dynode. Existing constructions of the entrance chambers of time photomultipliers ensure the maximum possible constancy of these times, independently of the direction of escape of the photoelectron from the photocathode [49J.

The resolving power Wopt (T) associated with the geometry of the radiator of a Cerenkov counter without focusing (see § 3) for Ro» 1 is, according to (5.13) and Figs. 15 and 21, determined by the velocity {3 of the particle and the refractive index of the radiator material:

- - - (cos-' a - 1)'" Wopt (T)::::::; Topt = 3t1 = 3/')...g~-SNl = 3 _ •

"g f3c~N

The situation is rather different in the range Ro« 1, where in accordance with (5.17) the resolving power w opt (T) is proportional to the thickness l of the radiator

Wopt max(T)=Topt = 1.5/1..~= 1.5l (cos-28 -1)/~c,

if l « 3/p.n {3.

The determination of the resolving power associated with the geometry of the radiator of a Cerenkov shower spectrometer is not a standard operation, since the approximation of the photoelectron-emission function f(t) (3.15) by an exponential is inapplicable to this case. The values of Topt and wfound graphically from Fig. 20 for the functions ej (4.6) and j W Ro (t) (2.21) relate

120 V. V. YAKUSHIN

to the recording of electrons or 'Y quanta with an energy of Eo = 500 MeV by a shower spectrometer with a radiator made of TF-1 glass of thickness l = 24 cm, with n = 1.7 and J1. = 0.01. If the area of the photomultiplier photocathode is small compared with that of the collecting face of the radiator, so that Ro «1, then, considering the illumination of the latter at the instant t> to as being uniform, we have

~Pt = 0,833/1..2 = 1.04 nsee; utnax = 0.69/1..2 = 0.863 nsee. (5.35)

If, however, the collecting face is completely covered by the photocathode, then Ro = 1700 and w = 0.67 '10-12 sec (Fig. 21). It is easy to carry out an analogous calculation for other parameters of the radiator or for another primary particle energy provided that, as before, we may consider the radiating components of the shower as homogeneous and moving at a relativistic velocity. Then for Ro « 1 the values of T opt and ware determined in units of 1/A2 in accordance with (5.35). With increasing energy Eo or average total number of photoelectrons Ro, the resolving power of the shower spectrometer improves (in the range of energies Eo between hundreds of MeV and several GeV the energy of the shower components expended on the formation of one photoelectron is, according to [19J, 0.3 MeV if the face is completely covered by the photocathode and f = 0.1).

Figure 22 shows the efficiency e1 (td max' 1') and resolving power w (1') of the spectrometer /

as a function of the resolving time T obtained from Fig. 20 in accordance with (5.31). The value of w (1') may also be approximately estimated (Fig. 22) from the expression w (1') = (1'2 + w2)! Thus if Ro « 1 then the width Wopt (1') at half the height of a coincidence curve with a resolving time Topt (e1 = 0.95e1 max) for a circuit with a Cerenkov shower spectrometer in one of the channels (the time fluctuations in the remaining channels of the circuit being negligibly small) is

Wopt (1') = 1.21w. (5.36)

In this case any improvement in the resolving power achieved by reducing T is inadmissible in experiments with poor statistics, since it is accompanied by a sharp fall in efficiency (see Fig. 22). The result (5.36) is associated with the fact that the maximum of the photoelectron-emission intensity l' (t) (3.14) does not coincide with the time origin t d max> to (see (5.30». However, in the range Ro » 1 a considerable improvement may be made to the resolving power, since the ratio Wopt (T)/W approximately coincides with that calculated for an exponential function f(t) (Fig. 21), Wopt (T)/W = 4.36.

'Zopt. w (rAt]

f~--~---r--~--~--~

o.O~r---+---~--~~~--~

o.wo~~--~--~--~--~--~ I 10 100 1000 14000

Fig. 21. Resolving time Topt and width w of the distribution 1 W R (t) for a

o Cerenkov shower spectrometer.

The resolving powers of shower spectrometers determined in [47, 501 are considerably worse than the foregoing estimates, even for Ro « 1, and are thus certainly limited by processes in the photomultipliers, electronic circuits, and so on.

Let us now consider the time characteristics of a recording apparatus with a photoelectron-emission function given by (3.2). Expression (3.2) is valid, for example, if the scintillation process takes place in two stages. As a result of the passage of the charged particle through the volume of the scintillator and the evolution of energy ~E therein, the first level is excited over a period short compared with the constant 11A1• Then the first level discharges on an exponential law with a time constant 1/A1; this leads to the population of the second level, characterized by a lifetime 1tA, on the discharge of which scintillation quanta are emitted. The function


W('rJ/W 5

e/tdmax ' T)

el(td max' T) .--------------------,1.0

w('r)/w r-------------"'---,

'I

1.5 0.75

3

1.0 0.5 Z

O'5~ ____ J_ ____ ~ __ ~~

{J o.Z5 I

T/w {l 3 0 2 3 If 'r/w T/w

Fig. 22. Resolving power w (T)/W and efficiency f-1(t d max' T) of a Cerenkov shower spectrometer as a function of the resolving time of the coincidence circuit. a) Ro« 1; b) Ro = 10; c) Ro = 100. 1) El (td max' T); 2) w(r)/w; 3) w(r)w from formula (5.33).

(3.2), generally speaking, describes any two exponential processes with constants Al and A following one after the other.

The value of the resolving time T opt in experiments with poor statistics may be determined, for a recording-apparatus efficiency equal to 95% of its statistic limit EI max' in accordance with (4.7) arid (4.9), from the expressions (Q = 1)

Ro« 1:

Ro» 1:

'topt As 'Topl 2

81 = Ro [1- (_A._ e - 2f1max T -~e - 2T1ma;) ] = 0.95Ro. A. - A.1 A. - A.1

81 = Ro [ 1 - (1 +

"t'opt ~ "::'opt Ro

_ 1 _ (_')..,_ - ROtl A _ ~ - R,t,) _ 0 95 81 - A. - ')..,1 e ').., _ A.1 e - . ,

81 = 1- (1 + TOEt )RO e ~~t = 0.95, /.. = A.!, Rotl

(5.37)

(5.38)

where 11 max and 11 correspond to an exponential with a constant A, i.e., according to (5.13) and (5.17) tl max = 0.5/A and tl = 1ARo. In expressions (5.37) and (5.38), for the function (3.2), Ro = ~N; for a photoelectron-emission function with intensity (3.8) Ro = ~g fN- from (3.6). The width of the probability-density function (2.21) for (3.2) is found by a graphical method. Figures 23 and 24 give the resolving time Topt and width w as functions of AlAI (Ro is a parameter) in units of the sum of the resolving times and widths relating to each of the stages of the process (3.2) individually. It follows from these relationships that for small energies ~E (Ro «1) the resolving time T opt is roughly twice the sum of these quantities for each of the components of the exponential processes. With increasing energy, however, the value of T opt exceeds this sum

122 V. V. YAKUSHIN

T opt Iii, ,[~+ M;'g J

1.0

R-O

0.1 0.3 1O 3.0 10 ;'/;',

Fig. 23. Resolving time Topt of a coincidence circuit for the function (3.2) in units of 1/(1 + )./)1.1) as a function of )./).1 (continuous curves are for )./).g = 0). Broken curves correspond to the function (3.16) for m = 3 and )'/).g = 0.4 (see § 3). Figures on the curves give the values of RD.

Ro=IOO

IL-~ ____ ~~ __ ~ __ ~ __ ~~ __ ~ 0.05 0.1 0.Z5 0.5 /.0 Z.O If. 0 10 ZO

A./A./

to a greater extent, so that the rate of fall of the resolving time becomes much slower. It should be noted that the relative value of the resolving time is associated with the ratio )./).1; the degree of this association intensifies with increasing RD. Thus in the recording of high-energy particles allowance for the second exponential process, even for )./).1 «1, greatly worsens the time characteristics of the recording apparatus as compared with the energy relationships shown in Fig. 16. It should be noted, however, that on further raising the ratio" the increase in the relative resolving times becomes slower (Fig. 23). On the other hand, if the value of T opt is kept constant as )./).1 varies, the recording efficiency falls in accordance with Fig. 25. All that has just been said in relation to the resolving time T is valid for the width~, with the simple difference that for Ro « 1 the relative width reaches 2.26 ()./).g = 1). Figure 24 shows the results obtained in (30) for Ro = 100. In the range Ro » 1 for a constant ratio )./).1 the width

1.57 w= --;:;==

VAAIRo (5.39)

(for )./).1 ~ 0.25 the accuracy of this expression is no worse than 10%), while for )./).1 = 0 according to (5.24) w = 0.694/). RD. The width w has a minimum for the highest sensitivity Q = 1 of the recording apparatus with a function (3.2) in analogy with (5.21); however, even for a very small value of )./).1 the w(Q) relationship in the region of the minimum is less sharp than for )./).1 =

o (Fig. 26, Ro = 100 [30]).

The value of the resolving power Wopt (T) for the function (3.2), determined in accordance with (5.31), or from the approximate formula (5.33), differs less from the width w than is the case for the exponential (wopt (T)/W = 4.42), i.e., on reducing T even a slight improvement in resolving power

Fig. 24. Width w of the distribution 1WRo(t) for the function (3.2) in units of 11(1 + )./).1) as a function of )./).1' Broken curve corresponds to [30J, Ro = 100. Figures of the curves give the values of RD.

can only be achieved at the expense of a sharp fall in efficiency. In this sense, the resolving powers of the recording apparatus are quantities of the same order, independently of the rate of access of information in the experiment.

The character of the relationships in Fig. 23 to 26 is determined by the fact that the photoelectron-emission intensity l' (t) for the function (3.2) vanishes at the time origin and rises with


t.O

0.75 t----~+--+_-----'......,

OL-___ ~_~_~

OJ o.S 0.5 /.0 ;'/;'1

Fig. 25. Efficiencye1 for a constant value of the resolving time T = T opt as a function of >..1A 1•

Topt• w. sec

w-G,----.---r----.---, NaJ-Tl

JU-16L-__ --'-___ "'--__ --'-_-=---""

0.1 1.0 to too tOOO /JE, keY

Fig. 27. Energy relationships for the resolving time Topt (continuous curves) and the width w (broken curves) of the distribution 1 WRo (t) for the function (3.2). Figures on the curves correspond to Tabl~ 3; "3" indicates a light collection of 0.02%. The dotted and dashed curves are obtained from [14].

lIJ '/O~ sec

/.0

O~t-ZL--5~----~tO~----~~~---~~

Fig. 26. Width w of the distribution 1 WRo (t) for the function (3.2) as a function of the sensitivity Q; >"/>"1 is a parameter (figures on the curves). Ro = 100; 11A = 3.10-9 sec.

increasing argument t. If Ro »1, then, in order to trip the electronic cut-off circuit, one uses a range of variation of the argument of the function f (t) corresponding to considerably lower intensity than in the case of an exponential function (the average total number of photoelectrons Ro is the same for both functions), and hence to a greater fluctuation of the interval preceding the emission of the Q-th photoelectron.

The energy dependence of the resolving time T opt of the coincidence circuit (e 1 =

0.95 e1 max) and the width w for the function (3.2) , corresponding to the emission of photoelectrons in a counter with organic scintillators, is obtained (see [17]) from Figs. 23 and 24 with due allowance for the data of Table 3, which is based on [27,28] (Fig. 27).

The energy scale corresponds to Fig. 16. The dotted and dashed lines are obtained from [14J and demonstrate the overestimate in the values of T opt and w, which is a result of the unjustified extrapolation of the proba,bility density (2.24) into the region of small energies ~E. In the energy range ~E 5$ 1 Me V

the advantages of organic scintillation materials fall off considerably, since allowance for the second exponential process necessarily affects the rate of improvement of the time characteristics of the counter with increasing energy. This feature does not, however, affect missed counts

124 V. V. YAKUSHIN

TABLE 3

Serial Optimum lA,. I 1/Al' Efficiency.

Scintillator Solvent cone .•

I relative to

No. g/liter nsec nsec NaI-Tl

1 Stilbene I 7.7 1.93 0.3 2 Paraterphenyl Toluene 6.0 2.4 0.8 0.22 3 » » 6.0 2.4 1.08 0.22 4 » POlystyrene 2.5 2.5 0.65 0.22 5 » Benzene. 6.0 2.4 0.64 0.22 6 PBD Toluene· 6.0 1.7 0.69 0.36 7 » I Anisole. 6.0 1.7 0.71 0.36 8 » I Benzene. 6.0 1.7 0.8 0.36 9 » Xylene . 6.0 1.7 0.54 0.36

Not e. No.3 is measured on excitation by ex particles of energy 4.77 MeV.

associated with the dead time of the counter. The energy dependence of the width w is qualitatively supported by experimental results [9, 511.

The resolving power of the recording apparatus is also associated with the time characteristics of the photomultiplier and the electronic cut-off circuit. In setting up physical experiments one often imposes severe demands on the recording systems employed. As a rule, one of the elements of a system satisfies the requirements imposed (high time resolution, low dead time, etc.) is a sensitive volume consisting of an organic scintillation substance. It is shown in [51] that, at the present state of development of scintillation technology, the characteristics of the scintillation process in organic scintillators are in fact responsible for the basic limitation to the resolving power of the recording apparatus.

According to theoretical estimates [30], it is found, for example, that for the scintillator Naton 136 PI./)"1 = 0.5; l/A = 1.6 '10-9 sec) Ro = 100, Q = 10 and a two-channel coincidence circuit the width of the probability density function (2.24) is w = 2.6'10-10 sec. The experimental value is w = 2.8'10-10 sec (using an XPI020 photomultiplier and an electronic cut-off circuit based on gallium arsenide tunnel diodes [51]). Increasing the supply voltage of the XPI020 by 500 V hardly made any difference to this value, although the parameters of the output pulse from the photomultiplier improved conSiderably for the higher voltage (the photocathode was illuminated by a pulsed light source in a separate experiment). The maximum resolving power of the XPI020 itself is expected to be no worse than w R;: 10-10 sec.

It is clear, in analogy with Figs. 23 and 24, that, as the energy AE or Ro falls, the width wand the resolving time T opt for the function (3.2) will not differ to any greater extent from the values determined experimentally, since the effect of the photomultiplier will be just as insignificant (see § 6). The results obtained under the same conditions with a 56A VP photomultiplier give w = 3.6'10-10 sec (the characteristics of the 56AVP are about the same as those of an FEU-36 [52]). The results obtained for Naton 136 are also valid for other examples of the most popular organic scintilla tors (see Table 3).

The time photomultipliers mentioned also ensure a high resolving power (agreeing with calculation) of a scintillation counter with a "slow" inorganic scintillator in recording high-energy particles. The recording of 'Y quanta corresponding to the annihilation of the p particles of Na22 (energy of each quantum AE = 511 keV) by reference to the total energy peak in a two-channel coincidence circuit with NaI-TI crystals in the channels gave a resolving power of w = 2 nsec and w = 1.9 nsec for the 56AVP and XPI020 photomultipliers respectively [40]. The very slight


improvement in resolving power achieved by using the XP1020, which has far better time characteristics than the 56AVP, indicates that the nature of the photomultiplier makes very little difference ~ven for an exponential function f (t) as far as w = 10-9 sec.

Analogous conclusions also hold in relation to a cut-off electronic circuit based on gallium arsenide tunnel diodes, used in experiments on the time selection of nuclear particles [53]. The experimental value of the width w determined in [51] for two-channel self-coincidence circuits with cut-off circuits based on gallium arsenide tunnel diodes in the channels (current pulses from a single photomultiplier were used) equalled 0.65'10-10 sec; similar results were also obtained in [9]. The values of w for other types of electronic circuits are also quite small [9].

With increasing energy ..6.E, the theoretical value of w for the function (3.2) becomes smaller (Fig. 35), and the time fluctuations in the photomultipliers and electronic cut-off circuits may give a more substantial contribution to the resolving power of the recording apparatus as a whole. An analysis of the relation between the time characteristics of the counters and the parameters· of the multipliers is given in § 6.

§ 6. Processes in the Photomultipliers and Resolving Power of

the Recording Apparatus

The values of w, T opt. and Wopt (T) were determined in § 5 without allowing for the photomultiplier, one of the main elements of the recording apparatus. However, the photomultiplier, which converts a very small light flux into an anode current pulse sufficient for triggering the electronic cut-off circuit, has some essential failings which in certain cases limit the feasibility of physical experiments. Let us consider the limitations imposed on the accuracy of time measurements in such experiments by the fluctuations in the time required to collect the photoelectrons and secondary electrons on the dynodes of the amplifying stages of the photomultiplier for an unlimited sensitivity of the electronic cut-off circuit, x = O. In subsequent calculations the photomultiplier is regarded as an ideally linear element of the recording system.

The processes taking place in the photomultiplier after the incidence of the light flux on the photocathode differ qualitatively from the processes in the sensitive volume of the counter considered in §§ 3 and 5. Whereas the transmission of information in the latter was effected at the expense of a partial loss of its carriers (absorption of light quanta in the substance composing the sensitive volume and their conversion into photoelectrons), in the photomultiplier the information carriers (the secondary electrons) undergo multiplication. A strict solution of the time distribution of the number of descendants in the m -th generation of information carriers, as applied to processes in the dynode system of a photomultiplier, may be obtained by a method based on the generating function for branched processes [15]. If the generating function (5.5) of the distribution (2.21) for the period up to the emission of the Q-th photoelectron by the photocathode is G1 (x), then the generating function Gm (x) of the distribution for the period up to the appearance of the Q m -th secondary electron at the m -th dynode is

(6.1)

where G1 (x). G2(x) •• .. are the generating functions of the distributions corresponding to the entrance chamber of the photomultiplier, the first amplification stage, and so on. The mathematical expectation and the dispersion of the distribution with the generating function G m(X) are determined in accordance with (5.6) and (5.7). An important property of the multiplication of information carriers is the retention by their m -th generation of the information laid up in the first generation, i.e., the dispersion calculated for the distribution of the total number of information carriers of the first generation in the majority of cases differs little from the dispersion for the m-th generation. Figure 28 gives the distributions of the total number of secondary electrons emitted in the various stages of the dynode system, from which it follows that, with increasing m, the width

126

J

Fig. 28. Distribution of the total number of secondary electrons in the various stages of the photomultiplier dynode system; m is the number of dynodes. Stage amplification (J = 4. The ordinate axis gives the normalized count rate.

V. V. YAKUSHIN

of the distributions increases only slightly as compared with that in the entrance chamber of the multiplier.

The mathematical apparatus of the generatingfunction method fails to produce expressions of a type convenient for calculation in relation to the multiplication of information carriers. Problems of this type are usually solved by cumbersome numerical methods [14, 39]. It is possible, however, to obtain another solution, avoiding the transformation (6.1), if we suppose that the number of information carriers remains constant in the various generations, but that the intensity of their generation (production) increases in an appropriate fashion. This approach is quite fruitful if we make certain assumptions regarding the form of the intensity function and the distribution of the number of information carriers in the generations. By using this method we may obtain an exhaustive explanation for the lack of agreement between the experimental results and the published theoretical estimates of the time resolution of the scintillation counter.

Let us turn to the time distribution of the secondary electrons appearing at the m-th dynode of the photomultiplier, assuming that the time distribution of the number of secondary electrons at the previous dynode is described by a <5 function. The literature only contains data relating to the distribution of the total number [54]; this has a greater dispersion than the Poisson distribution owing to the nonuniformity of the emitting surface of the dynodes and also perhaps to the properties of the actual secondary-emission process [21, 55, 56]. If, however, the nonuniformity of the secondary-emission coefficient (J is no greater than 10 to 15%, the distribution of the total number of secondary electrons is very little different from the Poisson distribution [55]. This conclusion is sufficient basis for assuming the validity of the distribution (2.21) for the period up to the appearance of the Q-th secondary electron at the m -th dynode. Analogous considerations may also be set forward in relation to the collection of the photoelectrons at the first dynode, since the non uniformity in the amplification factor of the photomultiplier at various points of the photocathode in the overwhelming majority of types of photomutliplier is no greater than 15%. The collection functions for the photoelectrons and secondary electrons at the dynodes may be regarded as exponential to a fair degree of accuracy, provided that the corresponding intensities at the time origin are maximal (see § § 3 and 5). The latter supposition is suffiCiently well-founded for processes in which the intensity is roughly proportional to the number of information carriers not yet realized. The time characteristics calculated for the assumed idealization of the processes in the photomUltiplier agree with experimentaL results for a wide range of recorded particle energies. It should be remembered, however, that the time characteristics of the photomultiplier thus calculated may appear rather better than is actually the case.

If the sensitivity of the electronic cut-off circuit is unlimitedly high, Q = 1, and the secondary-emission coefficient C1 > 1, so that the asymptotic form (2.24) of the distribution (2.21) is valid for any mean total number of photoelectrons Ro, then the fluctuations in the period up to the appearance of the first photoelectron and the first secondary electron at the first, second, ... , m-th dynodes are in accordance with (5.21) and (5.22)


Ro« 1:

Ro» 1:

0.5 -K crlKl ' . .. ,

1 1 KRo-' crlKIRo' -a-la-,K.-.-:":R'-o' ••• ,

(6.2)

1 (6.3)

where 'X., 'X. 1 , 'X.2, ••• , 'X.m _1 are the constants of the exponential functions representing the collection of photoelectrons at the first dynode and secondary electrons at the second, third, ... , m -th dynodes, 0" l' 0"2, ••• , 0" m-l being the amplification factors of the first, second, ... , and (m-l)-th stages. Expressions (6.2) relate to the emission of an average total number crt, a1 cr2' ... , a1 a2 •• , am-l of secondary electrons for a single photoelectron by the first, second, ... , (m-1)-th dynodes, and are reasonable valid (:e.16% for O"i ~ 3 according to [35]). As the number m of the stage increases, the fluctuation in the period preceding the appearance of the first secondary electron at the dynode of the next stage diminishes. This conclusion agrees quantitatively with [10].

Let us now suppose that the average total number of secondary electrons in each of the stages is constant and equal to Ro, but that the fluctuations in the period in question are described by (6.2) and (6,3) as before, It is clear that the simultaneous satisfaction of these conditions is quite possible if the process under consideration is effected in m stages without loss or multiplication of information carriers, if their number is represented by a time function constituting a series of exponentials with constants

Ro« 1: (6.4)

Ro »1:

(6.5)

Thus the determination of the time characteristics of the photomultiplier reduces to finding the values of w, 'T opt' and Wopt (T) for the distribution (2.21) with the secondary-electron emission function (3.16). Let us estimate the probability of recording one photoelectron (Ro «1) of a photomultiplier with dynode amplification factors 0"1' Oz, .•. , am_I' The desired quantity is clearly equal to the statistical limit of the efficiency (2.10) of the photomultiplier dynode system

Blmax = (1- e-a,) (1 - e-a,a,) ... (1- e-a,a, ... om-l)~, 1- e-a, .

The statistical limit of the recording efficiency of a photoelectron for 0" ~ 3 is no less than 93%. If Ro »1, then £.1 max = 1. Let US consider the asymptotic time characteristics of a photomultiplier and a scintillation counter with a photomultiplier in recording small (Ro« 1) and large (Ro » 1) energies.

Case Ro« 1

By using the distribution (5.24) and the function (3.17) with equal constants of the successive exponential processes (A = A g = A1 = "X, see § 3; multiplication of secondary electrons not considered) , we obtain the width wand resolving time T opt as functions of the number of exponentials

m (Fig. 29). The relative quantities Topt I ~ 1Dpt i and wi f Wi and the ratio Topt /w depend only

slightly on m. The resolving time of the counter is roughly equal to the sum of the resolving times calculated for each of the exponential processes separately. An improvement to the

128 V. V. YAKUSHIN

1A~------------~

5

OL-__ ~ ____ ~ __ ~ 1 Z 3 ~m

Fig. 29. Resolving time 'Topt and width w,

their relative values Topt I fl'oPt i , wi f Wi '

and the ratio 'T opt /w for a series of m exponentials with equal constants A = A1 =

Ag = 'X, Ro « 1.

&,/e,,max e, 1.0 1.0

If a

6:J 0.8

0.5 m-Z 3 If

0.8

0 Z 0.'1

lZJ o.Z

m-Z ~

0 0.5 1.0 1.5 Z,O 0 Z 7:/w 'Z'/w

Fig. 30. Relative efficiency e/e1 max as a function of the resolving time T/W of the coincidence circuit. a) U = 1, sequence of exponential functions with equal constants; b) 0' = 3; and c) 0' =04. Curves for a photomultiplier with Ro » 1.

resolving time w ('I') of the counter by reducing 'I' can only be achieved at the expense of a considerable reduction in efficiency (Fig. 30) (the resolving power according to (5.33) is w ('I') ::::!

...JT2 + w2). We remember that in recording low-energy particles the efficiency is, according to (2.10), decisive in relation to the choice OfT.

Let us now consider a photomultiplier with parameters 'X = 'Xi = 'X2 = . . . = 'Xm -1 and ui = ~ = . • • = Urn -i = 0'. For the dependence of 'T opt ' w, and T opt /w on m (Fig. 31) the general features of Fig. 29 are still characteristic; however, since the constants of the exponentials constitute a geometrical progression (6.4), for 0' ~ 3 the values of'Topt and w remain practically constant for all m ~ 3 (0' = 3 corresponds to an amplification factor of a 14-stage photomultiplier equal to ~ 5' 10-6). Thus the resolving tim.e of a coincidence circuit with a scintillation counter intended for recording low-energy particles in one of the channels is roughly equal to the sum of the resolving times calculated for the elements composing the counter. The width w of the distribution 1WRo(t) equals twice the sum of the widths Wi. If the constants 'Xi are quantities of the same order and Ui ~ 3, then for determining'Topt and w it is sufficient to confine attention to two or three stages of secondary-electron amplification. This conclusion is analogous to the experimental results of [56, 57] for the distribution of the total number of secondary electrons arising on the emission of a single photoelectron by the photocathode (see Fig. 28). The relative fluctuation of this distribution does not depend on the number of amplification stages [14]:

aa = a/ada -i),

if am > 1, where U1 and 0' are the amplifications of the first and subsequent stages. In the range Ro «1 the energy dependence of''T opt and w is asymptotic both for the whole counter and for each of its elements (see Figs. 16 and 27), since, in accordance with (5.23), in the overwhelming majority of particle-recording acts on the part of the photomultiplier photocathode, only single photoelectrons are emitted.

As a characteristic of the time properties of photomultipliers, one often uses the quantity t1j2 , i.e., the width of the anode-current pulse from a single photoelectron at half height [14, 39]


-<>- - - ---<>""'-- --.,.,.-o-.-.-.-o-.~-.

A-t 'ropt/IIT / 1'2 w-

6=1f OL-~--~----~----~

1/" 3,-------------------. 'ropt

lU

~pt!f~pt!

6=3 0

f Z 3 'fm

Fig. 31. Values of TOPbW,'t'opt l~l'oPt i,

W I f Wi, ToprW , and the width t1/2 of

the anode-current pulse for a photomultiplier with Ro« 1.

z

OL-~ ____ ~ ____ ~~ Z 3 Ifm

Fig. 32. Values of T opt and w in m-l

units of l/xa"2 '-JIR;; and the ratio T opt /w as function of the number m for functions (3.18). Horizontal broken line is an asymptote.

(Fig. 31). The value of t1/2 is practically constant for m 2:: 3 and a ;$ 3.

Case Ro» 1

The values of 'T opt and w in the range Ro » 1 are determined from the asymptotic form (2.24) of the distribution 1WR (t). The secondary-electron emission

o function J* (t) is (3.18) if

't'opt ~ (m + l)/Aruax,

w~(m + l)/Aruax, (6.6)

where Amax is the greatest constant in the sequence of exponentials considered. Inequalities (6.6) follow from the expansion of expres sion (3.16) in series in powers of A it. Considering (3.18) and (6.5), we have the following from (2.24) for 'X. = 'X.1 = 'X.2 = .•• ='X.m-1

and a1 = • •• = am -1 = a, (Q = 1)

m(m-l)

-( t )m-l lWR • (t) = Roxma 2 m~1 X

m(m-l) e [R -

X V2iWJ exp - ,~ x""a 2 X 2rrm ,2nm

m-l

x( ~rJ =BC rn e- c, (6.7)

where B is independent of the argument of function (6.7)

m-1

B = JI y'2nRo (m ~ 1) m-1 xa-~-

and m-1

--- xa --C - Ro [ -2- Ie Jm y"'2"W:;- m'

The width w is

(6.8)

where (C 1 - C2) is the half-height width of the function m-I

C--rne-c. Putting in (4.8)

81 = 1 - e-t·(~) = 0.95,

where J* (T) is described as before by (3.18), we have

m ";!3 2'V' 21m lOpt = -e- -m--'--1--- (6.9)

XCi -2- ";I"llo

130 V. V. YAKUSHIN

The ratio T optlw then is mjQ ~;-;-;- m ~

1Qpt/w = l' 3/(v (;2 - Y Cd·

Putting m »1 in (6.8) to (6.10) we have

0,907 w;.::::::-----,-

m-l

xcr-2- 'iRo m 1

'topt ;':::::: e m-l

XCl-2-~

'topdw;.:::::: 0.405 m,

(6.10)

The values of l' opt, w, and Topt /w are given in Fig. 32. With increasing number of exponentials m m-l

the ratio l' opt /w rises, while w expressed in units of 1/xa-2- no remains practically constant,

which indicates an increasing displacement in the distribution lWR (t) relative to the time origin o

and a more symmetrical shape of the distribution in question. In this case the resolving time l' opt is to be determined relative to the axis of symmetry of 1 WRo (t). However, for a limited number of exponentials (m :s 4) the ratio (6.10) changes very little (Fig. 32), i.e., the origin of the range of integration Topt (see § 4) may, as before, be considered as coinciding with the time origin of the function f* (t). Putting Amax = 'XO' m-1 and using expressions (6.8) and (6.9), we may rewrite inequalities (6.6) in the following form:

m-l

2";1 2nm (c1/m _ C1/m) C;-2-"':;--v- In 2 1 V RO>e m+l '

m-l

l/"lfo > m V'3 2~ cr-2-o:::?> e m+l (6.11)

The number Ro above which expressions (6.8) and (6.9)for wand Topt are valid, satisfies the conditions under which the asymptotic form (2.24) of the distribution lWRo(t) (Ro » 1) is valid. For lower numbers Ro, the resolving time l' opt and the width w only exceed the values calculated from (6.8) and (6.9) if the true form of the function f* (t) (3.16) deviates from the parabolic form (3.18). The smaller value of the intensity f*' (t) corresponding to (3.16) increases the fluctuation in the interval preceding the appearance of the first secondary electron at the m -th dynode of the photom ul tiplier.

Turning to expressions (6.8) and (6.9), it is not difficult to see that the minimum number of exponents m for which (for a given 0') the quantities Topt and w remain almost constant, depends on RD. In fact, the relative broadening of the distribution 1WRo (t) is the greater, the greater Ro and the smaller 0' (Fig. 33). On the other hand, a better resolving power of the photomultiplier corresponds to a greater amplification of the first stages and a faster collection of the photoelectrons and secondary electrons. The dynode-system potential distribution required for this refutes the recommendations of [10], which were based on the assumption that

(6.12)

where vm is the resultant fluctuation in the period preceding the appearance of the first secondary electrons on the m-th dynode, v, vi> v2' ... , vm-l are fluctuations calculated independently for each of the photomultiplier stages. The potential distribution for the minimum of the sum of disperSions (6.12) constitutes a decreasing function on passing from the first to the last stage. However, expression (6.12) is only valid for the dispersions of the total number of photoelectrons and secondary electrons.


w [!,Ix] 'ropt [!,Ix] !.O.-----.-----,-'r--.!-w~. £8

°PT/'/ If/~· ~3

0.8 t------F-"'-'--_+-----l/.G

O'6t----~---+-.~---l!.Z

O~ ___ ~ __ _L ___ ~O

! Z 3 Ifm

Fig. 33. Resolving time T opt (broken line), width w (continuous line), and ratio T opt Iw for a photomultiplier with a = 1, 3, 4; Ro = 49.

When quite a large amount of energy AE is evolved in the sensitive volume, the resolving power of the counter reaches a minimum value, varying very little with any further rise in AE. In fact, for an unlimited rise in Ro, the quantities Topt and ware determined by a larger and larger number of successive processes, including those having very little influence for a limited Ro (transient processes in the photomultiplier-dynode supply system, in the cut-off circuits, and so on). The asymptotic expressions (6.8) and (6.9), expressed on a double logarithmic scale, are almost independent of the argument Ro for m »1 (Fig. 34). Actually this situation may even exist for m :::: 4 owing to the development of nonlinear effects in the photomultiplier (a reduction in the amplification of the stages owing to an increase in interdynode space charge, etc.). Strictly speaking, the curves giving the dependence of T opt and w on Ro in accordance with (2.24) and (3.16) lie above the corresponding asymptotes and only merge with them when the amplification of the stages

increases without limit. The values of Topt and w (Fig. 34) are given in units of U'K; 'K is determined from t1/ 2 , in accordance with Fig. 31. An improvement in the resolving power of the counter in the range Ro » 1 is only possible at the expense of a considerable reduction in recording efficiency.

The results of the overwhelming majority of experimental investigations into the time resolution of the scintillation counter [8, 9, 40, 51, 52] only agree with the theoretical estimate [14, 25, 39] over a narrow range of recorded-particle energies. This circumstance is a result of the unjustified, arbitrary way in which the function describing the anode-current pulse from a single photoelectron was chosen. In [14], for example,

j *'(t)= 1.1125 {~ r_(t-2.2166)2]_~} .~ 32 exp 2 32 ' r O.5;t t,;. 0.5t,/,

L

(6.13)

where j*'(t} is a truncated Gaussian function, 0::::: t::::: 2.2166t1/2. The integral of the convolution of (6.13) with the function describing the form of the light pulse incident on the photocathode corresponds to w ,.., 1/Ro [25] and w,.., 1NRo [24, 39]. It would appear, however, that even on considering the mutliple sequence of identical processes taking place in the photomultiplier qualitatively, the value of w will essentially depend on the number of such processes. An additional failing of [14, 25, 39] is the extrapolation of the asymptotic distribution (2.24) to the region in which the electronic cut-off circuit is insensitive, Q/Ro ,.., 1. It is not hard to see that the values of T opt and w calculated in this way will be too high (see Figs. 13 and 16).

Figures 35 and 36 show the width wand resolving time T opt for the photomultipliers 56AVP (analogous to the FEU-36 [52].) and XP1020, for which we respectively take t1/2 = 2.25 nsec [14] and t1/2 =0.88 nsec [51]; 'K = 'Kt = ~ = ... = 'Km-I; 1) (Jl = (J2 = ... = (Jm-I = (J = 4.0; 2) (Jl = 8.0; 02 = ... = <Tm_1 = (J = 4.0. Thus in the range of large energies AE and for measurements with organic scintillators the photomultiplier is a potential source of fundamental limitations to the resolving power of the counter.

132 V. V. YAKUSIflN

1/2re~-, ____ .-__ ,-__ -, ____ ,-__ .-__ -, __ -.

a

" , a " ~

~

D,I "-, , , ,

, , , , ,

0,0" ,

3 10 30 100 300 3000 Ro

Fig. 34. Resolving time T opt (continuous line) and width w (broken line) in units of l/x for a photomultiplier with an anode-current pulse width A -t1/2 == 1.78/x(0" ==4); B-t 1/ 2 == 1.42/x (0"1 == 8; 0"2 == 0"3 == ••• == O"m-l == 4). a, b, c, d are asymptotes for m == 1, 2,3,4.

Topt. w, sec

1--1

3 ~ ~ r-- ----

.......... , " ~ ,,~

t'-.......... ' .......... t'-~

~ t' ........................

~:-... ..... ..... , .......... I~ -~ 2'" ........... _ R:::-

3

-- r-. .................... 3 10 30 100 300 1000 3000 Ro

Fig. 35

Figures 37 and 38 give the results of [8, 9, 51, 58] and the dependence of w on Ro, determined in accordance with the theory developed in this section. The width w for a twochannel coincidence circuit was determined from the expression [58]

Actually, since the time fluctuations in the channels are statistically independent, the disperSion of the period preceding the instant at which two events coincide equals the sum of the dispersions of the periods preceding the occurrence of each of the events individually in the channels. A characteristic feature of the figures presented is the asymptotic behavior of the width w, not only in the range Ro « 1, but also in the range Ro » 1, where the processes taking place in the photomultiplier have a greater effect; in the majority of cases it is sufficient to restrict consideration of the latter to the entrance chamber and two or three stages of secondaryelectron amplification.

Topt• w. sec 3~~.----,---,----.---,-----,----~--,

---10 30 100 300 1000 3000 Ro

Fig. 36

Fig. 35. Resolving time Topt (continuous line) and width w (broken line) for the photomultipliers 56AVP and FEU -36. l/x == 1.26 nsec. 1) 0"1 == 0"2 == ... == 4.0; t1/2 == 2.25 nsec [14]; 2) 01 == 8.0; 0"2 == 0"3 == ... == 4.0; t1/ 2 == 1.8 nsec.

Fig. 36. Resolving time T opt (continuous line) and width w (broken line) for the photomultiplier XPI020. l/x == 0.495 nsec. 1) 0"1 == 0"2 == ... == 4.0; t1/ 2 == 0.88 nsec [51]; 2) 0"1 == 8.0; 0"2 == 0"3 == ... == 4.0; t1/ 2 == 0.7 nsec.


III, sec

3 ~ -- ----:::--... ........ ~ r--..

...................... '"'X'-~ .... --~ ===--r-0 -- t:::: o -_ r-

1--

3

0.003 0.01 0.03 0.1 0.3 3 10 JE, MeV

Fig. 37. Width w for a two-channel coincidence circuit. In the channels are scintillation counters containing the organic scintillator Naton 136, with dimensions 2.5 x 2.5 cm2, and photomultipliers 56A VP (crosses) and XPI020 (circles). The parameters of Naton 136 are: Ih. 1.6 nsec, l/A 1 = 0.8 nsec [51], l/Ag = 0.13 nsec (see § 3), the conversion efficiency is 0.22 relative to NaI-Tl. The parameters of the photomultipl~ers correspond to Fig. 35 (t1/2 = 2.25 nsec) and Fig. 36 (t1/ 2 =0.88 nsec) , ""[=0.1. The emission of one photoelectron requires the expenditure of 1.5 keV of the energy evolved in the scintillator.

w,sec 10' , :::::.. -.........

3 ...........

6

3

10'" • 0.003 0.01

........

~~ o ~o 0 .........

~ 00

0.03 0.1 0.3

r--- --3 10 IlE, MeV

Fig. 38. Width w for a two-channel coincidence circuit. In the channels are scintillation counters containing the organic scintillator NE-I02 and an RCA6342A photomultiplier. The parameters of the NE-1U2 are: l/A = 4 nsec, l/A j = 1.6 nsec [58], the conversion efficiency is 0.20 relative to NaI-Tl. The parameters of the RCA-6342A are: 1/")(. = 1.14 nsec [58] ~nd tl/2 = 2.03 nsec in accordance with Fig. 35, ~ = 0.1. The emission of one photoelectron requires 1.5 keV. For the experimental data of [8] we take 2 keV/photoelectron (circles).

The close quantitative agreement between the majority of the experimental results and the theoretical relationships demonstrates the validity of the proposed theory for the time resolution of a scintillation counter.

From the practical point of view, the dependence of the width won the sensitivity Q of the recording apparatus is of considerable interest. With increasing Q and constant Ro, the width w falls to a minimum, the position of which is determined by the two competing factors in the distribution of (2.24). In this case, a constant value of the efficiency corresponds to a large resolving time T opt and hence a large width w (the ratio of Topt /w remains almost constant according to Fig. 32); an increase in the threshold Q, however, is accompanied by an increase in the intensity J*' (t) of the function (3.16) over a certain period of time. An increase in the constants of the exponentials or a fall in Ro displaces this period in the direction of the time origin, which corresponds to a greater sensitivity of the recording apparatus. This is why, when recording low-energy part icles with a counter using a "fast" photomultiplier, the width minimum practically coincides with Q = 1 (see [9] and [51]).

Conclusion

The analysis carried out in the foregoing article has provided us with a theory of the time resolution of a scintillation counter designed for recording nuclear particles by the coincidence method. We have shown that the recording of a nuclear particle takes place after the execution of a whole succession of processes in the counter, the intensities of these being fairly accurately described by exponential functions (the evolution of energy from the nuclear particle in the scintillator, the excitation and luminescence or de-excitation of the centers of luminescence, the propagation of the light to the photocathode of the photomultiplier from the sensitive volume of the counter, i.e., the scintillator

134 V. V. YAKUSHIN

or light guide, the collection of the photoelectrons on the first emitter and that of the secondary electrons on the subsequent emitters of the photomultiplier, and so on). We have obtained a relation for the resolving power of the counter (Le., the width of the curve of instantaneous coincidences at half its height) as a function of counter efficiency and the resolving time of the coincidence circuit containing the counter in question in one of its channels.

We have shown that, in experiments with a slow set of statistics, in which the counter recording of the nuclear particles takes place with maximum efficiency (2:: 95%), the resolving power and resolving time almost coincide. In experiments with a fast set of statistics, in which the fall in recording efficiency resulting from a reduction in resolving time is permiSSible, the resolving power of the counter approaches its limit, viz., the width of the distribution of the periods preceding the instant of recording the nuclear particle. The limit of resolving power is determined by the statistical character of the successive processes taking place in the counter.

As a result of correctly considering the time statistics of the processes, we have discovered that the resolving power exhibits asymptotic behavior for the recording of low-energy particles. We have confirmed the existence of an analogous asymptote for the recording of highenergy particles (observed experimentally earlier).

By considering the multiplication of the secondary electrons in the dynode system of the photomultiplier as a set of successive exponential processes, we have developed a statistical model for describing these; this illustrates the complex way in which the resolving power of the photomultiplier depends on the number of photoelectrons emitted by the photocathode. We have shown that, over a limited range of recorded-particle energies (a few hundred photoelectrons), the resolving power is determined by the entrance chamber and one or two stages of secondaryelectron multiplication; with increasing energy, one has to consider a larger number of stages (relationships are given for several types of timing photomultiplier).

It would appear that further developments of the theory should lie in establishing the relation between the resolving power and the sensitivity of the counter (the experimental estimates of [9, 51] indicate about a 30% improvement in resolving power). Experiments should be made to check the validity of the exponential approximation for the secondary-electron collection function and to determine its constants; experiments should also be made to find the asymptotes in the low-energy particle-recording range.

A correct choice of the optimum resolving time of the coincidence circuit enables us to set up physical experiments in which the useful information is collected slowly and high loads are placed on the channels of the coincidence circuit. The pulses of radiation from cyclical accelerators are of course made longer and the shielding of counters improved for the same purpose. However, as the energy of the recorded particles increases, the corresponding rise in the permissible levels of the loads in the channels may lead to missed counts, owing to the finite dead time of the channel, the poorer standardization of amplitude and pulse shape, and so on; the state of recording technology is the main obstacle to the use of the scintillation method in experiments with linear electron accelerators (radiation pulse width ,... 1 jJ.sec).

Due application of the theory should ensure a correct presentation of experiments relating to the recording of low-energy particles. A possible example of this type is an experiment relating to Compton effect at a nucleon in the case of low energy (,... 1 MeV).

Existing views on the low ionizing power of the Dirac monopole may in principle lead to a new attempt at observing this entity with a high-sensitivity scintillation counter.

LITERA TURE CITED

1. M. Brady and R. H. Deutsch, Phys. Rev., 78:558 (1950).


2. V. A. Petukhov, A. A. Komar, and M. N. Yakimenko, Joint Institute of Nuclear Research, Preprint, R. 283 (1959).

3. L. G. Rutherglen and R. M. Wabker, Nucl. Instr. Methods, 8:239 (1960). 4. G. Paic, J. Slaus, and P. Tomac, Nucl. Instr. Methods, 34:40 (1965). 5. C. D. Clark, J. Sci. Instr., 43(1):1 (1966). 6. R. Hofstadter and I. A. McIntyre, Phys. Rev., 78:619 (1950). 7. R. P. Grenier and T. K. Samuelson, Rev. Sci. Instr., 35(11):1575 (1964). 8. R. E. Bell, S. B. Jornholm, and I. C. Severiens, Kgl. Danske Videnskab. Selskab.,

Mat.-Fys. Medd., 32:12 (1960). 9. A. Schwartzschild, Nucl. Instr. Methods, 21:1 (1963).

10. V. I. Gol'danskii, A. V. Kutsenko, and M. I. Podgoretskii, Statistics of Counting in the Recording of Nuclear Particles, Fizmatgiz (1959).

11. A. V. Kutsenko, Pribory i Tekhn. Eksperim., No.1, p. 3 (1960) . . 12. H. L. Weisberg and S. Berko, IEEE Trans. Nucl. Sci., NS-11, 3:406 (1964). 13. T. Mayer-Kuckuk and R. Nierhaus, Nucl. Instr. Methods, 8:76 (1960). 14. L. G. Hymon, R. M. Schwarcz, and R. A. Schluter, Rev. Sci. Instr., 35(3):393 (1964). 15. W. Feller, Introduction to Probability Theory and Its Application [Russian translation],

IL (1952). [English edition: Second edition, Wiley, New York (1957).] 16. A. Ruark and L. Devol, Phys. Rev., 49:355 (1936). 17. V. V. Yakushin, Pribory i Tekhn. Eksperim., No.2 (1967). 18. V. V. Matveev and A. D. Sokolov, Photomultipliers and Scintillation Counters,

Atomizdat (1962). 19. E. M. Leikin, Dissertation, P. N. Lebedev Physics Institute of the Academy of

Sciences, USSR (1964). 20. I. R. Prescott and P. S. Takhar, Trans. IRE Nucl. Sci., 9(3) (1962). 21. E. Breitenberger, Progress in Nuclear Physics, Vol. 4, Pergamon Press, New

York (1955). 22. V. V. Yakushin, Pribory i Tekhn. Eksperim., No.1, p. 36 (1967). 23. R. F. Post and L. 1. Schiff, Phys. Rev., 80(6):1113 (1950). 24. R. Bell, R. Graham, and H. Petch, Can. J. Phys., 30:35 (1952). 25. E. Gatti and V. Svelto, Nucl. Instr. Methods, 30(2):213 (1964). 26. J. B. Birks, Scintillation Counters [Russian translation], IL (1955). [English edition:

Pergamon, New York.] 27. Y. Koechlin, Thesis, Paris (1961); CEA Report, No. 2194. 28. Y. Koechlin, A. Raviart, and M. Furst, Compt. Rend., 256:1096 (1963). 29. L. Pietri, Trans. IRE Nucl. Sci., 9(3):62 (1962). 30. M. A. El-Wahab and 1. V. Kane, Nucl. Instr. Methods, 15(2):15 (1963). 31. R. F. Post, Nucleonics, 20:56 (1953). 32. J. V. Jelley, Cerenkov Radiation and Its Applications [Russian translation], IL (1960).

[English edition: Pergamon Press, New York.] 33. V. V. Yakushin, Pribory i Tekhn. Eksperim., No.6 (1966). 34. K. E. Erglis and I. P. Stepanenko, Electronic Amplifiers, Fizmatgiz (1961). 35. V. V. Yakushin, Pribory i Tekhn. Eksperim., No.3, p. 93 (1965). 36. D. 1. Hudson, CERN, 69, Geneva (Aug. 15, 1963). 37. H. T. Davis, Tables of Higher Mathematical Functions, Vol. I, Indiana Principia

Press, Bloomington (1933). 38. H. B. Dwight, Tables of Integrals and Other Mathematical Data [Russian translation],

IL (1948). [English edition: third edition, Macmillan, New York.] 39. L. G. Hymen, Rev. Sci. Instr., 36(2):193 (1965). 40. K. W. Dolan, I. R. Hurley, and I. M. Mathiesen, Nucl. Instr. Methods, 39(2):232 (1966). 41. M. Bertolaccini, S. Cova, and S. Bussolati, Nucl. Instr. Methods, 37 (2):297 (1965).

136 V. V. YAKUSHIN

42. M. Forte, Electronique Nucleaire, Proc. Nucl. Electr. Conf., Paris (1964), p. 105. 43. R. E. Bell and M. N. Jorgensen, Nucl. Phys. 12:413 (1958). 44. V. G. Zolotukhin and G. G. Doroshenko, At. Energ., 18(3):287 (1965). 45. F. Reines and I. Kouffel, Proceedings of the Conference on Interaction Cosmic Rays

and High-Energy Physics, Case Institute of Technology, Cleveland, Ohio (Sept. 25-26, 1964), Pt. I, p. 48-65, Pt. IV, p. 1-4.

46. G. W. Clark, IRE Trans. Nucl. Sci., 7(2-3):164 (1960). 47. Yu. A. Aleksandrov, A. V. Kutsenko, V. N. Maikov, and V. V. Pavlov skaya , Pribory i

Tekhn. Eksperim., No.5, p. 45 (1965). 48. G. Luck, Nucl. Instr. and Methods, 8(2):249 (1960). 49. M. A. Vil'dgrube, Pribory i Tekhn. Eksperim., No.6, p. 91 (1961). 50. G. Davidson, Doctoral dissertion, MIT (1959). 51. G. Present, A. Schwarzschild, T. Spirn, and N. Wotherspoon, Nucl. Instr. Methods,

31(1):71 (1964). 52. M. Bonitz, W. Meiling, and F. Stary, Nucl. Instr. Methods, 29(2):309 (1964). 53. E. Gatti and V. Svelto, Nucleonics, 23(7):62 (1965). 54. R. Everhard and C. Gazier, J. Phys., 26:37A (1965). 55., I. R. Prescott, Nucl. Instr. Methods, 39(1):173 (1966). 56. G. T. Wright, J. Sci. Instr., 31:377 (1954). 57. M. Brault and C. Gazier, Compt. Rend., 256:99 (1963). 58. W. M. Currie, R. E. Azuma, and G. M. Lewis, Nucl. Instr. Methods, 13:215 (1961).

A CATHODE STAGE WITH AMPLIFIED FEEDBACK AND ITS APPLICATIONS

V. V. Yakushin

Introduction

A cathode stage with amplified feedback (otherwise known as a "White stage," Fig. 1), which constitutes an improved version of a stage with a cathode load ("cathode follower," Fig. 2), has a much lower output resistance and a greater transmission coefficient at low frequencies than a cathode follower [1, 2]. With increasing plate load Rp of the tube Lit these advantages of the white stage become greater; however, as will be shown by the calculations in the present article and their experimental confirmation, the best high-frequency characteristics correspond to a limited value of Rp. On further increasing Rp' the length of the leading edge of a pulse taken from the capacitive load of the stage is reduced very slightly, but at the same time the shape of the pulse is distorted (Fig. 3a-e). The high-frequency characteristics of a White stage loaded with a matched coaxial cable may in this case be worse than those of a cathode follower, especially for tubes with a small slope (transconductance) of the plate/grid characteristic.

A correct choice of the parameters of the White stage enables us to widen its range of application and to make a considerable improvement in the characteristics of electronic apparatus. The use of a White stage instead of a cathode follower enables us: a) to widen the pass band of the RC circuit used as load; b) in a scintillation counter, on transmitting pulses from a photomultiplier (Fig. 4), to improve the recording of nuclear particles; c) to increase the stability and depth of feedback and to widen the pass band of feedback amplifiers (Fig. 5).

The connection of a White stage to an amplifier, generator, etc., under certain conditions improves their high-frequency characteristics. We should notice the stability of the stage (better than that of a cathode follower) and the slight dependence of its characteristics on any spread in the nominal values of the components and tube parameters. We shall discuss the use of the White stage more fully below in the section "Some Applications of White Stages."

The results obtained in this paper enable us to determine the parameters of a White stage based on any tubes except triodes.

I. Transient Processes in Cathode Stages

The representation of cathode stages (see Figs. 1 and 2) by the equivalent circuit of Fig. 6 (series connection of the four-terminal networks I and II) and the introduction of the generalized parameters SRp, SRG, etc. (S is the transconductance of the plate/grid characteristic) enables us to make a comparative analysis of transient processes in such stages in generalized form.

137

138 V. V. YAKUSHIN

sg == screen Srid gc :: gdd-cathode gp = grid-plate c. pc :;:: plate-cathode ~ L'" load

Fig. 1. A cathode stage with amplified feedback.

gc = grid-cathode gp = grid-pla te

pc = plate-cathode L ~ load ch = c~thode-he8ler

capacity

Fig. 2. A stage with a cathode load.

~-

~----

... -.............. . a

~ - - - .................... .

b

c

d

- ~ -- ~- - -.... -.... ..........

e

Fig. 3. Transient characteristics of a White stage based on 6ZhllP tubes. a) Input pulse with amplitude 5 V; b to e) pulses taken from the capacitive load (CL = 5 pF) with successive increases in Rp: 15, 33, 150, and 1500 n. Parameters of the stage: C1/C2 = 0.8; C2 = 27 pF, S = 28 mA/V, SRL »1, RG= 910 n, Cg= 7.5 pF. Pulse parameters: tLW = 1.4 .10-8 sec; wC2/S= 1.25 for m = 100. One mark corresponds to 4· 10-9 sec. Taken through a cylindrical lens from the screen of a G5-U (GNI-1) display tube.

The operation of cathode stages at low frequencies has been considered in some detail in [I, 2]. The transmission coefficient Kw of the White stage

t Kw= 1

1 + SRL (1 + SRpl

and the ratio of the output resistance of the White stage to that of the cathode follower

Rout w/Rout CF = 1/(1 + SRp)

are shown as functions of the parameters in Fig. 7.

A CATHODE STAGE WITH AMPLIFIED FEEDBACK AND ITS APPLICATIONS 139

Fig. 4. Arrangement for transmitting a photomultipiier dynode pulse through a White sta,ge.

I

1-_____ _

+Eg

Fig. 5. Block diagram of a three-stage amplifier with negative feedback. 1), 2) Amplification stages; 3) White stage; Rfb, Cfb, elements of the feedback circuit.

r-------, r-~

1 RG t:: 1 1 N 1 ....:I

1.£ ~: ~ 1 .D u=: 1 1 1 1 I I --L ______ J

Fig. 6. Equivalent circuit of cathode stages. RG = internal resistance of the source of the input signal (generator); CG, capacity shunting the source of the signals; Z in' input impedance of the stage; CL, RL' elements of the load.

In the analysis presented, no account is taken of the inductance of the tube leads or phenomena associated with the finite time of flight of the electrons in the interelectrode spaces of the tubes. It is also supposed that any changes in the interelectrode potentials take place within the limits of the linearity of the plate/grid characteristics of the tubes. In this article we consider a White stage made from tubes of a single type. An analysis of the transient processes in cathode stages may conveniently be started from the four-terminal network II.

1. The Transient Function h II(t)

In the White stage, in addition to the ordinary feedback couplings, there are a number of feedback couplings via the interelectrode capacities of the tubes, the influence of which may be neglected under certain conditions.

The satisfaction of the inequalities (valid for pentodes and certain triodes)

and

where C1 is the total capacity connected to the plate of Ll (see Fig. 1) allows us to neglect the mutual coupling of the plate and grid circuits of Lt if the value of the static amplification factor SRp with respect to its plate is of the order of unity (at the initial instant of time the feedback with respect to the cathode current may be neglected). Actually the equivalent capacity connec-

ted to the control grid from the plate side, C~l~(l + SRp), is of the same order as CWo The

equivalent capacity connected to the plate from the grid side, C~~(l + l/SR p)' is the upper arm

140 V. V. YAKUSHIN

OL-__ ~ ____ ~ ____ ~ ______ ~ __ ~. 0.5

Fig. 7. Characteristics of cathode stages at low frequencies. Continuous curves: transmission coefficient of the White stage; broken curves: transmission coefficient of the cathode follower; dotted and dashed curves: relative rate of growth of voltage at the output of the four-terminal network II of the White stage at the initial instant of time. Horizontal line with an ordinate equal to unity is the asymptote for all the families of curves. The figures on the curves are the values of SRL •

of a capacitive potentiometer. The lower arm is shunted with the output resistance of Ll with respect to the plate. The condition SRp "" 1 is satisfied for the majority of cases in which the stage is used in practice (see later).

From analogous considerations, the satisfaction of the inequalities

and

(C~~ + C~~)/C2~1, where C2, the total capacity connected to the ,output of the stage, enables us to neglect the mutual coupling between the plate circuit of Ll and the output of the stage. It should be noted that the negative feedbacks through the interelectrode capacities cft cgJ, and Cg~ considerably reduce the rate of change of potentials in stages based on triodes.

Thus all the capacities of the stage may be reduced to two, C1 and C2, connected to the plate of Lt and the output of the stage:

C - C(l) + C(l) + C(2) + C(2) + C (1) 1 - gp pc gc gp w,

Cz = C~6+ C CU ) + rJ2) + C(2) + C1 + C(l) 'G (2) ch ~p pc L gc c(t) + C ' gc JG

where Cw is the capacity of the wiring of the amplified-feedback circuit (the wiring capacity of the output of the stage is allowed for in the load capacity CL ). c~12cG/(Ck~ + CG) is the value of the capacity presented to the output of the stage from the grid CIrcuit. When a pulse reaches the input of the stage, a pulsed potential difference arises on the capacity C£~ oWingto the finite growth time of the stage. If the length of the leading edge of the input pulse is vanishingly small, RG = 0 (see Fig. 6) and C2 »C£~, then tlw full amplitude of the input pulse falls on the capacity C£~ at the initial instant of time. Later C£~ is charged by the output current of the stage to a potential difference corresponding to the p''lge transmission coefficient. Thus in the load of the stage we should allow for the full value of the capacity C ~1c). In practice, the potential difference on C~1c) does not reach the peak value, and the time constant c~lRG is comparable with the growth time of the four-terminal network II, i.e., in the load, it is sufficient to consider a series connection of the capacities C£~ and CG. The inaccuracy committed in determining the last term in the sum (2) is not serious, since it consists of a small fraction of C2• It should be noted that ;he "creeping" of the pulse to the output of the stage through the potentiometer c~J, C2 reduces :he growth time of the stage.

Regarding the influence of the capacity c~1J on the input impedance of the stage, see the :ection entitled" Transient Function hI(t) " below.


Fig. 8. Equivalent circuit of the four-terminal network II of the White stage. Uin and Up are the voltages on the grid and plate of L j •

10

I --- -~-

{J.31~1~-r~-r~±==+==F~~t=t==

P.g3'c-:----:-'=---,-J,----,-L:--::'=--:~_=_-~_:___' {J.OOI {J.OOJ IJ.OI IJ.OJ 0.1 8.3 05 10

Fig. 9. Parameters of cathode stages at high frequencies. Continuous curves give the condition for the critical correction of the four-terminal network II of the White stage; the broken curves give the delay times; the dotted and dashed curves give the growth time for optimum overcorrection. The figures on the curves are the values of SRL .

For the equivalent circuit of the four-terminal network II (Fig. 8), set up with due allowance for C j and C2, the frequency dependence of the transmission coefficient is described by an expression derived from [3J:

(3)

where

A = (Ri + zp)!(l + /t)(/tzp + R i ),

ZL = Rt,/(1 + jwC2Rd, zp= Rp/(l + jwC1Rp).

Let us put (3) in the form

fl. RL K (p) = (1 + fl.) (1 + (1) (1 k (p) = Kok (p),

RL + R; 1 + (1

( 4)

where Ko is the static transmission coefficient of the four-terminal network II,

(5)

(6)

(7)

142 V. V. YAKUSHIN

t~W/t~ICF 100 _0'--_______________ ---., -------------------

---_._-_._._-_._._._-_._._ . . -.-

-------------------------

0.'1

az

o~----~----~----~----~----~--~

" 0 ..... - . .:::-..... - ----.:: ::':.--: = :---. =-"":" = .---: = :--~ =--..;-~

az

o z 3 5 G 0 2 3 5 fl SRp SRp

Fig. 10. Relative growth time of the voltage at the output of the four-terminal network II of a White stage. Dotted line: undercorrection for SRL »1; continuous curves: critical correction; dashed lines: overcorrection; dashed and dotted lines: overcorrection, with a peak limiter (growth time of limiter negligibly small). Short straight lines: asymptotic values; values finally attained beyond the limits of the figure indicated at the ends (Rp - co).

(8)

In these expressions, S is taken as being equal to the transconductance and Ri is the internal resistance of the tube. The roots of the denominator of expression (5) are

P1.2 = ;~2 (-1 ± ]/D) = Per (1 ± VD), (9)

where

and

P~r=- 2~2 (1+ S~L + Cl~~p)' Considering (7) and (8), from the equation D = 0 we obtain the condition for the critical correction of the four-terminal network II of the White stage (here and subsequently we assume JJ.» I):

(10)

A CATHODE STAGE WITH AMPLIFIED FEEDBACK AND ITS APPLICA TIONS 143

SRp cr= 1/ [( fl. +1 f -1J. (11)

SRp er Z 0.5 Y C2/C1 . (12)

The conditions of critical correction (10) and (11) in the range 0.001 ::: C/C2 ::: 10 are shown in Fig. 9.

The monotonic transient function corresponding to the critical correction [4] is

where

_ _ S 1 +SRp er Po - -1jal - - C SR C IC •

2 P er J 2

(13)

Allowing for (10) we have

Po = - ~ [1/SRL + CV C2/C1 + 1)2].

Per =- ~ (1 + 1jSRL + Y C2/C1). (14)

The transient function of the critically-corrected four-terminal network II for SRL » 1 is

h~er(t)=1-exp[- ~ (1 + YC2/C1)t] (1+ ~ YC 2/C1 t).

Analogous expressions may be obtained for a cathode follower by putting Rp = 0 in (5) to (8).

The delay time d\v of the monotonic transient function h ~ (t) [5] is

(15)

where

II C2 1 t dCF = S 1 + IjSR L

is the delay time of a cathode follower with one tube of the same type. To an accuracy of better than 4%

II II I (' SR p) tdW = tdCF 1-t- 1 + l/SR L •

if SRL ;=; 2. It should be remembered that in expression (15) the load capacities of the cathode follower CL and the White stage CL are related by CL = CL + CWo

The growth time ti\v of the monotonic transient function hM (t) is

t~~ = y'2it 11 bi - a~ + 2 (a2 - b2) =

(16)

We note that

144 V. V. YAKUSHIN

is the growth time of the cathode follower. To an accuracy of no worse than 3%

if SRL 2: 1. The relative growth times t~\.v cJtgCF of the critically-corrected four-terminal network II of the White stage, calculated from (16) with due allowance for (10) and (11), are shown in Fig. 10. With increasing capacity C2 and an active load impedance RL, the efficiency of the White stage relative to that of the cathode follower increases. For C/C2 - 0, allowing for (12), we obtain asymptotic formulas for the critically-corrected four-terminal network II

where

t~~ cr = 2 i C1C2/S'

t~~ cr = 3.54 f C1C2/S.

The pass band fo of the stage is determined from the condition

I k (jwb ) I = 1/112,

I k(jlub) 1= Vi + (~/Pd)2 / [1 + (~!P:r)21. For SR L » 1 we have, for example, to an accuracy of no worse than 5%, the following values for a critically-corrected White stage four-terminal network II:

t(cr) 0381 II b = . / tgW cr'

where

The exact value of it: t~w cr follows from Fig. 7.

The parameters of the transient function hUr (t) for a correction smaller than critical may be found from expressions (15) and (16). The transient function of the White stage four-terminal network II for a correction differing from critical [4] is

hII () 1 1 ( PI - Po t pz - Po ) w t = -- P2--ePl -Pl---eP,t , Po PI-P2 PI-P2

(17)

where Ph P2' and Po are determined from (9) and (13).

For overcorrection we rewrite the function h Ur (t) in the form

hVi (t) = 1 + A exp (P:r t) sin (rot - cp), (18)

where pt follows from (9),

A = ~ t '-']/',2 0'~j" -V 1 + ljSRp (19)


and

For C/C2 - 0, w = 0.5;C1Rp.

{arcsin (1/A),

<:p= Jt - arcsin (1/ A).

Let us determine the delay time for the transient function h~(t) for overcorrection in the following way:

'. II 1 ~ II' tdW = -u- thw (t)dt,

h w (t1) 0

where the first extremum t1 is a solution of the equation hU; (t) 0,

f l = (arctan I P; I + <:p )/co. (20)

The value of the first extremum

(21)

determines the normalizing factor in the expression for t~w (Fig. lla). Allowing for (9) and (17) we have

where

The growth time is

where

'.

II tgw=

Ij

~ (' (t - taw)2 h~' (t) dt hII (t ) J ' w ) 0

\ t2h~' (t) dt = exp (P~rtl)(L cos cot l + Q sin cot l ) + R, Ii

L = - R-2tl(bl-al)-t~,

Q = [t~ (2 :: -1) + 2tl(al-bl- P~ )-2(bl-al )(bl + P~ )- ;~ ] I Y D,

R = 2 (b~ - albl - b2).

(22)

(23)

It is not difficult to see that for t1 - co (monotonic transient function) expressions (22) and (23) coincide with (15) and (16). From the graphs of tgW/tgCF for overcorrection given in Fig. 10 we find that the minimum growth times tgW min approximately correspond to a two-fold overcorrection (m = SRplSRp cr ~ 2). With increasing ratio C/C2 and falling SRL the reduction in tdw min and tgw min as compared with the critical correction becomes insignificant (see Fig. 9). Further overcorrection leads to a rise in tgW' particularly sharp for C/C2 ;::: 2 and SRL S 4. When using the stage in conjunction with a peak limiter, the minimum growth time is achieved

146

0.0 b a

0.5

o.~

0.3

0." c I

I I d

0' I c,jCz oy ,/"

r-- V I / / V ~ .'

~ ---1-

~ Z.O

0.3

o.z

o

/ V

./ ~ 1.1 Z II 15m 1.1 Z

V. V. YAKUSHIN

--

ruCzIS '1.0

Z.O

1.0

/

1// f;; V

a!f

az

at ruCzIS c 1f..0

1.0

/' If/

Z.O

a C,/C? OJ -

as jjf-

r-.n-Z.O _

Ci7fJ7. OJ --

as .-

J]/ II-~ t--.... 0.1

~5 ___

15m

lJ.lf

az

0.1 t

z.~

z 8

b 1l17~

/ 0.1

1// as --

10- ---rw----

Fig. 11. Value of the first extremum of the function h~ (t). a) - d) Values of SRL respectively » 1, 4, 2, and 1.

Fig. 12. Frequency of the oscillations in the transient function hUr (t) in units of s/C2•

Notation as in Fig. 11.

for m = 2 to 4 (see Fig. 10; the level of limitation is taken as equal to h U, (t) for t = tgW cr'

Overcorrection is accompanied by oscillations of the transient function, the frequency of these being presented as a function of the degree of overcorrection in Fig. 12a-d, in accordance with (19).

If the total amplitude of the oscillations (from peak to peak) at a moment of time tn constitutes the n-th part of hU, (t1),

[~(t,,) - h{J(t~)l/~ (tl) = n,

where hU, (tn) and hU, (th) are the values of hU, (t) at the corresponding extrema (see formula (18»,

IT • hw(tn ) = 1 + A exp (Per tn ),

II I * , hW(tn) = 1- A exp (Per tn),

then the oscillation attenuation time (Fig. 13a-d) is


With falling SRL, the oscillation attenuation time, value of the first extremum ha,(t1), and the oscillation frequency w also fall until the oscillations are broken off (C/C2 = 2.0, SRL = 2, m »1; SRL = 1, m> 6). The corresponding monotonic transient function has an unacceptably large value of %v (see Fig. 10).

The rate of voltage growth at the output of the four-terminal network II of the White stage at the initial instant of time, obtained from (17),

II' S , [hw (t)]t=o = al/b2 = O2 [1 + 1/SRL (1 + .)Rp)]

is no greater than the analogous value for the cathode follower (see Fig. 7). However, for t> ° the action of the amplified feedback leads to a comparatively faster growth of h~ (t). This feature is manifested by the fact that there is a smaller gain in the delay time than in the growth time (see Fig. 9) for optimum overcorrection (m ~ 2).

It follows from the foregoing arguments that even a slight overcorrection (m> 2), which slightly reduces talr and t~w ' worsens the other characteristics of the stage, and as already noted is only suitable for low-frequency working.

In constructing a White stage, any spread in the nominal values of the components and tube parameters will not lead to any serious deviation of the characteristics from the calculated values (see graphs).

2. Transient Function hI (t)

Let us consider the critical,correction of the input circuits of cathode stages with an active impedance RG (see Fig. 6) connected in series with the control grid. The solution of this limited problem for the input circuits nevertheless enables us to make some general recommendations regarding the use of the stages.

The input impedance of the White stage [6] is

where (Fig. 14)

z,Zo + 2a (Zl + Z2) + SZIZ2Za Zin= zo+Za+ Z4+ SZa(Zl+ Z2) ,

Zl = 1IPC~~: Z2 = RL/(l + pRLC2 ),

Za = Rp/(l + pRpC1 ): Z4 = l/pC~~.

(24)

The transmission coefficient of the four-terminal network I (see Fig. 6) obtained from (24) in accordance with Fig. 14 is

(25)

where the quantities a l' a2, b1, b2 and b3 are determined from the expressions

(26)

(27)

(28)

148 V. V. YAKUSHIN

a tn slcz mr-----r-------~------7Z.=V

(1/ ;;1;; /,Jl

b

'Z.O

0.5,'0.

C~C;,j.O ~----~--------_4__r, 'lO ,

," a~/aj1J.1 " /~JO/(

", ,'>;" ,/ , '//'0.; Z.q,,·

" ;/' ,'/-

d

~~'~~ 8~----~------~--~--~

sit =1

0.5;0.1 ---q~----~--,'-,-'r-~~~'

~'

o~--~--------~----~ I Z 6 15 ill I

';/~;:!i /" ,,0.( __ -"-y , -

,,' "",/ Z~> >--o.~/. .,.' ~"",/ /--........... :' --:7.'1:'!7~~' D!.:o.

~:;;>-: ~.", ",- _-0.5_

~~ ... ./' --- -""'-

",",,"-.c:zo-~ - - . 0.5 = -: --~Z.O=I ;~

z 15m

Fig. 13. Attentuation time of the oscillations in the transient function. Notation as in Fig. 11. Solid lines, n = 0.3; long dashes, n = 0.1; dash-dot lines, n = 0.03; short dashes; n = 0.01.


c = 1 + SRd1 + SRp). (31)

Expressions (26) to (31) are valid for Cg1 « CG, C/C2 S 2, SRL ~ 1 and SRp S 10. Analogous results may be obtained for the input circuit of the cathode follower by putting Rp = 0 in (25) to (31). It should be noted that the static transmission coefficient of the four-terminal network I of the cathode stages equals unity.

The condition of critical correction for the four-terminal network I of the White stage

[(b 2/ba)2 - 3bl /bala = 0.25 [2 (b2/ba)3- 9b1b2/b; + 27/baJ2

corresponds to two solutions of the equation (see (25»

1 + pbl + p2b2 + paba = 0

(32)

(of the three real roots two coincide [7]). Expression (32) is an equation of the fourth degree in SR G cr. The values of the generalized active impedances SRG cr satisfying (35) (the growth time is a real number) have been calculated on an electronic computer for a wide range of combinations of the five parameters: Cgc IcG , Cgc IC2 and C/C2 for three values each (2.0, 0.5, 0.1), SRL for four values (» I, 4.0, 2.0, 1.0), and m for five values (1.0, 1.3, 2.0, 4.0, 11.0). The values of SRG cr for combinations with intermediate values of the parameters may be determined from relationships plotted point by point from the data of Tables 1 to 4.

The critical-correction condition of the four-terminal network I of the cathode follower

SR G CF cr (33)

where

is obtained in analogy with (10) (see Tables 1 to 4, m 0). For an unlimited increase in C2, the value of SRG CF cr increases in proportion to C2• The condition for critical correction of the four-terminal network I of the White stage for SRL »1 and C/C2 ~ 0.1 is given to an accuracy of no worse than 10% by the expression

SR SRG CF cr G cr= 1 +SR p ,

valid for SRp S SRp cr (the values of SRG cr and SRG CF cr correspond to the same set of parameters C~~/C2 and c~~/cG). The values of SRG cr and SRG CFcr are the minimum values for which the transient function hI (t) is still monotonic. The transient function h~ cr(t) corresponding to critical correction [4] is

150

Fig. 14. Diagram to illustrate the determination of the input impedance of the stage.

V. V. YAKUSHIN

where

Pa = - 2 yq - b2/3ba; P1.2 = yq - b2/3ba; Po = -1/a2;

q = (b2/3ba)a- bIb2/6b~ + 1/2ba.

The transient function for the cathode follower hbF cr(t) is obtained by analogy with h~ cr(t) (see (12».

The delay time of the monotonic transient function h{Y (t) is

I C2 c~lj ( C G 1 ) tdW = bi - al = S C2 SRG ck1J + 1 + SRd1 + SRp) . (34)

The growth time of the monotonic transient function is

For the cathode follower we obtained the following from (34) and (35) with Rp = 0:

(36)

1 Cge 1--...",.-

I 2 Cge C2 SR L l.-.

(tdCF ) -2-SRG (1+1jSRLl' (37)

For an unlimited rise in C2 the values of t dCF and tgCF rise in proportion to C2• The values of tgcF ,tgw in units of...f2ir cls are given in Tables 1 to 4 for the corresponding sets of parameters and critically-corrected input circuits of the stages.

With increasing C2, RL , and m (up to m ,.., 2 to 4) the effiCiency of the four-terminal network I of the White stage increases by comparison with the cathode follower (see the tables), as also occurred for the four-terminal netwot'ks II (see Fig. 10). Further increasing the overcorrection of the four-terminal network II leads to a rise in the delay and growth times of the critically-corrected input circuit, and hence of the stage as a whole. The worsening of the characteristics of the four-terminal network I is associated with a considerable rise in the initial amplitude of the oscillations in the four-terminal network II (see Fig. 11), with a simultaneous rise in t~w and t~w (see Fig. 10). However, for certain uses of the stage the worsening of the characteristics of the input circuit is unimportant, and the advantages of the four-terminal network II of the White stage (as compared with the cathode follower) are realized almost completely (for example, in the transmission of pulses from a current source).

It is well known that the input impedance of a cathode follower with an RC circuit as load has a negative active (resistive) component [8]. The active impedance SRG CF cr, included for

Ci/

C2

2.0

1 C

gc

2.0

0.5

0.1

C;

c1c

2.0

I 0.5

I 0.

1 2.

0 I 0

.5 I 0.

1 2.

0 I 0

.5 I 0

.1

2.0

CG

13

.3

1.6

6

0.2

3

43.1

4.

5·J

0.5

0

202

19

.5

l.E

9

13

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13

.3

6.6

4

4.32

1

0.8

4

.50

2.

50

111.

1 3.

89

1.89

1

3.3

m

=O

11

.1

6.11

4.

22

8.5

3

3.9

7

2.40

7.

85

3.3

5

1.79

11

.1

0.8

3

0.9

2

0.98

0

.79

0.

88

0.96

0.

78

0.86

0

.95

0

.83

1u

.9

1.3

5

0.17

3

5.2

3

.59

0

.39

16

4 1

5.4

1.

51

8.5

9

10.9

5

.40

3

.70

8.

80

3.59

1.

95

8.21

3.

10

1.51

8

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m

=

1

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4

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3.

62

6.95

3

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1.

87

6.34

2.

67

1.4

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77

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3

10

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0.16

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35

154

14.0

1

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1 7

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8

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31

1.7

3

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2.

80

1.0

51

7.6

9

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13

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91

1.6

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85

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28

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0.

30

130

10

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9

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0

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5

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60

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71

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31

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96

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1.

00 1

1.0

0

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0

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;

TA

BL

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S

RL

=

OC

2.0

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o

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0

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6

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3

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4

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1

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1

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>

() >

>-3 ::r: o t:1

trj

[J).

>-3 >

o trj ~ ~ ::r: >

~

"d ~

.-.

"rj .-.

trj t:1

"rj

trj

trj t:1 ~ (

)

l:"::: > S .-.

>-3

[J).

>

"d

"d ~

()

>

>-3 o 63 ....

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156 V. V. YAKUSHIN

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Fig. 16. Condition for critical correction of the four-terminal network II of a White stage based on Soviet-made tubes. Notation and parameters as in Fig. 15.

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Fig. 17. Reduction in the growth time of the voltage on the capacity CG for critically-corrected cathode stages SRL » 1. Notation and parameters as in Fig. 15.

critical correction of the input circuit, neutralizes this component with respect to the input of the stage. Thus the value of the resistance SRG CF cr under otherwise equal conditions (Cgc ' C2,

and RL) characterizes the phase shift in the stage or the delay in the reaction of the stage to the input pulse. If SHe is insufficient (smaller than SRG CF cr )

there may be oscillations of the transient function hh (t) until generation occurs. The White stage, which requires smaller resistances SRG cr

for the critical correction of its input circuit, is stabler than the cathode follower (for corresponding sets of parameters and m s 2).

In a certain range of parameters the active component of the input impedance of cathode stages is positive. Hence the critical correction of their input circuits by means of an active impedance in the control-grid lead is impracticable. In fact from (24) we obtain an expression for the frequency

A CATHODE STAGE WITH AMPLIFIED FEEDBACK AND ITS APPLICA TIONS 157

dependence of the active component of the input admittance of a White stage loaded with an R L C2 circuit:

where

For a cathode follower (Rp = 0)

R v 2C R Cgc-C2SRL eLinCF=W gc L(1+SRL)2+[wRL(CgC+C2)P'

The active component of the input admittance of the cathode follower is always positive if

(38)

(39)

C2SRL < Cgc ' However, over a certain range of frequencies and a particular range of parameters of the White stage ReYin W > 0 for C2SRL > cgb, if the denominator of expression (38) is negative. For C2SRL »Cgc expression (39) coincides with the approximate expression for the cathode follower derived in [8]. A negative input resistance of the cathode stages connected to the capacity CG from the side of the stage in a considerable number of cases reduces the growth times tgCF and tgW as compared with the growth time tgG of the voltage on the capacity CG of the integrating RGCG circuit in the absence of the stage, tgG = &RGCG. This property may be used in order to improve the characteristics of electronic systems when SRG = SRG cr (the values of tgcdtgG and tgW /tgG are given in the tables). However, the growth time of the voltage at the output of the stage as a whole is as a rule greater than tgG , which limits its use as a series link in the circuits. The "reduction" in the capacity CG for SRG > SRG cr may be found from formulas (35) and (37).

Figure 15 gives the values of the active impedances RG cr ensuring minimal growth times of the monotonic transient function hI(t) for a cathode follower and a White stage made from Soviet-produced tubes, while Fig. 16 gives the anode load of tube L1 (see Fig. 1) for a criticallycorrected four-terminal network II of the White stage; the stages are loaded with an RLCL circuit (CG = 7.5 pF, Cw = 5 pF, RL = 75 and 150 n, SRL »1). Figure 17 shows the effect of the "reduction" in the growth time of the voltage on the capacity Cc for SRL »1. (The four-terminal networks I and II of the cathode stages are critically corrected). For RL < 150 n the values of t gCF and tgw exceed tgG for almost any Soviet tubes, owing to the shunting action of RL. For overcorrection of the four-terminal network II of the White stage the effect of the "reduction" of CG changes very little (see tables).

3. Complete Characteristics of the Cathode Stages

The total delay and growth times of the stages are determined from the expressions

td = ta+ t~I,

tg = -V (t~l + (t~)2 ,

(valid to a fair accuracy for circuits with correction), with the aid of formulas (15), (34) and (16), (35) or the graphs of Fig. 10 and the tables. Figure 18 shows graphs of the total growth times of critically-corrected cathode followers and White circuits made from Soviet-produced tubes loaded with an RLC L circuit with the parameters of Figs. 15 and 16.

The values of the first throw and the attenuation time of the oscillations of the output voltage of a White stage with an overcorrected four-terminal network II are determined with an

158 V. V. YAKUSHIN

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accuracy sufficient for practical calculations from the graphs of Fig. 12 if we take tgw == tCF (SRG ::: SRG cr ).

If the value of SRG is fixed and exceeds all possible values of SRG cr, then the growth time of the input circuit of the White stage improves with increasing overcorrection of the four-terminal network II (see below).

It should be noted that sometimes the quality of the operation of a White stage may be conveniently determined by reference to the amplitude, shape, and duration of the pulse taken from the plate load of tube L1• A reduction in the amplitude, duration, and oscillations of the pulse accompany any improvement in the characteristics of the White stage.

II. Some Applications of White Stages

1. Transmission of Negative Pulses

In transmitting a pulse of negative polarity with cathode stages there is a displacement of the working point on the plate/grid characteristic of the L j • The extent of the displacement (bias) is determined by the amplitude and the length of the leading edge of the input pulse and by the stage growth time. In a White stage the displacement of the working point of tube L j is considerably reduced by the action of the amplified feedback, but not entirely removed. The fall in the transconductance of the plate/grid characteristic during the action of the pulse causes an increase in the stage growth time (see the negative-voltage drops in Fig. 3b-e). A certain improvement in the characteristics of the White stage on transmitting negative pulses is achieved by increasing the correction, provided that this does not exceed the optimum value.

2. Raising the Sensitivity of a Scintillation Counter

In a number of cases the method of transmitting the voltage pulses arising in the collector load of a photomultiplier through a matched coaxial cable is inapplicable owing to the considerable ballistic error in measuring the amplitude, although it is convenient in view of the short dead time of the recording channel [9]. In addition to this, the sensitivity of the scintillation counter is conSiderably reduced by this method. The transmission of the dynode pulse of the photomultiplier by a cathode follower [10] is conSiderably improved by using a White stage (see Fig. 4). The circuit may effectively be used with spectrometric photomultiplier having a considerable capacity of the collector output. The value of the resistance RG is a compromise between the opposing demands of minimum ballistic error and minimum dead time. The circuit allows undistorted transmission of an extended spectrum of pulses if the value of RG ensures at least critical correction of the input circuit of the stage. With falling stability of the input circuit (RG < RG cr) there must be a contraction in the amplitude range of the pulses analyzed, since the spectrum is not reproduced accurately in the range of small amplitudes. The circuit operates in the following way. The feedback voltage coming from the output of the stage prevents variations in the potential difference on the capacity of the dynode output and hence on the last interdynode gaps of the photomultiplier, thus ensuring a high degree of linearity of the counter. This "reduction" in the output capacity and the practically complete integration of the current in the pulse raise the sensitivity of the scintillation counter and enable a direct analysis of the pulse spectrum to be carried out without additional amplification.

3. Use in Amplifiers with Negative Feedback

The negative phase shift occurring in the open feedback loop of an amplifier at high frequencies, increasing with frequency, leads to a change in the sign of the feedback at a frequency w- lT • corresponding to a total phase shift of -1I" in the loop. As a result of the inadequate weakening of the amplification in the feedback loop at this frequency, the amplitude/frequency characteristic of the closed feedback loop rises until the amplifier is excited [11]. The use of a White stage instead of a cathode follower in such amplifiers (see Fig. 5) enables us to raise the frequency

160 V. V. YAKUSHIN

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I I , I , !-a1 :-az i-a35 !-a5z , '1(w)=-aOl

0.01 0.1 az 0.3 0.5 1.0 1.5 z.s w/wz

Fig. 19. Correction SRp of the four-terminal network II of the White stage for a phase shift cp (w) at a relative frequency of w/ W2; C1/C2 is a parameter.

w-1T • particularly in a three-stage amplifier [12], and thus reduce the amplification in the feedback loop at this frequency. On the other hand, on making this substitution the depth of feedback may also be increased in an absolutely stable amplifier.

The phase-frequency characteristic of a White stage with a capacitive load (SRL »1) follows from the expression for the transmission coefficient of the four-terminal network II (5) if the input circuit of the stage is considered as the load of the previous stage of the amplifier:

00 (Cl SRp 1) (

00 Cl SRp ) 002 -c; 1 + SRp + 1 + SRp IP (w) = arctan -- ----- -- arctan

002 C2 1 + SRp ( 00 )2 Cl SRp 1 - w;- -C-2 1 + SRp

(40)

where W2 s!C2• From the phase-frequency characteristics (40) we determine the values of the overcorrection SRp ensuring the required phase shift cp (w)at a relative frequency of WlW2 for various stage parameters (Fig. 19, SRL »1). It is not .hard to determine the phase-frequency characteristic of the input circuit of the White stage if the correction of the input circuit is practically absent, i.e., SRG »SRG cr. On satisfying this condition, the growth time tgW is a linear function of RG and differs very little from the growth time of the stage as a whole (Fig. 20). Hence the phase-frequency characteristic of the input circuit differs very little from the characteristic of the integrating RG CG circuit, the capacity of which is determined from the equation

tlG = t~w, where ttG = & RGC~, t~w is calculated from (35). We have

t ~w~ "Jf2it taw and

C(l) ,. gc G:;~ CG+ 1 + sit (1 + SRp)

For SRL » 1 the minimum value of the capacity CG* . = CGo mm

(41)

A CATHODE STAGE WITH AMPLIFIED FEEDBACK AND ITS APPLICA TraNS 161

tg. nsee

8 I~O

100

GO

Rp=3,9kO RL=IZ50

0 I Z 3 ~ 5 8 RG 0

tg• nsee

28

\ \ ,

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0 2

b

3

____________ 1

RdllCQ RC l250

8 8 10 p. Q

e -,~I Rd500Q Rp =3,9kO

o ISO 300 ~50 SOO 750 RL. 0

Fig. 20. Growth time of cathode stages based on 6ZhlP tubes as a function of the resistances RG (a), Rp (b), and RL (c). The relationships obtained in [13J are shown by the broken line. CG = 7.5 pF, CL = 5 pF. 1), 3) White stage; 2), 4) cathode follower; 5) White stage with critically-corrected four-terminal network II (Rp er = 80 Q); 6) White stage based on 6AK5 tubes connected as triodes (experimental relationship obtained in [13]).

The condition SRG » SRG cr may be satisfied in the majority of amplifiers constructed: for example, in an amplifier with a small capacitive load (C/C2 2:: 0.05; Cge /C2 2:: 0.5), in the presence of a separating (buffer) cathode stage; or in an amplifier with a large dynamic range, or with a considerable depth of feedback due, in particular, to the size of the plate lead of the previous stage of the amplifier.

The rises in the amplitude/frequency characteristic of the White stage obtained fro'm (5), which are needed in order to improve the degree of attenuation at the frequency W -IT in the feedback loop

lk(jw)l= / ( 00 )2 ( C1 SRp )2 t 1 + 002 Cz 1 + SRp (42)

are presented in Fig. 21 for the frequency W-'/r' the phase shift, and the stage parameters corresponding to Fig. 19.

4. Cathode Stages Based on 6ZhlP Pentodes

Figure 20 shows the growth times of a cathode follower and a White stage based on 6ZhlP pentodes as functions of RG, Rp' and R L, varying over wide limits. The relationships were obtained in accordance with the foregoing general analysis. The results of a partial analysis of cathode stages based on 6AK5 pentodes (analogs of 6ZhlP's) obtained in [13J are shown by the broken curves. The differences existing may be explained in the following way. As already noted, on increasing RG without limit the input circuit of the cathode stages differs very little from the integrating RGCG circuit, the capacity of which is determined from formula (41). It is not diffi-cult to see that, for curve 3 of Fig. 20a, C~ = 7.8 pF. The minimum value of C~ min = CG = 7.5 pF.

162 V. V. YAKUSHIN

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The following qualitative considerations lead to the same quantitative result. As RG rises without limit, the growth time of a pulse falling on the control grid of tube Ll (Figs. 1 and 2) also increases without limit, i.e., the pulse potential difference on the capacity C~~ is reduced practically to zero owing to the comparatively short growth time of the stage if the output current of the stage is not shunted into an ohmic load (SRL »1). In this case the stage is in essence isolated from the input circuit (its input resistance is fairly large) and the capacity of the RGCG circuit is a minimum, C~ min = Cc (as SRL - 1 this can only rise). From curve 1 of Fig. 20, however, we find that C~ = 0.6CG. On the basis of the foregoing argument, the results of [13] relating to a White circuit based on 6AK5 pentodes, would appear unreliable,

5. Characteristics of Cathode Stages Based on Triodes

and

With increasing capacity CG , Ch and C2 the inequalities

C~~!CG~1,

Cg~!Cl~1,

(q~ + q~)!Cl ~ 1

(see section on the "Transient Function hII(t)") are also satisfied for triodes, i.e., the results of the foregOing analysis are completely applicable to triode-based cathode stages. However, of


the inequalities listed, only the last has to be satisfied in order to be able to calculate triodebased stages. In this case, in fact, the growth times of critically-corrected pentode- and triodebased stages practically coincide and rise in proportion to C2 (see Fig. 18), since the rate of change of the potentials on the elements of the four-terminal networks I and II diminishes (the transconductances of the plate/grid characteristics of the pentodes and triodes are the same). For the same reason the growth times of pentode- and triode-based stages agree for RG » RG cr

(Fig. 20).

Conclusion

On the basis of the generalized analysis presented in this article, we have made some recommendations on the practical uses of the White stage. We have developed recommendations regarding the choice of parameters giving a monotonic transient characteristic for the stage with a minimum growth time. We have shown that heavy overcorrection of the stage, while improving its low-frequency characteristics, considerably worsens those at high frequencies. Experimental verif~cation has shown good agreement between the actual and calculated stage characteristics up to growth times of the order of a few nanoseconds.

LITERA TURE CITED

1. C. M. Hammack, Electronics, 19:206 (1946). 2. B. Y. Mills, Proc. Inst. Radio Engrs., 37:631 (1949). 3. A. E. Voronkov, L. N. Korablev, I. D. Murin, and I. V. Shtranikh, A Fast-Acting Multi

channel Amplitude Analyzer, All-Uniop Institute of Scientific and Technical Information, Izd. Akad. Nauk SSSR (1957).

4. I. I. Teumin, Handbook on Transient Electrical Processes, Gos. Izd. Lit. po Voprosam Svyazi i Radio (1951), p. 309.

5. W. C. Elmore, J. App!. Phys., 19:1 (1948). 6. N. F. Moody, W. J. Battell, W. D. Howell, and R. H. Taplin, Rev. Sci. Inst., 22:557 (1951). 7. 1. N. Bronshtein and K. A. Semendyaev, Handbook on Mathematics, GITTL (1953), p. 138. 8. K. E. Erglis and 1. P. Stepanenko, Electronic Amplifiers, Fizmatgiz (1961), p. 179. 9. V. O. Vyazemskii, I. I. Lomonosov, A. N. Pisarevskii, Kh. V. Protopopov, V. A. Ruzin,

and E. D. Teterin, The Scintillation Method in Radiometry, Gosatomizdat (1961), p. 182. 10. R. K. Ashmore and L. F. Collinge, Indian J. Phys., 31:5 (1957). 11. H. W. Bode, Network Analysis and Feedback Amplifier Design [Russian translation]

IL (1948), p. 532. [English edition: Van Nostrand, Princeton, New Jersey (1945).] 12. W. C. Elmore and M. Sands, Electronics: Experimental Techniques [Russian translation]

IL (1951), p. 166. [English edition: McGraw-Hill, New York (1949).] 13. M. Brown, Rev. Sci. Instr., 31:4 (1960).

WILSON CHAMBER FOR STUDYING

PHOTOMESON PROCESSES

v. P. Andreev, Yu. S. Ivanov, R. N. Makarov,

V. T. Zhukov, V. E. Okhotin, and I. N. Usova

Introduction

The use of chamber methods (Wilson, bubble, and diffusion chambers) in electron accelerators is usually limited either owing to the long dead time, as in the case of Wilson chambers, or owing to the unacceptably large ion overloadings, as in the case of the diffusion and (particularly) bubble chambers.

Nevertheless, a number of physical investigations, particularly the study of many-particle reactions, may reasonably be carried out by the chamber method. For studying photonuclear reactions, Wilson chambers have been used for practically the whole period of operation of electron accelerators, even up to the present time [1-3}. The use of these chambers for studying photomeson processes, however, has for some time been considered doubtful, since the cross sections of photomeson processes are much smaller than those of photonuclear reactions, and also the greatest number of photons in the brems spectrum belong to the low-energy range, not extending to the threshold of the photogeneration of mesons. If the Wilson chamber is to be used for studying photomeson processes also, its efficiency must be greatly increased. The simple replacement of the Wilson chamber by, for example, the bubble chamber, which has a greater density of nuclei, cannot give the desired result. The reason is as follows.

The main obstacle to raising the efficiency of the Wilson chamber by increasing the intensity of the 'Y radiation or the density of the nuclei is the background of electrons and positrons formed by the photons inside the chamber itself and in its entrance window. The cross sections of the Compton process and the process of pair formation (which are mainly responsible for this background) is on average two orders greater than those of the photomeson processes, while the number of photons with energies smaller than the meson-formation threshold for accelerators with maximum energies of 500 to 1000 MeV is about 10 times greater than the number of photons able to form mesons. Hence the limiting permissible ionic loading on the chamber when operating with a pulsed photon beam is determined almost solely by the extent of this background. The condition for the operation of the chamber in such a beam may be written in the form of the following relationship:

(1)

where I is the maximum acceptable intensity of the 'Y radiation, Nn is the density of nuclei of the substance in the chamber, and A is a quantity inversely proportional to the Z2 (Z == atomic number) of the substance in the working volume, depending on the conditions of the experiment,

164

WILSON CHAMBER FOR STUDYING PHOTOMESON PROCESSES 165

wf(w), re!. units

ZOr-----__

18

IG

I~

IZ

10

50 100 ZOO ¥OO GOO W,MeV

Fig. 1. Effect of a "ruggedizer" on the spectrum of a synchrotron. 1) Photon spectrum from the synchrotron target; 2) to 6) theoretical photon spectrum from the synchrotron target behind 50 cm and 1, 2, 3, and 4 m of a lithium hydride "ruggedizer"; broken line: experimental curve measured behind 2.5 m of "ruggedizer."

N, electrons/ems

IWO

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800

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namely, the degree of purity of the beam, the thickness of the substance composing the entrance windows of the chamber etc., and (chiefly) the acceptable ionic loading of the chamber. If the intensity of the electron accelerator is high enough for this relation not to be satisfied (this requires 10 5 to 106 electrons in the orbit in a single pulse), it is impossible to increase the mesonrecording efficiency by substituting a bubble for a gas chamber, since in so doing one must reduce the intensity by the same number of times as the density of nuclei is increased. For the efficiency with which the chamber may be operated, it is quite impossible to study photomeson processes in it.

The introduction of filters made of a light material in order to change the brems spectrum of electron accelerators opened new possibilities in relation to the use of the Wilson chamber for electron accelerators. The idea of using such filters (which have become known as "ruggedizers") was simultaneously proposed in our laboratory by A. A. Komar, V. A. Petukhov, and M, N. Yakimenko and in the Cornell Laboratory by Cocconi and his colleagues [5].

The use of such a filter for relatively increasing the number of high-energy photons in the brems spectrum is based on the way in which the absorption coefficient of photons in light materials varies with their energy. For light substances, the photon-absorption coefficient passes through a minimum at an energy of the order of 100 to 500 MeV and subsequently rises very little. As a result of this, photons with energies up to 10 MeV are absorbed almost completely after passing through such a "ruggedizer," While those with energies of over 100 MeV are hardly affected. The reduction in the number of low-energy photons greatly reduces the the electronpositron background of the Wilson chamber and enables the intensity passed through it to be increased.

The effect of different "ruggedizers" on the spectrum of 'Y radiation from the target of an electron accelerator is shown in Fig. 1, which presents the results of theoretical calculations of the change in the spectrum of brems radiation after passing through the "ruggedizer." These calculations were carried out without allowing for the cascade process in the "ruggedizer." The presence of the cascade process leads to a certain rise in the number of photons in the middle part of the spectrum.

166 V. P. ANDREEV ET AL.

The development of the cascade process may be reduced by placing the "ruggedizer" in a magnetic field, which deflects the secondary electrons formed by the photons from their original direction and brings them out of the "ruggedizer" unit. According to an approximate estimate of M. N. Yakimenko and V. N. Maikov [6], on placing a lithium hydride "ruggedizer" in a 5-kG magnetic field, only about 10% of the total number of secondary electrons can radiate photons in the direction of the original beam.

However, in considering the prospects of setting up experiments in the Wilson chamber, we are not so much interested in changing the photon spectrum as in changing the number of background electrons and positrons in the working space of the chamber. The number of electron traces formed in 1 cm3 in the Wilson chamber may be calculated from the formula

Nel =Nnpa~[adW)+2.5ap(W)lf(W)dW, (2)

where p is the pressure of the gas in the chamber, O"e (W) is the Compton-scattering cross section, O"p (W) is the pair-formation cross section, and a is a normalizing factor relating the theoretical photon spectrum j(W) with the true number of photons falling into the Wilson chamber in one pulse of 'Y radiation. In the case in which 1.108 photons per pulse pass through the chamber in the absence of a "ruggedizer" in front of the latter, this factor is a =: 6470. The coefficient of 2.5 in front of up (W) in formula (2) allows for the fact that both the electrons and the positrons of the pairs form tracks in the Wilson chamber, so that the formation of a pair at a proton leads to the formation of two tracks and the formation of pairs at an electron to the formation of three tracks.

The calculation was carried out for a chamber filled with hydrogen.

The change in the number of background tracks in different lengths of "ruggedizer" (LiH) is shown in Fig. 2. Table 1 shows the number of tracks in 1 cm3 of hydrogen in the chamber with p = 2 atm and Wmax =: 700 MeV, and also the number of acts of Compton scattering and pair formation for "ruggedizers" of various lengths; Table 2 shows the number of tracks created by photons in various parts of the spectrum. We consider that these data may prove a great help in chOOSing the conditions for setting up an experiment. In particular these tables show the advantages of "ruggedizers" made from lithium hydride over those composed of grapl).ite .

. We see from the graphs and tables that, for the same number of background tracks in the working volume of the chamber, the presence of a "ruggedizer" enables us to increase the intensity passed through the Wilson chamber by at least 1.5 orders. Under these conditions the study of photomeson processes with a Wilson chamber becomes perfectly feasible.

If in fact we consider that the solid angle embraced by the chamber is "'0.5 sr, then for a working pressure of 2 atm in a chamber filled with hydrogen a case of photogeneration with a cross section of the order of 10-29 cm2 will be observed every 30 photographs. The number of electron-positron tracks will be about 60. In this case in order to obtain data with a 10% statistical accuracy for an energy range of "'20 MeV we shall require 75,000 photographs, which would appear to be a reasonable figure.

The question as to the possibility of setting up experiments in the Wilson chamber was particularly important in planning work on the Physical Institute 680-MeV accelerator, since the repetition frequency of the intensity pulses of this accelerator was fairly low (10 pulseshnin). However, in order to make the use of the chamber method really profitable the "dead time" of the chamber had to be reduced to between 6 and 12 sec.

On working with a Wilson chamber in the "classical" manner (rapid adiabatic expansion with a subsequent comparatively slow compression), the "dead time" of a medium-size chamber reached as much as 2 min. The introduction of a system of rapid compressions and slow expansions enabled the "dead time" of medium-size chambers to be reduced to between 30 and 40 sec


TABLE 1.

e Number of I

e Number of

\

Number of Type of

Number of acts of Type of acts of "rugged ...

-5' electron Compton -5' electron Compton " and posi· scattering tlrugged_ " and posi .. scattering

izertl a and pair izer" a • • tron tracks and pair ...l tron tracks formation ...l I formation

Lithium 0 1420 1290 Lithium 2 110 72 hydride 0.;; 430 360 hydride 3 64 35

1 260 200 4 40 23 Graphite 0.5 50 32

TABLE 2.

Number of tracks created by photons in different spectral ranges

Spectral Length of LiH "ruggedizer", m Graphite range, MeV --- Spectrum

I 0.5 I I 3 I 4 0.5 from [4]

0 2

0.1-1 790 90 0-3 (I 0 II 0

1-10 414 166 18.5 5.7 1.9 8 15 10-140 155 122 58.4 35.7 22.4 26 50

140-320 31 27 16 11.3 8.2 6 16.5 320-700 31 26 16 11.3 7.7 5 7.7

[7-10]. Further reduction in the "dead time" could only be achieved by operating the chamber in the so-called "hypercompression" manner [11-14]. Under these conditions, immediately after the working expansion there was a rapid, almost adiabatic compression, in which the pressure in the working volume became 15 to 30% higher than it was before the working expansion. Thus the gas became superheated and the drops formed in the chamber during the working expansion evaporated, so that the working volume lost none of its vapor. Then a slow expansion took place, the pressure of the gas in the working volume falling to a level slightly below the working value. After the slow expansion there was a pause during which the temperature and pressure in the chamber gradually came back to their working values.

The length of the cycle for medium -size chambers working with hypercompression may be reduced to between 10 and 15 sec. Experience showed that the reduction in the length of the cycle had no effect on the quality of the tracks and the stability of the chamber operating conditions.

In working with the PhYSical Institute S-680 accelerator, the reduction of the working cycle of the chamber to 12 sec made it convenient to use the chamber in a photon beam. In this case, in fact, in order to obtain the result mentioned earlier with a 10% statistical accuracy, some 250 h work-on the accelerator were required, or on the basis of a 120-h working week, two to three weeks.

1. Description of the Apparatus

The arrangement of the apparatus in the experimental room is shown in Fig. 3. The beam of photons from the accelerator target is collimated and shaped by collimators 1 and 2, passes through several purifying (scavenging) magnets 4, which free it from electrons, and passes into the "ruggedizer" 3, which is placed in a magnetic field formed by the permanent magnets 4a. The Wilson chamber 7 with the main beam-scavenging system is placed behind a lead and concrete shield 6. In the center of this shield is a lead cube 5, 30 x 30 x 30 cm in size, in the middle of

168 V. P. ANDREE V ET AL.

~ ~

'\ r f 'I

'I

'I 19.5

25 If

Fig. 3. Geometry of the arrangement of the Wilson chamber in the synchrotron 'Y beam. 1), 2) collimators, 3) "ruggedizer," 4), 4a) scavenging magnets, 5) lead cube with working collimator, 6) lead-concrete shielding, 7) Wilson chamber, 8) relative monitor. Dimensions are given in meters.

Fig. 4. Schematic sketch of the Wilson chamber.

which is a cylindrical aperture 40 mm in diameter. The cube is placed on a platform which may be moved in the horizontal and vertical directions by means of screws. In addition to this, the cube may execute a rotation around vertical and horizontal axes intersecting in the middle of the aperture in the front (from the point of view of the beam) face of the cube. This rotation is also executed by means of screws. This construction enables the axis of the cylindrical aperture in the cube to be set rapidly along the axis of the beam to an accuracy of ± 8'. In the cube aperture are interchangeable collimator sleeves with apertures of different diameters and a configuration enabling the intensity of the 'Y beam passing through the chamber to be varied over a wide range without altering the intensity of the accelerator.

Behind the cube is a scavenging electromagnet with a vacuum tube connected to the intermediate "sluice" tube directly adjacent to the Wilson chamber. We shall consider the beamscavenging arrangements in more detail later.

The Wilson chamber is placed between the poles of an electromagnet. The distance from the synchrotron target to the Wilson chamber is about 30 m. In addition to this, the room contains: a photographic system including an arrangement for the remote control of the films, a system for illuminating the working volume of the Wilson chamber, including IFP-500 flash lamps together with their supply circuits and light guides, a pneumatic system with a gas holder 0.9 m3

in capacity and a vacuum pumping system, a control system for all the working operations, and a relative monitor with a preamplifier.


Let us consider the individual components of the apparatus in more detail.

a. The Wi Iso n C ha m b e r . A distinguishing feature of the Wilson chamber is the simplicity of its construction and its small height, despite its considerable volume for such chambers; the working (photographed) part of the chamber has a diameter of 30 cm and a depth of 6 cm, the total height of the chamber being 11 cm. This realization was achieved as a result of stabilizing the gas pressure in the lower space of the chamber and regulating the expansion coefficient by means of a controllable change in this pressure; in this way the traditional lattices and mechanisms for controlling the expansion coefficient in the lower space proved unnecessary.

A schematic sketch of the chamber is given in Fig. 4. The chamber constitutes a rectangular brass compartment 1 made of a whole piece of metal, with windows 2 and 3 for illuminating and photographing the working volume, and also with apertures allowing the 'Y beam to enter and leave, 4 and 5.

The diaphragm 6 separating the upper and lower spaces of the chamber constitutes a Duralumin disc 30 cm in diameter, with a rubber sheet lapped around the perimeter. This sheet is fixed to the body of the chamber by means of a framework. The whole diaphragm is covered with black velvet. At the instant of taking the photograph, the velvet together with the diaphragm lies on the bottom of the chamber and is not exposed.

At the corners of the chamber are taps for filling the working volume with gas and emptying it, a reservoir for injecting the working mixture into the chamber, and also electric leads for applying the scavenging electric field. The scavenging field of 50 V/cm is applied between the Duralumin diaphragm and the inner metallized surface of the top glass of the chamber. In order to ensure reliability in the application of the scavenging field, its supply system inside the chamber is doubled; the second leads may be used for rapidly adjusting the timing of the application and removal of the field. The bottom of the chamber is a rubber sheet 7 lying on the pole of the electromagnet. The chamber is clamped to the pole with four screw jacks 8; this ensures reliable hermetization of the lower space.

In order to facilitate the motion of the gas in the lower space during the working expansion and hypercompression, rectangular rubber "cups" are stuck to the rubber sheet. The gas passes into the lower space through pockets made in the side walls of the chamber (not shown ill the figure). For joining to the pneumatic system the pockets are opened on the outside of the chamber. This arrangement for bringing the gas into the lower space contributes to the reduction in the total height of the chamber

The Wilson chamber is placed between the poles 9 and 10 of the electromagnet. The magnetic field is about 8000 G for the gap used (250 mm). The uniformity of the magnetic field at the chamber location is ± 0.5% or better. This high degree of uniformity of the magnetic field can be achieved because the working volume of the chamber is photographed, not through an opening in the upper pole of the magnet, but by means of a periscope system consisting of two mirrors (see later).

The magnetic field is stabilized by regulating the current; the stability is ± 1 % or better. The constancy of the magnetic field during operation of the apparatus may also be periodically checked by means of a IMI-3 magnetometer, the sensitive element of which is fixed to the upper pole of the magnet. The magnetometer is calibrated by the nuclear-resonance method.

The chamber is filled with the working gas (hydrogen or deuterium) to a pressure of 2 atm. Before this the chamber is evacuated to a pressure of the order of 10-1 mm Hg and filled with a working mixture of 60% C2HpH + 40% H20. In this way the reservoir filled with the working mixture is completely freed, so that a controlled amount of mixture can be injected into the


Fig. 5. Schematic sketch of the photographing system.

chamber. The proportion of nuclei from this mexture added to the working gas in the chamber equals .... 3%.

After injection of the working mixture the chamber is filled with gas to the required pressure; under these conditions it may operate for at least three months without further addition of working mixture. About twice a month the pressure in the upper space of the chamber is checked and small quantities of working gas are added in order to compensate the diffusion of gas through the Lavsan films in the entrance and exit windows and through the rubber diaphragm. The change in gas pressure over two weeks is about 5% of the working pressure.

b . C 0 IIi mat ion and S c a v eng in g 0 f the 'Y Be am. The intensity of the 'Y beam which may be passed through the Wilson chamber depends to a large extent on the scavenging of the proton beam; special attention was therefore paid to this question.

In front of the shielding wall (wall 6 in Fig. 3) was a lithium hydride "ruggedizer" poured into copper tubes each 50 mm in diameter and 50 cm long. The tubes had entrance and exit windows closed with Lavsan films 20 jJ. thick and hermetized. This construction ensured that the lithium hydride should be preserved for the whole period required for the experiments. The "ruggedizer" was placed in a magnetic field of 2.5 kG, which reduced the effects of the cascade process. The positioning of the "ruggedizer" at several meters from the working collimator served the same purpose.

The spectrum of the photon beam after passing through the "ruggedizer" can only be calculated approximately. Hence an experimental determination of this spectrum is required. This was effected with the help of the Wilson chamber, which in this case operated with a sharply reduced intensity, so that in one intensity pulse no more than 10 electron-positron pairs were formed in the working volume of the chamber. In all, about 15,000 cases were analyzed; this enabled the spectrum to be determined over a 20-MeV range to an accuracy of ± 5%. The experimental spectrum is shown in Fig. 1 by the broken curve. The electrons emerging from the "ruggedizer" were removed from the 'Y beam by permanent magnets with a field of 2.5 kG. Since the "ruggedizer" was placed about 3 m from the working collimator, this degree of scavenging was sufficient.

Immediately following the working collimator is a vacuum tube 2 m long situated in the field of a scavenging electromagnet (l = 50 cm, H = 12,000 G). The back (relative to the 'Y beam) end of the vacuum tube is connected to a tube 40 cm long filled with hydrogen at a pressure of 0.5 to 0.8 atm. This tube serves as a kind of "sluice" between the working volume of the chamber and the vacuum tube enables the thickness of the Lavsan film at the entrance window of the chamber to be reduced to 10 jJ. instead of the 40 jJ. which would be required if the vacuum tube were directly connected to the chamber. Since the Wilson chamber is moved to the rear end of the magnet


Fig. 6. General arrangement of the Wilson chamber in the magnet.

pole (see Fig. 3), the sluice tube lies between the poles of the magnet over a distance of 20 cm, i.e., in a field of 8000 G. The rest of the tube, with the entrance window, lies in the magnetic fringing field, the field strength at the position of the entrance window of this tube being 3200 G. With this geometry, secondary electrons with energies up to 400 Me V are completely removed from the proton beam and are absorbed by the lead shield situated directly behind the wall of the Wilson chamber. High -energy electrons, which make up about 20% of the total number of secondaries, are about half-removed from the beam, depending on the point of their formation.

Thus the hydrogen-filled sluice tube 40 cm long with 10 J1. of Lavsan at the entrance corresponds to a Lavsan film 8 J1. thick in the entrance window of the chamber, so far as the number of electrons and positrons falling into the working volume are concerned. The total effective thickness of material in front of the working volume corresponds to 18 J1. of Lavsan; this enables the intensity of "y radiation which may be passed through the chamber to be more than doubled.

c. Illumination and Photographing of the Working Volume of the C ham b e r. The working volume of the Wilson chamber is photographed through a periscope system of two aluminum mirrors.(Fig. 5). This enables us to place the photographic apparatus in the magnet gap and makes it unnecessary to photograph through a hole in the upper polepiece. This improves the uniformity of the magnetic field, facilitates adjustment of the chamber, and allows the "sluice" system described in the foregoing to be employed.

The two mirrors are firmly fixed in an opaque metal casing. The casing has a special flat part for fixing the photographic camera, and both are attached together to the top of the Wilson chamber. The general form of the chamber with the photographic system is shown in relation to the magnet in Fig. 6.

Photographs are taken with a stereo camera, using a base of 86.5 cm, on aerial-photography film of high sensitivity; the scale is 1:12.5. The camera contains two "Helios-42" objectives with a focal length of 27 mm. The optical axes of the objectives are inclined in such a way as to intersect in the center of the chamber; the angle of convergence is 11.5°. The film in the film channel is arranged perpendicular to the optical axes of the objectives.

The camera uses an electromagnetic system for holding the films in place. The electromagnet controlling this system is taken outside the magnetic field. The film is clamped with the help of a Cardan shaft and a system of levers. The pressure is only applied by the control system when photographs are being taken; it does not prevent the movement of the film. The cassettes of the camera are designed for 35 m of film. A frame counter is mounted in the body of the periscope system. For numbering, the reflection in the upper glass of the chamber of the figure on the frame counter is used [15].

The film is wound into the camera and the drum of the counter is moved by means of electric motors lying outside the magnetic field. The force is transmitted to the camera and the frame counter through Cardan shafts. For the sake of film economy, the film is moved in such a way that the gap between two frames of a stereo pair is occupied by two more frames from the following two stereo pairs.


atm

Fig. 7. Pneumatic system of the Wilson chamber. 1) Compressor, 2) gas holder, 3) reducer, RC-1 and RC-2 receiver cylinders, 4 and 5) mercury manometers with photographic detectors, Hj and H2) valves for filling the receivers, C j and C2) valves for eliminating the excess pressure from the receiver cylinders, Rj ) needle valve controlling the rate offilling of receiver RC-1, R2)

tap connecting receivers RC-1 and RC-2, R3) tap connecting receiver RC-1 with the lower space of the Wilson chamber 6, MV) mechanical valve, SEV -1 and SEV -2) slow-expansion valves, RFV) rapid filling valve, R4) tap regulating the rate of slow expansion, R5 and Rs) taps connecting the receiver cylinders RC-1 and RC-2 to the corresponding mercury manometers, EV) exhaust valve.

lllumination of the working volume of the chamber is effected by means of two IFP-500 flash lamps connected in series; the total power of the flash is 1000 J. In order to reduce the exposure of the walls and bottom of the chamber, light guides are used to shape the light beam. The light guides are rectangular frames carrying black paper strips stretched out parallel to the light beam. The width of the strips is 15 cm and the distance between them is chosen in such a way as to reduce the exposure of the walls and upper glass of the chamber to a minimum. This way of shaping the light beam is much simpler than that described in [16]; moreover, the fluctuations in the illumination of the working space (the part photographed) are no greater than 10%.

The use of a periscope system of mirrors led to the appearance of highlights on the stereo photographs (see, for example, Fig. 15, p. 180). The light guides, which reduced the illumination of the sides and top glass, also reduced this highlighting so much that it in no way interfered with the measurement of the tracks.

d. P n e u mat i c S Y s t em. As mentioned earlier, in the present construction of the Wilson chamber there were no grids or lattices restricting the motion of the diaphragm. The motion of the diaphragm was limited from below by the bottom of the chamber (magnet pole); the upper position of the diaphragm, and hence the expansion coefficient, is determined by the ratio between the gas pressure in the working and lower spaces of the chamber. On the one hand, this makes it easy to pass from one mode of operation of the chamber to another; on the other hand, the absence of a lattice imposes quite rigorous demands on the stability of the pressure in the lower part of the chamber, since it is in fact the stability of the pressure which determines the good quality of the tracks in the chamber. This demand was satisfied by using receivers with a volume ("'100 liters) many times exceeding that of the lower part of the chamber. This ensures


the desired inertia of the whole system for small changes in the amount of gas contained (i.e., in the exhaust operation).

The arrangement of the pneumatic system is shown in Fig. 7; it is assembled in a separate cabinet in the immediate vicinity of the Wilson chamber and connected to the latter with a Durite hose 20 mm in diameter.

The system is supplied from a diaphragm compressor 1, which pumps air into the 0.9-m3

gas holder 2. By means of a contact manometer and relay system, the pressure in the gas holder is held within the limits of 3 to 3.5 atm throughout the whole period of operation of the apparatus. The air passes through a reducer 3, controlling the rate of air flow, from the gas holder to the receivers. In one of the receivers (RC-1) the pressure determining the expansion coefficient is established. The pressure in the other receiver (RC-2) determines the coofficient of hypercompression. The pressure in the receivers is stabilized in the following way.

The pneumatic system includes two mercury manometers. One arm of the manometers communicates with the atmosphere (the atmospheric pressure is the standard); photodetectors are placed on the other (these are not shown in the figure). The upper photodetector controls the filling valve and the other the emptying valve. The pressure in the receiver is determined by the position of the upper detector on the arm of the manometer; the lower detector removes the gas in case of an accident.

The signal from the photodetectors falls on to a Schmidt circuit, the positive supply terminal of this system being grounded. One of the anodes (plates) of the circuit is galvanically connected to the grid of a triode, which has a relay in its plate circuit. The triode is cut off while current is flowing through the corresponding anode of the Schmidt circuit. When the Schmidt circuit trips the triode opens, the relay operates and the valve is turned on.

The filling valves are opened when the light flux falling on the photoresistance is cut off by the mercury rising in the arm. The scouring valves Ci and C2 open when the mercury is allowed to fall below the lower detector. The inertia of the valves and the mercury itself may lead to the mercury "slipping" under the level of the upper detector, which produces instability in the expansion coefficient. In order to avoid this, the filling of the receiver RC-1 is effected through the needle valve Ri , which controls the rate of flow of the air. Since the gas flow in this receiver is slight, the latter is able to fill even with a low rate of air flow. The pressure in the receiver RC -1 is kept constant to an accuracy,of ± 0.1 % during the whole period of operation of the apparatus, and this ensures a fair stability of the expansion coefficient.

The flow of gas from receiver RC-2 is considerable, and its rate of filling must be much higher. This leads to instability in the coefficient of hypercompression (of the order of ± 2%). However, experience shows that this instability has no effect on the quality of chamber operation.

If necessary the two receivers may be connected by means of the tap ~.

The air passes from the receiver RC-1 through the tap ~, the mechanical valve MV, and the slow-expansion electromagnetic valve SEV-1 into the lower part of the chamber. The purpose of these valves is to prevent air from passing from receiver RC-1 to receiver RC-2 when the rapid filling valve RFV is open (see Fig. 7).

The hypercompression pressure is supplied to the lower space of the chamber from receiver RC-2 through the RFV valve. The slow expansion is effected by the valve SEV-2 through the needle valve R5, which controls the rate of slow expansion. The valve SEV-1 meanwhile cuts off the lower part of the chamber from receiver RC -1.

The electromagnetic exhaust valve serves to produce the rapid working expansion. The valve is placed outside the mRgnetic field and connected to the lower volume with a Durite hose


Fig. 8. Arrangement for filling the working volume of the Wilson chamber. 1) Upper volume of the chamber, 2) lower volume of the chamber, 3) tap connecting the upper and lower volumes while the chamber is being evacuated, 4) tap connecting the lower volume of the Wilson chamber to the vacuum receiver, 5) tap for filling the upper volume with the working gas, 6) spiral tube, 7) standard man-0meter' 8) vacuum gage, 9),tap cutting off the standard manometer, 10) vacuum tube, 11) "sluice" tube, 12) tap closing the" sluice" tube, 13) vacuum receiver, 14) backing pump, 15) cylinder containing the working gas.

25 cm in diameter. The working expansion rate is mainly determined by the length of this hose; for the hose length used, the duration of the working expansion is roughly 100 msec. The arrangement for filling the upper volume of the chamber is shown in Fig. 8.

Before filling the chamber with the working gas, the upper space of the chamber is completely evacuated. Since there is no limiting grid or lattice in the chamber, the lower volume 2 has to be evacuated at the same time. So that the rate of evacuation of the upper volume should not exceed that of the lower volume, evacuation of the lower volume takes place through the tap 4 and the upper volume is connected to the lower with the tap 3. In so doing the lower volume is first cut off from the pneumatic system.

After injecting the working mixture, the chamber is filled with the working gas through the tap 5 until the required pressure is reached. In order to scavenge this gas from possible impurities, a spiral tube

6 is placed in front of the working volume and cooled with liquid nitrogen. The pressure in the upper volume is determined from the standard manometer 7 to an accuracy of ±0.5%.

e . Con t r 0 I S Y s t em. The control system specifies the sequence and duration of the operations of the pneumatic system, the switching on and off of the scavenging field, the illumination and photographing of the working volume, and so on, and also ensures the necessary synchronization of the operation of the chamber with the operation of the accelerator.

The magnetic field of the accelerator [17] increases linearly to a value corresponding to the maximum electron energy, after which it remains constant for ..... 0.5 sec. The electrons can either be thrown instantaneously onto the. target at any point of the magnetic plateau or in stages over the whole extent of the plateau, thus giving a photon beam drawn out in time. This mode of operating the accelerator was chosen in order to reduce the load on the nuclear radiation counters. The Wilson chamber is operated with an instantaneous throw of electrons onto the target, since clear tracks are only obtained over an interval of the order of 20 msec. The chamber operates with a cycle of 12 sec, i.e., it uses every other pulse of 'Y radiation; the missed cycle is used for other physical experiments; the accelerator may for this purpose have alternate cycles with an extended and a nonextended 'Y beam. In the cycle in which the working expansion of the Wilson chamber is to take place, a pulse falls on the control system from the magnetic-field integrator, triggering the control circuit and constituting the synchronizing pulse between the Wilson chamber and the accelerator.

The control circuit is triggered by two command pulses (see Fig. 10, giving the block diagram). One of these is a pulse from the magnetic-field integrator, which falls on the circuit input as the magnetic field reaches the plateau. This pulse triggers the blocks (units) determining the time of the working expansion, the throwing of the electrons onto the target, and certain


t, sec o 2 J ¥ 5 G 7 :~~I--~I ~I--~I~I--~I~I ' Synchronlzlng pulse .

Throw delay, 200 msec

I i r I

~: EV delay, 10 to 100 msec

/1 (I EV, -100 msec

l,i-+., RFV delay, 0.45 to 1.0 sec

I I I I I RFV, 0.2 to 1.0 sec I I

SEVl;~ I I I i I I I

I I I r I I I ~ Removal of the scavenging field, 400 msec

III. I Light delay. 20 to 200 msec

I I I I I ~ Light

rh Clamping of film, 500 msec

: I Illumination of frame counter, 2.0 sec I

I I _.h Movement of film, 300 msec

Fig. 9. Diagram indicating the operating times of the apparatus. EV, exhaust valve; SEV, slow expanding valve; RFV, rapid filling valve.

other operations, for example, the clamping of the film and the removal of the scavenging field. A photograph of the working volume is taken shenever a pulse from the relative monitor (a 'Y pulse) falls on the circuit input in addition to the pulse from the integrator. This pulse only arrives when the intensity of the photon beam is greater than a prearranged limit [18]. As a result of this, no photograph is taken in the absence of the 'Y beam or in the case of very low intensity. If the chamber has to be adjusted with the accelerator not working, the control circuit is triggered by a pulse from an internal generator.

Figure 11 shows the main circuit of the control system. The principal element is a set of univibrators, the pulse lengths of which determine either the time displacement relative to the synchronizing pulse for various operations, or else the duration of the operations.

Let us consider the operation of the control circuit in more detail.

The working expansion of the Wilson chamber should occur 50 to 60 msec before the photon beam passes through the chamber. This interval should be maintained very precisely. It is

"Throw" pulse

Synchronizing pulse f RFV r- EV delay r-- EV (L2.) t----+- delay

(L2D) (L22 )· Throw delay (L.)

1 Removal of

SEV (L2.) scavenging - RFV field (L., Ls) (L2,)

Internal generator Clamping

'--~ of synchro- '-+ of film nizing (L7 ) pulses (L6)

y Illumination Move of frame r-- film counter (La) (L9)

'--+ Light r-- Light delay (L" to L 13 ) (L.D)

Fig. 10. Block diagram of the control system. Notation as in Fig. 9.


L216N15P L24 6N15P

Fig. 11. Main circuit of the control system.

+150v +JOOv -JOOv

precisely in order to ensure the necessary accuracy that the synchronization is based on a pulse from the integrator arising at the instant at which the magnetic field comes out onto the plateau, i.e., at the moment at which acceleration of the electrons ceases. This minimizes the pulse lengths of the" EV delay" (EV = exhaust valve) and "throw delay" univibrators (see block diagram, Fig. 10), which specify this interval of time. The length of these pulses is of the order of 100 to 200 msec, and in order to ensure its stability to within 5 or 10 msec their amplitude had to be increased as much as possible.

The onset of the working expansion is determined by the "EV delay" block which is directly triggered by the synchronizing pulse. Since the duration of the working expansion is "'100 msec, the 'Y beam must pass through the chamber roughly 150 to 180 msec after the pulse for producing this expansion has been given. In practice the interval between the arrival of the synchronizing pulse and the throwing of the electrons onto the target is set constant and equal to "'200 msec. This interval is specified by the "throw delay" block. The moment for the onset of the working expansion is selected experimentally. For this purpose provision is made in the "throw delay" block for varying the period between 10 and 100 msec in steps of 10 msec.


The "EV" block (see Fig. 10) ensures the open position of the exhaust valve, simultaneously cutting off the lower volume from the receiver RC-l.

Control of all the remaining operations of the pneumatic system is effected by a chain of univibrators, each of which is triggered by the end of the pulse from the previous univibrator. The principle of operation of this part of the circuit may be seen on comparing Figs. 9 and 10.

The clamping of the film and the removal of the scavenging field are effected simultaneously by the synchronizing pulse. The film-clamping univibrator determines the time during which the film remains clamped, which is about 400 msec. The scavenging field is fed to the chamber directly after photographing.

The working volume is photographed 70 to 180 msec after the passage of the 'Y beam, depending on the conditions of the experiment. The instant of photographing (flash from the flash bulbs) is established by means of the light delay univibrator. The system enables the delay time to be varied from 20 to 200 msec in steps of 20 msec. For visual observation of the tracks when adjusting the chamber there is a bias lighting system (Ls in Fig. 11) triggered by a 2-sec synchronizing pulse.

Tubes L14 to L19 serve to transmit the control pulses to the corresponding executive elements of the pneumatic system and the system taking the photographs. The plate circuits of these tubes contain relay windings and their grids are galvanically connected to the plates of the corresponding univibrators. Since the plates of the univibrators are grounded, tubes L14 and L19 operate in the switch (gating) mode.

The relays are not shown in the main circuit. These are mounted directly in the cabinet with the pneumatic system and are connected to the control system by a multicore cable. The executive elements cannot be directly switched on without operating the relays, as they require too much current.

2. Operation of the Wilson Chamber in a High-Intensity Photon Beam

As already mentioned, the main condition governing the suitability of the Wilson chamber for studying photomeson processes in the synchrotron was that of increasing the intensity of the 'Y beam which could be passed through the chamber in one pulse (still preserving the equality of the tracks) as far as possible, provided that the required traces could be identified clearly on the background of electron-positron tracks.

On operating the Wilson chamber in a photon beam without a "ruggedizer," the secondary electrons from the soft photons create a "column" in the chamber where the 'Y beam passes; the column cannot be removed with a magnetic field, since it is created by secondary electrons of such low energies that the radius of curvature of their trajectories is less than the width of the 'Y beam passing through the chamber. The presence of this "column" severely limits the permissible 'Y beam intensity, since it impedes identification and analysis of the desired tracks. If the soft photons in the 'Y beam are removed with a "ruggedizer," this column is absent, since the electrons of higher energies have a much greater radius of curvature and may be removed from the beam by a magnetic field (see, for example, photographs a and b in Fig. 12). In addition to the general relative reduction in the number of photons with energies lower than the threshold of the photogeneration of pions, this enables us to increase the permissible energy of the photon beam by more than 10 times. The use of a "sluice" tube enables us to increase the intensity of the photon beam by another factor of two.

The mode of operation also offers some further possibilities of increasing the efficiency of the Wilson chamber for working in an accelerator. Let us consider this question in more detail.


Fig. 12. Photographs of tracks in the Wilson chamber. a) Working in a nonruggedized 'Y beam with ~105 photons passing through the chamber per pulse; b) working in a ruggedized beam with "'2.5.107 photons passing through the chamber per pulse.

The duration of the working cycle of the Wilson chamber is determined by the time which the drops, formed in the previous expansion, take to evaporate completely. It is furthermore essential that the temperature of the gas in the chamber should be restored and the whole volume saturated with vapor; otherwise tracks in different parts of the chamber will have different densities. Figure 13 shows the variation in the temperature and pressure of the Wilson chamber under various operating conditions. In the case of mode a, the temperature of the gas rises very little above its initial value during the whole working cycle (Le., the temperature of the chamber walls). The evaporation of the drops takes place slowly and fine drops remain in the working volume for a long time; in subsequent expansion, these may produce background (noise). The presence of a large number of such unevaporated drops leads to clouding of the chamber, and in order to avoid this the working cycle of the chamber has to be increased to several minutes. * It would appear most reasonable to reduce the working cycle by creating conditions for the faster evaporation of the drops. This may be achieved, for example, by two adiabatic compressions, between which a slow, practically isothermal expansion takes place [10] (Fig. 13b). As a result of

*Here and subsequently all the figures pre-sented relate to chambers of medium size, Le., a volume of the order of 10 to 20 liters .

.-________________________ ~at~m~------------------__.

10

o 10 ZO 30 '10 50 GO 70 80 90 100 110 lZU 130 t,sec

Zhr-------------------.

o 10 ZO 30 If{} 50 GO 70 80 90 100 110 IZ0 m t,. sec

Fig. 13. Changes in the temperature and pressure of the working volume of the Wilson chamber operating in various conditions. a) Ordinary "classical" mode; b) mode with two adiabatic compressions; c) mode with hypercompression.


Fig. 14. Oscillogram of the change in gas temperature on working with hypercompression. Ts = gas temperature at the start of the working cycle, Tw = temperature of the chamber walls. a) Ts < Tw , b) Ts > Tw.

this expansion, in the second adiabatic compression the temperature of the gas is above that of the chamber walls, and this greatly accelerates the evaporation of the drops. With a cycle of this kind the period was reduced to between 30 and 40 sec without diminishing the irt ensity of the "y beam, thus increasing the efficiency of the Wilson chamber by roughly a factor of four. Further improvement to the efficiency was limited by the presence of the "column" mentioned earlier (low-energy electrons) and also by the inadequate rate of evaporation of the drops, as a result of which, on increasing the intensity of the "y beam, the drops failed to evaporate, and in subsequent expansions the chamber was clouded. In order to restore working conditions after this had happened, at least ten working expansions were required with the "y beam switched off. A Wilson chamber has operated in the Physical Institute S-250 synchrotron for anumber of years in the mode speCified with a mean "y beam intensity of~1.5 . 105 photons per pulse [lJ.

A still more effective method of accelerating the evaporation of the drops and reducing the working cycle to "'10 sec is the socalled hypercompression, in which the pressure in the chamber after the working expansion is increased adiabatically to 110

or 120% of its initial value. This sharply raises the gas temperature, which becomes appreciably greater than that of the walls. The amount of heat brought into the chamber during the hypercompression is so great that all the drops formed in the working cycle evaporate in a few hundred milliseconds.

An illustration of this mode of operating the chamber is presented in Fig. 13c.

The adiabatic hypercompression is usually carried out some time (""500 msec) after the end of the working expansion. The pause is due to the inertia of the mechanical components of the pneumatic system and also to the conditions of photography. After the end of the hypercompression there is a fairly slow expansion, lasting 2 or 3 sec. During this the temperature in the chamber falls sharply and becomes slightly lower than its initial value. In the pause following the slow expansion, the gas temperature rises as a result of heat transfer from the chamber walls.

This temperature mode of the chamber may be clearly seen from the oscillograms of Fig. 14a and b. In order to obtain these oscillograms, a resistance (filament of a 220 V, 25 W glow lamp 25 p. in diameter) was placed in the working volume of the chamber; this formed one of the arms of a Wheatstone bridge. The inertia of the resistance was small, and the signal taken from it represented the true changes in temperature of the chamber quite closely.


Fig. 15. Photograph of electron and positron tracks in the Wilson chamber. The chamber operated with hypercompressions and a cycle of 12 sec. The intensity of the photon beam was 1 .105 photons.

The oscillograms clearly show the sharp rise in gas temperature during the hypercompression. The slight rise in temperature after the working expansion and before the hypercompres,... sion is evidently associated with heat transfer from the chamber walls and also with the heat released in the condensation of the drops. The fairly slow fall in temperature during the slow expansion and the rise after the end of this expansion may also be seen quite clearly. The mark gives the initial (chamber-wall) temperature.

Complete restoration of the temperature in the chamber takes 17 to 20 sec. However, experience showed that the chamber could operate stably, with good quality tracks, even when the length of the working cycle was shorter than this. Under these conditions the gas temperature before the working expansion is constant in each cycle, though differing from the temperature of the chamber walls. Depending on the conditions of operation of the chamber (the degree of hypercompression, the duration and velocity of the slow expansion), the temperature mode may take various forms. These may be seen from the oscillograms of Fig. 14, which differ as regards the depth of the slow expansion; the temperature of the gas at the end of the working cycle may be either below (Fig. 14a) or above (Fig. 14b) that of the chamber walls.

The chamber operated with a gas temperature 1 to 2° below that of the walls. If not, the glass of the chamber might become clouded. We may judge the quality of the tracks obtained in the chamber under these conditions from Fig. 15, which shows a photograph of electron tracks produced in the chamber by a low-intensity 'Y beam. The photograph clearly shows spirals formed by the tracks of 0 electrons.

As a result of the rapid evaporation taking place during hypercompression, even if the chamber suffers chance clouding (for example, owing to a badly-timed expansion or too great an expansion coefficient) the working conditions are restored in the chamber after two or three cycles, with the intensity of the 'Y beam unaltered. This enabled us to vary the intensity of the

WILSON CHAMBER FOR STUDYING PHOTOMESON PROCESSES 18]

Fig. 16. Example of the establishment of a discrimination mode of the electron tracks in the chamber with a high intensity of the photon beam. The figure shows photographs of the working volume of the chamber in three successive expansions: a) With zero intensity of the photon beam; b) with a photon-beam intensity of "'0.5.107 photons; since this was the first working expansion at this intensity, the discrimination mode was not established; c) second working expansion at the same intensity; discrimination mode of the electron tracks established.

'Y beam passing through the chamber over quite wide limits so as to establish its value experimentally, without any clouding of the chamber as a result of excessive ionic loading. The following aspects were also clarified.

Since, in the presence of a "ruggedizer," the "column" of slow electrons marking the passage of the 'Y beam is absent, while owing to the high rate of evaporation of the drops no clouding of the chamber occurs, the limiting sensitivity is in practice set only by the requirement that the cases of interest should be clearly identified and that the conditions for analyzing these on the background of electron-positron tracks should be good. With increasing intensity these conditions worsen; however, on passing to still greater intensities ("'1 to 2'107 photons per pulse) the depletion of the working volume with respect to condensed vapor starts playing an appreciable part, and this leads to discrimination of the electron tracks. This state of affairs in the chamber develops automatically and remains unaltered for the whole period of operation of the chamber, provided that the intensity of the 'Y beam remains over "'0.5.107 photons per pulse. The mechanism of this phenomenon has not yet been studied in detail. It would appear that the depletion of the working volume with respect to condensed vapor occurs on account of the fact that a small proportion of the drops fall to the bottom of the chamber in the pause between the working expansion and the hypercompression and also during the slow expansion, and cannot evaporate during the working cycle. The proportion of these drops is small, and in the case of a low intensity, has no effect on the operating conditions of the chamber. However, for an extremely high intensity, so many drops are formed in the working volume of the chamber that the loss of even a small


Fig. 17. Photograph of tracks in the chamber with a photonbeam intensity of about 3 . 107 photons and discrimination of the electron tracks.

Fig. 18. Reactions (y2p), (y3p) and (yn) (lower short track) at impurity nuclei and a recoil proton from a 7r 0

photogeneration reaction in hydrogen. Intensity of the photon beam about 1.5.107 photons.

WILSON CHAMBER FOR STUDYING PHOTOMESON PROCESSES

Fig. 19. Case ~f inelastic rr- photogeneration at an impurity nucleus. Intensity of the photon beam about 1 .107 photons.

183

proportion seriously depletes the working volume of its vapor. The tracks of strongly-ionizing particles are then sharply visible and maybe easily identified and analyzed. Examples of this appear in Figs. 16 to 19. The intensity of the photon beam is not the same for all the photographs shown. Figure 16 gives an example of a case in which this state of affairs is established in the chamber; photograph a corresponds to the absence of a 'Y beam; b to the first working expansion at high ("'0.5'107 photons) intensity, and c to a second expansion at the same intensity. We see from this figure that, if the chamber operates in the discrimination mode, then for the number of electron tracks in the chamber indicated in Fig. 16b, the analysis of the desired reaction is extremely difficult. Figure 17 shows a photograph for a considerably greater intensity (about 3'107

photons) but with discrimination of the electron tracks. Seven photonuclear reactions occurred in the volume of the chamber during this pulse, several tracks intersecting in the middle of the chamber. Despite this, all the tracks, including the short ones relating to recoil nuclei, are clearly visible.

Figures 18 and 19 represent characteristic examples of how easily visible nonrelativistic particles are for an intensity of 1 to 2'107 photons in the pulse. Figure 19 presents a photograph of an inelastic 'Ir- photogene ration at an impurity nucleus. The track is clearly seen.

Nothing can yet be said regarding the operating conditions of the Wilson chamber at still higher intensities, since the present investigation was carried out with the maximum diameter of the collimator and the intensity of the 'Y beam was restricted by that of the accelerator itself.

In order to verify our conclusions and the operation of the system as a whole, we obtained a series of about 25,000 stereo photographs in hydrogen. We considered the reaction y + p -p + rro. The desired cases were selected by reference to recoil protons. The frequency ofthese cases agreed with calculation, about one case for every 20 frames. The statistical accuracy of the experimental data was low; however, the resultant Gtot = f (W) relationship agreed closely with published data.

At the present time 75,000 stereo photographs obtained in deuterium are being analyzed in order to study the 'Y + d - d + 'lr0 reaction in the region of the second resonance.


In conclusion, the authors offer their sincere thanks to 1. M. Obodovskii, Matei Florek, and E. D. Starostin, and also to V. A. Dugin and V. V. Svetlov, who took part in making and adjusting the apparatus at various stages.

LITERATURE CITED

1. A. N. Gorbunov, Trudy FIAN, 13:174 (1960). 2. J. F. Wright, D. Morrison, J. Reid, and J. Atkinson, Proc. Phys. Soc., A69: 77 (1956). 3. A. T Varfolomeev and A. N. Gorbunov, Zh. Eksperim. i Teor. Fiz., 47:30 (1964). 4. V. A. Petukhov, A A. Komar, and M. N. Yakimenko, The Compton Effect and the Limits

of Applicability of Quantum Electrodynamics, Preprint of the Joint Institute of Nuclear Research, R-383 (1959).

5. J M. Sellen, G. Cocconi, V. T. Cocconi, and E. Hart, Phys. Rev., 113:1323 (1959); 115: 678 (1960).

6. V. N. Maikov and M. N. Yakimenko, Report [in Russian] (1961). 7. C. Emigh, Rev. Sci. Instr., 20:279 (1949). 8. G. Evans, N. E. Fancey, J. D. Norbury, and A. A. Watson, J. Sci. Instr., 41:770 (1964). 9. M. 1. Daion and V. M. Fedorov, Zh. Tekhn. Fiz., 25:771 (1955).

10. Yu. S. Ivanov and A. 1. Fesenko, Pribory i Tekhn. Eksperim., No.3, p. 36 (1959). 11. E. R. Gaerttner and M. Y. Yeatter, Rev. Sci. Instr., 20:588 (1949). 12. J. Walker, G. Tagliafferre, J. C. Bower, and D. W. Hadley, J. Sci. Instr. 33:113 (1956). 13. 1. P. Yavor and A. P. Komar, Zh. Tekhn. Fiz., 27:868 (1957). 14. A. P. Andreev, Pribory i Tekhn. Eksperim., No.4, p. 53 (1959). 15. Yu. S. Ivanov and V. V. Yakushin, Pribory i Tekhn. Eksperim., No.3, p. 146 (1960). 16. A. M. Rezkyan and A. M. Miatsakanyan, Pribory i Tekhn. Eksperim., No.6, p. 115 (1960). 17. A 680-MeV Electron Accelerator, Collection of Articles, Moscow (1962). 18. V. P. Andreev, T. 1. Kovaleva, and 1. N. Usova, Tr. Fiz. Inst. Akad. Nauk, 40:215 (1968).

RELATIVE MONITOR FOR A WILSON CHAMBER

V. P. Andreev, T. I. Kovaleva, and I. N. Usova

A relative monitor measuring the flow of energy in a photon beam passing through a Wilson chamber should satisfy the following requirements.

1. It should ensure accuracy of the relative intensity measurements (± 2% or better) with continuous round-the-clock working for 10 days or more.

2. It should only measure the intensity of the working 'Y radiation pulse, i.e., the pulse of 'Y radiation during which the working expansion of the Wilson chamber takes place.

3. It should ensure the possibility of systematically checking the constancy of the working expansion of the Wilson chamber.

In addition to this, one more requirement appeared in the course of the work in relation to the operating conditions of the Wilson chamber. This related to the triggering of the system for photographing the working volume of the Wilson chamber, i.e., the flashes from the flash bulbs and the movement of the film should only occur in cases in which the intensity of the 'Y radiation in the beam exceeds a certain prearranged value. It should be possible to change this triggering position in accordance with the experimental conditions very simply, without introducing any changes into the monitor circuit or the control circuit of the Wilson chamber.

The block diagram of the monitor used in the present investigation is shown in Fig. 1.

The energy flux of the 'Y radiation passing through the Wilson chamber is determined from the ionization created in a plane-parallel ionization chamber. * The main requirement in the construction of this chamber was that the time for collecting the ions should be reduced as much as possible. It is already known [I) that this time, which is determined by the mobility of the ions (u"" 1.5 cm/sec for air), is quite large; it is expressed by the formula

[2 T-- uV ' (1)

where V is the potential on the chamber in volts and l is the distance between the electrodes in centimeters, and can only be made smaller by reducing the distance between the electrodes in the chamber.

The chamber constitutes a set of Plexiglas and Duralumin rings held together by three bolts. The rear wall of the chamber is the collecting electrode, a Duralumin plate 1 mm thick. This is separated from the high-voltage electrode by two polished Plexiglas rings with a Duralumin guard ring between them. The Plexiglas rings have cuts into which the electrodes are fitted. With this construction the depth of the working volume was 0.52 cm, and on applying

'" The use of an integral ionization chamber was difficult owing to the low pulse repetition frequency of the accelerator y radiaLon (once in 6 sec).

185

186 V. P.ANDREEV, T.I. KOVALEVA,ANDI. N. USOVA

Fig. 1. Block diagram of the relative monitor. 1) Ionization chamber with preamplifier, 2) amplifier, 3) amplitUde/time conversion unit, 4) counting system, 5) monitor key, 6) control pulse from the Wilson chamber control-circuit.

800 V to the chamber the collection time was about 210 jJ.sec.

The front wall of the chamber served simultaneously as a high-voltage electrode and a converter. The thickness of this wall was chosen experimentally so that the signal at the input of the preamplifier should be no less than 5 mV.

The chamber was filled with air at atmospheric pressure. The working volume was hermetically sealed so as to eliminate any effects from fluctuations of atmospheric pressure.

The voltage pulse taken from the collecting electrode of the ionization chamber was amplified by a preamplifier with an amplification factor of

20 arranged in the immediate vicinity of the ionization chamber, and was then taken through a long cable to the input of the main amplifier (K'" 100). The pulse from the output of the main amplifier passed to the input of an amplitude/time converter (see Fig. 1) which converted this pulse into a pulse train, the number of pulses in the train being proportional to the amplitude of the input pulse. The number of pulses in the train was counted by a counting system.

In order to ensure that the monitor should only record working pulses of y radiation with intensities greater than a prearranged level, we used the circuit represented in Fig. 1 as the "monitor key" unit. This circuit carries out amplitude discrimination and time selection of the pulses from the output of the amplifier. Pulses from both the amplifier and the Wilson chamber control circuit are fed to the input of this unit. A pulse only arises on the output of the "monitor key" unit when the pulse from the amplifier exceeds the discrimination threshold and is associated with the instant of time specified by the pulse from the Wilson chamber control-circuit.

Let us give more detailed consideration to the individual components of the monitor. As preamplifier we used the demountable TV section of a USh-2 amplifier. The main circuit of the principal amplifier appears in Fig. 2. This amplifier is described in [2].

Circuit of the "Monitor Key" Unit

Figure 3 shows the main circuit of the "monitor key" unit. The operation of this unit takes place in the following way. A pulse from the amplifier passes through a cathode follower (L1) to a Schmidt circuit (L.!), playing the part of a discriminator. The threshold of discrimination may be varied by means of a potentiometer in the grid circuit of the cathode follower. A pulse from the Wilson chamber control-circuit triggers the univibrator formed by L:J.

The positive rectangular pulses from the Schmidt circuit and this univibrator fall on a coincidence circuit based on a 6Zh2P pentode (Lfi) cut off with respect to two of the grids. If these pulses overlap in time, a negative pulse arises in the plate circuit of Ln. The Ls stage changes the polarity of this pulse, so that a positive pulse is formed at the circuit output (y pulse). This pulse trips the system for photographing the Wilson chamber [3] and the univibrator (Ls, Fig. 4) of the amplitude/time converter circuit, which "allows" the train of pulses to appear at the output of this circuit.

Amplitude/Time Conversion Circuit

The main circuit of the amplitude/time converter is shown in Fig. 4. The stages formed by the tubes Lt to L4 convert the pulse from the amplifier into a negative rectangular pulse, the

RELA TIVE MONITOR FOR A WILSON CHAMBER

fJ.001

...GO 91K :.: ~

IJOV

MN-5

I +250 v plug 1-4 --

-2~0 V plug 1-5 L-________ ~--------_+~~--~~~~

Ls 6P t"5P L6 6P 15P LI 6Zh5P L2 6ZhiP Lg 6ZhiP L4 6P15P

Fig. 2. Main circuit of the principal amplifier.

• +JOOV L2 6N15P

:.: k?

Lg 6N15P L4 0.5 6NIP

Ls 6Zh2P L6 0.5 6NIP

~ :.::.: :.: ~ ~ ~ ~

~--~~--------~--~~~--~-'~--~~~--~-----+--~~ 1 ConverSion CIrcuit

Fig. 3. Main circuit of the "monitor key" unit.

187

length of which is proportional to the amplitude of the input pulse [4]. The pulse from the amplifier passes through a cathode follower (L1) and double diode (~) and charges the capacity C to its peak value. Then follows a practically linear discharge of the capacity C through the resistance R and the pentode Ls. During this discharge the potential of the screen grid of the pentode increases, so that a positive rectangular pulse may be taken from the screen. This pulse is amplified by half of tube L4• The second half of this tube is connected as a diode, which "truncates" the negative throw arising in front of the positive pulse in the course of charging the capacity C.

Subsequent operation of the circuit is determined by the arrival of the" 'Y pulse" from the "monitor key" unit. This pulse triggers the univibrator (L fi), at the output of which a pulse accordingly appears; the length of this pulse is taken well above the maximum length of the pulse coming from L4• The negative pulses from tubes L4 and L5 fall on the input of a coincidence circuit (Ls) composed of two cathode followers with a common load. In order to improve the shape of the output pulse from the coincidence circuit, the signal is taken from a silicon stabilotron. The bad shape of the pulse in the absence of the stabilotron is due to a certain differentiation of the long pulses on the inputs of the coincidence circuit.

The pulse at the output of the coincidence circuit coincides in time with the pulse arriving from the tube L4, which constitutes the working pulse of the monitor. No distortion of this pulse

188 V. P.ANDREEV, T. I. KOVALEVA,ANDI.N. USOVA

Fig. 4. Main circuit of the amplitude/time converter.

N m&r---------------------~

!OO(J

51J(J 751J IOO(J V, mV

Fig. 5. Calibration of the amplitude/time converter together with the amplifier.

by the pulse from the univibrator occurs because of the automatic time displacement of the leading edges of these pulses; the leading edge of the pulse from the univibrator corresponds to the instant of time at which the value of the pulse from the main amplifier reaches its t h res hoi d value; the leading edge of the pulse from L4 corresponds to the instant of time at which the pulse from the amplifier reaches its pea k value. The growth time of the pulse from the amplifier is ""150 p.sec. As already mentioned, the length of the pulse from the univibrator is much greater than that from the plate circuit of L4.

The negative pulse from the coincidence circuit cuts off the half triode 4, which in the open state shunts the sinusoid generator (La). The characteristic frequency of the generator is 600 kclsec. This value was chosen in order to obtain the desired accuracy when measuring pulses of small amplitude. With this generator frequency, for each pulse of medium-intensity 'Y radiation there is a reading of several hundred pulses. This ensures good accuracy in measuring the amplitude of each pulse.

Calibration of the Relative Monitor

The linearity of the amplitude/time converter and amplifier taken together was checked with the help of a G5-2A generator. The error in setting the amplitude of the pulses from this generator was ±1.5%. Figure 5 shows a graph of the linearity of the amplitude/time converter together with the amplifier. Deviations from linearity on changing the amplitude of the pulse on the input by a factor of 15 are no greater than ± 2 %, the error of the generator itself being included in this error.

RELATIVE MONITOR FOR A WILSON CHAMBER 189

The absolute calibration of the relative monitor with 'Y radiation was carried out with the help of a quantometer [5]. Unfortunately, owing to the low sensitivity of the measuring system of the quantometer, it was impossible to calibrate the relative monitor by reference to single pulses of 'Y radiation. In calibrating with a series of 'Y radiation pulses (30 to 50 pulses in each series) the spread in the average value for each series was no greater than + 210•

The authors wish to thank A. V. Kutsenko for great help in the initial stages of the work on the relative monitor and also E. M. Leikin, Yu. M. Aleksandrov, and V. F. Grushin for enabling them to carry out the absolute calibration of the relative monitor with a quantometer.

LITERATURE CITED

1. V. 1. Veksler, L. P. Groshev, and B. 1. Isaev, Ionization Methods of Studying Radiations, GTTI, Moscow (1949).

2. V. O. Vyazemskii, 1. 1. Lomonosov, A. N. Pisarevskii, Kh. V. Protopopov, V. A. Ruzin, and E. D. Teterin, The Scintillation Method in Radiometry, Gosatomizdat (1961).

3. v. P. Andreev, Yu. S. Ivanov, V. T. Zhukov, R. N. Makarov, V. E. Okhotin, and 1. N. Usova, Tr. Fiz. Inst. Akad. Nauk, 40:192 (1968).

4. A. A. Sanin, Electronic Apparatus in Nuclear Physics, Fizmatgiz (1961). 5. R. Wilson, NucI. Instr., 1:101 (1957).

experimental physics: methods and apparatus

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