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Compton Scattering

MIT Department of Physics(Dated: August 27, 2013)

You will observe the scattering of 661.6 keV photons by electrons and measure the energies of thescattered gamma rays as well as the energies of the recoil electrons. The results can be comparedwith the formulas for Compton scattering, considering the quantized photon like a particle.

PREPARATORY QUESTIONS

Please visit the Compton Scattering chapter on the8.13x website at mitx.mit.eduto review the backgroundmaterial for this experiment. Answer all questions foundin the chapter. Work out the solutions in your laboratorynotebook; submit your answers on the web site.

SUGGESTED SCHEDULE

Day 1: Familiarize yourself with the equipment and

do SectionIII.1.

Day 2: Repeat SectionIII.1 and do SectionIII.2.By the end of this session you should have verifiedthat Escat + Erecoil 662 keV for at least twoangular positions.

Day 3: Perform SectionIII.3.

Day 4: Repeat portions of the experiment that needimprovement.

I. INTRODUCTION

By 1920 the successes of the quantum theories of black-body spectra (Planck, 1901), the photoelectric effect(Einstein, 1905) and the hydrogen spectrum (Bohr, 1913)had established the idea that interactions between elec-tromagnetic radiation of frequency and matter occurthrough the emission or absorption of discrete quanta ofenergyE= h. The next crucial step in the developmentof the modern concept of the photon as the particle ofelectromagnetic radiation was taken by Arthur Comptonin the interpretation of experiments he initiated in 1920to measure with precision the wavelengths of X-rays scat-tered from electrons in materials of (low atomic number)[1, 3]. The phenomena of X-ray scattering had alreadybeen studied intensively. It was known that the penetrat-ing power of X-rays decreases with increasing wavelengthand that X-rays are less penetrating after being scatteredthan before, which indicated that the scattering processsomehow increases their wavelength.

Comptons idea was to use the recently developed tech-nique of high-resolution X-ray spectrometry, based onmeasurement of the angle of Bragg reflection of X-raysfrom crystals, to measure precisely the wavelengths ofthe scattered X-rays[4]. Irradiating a carbon target with

an intense collimated beam of monochromatic molybde-num K X-rays and using an ionization chamber as thedetector in his spectrometer, Compton found that thespectrum of scattered X-rays had two distinct spectrallines, one at the wavelength of the incident X-rays andanother at a wavelength that was longer by an amountthat depends on the angle of scattering. The scatter-ing without a wavelength shift was readily explainedby the classical theory of coherent scattering of electro-magnetic waves from electrons bound in atoms. How-ever, the classical theory provided no explanation of thewavelength-shifting incoherent scattering process. Thephenomenon of Bragg reflection used in Comptons mea-surements was a clear demonstration of the wavelike char-acter of the X-rays. Nevertheless, Compton put forwardthe apparently contradictory idea that X-rays, known tobe electromagnetic radiation of very short wavelength,interact with electrons like particles of zero rest mass sothat their energy E= h= hc/ and momentum p arerelated by the relativistic equation for particles of zerorest mass, namely p = E/c. He calculated accordingto relativistic mechanics the relations between the initialand final energies and momenta of an X-ray quantumand a free electron involved, like billiard balls, in an elas-tic collision. In this way he arrived at the formula forthe Compton shift in the wavelength of incoherently

scattered X-rays, namely (see[2], AppendixA)

= h

mc(1cos ), (1)

where m is the mass of the electron and is the an-gle between the trajectories of the incident and scatteredphoton.

The agreement between Comptons experimental re-sults confronted physicists and philosophers with the con-ceptual dilemma of the particle-wave duality of electro-magnetic radiation. In particular, how was one to un-derstand how each particle of light in a Young interfer-ence experiment goes simultaneously through two slits?The dilemma was further compounded when, in 1924,de Broglie put forward the idea that material particles(i.e.electrons, protons, atoms, etc.) should exhibit wave-like properties characterized by a wavelength related totheir momentump by the same formula as that for lightquanta, namely = h/p. In 1927 the wave-like propertiesof electrons were discovered by Davisson and Germer inexperiments on the reflection of electrons from crystals.A particularly interesting account of these developmentsis given by Compton and Allison in their classic treatise[5].

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I.1. Experimental Goals

In this experiment you will measure:

1. the energies of Compton-scattered gamma-ray pho-tons and recoil electrons,

2. the frequency of scattering as a function of angle ,and

3. the total cross section of electrons for Comptonscattering.

The results of the energy measurements will be comparedwith the predictions of Compton kinematics. The re-sults of the scattering rate and cross section measure-ments will be compared with the predictions of the clas-sical (Thomson) theory of X-ray scattering and with theKlein-Nishina formula derived from relativistic quantumtheory.

The experimental setup employs a radioactive sourceof 661.6 keV photons from 137Cs, and two NaI scintilla-tion counters. One counter serves as the scattering targetand measures the energy of the recoil electrons; the othercounter detects the scattered photons and measures theenergy they deposit. Both the target and scatter coun-ters have a scintillator consisting of a 22 cylinder ofThallium-activated sodium iodide optically coupled to aphotomultiplier. A 661.6 keV photon traversing sodiumiodide has about equal probability of undergoing photo-electric absorption and Compton scattering. A photo-electric absorption event is a quantum mechanical tran-sition from an initial state consisting of an incident pho-ton of energyEand a neutral atom to a final state. Thisevent therefore consists of an excited ion with a vacancyin one of its inner shells where the electron that was for-

merly bound and a free photoelectron with kinetic en-ergyEW, where Wis the binding energy of the electronin the neutral atom. The energetic photoelectron loses itsenergy within a small fraction of a microsecond by multi-ple Coulomb interactions with electrons and nuclei in thesodium iodide crystal, and a certain fraction of the lostenergy is converted into scintillation light. Meanwhile,the ion from which the photoelectron was ejected losesits energy of excitation through a cascade of transitionsthat fill the vacancy in its electronic structure and giverise to lower energy photons or electrons that, in turn,lose their energy in the crystal and generate additionalscintillation light.

The total amount of scintillation light produced isclosely proportional to the total amount of energy dissi-pated in all these processes. As a consequence, the pulseheight spectrum of a NaI scintillation counter exposed tomonoenergetic gamma rays shows a distinct photoelec-tric peak which facilitates accurate calibration in termsof pulse height versus energy deposited. The counterscan then be used to measure the energies of scatteredphotons and recoil electrons. The following backgroundtopics should be studied in [2,6]:

1. Kinematics and theory of Compton scattering;

2. Properties of 137Cs;

3. Scintillation counters and the interpretation ofpulse height spectra (see preparatory question 3);

4. Radiation safety;

5. Parameter and error estimation[6].

II. EXPERIMENTAL ARRANGEMENT

The experimental arrangement for the Compton ex-periment is shown schematically in Figure 1. The 2x2cylindrical recoil electron or target scintillator de-tector (Canberra model 802-3/2007), is irradiated by abeam of 661.6 keV photons emitted by about 100 Ci(1 microgram!) of 137Cs located at the end of a hole ina large lead brick which acts as a gamma-ray howitzer.If a photon entering the target scintillator scatters froma loosely bound, effectively free, electron, the resultingrecoil electron may lose all of its energy in the target,causing a scintillation pulse with an amplitude propor-tional to the energy of the recoil electron. If the scatteredphoton emerges from the target scintillator without fur-ther interaction, and if its trajectory passes through theNaI crystal of the scattered photon detector, it willhave a substantial probability of depositing all of its en-ergy by a single photoelectric interaction or by a sequenceof Compton scatterings and photoelectric interactions.This will produce a scintillation pulse that contributesto the photopeak of the pulse height spectrum. Themedian channel of the photopeak in the multichannel an-alyzer (MCA) display is a good measure of the median

energy of the detected photons. If the scattered photonundergoes a Compton scattering in the scatter scintilla-tor and then escapes from the scintillator, the resultingpulse, with a size proportional to the energy of the Comp-ton recoil electron, will be registered in the Compton re-coil continuum of the spectrum. Several experimentaldetails are considered in[711].

In the experimental arrangement described by[2] (pg.256) the target is inert material (not a detector). Thescatter counter suffers background due to gamma-rayphotons leaking through the lead from the 137Cs source,cosmic rays, and radioactive isotopes in the environ-ment which is suppressed by the use of very heavy

(200 lb) lead shielding around the source and the scat-ter counter. Without such shielding and without anyelectronic tricks the pulses produced by Compton scat-tered photons would be buried in the background. Ina better way, the Compton scattered signal can be dis-criminated from the background using time coincidencetechniques, as described in the following paragraphs.

In our experiment (see Fig. 1) both the recoil electrondetector and the scattered photon detector are scintil-lation detectors. Circuits are arranged so that a pulse

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FIG. 1. Schematic diagram of the experimental arrangement for measuring Compton Scattering.

from a detector is accepted for pulse height analysis bythe multichannel analyzer (MCA) only if it is coincident

(within a fraction of a microsecond) with a pulse from theother detector. Coincident pulses occur when a gammaray photon produces a Compton-recoil electron in the re-coil electron detector, and the Compton-scattered photonis subsequently absorbed in the scattered photon detec-tor. The two nearly simultaneous pulses produced in suchan event are amplified and fed through inverters (Ortecmodel 533) to constant fraction discriminators (CFDs)(Canberra model 2126) whose threshold is user selectablebetween -50 mV and -1 V through a 10-turn potentiome-

ter. (The inverters are required because the amplifiers aredesigned to put out positive pulses, whereas the CFDs

are designed for negative input pulses.) CFDs employa historically interesting method to achieve precise tim-ing described in AppendixB.Your experiment does notrequire this accuracy, and it is superseded by modernelectronics.

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FIG. 2. Schematic representation of the proper time rela-tion between the photomultiplier tube signal (amplified andinverted), whose pulse height is to be measured, and the coin-

cident gate pulse produced by the Gate and Delay GeneratorNIM module. The coincident gate pulse needs to arrive beforethe peak of the pulse and persist for 0.5 s after the pulsepeak.

II.1. Coincidence Techniques

A true Compton event will produce coincident (within2 s) signals in both detectors. If the logic (square) dis-criminator outputs overlap, the coincidence (AND) pro-duces an output YES signal to the gate generator, whichin turn notifies the MCA in the computer to analyze the

pulse heights generated by this event. Most likely it isa Compton event, however there is a chance that twodifferent background events accidentally occurred withinthe overlap time 2given by the discriminator outputs.From a scatter counter with rate ns and target counterwith ratent the accidental rate

nacc = 2nsnt (2)

is small unless you have noise increasing ns or nt. Agood method to set up the coincidence uses a 22Na sourceplaced in between the two detectors. But for Comptoncoincidences, consider that for small , the recoil (target)energy and hence signal are small. If the signals are below

your discriminator threshold, you fail to see coincidentCompton events at small .

II.2. Gating the MCA

In principle, the gate should tell the MCA ahead oftime that a good event is coming. With the presentelectronics, this is not possible so we try to catch themajority of the signal (see Fig.2) including the peak. The

analog pulse shown is derived from the unipolar outputof the amplifier. The bipolar has positive and negativepeaks and is more tricky to use. Nuclear physicists useit at high counting rates.

When you think you have everything working correctly,and before you commit to a long series of measurements,check the performance at the extreme values of, say 0

and 150. At both positions the peaks in the spectra due

to the scattered photons and the recoil electrons shouldstand out clearly. If they do not, then more adjustmentsmust be made to achieve correct operation of the coin-cidence gate logic. In coincidence mode, the MCA his-togram is not updated unless an appropriate gate pulsearrives at the gate input during the rise of the signalpulse. During the dead time required for each pulseheight analysis ( 8 s for the MCAs used in JuniorLab) the input to the pulse height analyzer is held closedby a disable gate so that the arrival of a second pulsewill not interfere with the current analysis. More detailsof the specific MCAs (Perkin-Elmer model Trump-PCI)used in Junior Lab are available in the Junior Lab FilingCabinet or in the online e-library.

III. EXPERIMENTAL PROCEDURE

Keep the lead door of the gamma-ray howitzer closedwhen not in use. The purpose of the procedures describedbelow is to acquaint you with the operation of the equip-ment, particularly the coincidence scheme which permitsthe experiment to achieve a good signal/background ratiowith a minimum amount of shielding. The steps belowsuggest a sequence of tests and adjustments to achievethat purpose, but you should feel free to devise your own.

III.1. Experimental Calibration

Set up and test the experimental arrangement with511 keV annihilation photons from a 22Na calibrationsource. 22Na, a radioactive isotope of sodium, decays toan excited state of 22Na by emission of an anti-electron,i.e.a positron. The excited neon nucleus, in turn, decaysquickly to its ground state by emission of a 1.27 MeVphoton. Meanwhile, the positron comes to rest in thematerial in which the sodium isotope is embedded, com-bines with an electron to form an electrically neutralelectronic atom called positronium (discovered in 1947

by Prof. Martin Deutsch of MIT). The latter lives about107 s until the electron and positron annihilate yield-ing two 511 keV (= mec2) photons traveling in oppo-site directions. Thus 22Na is a handy source of pairs ofmonoenergetic photons traveling in opposite directionswhich can be used to test a coincidence detector system.

1. Set up the NaI recoil electron and scattered pho-ton scintillation detectors as illustrated in Figure

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1. Turn on the power to the NIM bin, HV sup-plies, MCA and oscilloscope. Keep the door on the137Cs source howitzer closed to reduce the intensityof 661.6 keV photons at the detectors.

2. Adjust the gain of the scattered photon detectorand calibrate it. Apply high voltage to the scat-ter counter photomultiplier (+1000 VDC). Usingthe Maestro-32 MCA software, turn off the co-incidence gate requirement on the MCA. Place the22Na calibration source near the scattered photondetector and examine the output of the scatteredphoton amplifier signal (following the pre-amplifierAND the amplifier in series) with the oscilloscope.Adjust the gain so that the amplitude of the pulsesproduced by 511 keV annihilation gamma rays isabout +7 volts. (You might see that a few veryhigh energy pulses saturate the amplifier result-ing in square peaked signals. Since the Comp-ton scattering experiment will measure only ener-gies< 661.6 keV, it is okay to discriminate againstthese pulses.) Determine the gain setting of the pri-

mary amplifier at which non-linear response begins,as indicated by a flattening of the top of the pulseas viewed on the oscilloscope. Then reduce the pri-mary amplifier gain sufficiently to assure a linearresponse up to energies safely above the 661.6 keVenergy of the 137Cs photons.

Now feed the output pulses from the amplifierto the input of the MCA and acquire about 30seconds of pulse height data using the acquisi-tion presets sub-menu. Examine the Energy spec-trum/histogram of pulses from the scattered pho-ton detector and adjust the gain of the amplifierso that the median channel of the photopeak for

511 keV photons is 2/3 of the full scale MCArange (check that this is set to 2048 channels). Cali-brate the counter with other plastic rod calibrationgamma-ray sources. A good set consists of 137Cs(661.6 keV), 22Na (511 keV), and 133Ba (356, 302,and 81 keV plus other lines) each of which pro-duce easily identified photopeaks. Be aware thattoo high a counting rate will spoil the spectrum byoverlapping signals and amplifier saturation.

Write in your lab notebook a succinct de-scription of what you are doing along withthe data so that you can later recall exactlywhat you did and repeat it quickly. You will

have to redo the calibration at the start ofeach lab session because other users will havechanged the settings. When you have identi-fied the photopeaks and are confident about thecounting rates, measure the median channels of thephotopeaks and plot the average median photopeakchannel numbers versus the photon energies on mil-limeter graph paper to provide yourself with a cal-ibration curve. The simplest way to estimate therandom error in a measurement of the median is to

repeat it five or more times and then evaluate theroot mean square (RMS) deviation of the successivemeasurements from their mean value.

3. Adjust and calibrate the recoil electron target de-tector in the same way.

4. Send the amplified recoil electron and scatteredphoton signals to the Inverter module and then

send the resultant negative polarity pulses into thetwo discriminators. Examine the positive outputpulses generated by the units.

5. Send the two discriminator outputs into channels1 and 2 of the coincidence module and examineits output. (It is easiest if you temporarily use asingle detector signal to trigger both discriminatorsfor this test.)

6. Send the output from the coincidence circuit intothe gate generator unit and explore its operation.Adjust the delay of the delay amplifier so the coin-cidence gate pulse arrives at the MCA gate input

during the rise of the analog signal pulse (see Fig-ure2). A good way to verify the action of the gatepulse is to feed the pulses from one of the amplifiersto both CFD inputs. This will ensure that a coin-cidence pulse is generated at relatively high rates.(Students should play with the gate generator con-trols, specifically delay and width, to verify the op-eration of the MCA card coincidence circuitry asdescribed in Figure2.)

7. Test the coincidence logic. Place the 22Na sourcebetween the two scintillators so that pairs of an-nihilation photons traveling in precisely opposite

directions can produce coincident pulses in the twocounters. For this test it is interesting to reducethe gain by a factor of 2 so that the photopeak ofthe 1.27 MeV photons can be seen on the MCAspectrum display. Compare the spectrum with andwithout the coincidence requirement, and observethe effect of moving the source in and out of thestraight line between the scintillators in order toobserve hits by a pair of annihilation photons. Takecareful notes and be sure you understand what youobserve. Do not hesitate to ask for help if you needit. In each of the above steps it is very usefulto sketch a picture (or generate a bitmap us-ing the Agilent digital oscilloscopes) of the

waveforms taking particular note of theiramplitudes, rise and fall times and temporalrelationships with other signals, e.g. gates.

III.2. Compton Scattering Using 137Cs

This section details the measurement of the energiesof the scattered gamma rays and the recoil electrons asfunctions of the scattering angle .

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1. Align the primary gamma-ray beam from the 137Cssource in the howitzer. Position the howitzer andthe scatter counter so that the source (at the insideof the hole in the lead howitzer) and the scattercounter scintillator are at opposite ends of a diam-eter of the round platform, with the source 40 cmfrom the center of the scintillator. (Keep carefulnotes about the exact geometry you choose for the

setup so you can return to it if necessary at a latertime.) Open the door of the howitzer and mea-sure the rate of pulses in the photopeak from thescatter counter as a function of its position angle.In other words, measure the profile of the beamand determine = 0. Check that the beam profileis symmetric about a line from the source throughthe center of the turntable to insure that the tar-get counter will be uniformly illuminated when itis placed at the center.

2. Adjust the discriminators of the CFDs to insurethat they will generate coincident pulses whenevera Compton scattering occurs in the target scintil-

lator with a scattering angle in the range you wishto cover. Position the NaI target counter over thecenter of the turntable at a height such that theaxis of the beam intersects the mid-plane of thescintillator. For a preliminary test, connect the (in-verted) output of the target amplifier to the inputsof both CFDs. Simultaneously connect the outputof the target amplifier to the INPUT BNC connec-tor on the MCA card in the PC. Connect the out-put of the coincidence circuit to the gate generatormodule and the output of the gate generator to theGATEBNC on the MCA card. Using theMaestro-32 software package, set the discriminators as low

as possible without getting into the rapidly risingspectrum of photomultiplier noise such that the ac-cidental coincidence rate would rival the true coin-cidence rate. Start a spectrum accumulation andtest the effects of changing the delay of the gatepulse with respect to the signal pulse.

3. Measure the median energy Eof scatteredphotons and the median energy Ee of recoilelectrons as functions of the position angle

of the scattered photon detector. With boththe source in the howitzer and the scintillator of thescattered photon detector located40 cm from therecoil electron detector, set the position angle of the

scatter counter at 90. With the coincidence re-quirement enabled, accumulate a spectrum of scat-tered photon detector pulses on the scatter MCAand a spectrum of recoil electron detector pulses onthe target MCA and measure the median channelof the pulses in the photoelectric peaks. Repeatthis dual measurement at several position anglesfrom just outside the primary beam profile to asnear to 180 as you can get. A good strategy is totake data at widely separated position angles, say

90, 30, 120, 150 and then fill in when you havetime. Plot the raw results (median energy againstposition angle) as you go along to guide your mea-surement strategy. It is wise to check the calibra-tion between each measurement by switching the(software based) gate setting to OFF and record-ing in your notebook the median channels of thephotopeaks produced by 661.6 keV photons from

the 137

Cs calibration source, the 511 keV photonsfrom 22Na, and the 356 keV and 81 keV photons of133Ba.

Record the integration time of each measurementto find the rate as a function of to later comparewith Thomsons classical prediction.

III.3. Scattering Cross Section Measurements

In this part of the experiment, you will measure thetotal scattering cross section per electron in plastic. Theplastic scintillator blocks used as absorbers in this mea-surement are made from polyvinyltoluene (C10H11, den-sity 1.032 g/cm3, www.bicron.com) composed of almostequal numbers of carbon and hydrogen atoms. The mosttightly bound electrons are in the K-shells of the car-bon atoms where they are bound by 0.277 keV, which issmall compared to 661.6 keV. In fact, many other materi-als can be used providing approximately free electrons.This material resembles most modern plastic scintilla-tors. You will determine their quantum efficiency, e.g.how often they see -rays. Place the scatter counter at0 and remove the target counter from the beam. Mea-sure the counting rates in the 661.6 keV photopeak ofthe NaI scatter counter with no absorber and with three

or more different thicknesses of plastic scintillator placedjust in front of the exit hole of the howitzer. Determinethe thicknesses (in g/cm2) of the scintillator blocks bymeasuring their dimensions and mass. Plot the natu-ral logarithm of the measured rate as a function of thethickness and check the validity of your data by observ-ing whether the data points fall along a straight line asexpected for exponential attenuation. Question: Why isit better to place the absorbers as near as possible to thesource rather than just in front of the detector?

IV. ANALYSIS

1. Plot Etarget+ Escatter vs. .

2. Compare your results on the angular dependenceof energies of the scattered photons and the re-coil electrons with the predictions of the Comptonformula. Since the relation between the scatteredphoton energy E and the scattering angle , isnon-linear, one can anticipate that a plot of the
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measured energy versus position angle could be fit-ted only by a curve. However, an easy manipula-tion of the Compton formula yields a linear rela-tion between 1/E and 1-cos . Plot 1/E

against(1 cos ) with appropriate error estimates. Onthe same graph plot the calculated curve of 1/Eagainst (1 cos ) based on the Compton formula.Do a similar analysis of the recoil electron data on

a separate graph. Plot 1/Ee against a function of such that the expected relation is linear. Com-ment on the degree of conformance of the predictionto your experimental results and discuss possiblecauses of any systematic differences.

3. Plot the measured rate as a function ofand com-pare to the classical Thomson prediction.

4. Compute the total interaction cross section perelectron. The attenuation of a collimated beamof particles by interactions in a slab of materialof thickness x (in centimeters) is described by theformula

I(x) = I0ex, (3)

where I0 is the initial intensity and is the to-tal linear attenuation coefficient (cm1). In plasticscintillator the attenuation of 661.6 keV photons isdue almost entirely to Compton scattering. Underthis circumstance the total attenuation coefficient

is related to the Compton scattering cross sectionper electron, which we call total, by the equation

total =

ne, (4)

whereneis the number of electrons/cm3 of the ma-

terial. To find from your data, you should fit anexponential directly to the data, determining the

best-fit parameters and their errors. Alternatively,you may plot the natural log of the measured val-ues of I/I0 against x . Then determine the slope(and its error) of the straight line that best fits thedata by linear regression. In either case, calculatethe number of electrons/cm3 of scintillator (CH)n,and find total. Compare to the classical Thomsoncalculation and to the theoretical result of Klein-Nishina [12].

IV.1. Suggested Theoretical Topics

1. Relativistic mechanics and the Compton shift.

2. Derivation of the Thomson differential and totalcross section for unpolarized photons.

3. Discuss the correct Klein-Nishina prediction.

[1] A. Compton, Phys. Rev. 21, 483 (1923), brother of K.T.Compton, MIT President.

[2] A. Melissinos and J. Napolitano,Experiments in ModernPhysics, 2nd ed. (Academic Press, Orlando, 2003).

[3] A. Compton, Phys. Rev. 22, 409 (1923).[4] A. Compton, Nobel Prize Lecture(1927).[5] A. Compton and S. Allison, X-Rays in Theory and Ex-

periment (Van Nostrand, Princeton, 1992) qA481.C738Physics Department Reading Room.

[6] P. Bevington and D. Robinson, Data Reduction and Er-ror Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2003).

[7] G. Knoll, Radiation Detection and Measurement, 3rd ed.(John Wiley, Orlando, 1999) ISBN 0471073385.

[8] R. D. N.H. Lazar and P. Bell, Nucleonics 14, 52 (1956).[9] R. Garner and K. Verghese, Nuclear Instruments and

Methods 93, 163 (1971).

[10] R. Hofstadter and J. McIntyre, Physical Review 78, 619(1950).

[11] J. Higbie, American Journal of Physics 42, 642 (1974).[12] D. J. Griffiths, Introduction to Quantum Mechanics

(Prentice-Hall, 1995).

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P

h /c

h /c

FIG. 3. Schematic representation of Compton scattering.

Appendix A: Derivation of the Compton Scattering

Formula

Consider the interaction between a photon with mo-mentum vector p(and zero rest mass) and an electroninitially at rest with rest mass me as depicted in Figure

3. Callp and pe the momenta of the photon and theelectron after the interaction, respectively. By conserva-tion of momentum, the relation between the initial andfinal momenta is

p=p

+ p

e. (A1)

Rearranging and squaring both sides we obtain

p2+ p2

2p p

=p2

e. (A2)

The total relativistic energy E and momentum p of aparticle are related to its rest mass m by the invariantrelation (p p)c2 E2 = m2c4. By conservation ofenergy,

p+ mec= p

+

m2ec

2 +p2e. (A3)

Rearranging and squaring both sides, we obtain

p2+ p2

2pp

+ 2mec(p p

) + m2

ec2 =m2ec

2 +pe2.

(A4)Subtraction of Equation (A2) from Equation (A4) andrearrangement yields

mec(p p

) = pp

p p

. (A5)

Finally, dividing both sides by mecpp, we obtain forthe relation between the energies of the incident and thescattered photons the equation

1

E

1

E=

1

mec2(1cos ), (A6)

where is the angle between their final momentum vec-tors.

Appendix B: Introduction to Timing

A timing instrument marks or measures the preciseoccurrence of nuclear events. In practice, the release ofelectrons, which result from the detectors absorption ofa particles energy, occurs after the actual event; howlong after is dependent upon the detector and how it isemployed. The electronics processing this output further

modifies its characteristics. All of these factors must beaccounted for, in addition to the problem of noise, whenmaking precise and highly resolved timing measurements.

Sharp timing (2overlap) reduces the number of ac-cidentals. The primary limitations in this (and other)experiments is the time slew. A pulse of lower amplitudereaches the threshold of the discriminator later. The co-incidence time has to be enlarged to ensure overlap withthe other normal (earlier) signal. The time slew is largebecause the rise time our amplifiers is slow. To over-come timing inaccuracies from slew and differing pulseshapes, the constant fraction technique is used.

FIG. 4. Manipulation of input signal for constant fractiondiscrimination.

Constant fraction timing involves the inversion, delayand recombination of the signal to create a zero crossingmark. However, rather than adding the inverted and de-layed signal to the original to achieve cancellation, somefractional part of the original is used to insure baselinecrossover. As is evident in Figure4, the crossover pointis independent of amplitude of the input signal. The con-stant fraction discriminator (CFD) fixes its reference atthis point.

1. Constant Fraction Discrimination

CFDs have an advantage over simple discriminatorsin experiments that require the detection of coincidenceswith high time resolution. A simple discriminator pro-duces a logic output pulse that starts at some fixed time

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after the voltage at the input rises above a set value.Therefore pulses of similar shape but different peak am-plitudes trigger a simple discriminator at different timesrelative to the occurrence times of the events that gen-erate the input pulses. A CFD also produces a logicoutput pulse if the voltage of the input pulse exceeds aset value; its special virtue is that the output pulse startsat a fixed time after the input pulse has risen to a cer-

tain constant fraction of its peak value, independent ofthe magnitude of that peak value. If the input pulseshave a fixed shape, but different amplitudes, then thedelay between the event that produces the input pulseand the logic output pulse from the CFD will be a con-stant, which can be adjusted by the WALK control. Thepositive +5 VDC outputs of the CFDs are fed to a coin-cidence circuit that produces a logic pulse when all of itsenabled input ports see pulses within the selected resolv-ing time (user selectable within two ranges: fast coin-cidence 10100 ns or slow coincidence 1001000 ns).The output pulses from the coincidence circuit are fedto a gate generator and delay module, after which, thesuitably stretched and delayed gate signal is sent to theGATEinput of the MCA.

The amount of delay and the gate width in the delaymodule should be adjusted to meet the requirements ofthe coincident gate mode of the MCA. This requirementis that the gate pulse begin before the peak of the analoginput signal and last for at least 0.5 s beyond the peak.The coincidence gate requirement suppresses backgroundevents and makes the Compton scatter events stand outclearly in the resulting pulse height spectrum without theuse of heavy shielding.

Appendix C: Scintillation Counter Efficiency

Calculations

A significant complication in the analysis of the dataon the differential cross section is the fact that the targetis not thin, i.e. the mean free paths of the incidentand scattered photons suffer substantial attenuation asit passes through the target material, and the scatteredphotons suffer substantial attenuation in their passageout of the target. The effects of these attenuations onthe counting rate obviously depend on the position ofthe detector. A count must therefore be taken of thesethick target effects in order to derive accurate values ofthe differential cross section from the measured counting

rates at various scattering angles.As in all cross section measurements, the necessarily fi-

nite sizes of source, target and detector introduce into theanalysis geometrical factors that require laborious mul-tidimensional integrations for their evaluations. In ourexperiment we have a divergent beam of photons from aradioactive source of finite size that can interact at var-ious depths in the target within certain ranges of solidangle to produce scattered photons that traverse the de-tector would be a convolution of the differential cross

section with geometrical and attenuation factors repre-sented by a multiple integral over at least ten variableswith hideous limits of integration. The only practicalway around this kind of mess in particle physics experi-ments where the highest possible precision is required isthe Monte Carlo technique in which the sequence ofphysical processes from emission to detection is sampledby use of appropriate distribution functions to represent

the various possible processes, and random numbers toselect the outcome of each process.Here we outline an approximate analytical treatment

of our problem based on several simplifying assumptions.The formulation is suitable for numerical computation ona modest computer. We will consider only single scat-tering events. We call R and H the radius and length,respectively, of the cylindrical scintillators which are thesensitive elements of the target and scatter counters. Weassume that the source is so far from the target scintil-lator that the incident beam may be considered uniformand parallel (i.e. unidirectional). We assume that thedistance D of the scatter counter scintillator from thetarget is so large that the detected portion of the scat-tered radiation may be considered to comprise a parallel(but not uniform) beam.

Consider the scatterings that occur within a solid ele-ment of volumerddrdz at a position in the target scin-tillator with cylindrical coordinates r, , z. As can bededuced with the aid of Figure5, the distance x of thiselement from the point of entry of the incident photonsinto the target scintillator is given by the expression

x= R{[1(qsin)2]12 qcos}, (C1)

where q = r/R. Similarly, for photons scattered fromthe pointPat the angle , the distancey of the element

from the point of exit is

y= R{[1(qsin)2]12 qcos}, (C2)

where = + . We call the total linear at-

FIG. 5. Diagram of the Compton scattering geometry.

tenuation coefficient (units of cm1) of the target plastic

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scintillator for the incident photons, () the total linearattenuation coefficient of the plastic for photons scatteredat angle , and () the total linear attenuation coeffi-cient of the scatter counter NaI scintillator for those scat-tered photons. The efficiency for detection of scatteringevents that occur at a given position in the target scin-tillator is exp(y)[1 exp(H)] which represents theprobability that the scattered photon will escape from

the target counter multiplied by the probability that thescattered photon interacts in the scatter counter.We must now find the average of this efficiency over

the target scintillator. Recalling the assumption that theincident beam is parallel and uniform, the weighting fac-tor is I0exp(x) which is the intensity of the incidentbeam at the point of interaction inside the target scin-tillator. We can now write for the average efficiency theexpression

() =

exp(x)exp(y)[1exp(H)]dV

exp(x)dV , (C3)

where the integrals are computed over the volume of thetarget scintillator. We note that 1 as 0, 0.

The total attenuation coefficients are functions of theenergy of the scattered photon and the energy of thescattered photon is a function of the scattering angle.According to the Compton theory,

E() = E0

1 + (E0/mc2)(1cos ). (C4)

In the energy range from 0.2 MeV to 0.7 MeV, the to-tal attenuation coefficient in sodium iodide can be fairlyrepresented by the formula

=

0.514

E100 keV

0.368+ 5.51

E

100 keV

2.78 cm1,

(C5)and in plastic by the formula

= 0.177

E

100 keV

0.37

cm1. (C6)

The quantity can be evaluated as a function of bynumerical integration.

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