detection efficiency of time-of-flight energy elastic recoil detection analysis systems

Detection e�ciency of time-of-¯ight energy elastic recoil detectionanalysis systems

Yanwen Zhang a,*, Harry J. Whitlow a, Thomas Winzell a, Ian F. Bubb b,Timo Sajavaara c, Kai Arstila c, Juhani Keinonen c

a Department of Nuclear Physics, Lund Institute of Technology, Box 118, Lund S-2100, Swedenb Department of Applied Physics, Royal Melbourne Institute of Technology, GPO Box 2476V, Melbourne 3001, Australia

c Accelerator Laboratory, P.O. Box 43, FIN-00014, University of Helsinki, Finland

Received 2 October 1998; received in revised form 17 November 1998

Abstract

The detection e�ciency of recoils with masses ranging from H up to Nb at energies from 0.05 to 1 MeV per nucleon

has been investigated for Time-of-Flight Energy Elastic Recoil Detection (ToF-E ERD) systems. It is observed that the

detection e�ciency for the ToF-E detector telescope depends on the stopping power in the carbon foils, which in turn

relies upon the recoil mass and energy. Furthermore, the limits of this behaviour depend on the setting of the dis-

criminator thresholds. The detection e�ciency of a time detector could be ®tted to a universal curve that can be de-

scribed by a simple empirical formula as a function of recoil electronic stopping power in the carbon foil. This formula

can be used to predict the detection e�ciency by recoil energy for N, O and other elements, for which it may not be easy

to prepare suitable reference samples containing only that element. Ó 1999 Elsevier Science B.V. All rights reserved.

PACS: 34.50.Bw; 34.80.Kw; 41.75.-I; 73.50.Bk

Keywords: Detection e�ciency; ToF-E ERD; Secondary electrons; Stopping power; Carbon foil time detectors

1. Introduction

Since the mid-1980s, Time-of-Flight EnergyElastic Recoil Detection Analysis (ToF-E ERD)has undergone rapid development by di�erentgroups, for example, in France [1±3], Canada [4±7], Sweden [8±15], Finland [16±18], Australia [19±

21], USA [22,23], Japan [24,25], Germany [26] andKorea [27]. As one of the most powerful ion beamanalysis techniques, Elastic Recoil Detection(ERD) analysis combined with a multi-dispersivedetector telescope, for example a Time-of-FlightEnergy (ToF-E) detector telescope, provides si-multaneous and quantitative elemental depth dis-tributions of light, medium and heavy masselements in both light and heavy matrices. Withthe increasing development of multi-layer thin ®lmmicroelectronic devices, thin-®lm magnetic storage

Nuclear Instruments and Methods in Physics Research B 149 (1999) 477±489

* Corresponding author. Tel.: +46-46-2227682; fax: +46-46-

2224709; e-mail: [email protected]

0168-583X/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 9 6 3 - X

devices, metallized polymers, advanced ceramicmaterials, and tribo-coatings, etc., there is a rap-idly increasing need to measure depth pro®les ofelements in surface layers from 10 to 1000 nmthickness. For quantitative analysis of such struc-tures with ToF-E ERD, four factors, the recoilenergy calibration [8,11,12], recoil mass calibration[12±15], mass resolution [10,13] and the detectione�ciency [9,16,27±30], need to be established inorder to extract accurate depth information fromERD spectra.

ToF-E ERD measurements are based on twothin carbon foil time detectors and one silicon di-ode charged-particle energy detector. The responseof the energy detector is a function of mass andenergy of the detected particles [12]. In the case ofheavy ions, the pronounced non-linear response ofsilicon detectors leads to distortion of the massdata from lines of constant mass which can resultin signi®cant overlap between near-lying elements.In order to overcome this di�culty, some of ushave previously developed a multivariate method[12] for the energy and the mass calibration, whichresults in straight mass lines over the energy rangeof interest for ToF-E ERD.

Some studies have reported on detection e�-ciency study of ToF-E ERD methods [9,16,27±30].However, most of these measurements were car-ried out for a small number of light elementswithin a narrow low energy range. In this work weconsider not only the single time detector detectione�ciency but also the overall telescope detectione�ciency and the relative telescope detection e�-ciency of the whole ToF-E detector telescope. Theoverall telescope detection e�ciency, g0total is thatfraction of recoils incident within the solid angleinterval DX spanned by the ToF-E detector tele-scope that are correctly registered (Fig. 1). Correct

registration of a recoil requires that both the twotime detectors and the silicon energy detectorregister the particle in question. It then followsthat g0total is the product:

g0total

� Number of recoils registered by all three detectors

Number of recoils in the solid angle subtended by the telescope

� g01g02g0E ; �1�

where g01; g02 and g0E are the overall detection e�-

ciencies for the ®rst and the second carbon foildetectors and the silicon charged particle detector,respectively. A prime here is used to denote theoverall telescope detection e�ciencies. g0total is lessthan 1 because of the ®nite probability that a recoilwill impinge on the time detector grids (Fig. 2) orbe scattered out of the solid angle by the carbonfoils [18,29,30]. The e�ciency g0E of a siliconcharged particle detector is essentially 1. Then,

g0total � g01g02: �2�

Unlike silicon detectors, the e�ciency of the car-bon-foil time detectors is often considerably lessthan unity. This is particularly true for light ele-ments, like hydrogen [16], because the detectione�ciency of a time detector relies upon statistics of

Fig. 1. Schematic illustration of ToF-E ERD set-up.

Fig. 2. Electron-mirror carbon-foil time detector (after Busch

et al. [31]).

478 Y. Zhang et al. / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 477±489

secondary electron emission from the thin carbonfoil. The variation in detection e�ciencies for lightelements is so large (�0.1±0.6) that it cannot beneglected for accurate determination of light ele-ment concentration in medium or heavy matrices.

The overall telescope detection e�ciency g0total isimportant when the elemental concentrations aredetermined from the number of the incident ions,solid angle of the detector telescope and absolutecross sections. For practical measurements it isoften more convenient to use the relative telescopedetection e�ciency, gtotal. This is the fraction ofrecoils actually impinging on the energy detectorthat are correctly registered,

gtotal

� Number of recoils registered by all three detectors

Number of recoils registered by the energy detector

� g1g2 � g2: �3�g1 and g2 are the relative detection e�ciencies ofthe two time detectors. Assuming that the two timedetectors are identical, gtotal can be explained as thesquare of g, the relative detection e�ciency of atime detector. In other words, gtotal is the proba-bility that the signals produced by a particle whichcompletely transverses the two time detectors areregistered by the associated discriminators. Forsilicon and lighter recoils, the detection e�ciencyreduction caused by scattering in the ®rst carbonfoil can usually be neglected.

The objective of this work was to investigate therelative importance of the factors governing thedetection e�ciency of ToF-E ERD systems anddevelop a simple universal calibration law that canbe used to determine the detection e�ciency as afunction of recoil atomic number and energy usinga restricted number of reference samples.

2. Experimental

The ToF-E ERD measurements have been car-ried out at the Tandem Accelerator Laboratory ofUppsala University and the Accelerator Labora-tory of the University of Helsinki. The following isa description of the experimental set-up used inUppsala. A schematic representation of the ex-perimental set-up is shown in Fig. 1. The Uppsala

6 MV EN-tandem van de Graa� accelerator wasused to produce 60 MeV 127I11� ions. The ToF-Edetector telescope consisted of two carbon-foil timedetectors [31] (Fig. 2) separated by a 437.5 mm¯ight length (L) followed by a silicon energy de-tector. The carbon foils were 4 lg cmÿ2 thick(2´1017 atoms cmÿ2) which corresponds to 17.6 nmassuming bulk graphite density. The energy detec-tor, placed 25 mm behind the second time detector,was a 280 lm thick ion implanted silicon chargedparticle detector with a 10´ 10 mm active area(SINTEF, Oslo, Norway) and an 8 mm diametercollimator placed immediately in front of it. Thiscollimator de®ned the solid angle subtended by thedetector telescope. The recoiled target atoms weredetected at an angle /� 45° relative to the incom-ing beam direction. The incident angle h1 of pri-mary ions and exit angle h2 of recoils were both67.5° to the sample surface normal.

The electronics system consisted of standardNIM electronics and is shown schematically inFig. 3. The negative anode signals from the timing

Fig. 3. The set-up of the electronics system. HV: high voltage,

T1: ®rst time detector, T2: second time detector, E: energy de-

tector, PA: preampli®er, AMP: ampli®er, CFD: constant frac-

tion timing discriminator, DISC: discriminator, CD: cable

delay, TAC: time-to-amplitude converter, COINC.: multiple

coincidence, D&G: delay and gate generator, DELA: delay

ampli®er, ADC: analogue-to-digital converter.

Y. Zhang et al. / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 477±489 479

detectors produced logic pulses after +26 dB pre-ampli®ers (PA) using constant fraction timingdiscriminators (CFD). The bias voltage appliedacross one micro-channel plate (MCP) (PhilipsG12-25SE) was nominally 800 V. The actual valuemight vary by �20 V depending on the MCP re-sistance (200±750 MX). The exact MCP biasvoltages and resistances are di�cult to measurebecause of the high resistance and non-ohmic be-haviour of the MCPs. The thresholds of the CFDswere set to ÿ50, ÿ125 and ÿ250 mV for di�erentmeasurement sequences. The output signal fromthe ®rst time detector discriminator (T1) was de-layed in a coaxial cable and fed together with thesignal from the second time detector (T2) to atime-to-amplitude converter (TAC) in an inversestart-stop mode in order to minimise the deadtime. A conventional preampli®er-main-ampli®erspectroscopy chain was used for signals from theenergy detector.

Measurements were carried out in two modes:(i) ``Normal mode'' where the data were readout via the ADCs when signals were registeredwithin 0.25 ls in the two time detectors andthe energy detector (dashed lines in Fig. 3 con-nected).(ii) ``Calibration mode'' where the data wereregistered when a particle produced a signal inthe energy detector only (dashed lines in Fig.3 disconnected). In this setting, a null signalfrom the TAC indicated that the recoil hadnot generated a signal in one, or both, time de-tectors.The coincidence signal triggered digitisation of

the ToF and energy signals for each recoil by twoanalogue-to-digital converters (ADC) and theconversion values were stored in list mode in asequential computer ®le. The maximum countrates were a few hundreds per second, consider-ably below the level (�2´105 sÿ1) where the MCPe�ciency is expected to become count-rate de-pendent [32]. In normal ERD setting, only theevents that were registered in both time detectorsand the energy detector were recorded by the dataacquisition system. In the calibration mode, allrecoils that passed through the ToF telescope andproduced a signal in the energy detector were re-corded. The work of detection e�ciency for dif-

ferent particles was measured in calibration mode.Data analysis was carried out o�-line using CERNPhysics Analysis Workstation (PAW) [33] andTurbo Analysis Support (TASS) software [34].The stopping powers for the recoils in the carbonfoil were obtained using ZBL stopping powers[35].

The system used for the complementary mea-surements of H, Li and B in Helsinki is very sim-ilar to that described above [16,17]. The ¯ightlength, L, was 684 mm and the detection angle /relative to the ion beam direction was 40°. Theprojectiles were produced using the 5 MV EGP-10-II tandem accelerator. Instead of measuring therecoiled target atoms generated by much heavierion beam, scattered light projectiles from heavyelemental targets were detected. Lithium and hy-drogen projectiles were scattered from Cu targetsand boron isotopes were scattered from Ta targets.The thickness of the carbon foil was 21.6 lg cmÿ2

in both time detectors for boron measurements,22.9 lg cmÿ2 in the ®rst and 21.6 lg cmÿ2 in thesecond time detector for hydrogen and lithiummeasurements. In this system a higher voltage,6200 V, is applied to the time detector and 900 Vacross the channel plates. Negative anode signalsfrom the time detectors were fed directly to thediscriminators without intermediate preampli®ca-tion. As the amplitude of the MCP signals for thelight recoils is low, the detection e�ciency of theToF system is sensitive to the threshold of thetiming discriminator. Therefore, the threshold wasset at the lowest possible value of ÿ10 mV.

For the ERD measurements in Uppsala, thethick calibration standard samples speci®ed inTable 1 were chosen to give isolated pure elementsignals with little or no overlapping signals fromscattered projectiles or other elements on two-di-mensional plots of time versus energy (e.g. Fig. 4).The metallic Li sample was produced by scrapingo� the oxide layer with a scalpel from a 3 mmdiameter, 99.7% purity Li wire, before hammeringit ¯at against a polished steel anvil. The Li samplewas quickly transferred to the goniometer and thetarget chamber set under vacuum within a fewseconds. It was found that with practice, a ¯at,visibly clean Li surface with a minimal oxide layercould be achieved.


3. Results and discussion

3.1. Analysis of data

A two-dimensional Time vs. Energy plot isshown in Fig. 4. The energy calibration was per-formed using the multivariate procedure suggestedby El Bouanani et al. [12]. To determine gtotal as afunction of energy, the events in a certain energyinterval which have both ToF and energy infor-mation were counted as YieldTrue (Fig. 4), whilstthose which had null time information werecounted as YieldFalse. The relative ToF telescopedetection e�ciency, gtotal, at that energy is thengiven by

gtotal �YieldTrue

YieldTrue �YieldFalse

� g1g2 � g2: �4�

Fig. 4. Time vs. energy data for the Cu sample for ÿ125 mV discriminator threshold.

Table 1

Reference samples

Atomic number Symbol Sample description

3 Li See text

4 Be Metal foil, 0.05 mm

6 C Graphite

12 Mg >99.9%, metal foil,

0.05 mm

13 Al >99.9%, metal foil,

0.5 mm

14 Si Crystal, Si (100)

25 Mn Metal, 1 mm

26 Fe >99.9%, metal foil,

0.5 mm

29 Cu >99.9%, metal, 2 mm

32 Ge Crystal, Ge (100)

41 Nb >99.9%, metal foil,

0.5 mm


In light element measurements, the elementsignals were well separated over a wide energyrange. For medium heavy elements, scattered 127Iand other signals that could be attributed to 12Cand 16O were observed from some samples. Thesecan be seen in Fig. 4, which is a two-dimensionaltime vs. energy plot for the Cu sample measuredwith ÿ125 mV CFD threshold. In addition to thedominating Cu signals, weak signals that are as-sociated with C, O, Al and scattered 127I can alsobe seen. The 12C and 16O signals, which originatefrom surface contamination, are only signi®cant insurface peaks and were avoided in subsequentanalysis. The weak Al signals are associated withsecondary recoils from the Al sample holder re-sulting from impingement of energetic recoils andscattered projectiles from the sample. The signalsof lighter recoils from the bulk are small comparedto the major element signals, and consequentlythey had a negligible e�ect on gtotal and were ne-glected. It is worth noting that a week 127I signalwas observed even for medium heavy targets (Cu,Ge) which are lighter than the minimum mass(�90 u) that can scatter 127I at an angle ofhmax� 45° to the incident beam direction. Thissignal can be attributed to 127I ions that have un-dergone plural and multiple scattering. To avoidthe interference of lighter elements or 127I, the re-coil signal was chosen from energy intervals wherethe recoil data were free of scattered 127I. This isillustrated in Fig. 4, where the signals of bothscattered 127I and light contaminants lie on the lowenergy side of the dotted line. Only data from thehigh energy side, indicated by the arrow, where thesignal is free of contamination, was used for sub-sequent analysis.

The relative telescope detection e�ciencies fordi�erent elements and CFD discriminator settingswere evaluated by the methods described above.Fig. 5 presents the energy dependence of gtotal forCFD threshold settings of (a) ÿ50 mV, (b) ÿ125mV and (c) ÿ250 mV. The most striking feature inFig. 5 is that the relative telescope detection e�-ciency, gtotal, depends on the recoil species, ener-gies and CFD thresholds. Considering ®rst theenergy dependence of the relative detection e�-ciency gtotal, it may be seen that for low mass re-coils the detection e�ciency decreases with

increasing energy whilst for higher mass recoils itis largely energy independent when the relativedetection e�ciency is close to the limiting value of1 (e.g. Al±Nb in Fig. 5a). Fig. 5 also illustrates thegeneral trend that the detection e�ciency increaseswith increasing recoil mass. For given recoil spe-cies and energy, Fig. 5 also shows that the detec-tion e�ciency increases as the CFD thresholddecreases.

3.2. Factors governing the detection e�ciency g of acarbon-foil time detector

The relative detection e�ciency g of a carbonfoil time detector, such as the electron-mirror typeused here [31] (shown in Fig. 2), is governed by therequirement that su�cient secondary electronsmust be produced to ensure that the CFD

Fig. 5. The relative telescope detection e�ciency gtotal vs. recoil

energy. The discriminator thresholds were set to (a) ÿ50 mV,

(b) ÿ125 mV and (c) ÿ250 mV.


threshold is exceeded. g is thus determined by, thestatistics of secondary electron production in thefoil, the transmission probability for secondaryelectrons from the foil to the entrance plane of theMCP pair, the quantum e�ciency of the ®rstMCP, MCP-pair gain and discriminator thresh-old.

The basis of the detection process is the emis-sion of secondary electrons from a carbon foil asthe energetic recoil passes through it [36±41]. Thenumber of secondary electrons is a distributedquantity. In particular whether a particle is regis-tered or not is determined by the probability thatthe number of electrons produced exceeds thenumber corresponding to a signal that exceeds theCFD threshold. Secondary electrons can be pro-duced by potential emission or kinetic emissionprocesses [38,39]. The former takes place if theionisation potential of the ion exceeds twice thetarget work function. Kinetic emission, whichoriginates from ion±electron and electron±electroncollisions as the ion penetrates the foil, predomi-nates at energies exceeding 2´105 msÿ1 (208 keVper nucleon). According to the Sternglass theorythe secondary electron emission scales accordingto the energy deposited in electronic processeswithin the foil [41].

The transport of electrons from the foil to theMCP is limited by the transparency of the grids theelectrons pass through and the loss of electronsthat impinge on the sides of the ®eld-free regionetc., associated with defocusing due to non-uni-formity in the grids and the foil etc. In our detec-tors (Fig. 2) the optical transparency is estimatedto be �0.85. Starzecki et al. [42] have investigatedthe ®eld strengths in electron-mirror time detectorsand found that provided the acceleration andelectron mirror ®elds are su�ciently large, thedetection e�ciency saturates at a constant value.This suggests that the transport e�ciency ap-proaches the limit set by the optical transparencyof the grids. The electric ®elds used here are all inthe saturation region [42] and we may assume thatthe transmission of electrons is constant.

The ®rst MCP will chie¯y limit the quantume�ciency of the MCP pair. This is because the gainis su�ciently large to ensure that every incomingelectron will give rise to �300 electrons that im-

pinge on the entrance plane of the second MCP.The quantum e�ciency of the ®rst MCP is mainlygoverned by the fraction of the active area coveredby channels (�40±60%), the probability that anincident electron impinging on a channel creates atleast one secondary electron in the channel and theprobability that an electron impinging betweenchannels creates secondary electrons that arecaptured into channels [43]. The largest e�ect ofthe MCP bias voltage is found for the detection ofhydrogen. Gujrathi and Bultena [44] have foundthat for a speci®c proton energy the detection ef-®ciency increases non-linearly and shows a ten-dency of saturation at higher bias voltage. Forexample, over 100% increase in e�ciency, could beobtained for 3 MeV protons when 1000 V biasvoltage was used instead of 900 V, and a further20% increase could be gained when 1100 V wasapplied. Similarly the proton e�ciency appears toincrease at reduced energies. It has also been notedthat the e�ciency could be increased to more than90% by coating the carbon foil by a thin layer oflow density MgO [44]. For the ToF system inHelsinki, we have observed that applying a voltageof 990 V across the MCP results in a �50% in-crease in the detection e�ciency for hydrogencompared to 900 V. However, the e�ect is of lessimportance for heavier elements with high detec-tion e�ciency.

3.3. Correlation with the stopping power in carbonfoils

From the previous sections, it is clear that theonly factor governing the detection e�ciency of atime detector with ®xed bias voltage and thresholdis the production of secondary electrons as therecoil traverses the foil, which in turn depends onthe recoil atomic species and energy [35]. Thetheory of secondary electron production bySternglass [41] predicts that the electron yieldshould be proportional to the electronic stoppingpower dE/dx. Experimental measurements byClerc [45], Clouvas [46] and Rothard [47] on themean number of secondary electrons emitted froma thin carbon foil for di�erent ions and di�erentenergies support such a dependence. The particleenergies in their simulations and experiments are


greater than for the recoils studied here. Conse-quently the detection e�ciency of a time detectoris anticipated to be correlated with the electronicstopping power in the carbon foil. Figs. 6±8compare the detection e�ciency g for one timedetector telescope with the electronic stoppingpower in the carbon foil by assuming that the twotime detectors are identical. Figs. 6 and 7 aremeasured using the set-up in Uppsala with CFDthresholds of ÿ50 and ÿ125 mV, respectively.Additional measurements for B, Li and H (Fig. 8)have been conducted in Helsinki. The data can beclassi®ed into two groups according to the di�er-ent correlation between the detection e�ciencyand stopping.

The ®rst group is constituted of recoils wherethe detection e�ciency g is considerably less than 1

and strongly correlated with the particle energyand the electronic stopping power. This groupconsists of Li, Be and C in Fig. 6; Li, Be, C, Al andSi in Fig. 7, H and high energy Li and B recoils inFig. 8. It is worth noting that the detection e�-ciency for H is small for a single time detector(<0.35 at energies of 1 MeV or greater in Fig. 8). Italso emphasises that the measurement of H isoptimal where the recoil energy is low (<1 MeV)and that accurate detection e�ciency values arecritical for quantitative analysis of H contents inmaterials.

The second group is constituted of all the otherrecoils that do not belong to the ®rst group asshown in Figs. 6±8. In this group, the electronicstopping power is larger (6±8 keV/nm). g is close tounity, largely independent of the particle energy

Fig. 6. Calculated electronic stopping power in the carbon foil

and the detection e�ciency g for a single time detector vs. the

recoil-energy for ÿ50 mV CFD threshold.

Fig. 7. Calculated electronic stopping power in the carbon foil

and the detection e�ciency g for a single time detector vs. the

recoil-energy for ÿ125 mV CFD threshold.


and weakly dependent on energy deposited inelectronic stopping. In this case, the detection ef-®ciency g is constant within a few percent althoughthe electronic stopping power in a carbon foil maychange by as much as 30%. For example, Fe andCu in Fig. 7 exhibit 30% variations in the stoppingpower over the energy range of interest, whilst g isconstant (0.94�0.01). Similar behaviours for Band Li are shown in Fig. 8. The detection e�ciencyis rather constant and close to unity, 0.98�0.02 forB within the energy region of the whole measure-ment and 0.95�0.01 for 6Li and 7Li within theenergy region of 0.8±3.0 MeV, although the elec-tronic stopping power varies by 30%. Figs. 6 and 7also show that for the group two recoils g increasesslowly towards 1 with increasing recoil mass.Comparison of these ®gures also reveals that theaverage value of g for the group two recoils issmaller for ÿ125 mV CFD threshold than for ÿ50mV. The dependence of detection e�ciency on

threshold level was signi®cant for the set-up inUppsala. This is probably attributed to di�erencein the operating gain of the MCP-pair and di�er-ent noise pedestals.

For the data presented in Figs. 6 and 7, thestrong correlation of g for the ®rst group of recoilsre¯ects the fact that the stopping power in the foilare small (<4 keV/nm) and insu�cient number ofsecondary electrons are produced to ensure a sig-nal that exceeds the CFD threshold. Then g isrelated to the stopping power in the carbon foil.For the second group of recoils, where the elec-tronic stopping power is rather large, the correla-tion between g and stopping power is weak eventhough the detection e�ciency is less than 1 anddependent on the CFD threshold. The implicationof this is that the probability of producing a suf-®cient number of secondary electrons to exceed theCFD threshold is only weakly dependent onstopping power over the energy range measured.The shape of the distribution of the secondaryelectron yield governs this probability. Althoughthe mean number of secondary electrons followsthe electronic stopping power according to theSternglass theory [41], the probability to exceed agiven threshold value need not exhibit the samebehaviour.

Moreover, the data presented in Figs. 6±8 alsoindicate that higher detection e�ciency for lighterparticles could be obtained by the ERD system inHelsinki. This is most probably due to the lowerdiscriminator setting (ÿ10 mV), the higher biasvoltage applied across the MCP and di�erentelectronics used in Helsinki. The di�erences in thedata from Helsinki and Uppsala systems mightalso be attributed to spreading of the MCP am-pli®cation factors. This may arise from the di�er-ences between the MCP resistances and may leadto departures from the nominal potential di�er-ence applied to each MCP.

3.4. Empirical description of the relative detectione�ciency

Figs. 9 and 10 show the detection e�ciency fora single time detector g for recoils plotted againstthe electronic stopping power in the carbon foil(Se). Clearly, the data points are clustered along a

Fig. 8. The calculated electronic stopping power in the carbon

foil and the time detector detection e�ciency g of a single time

detector of B, Li and H vs. recoil-energy for ÿ10 mV threshold

measured in Helsinki.


continuous curve. This con®rms that the relativedetection e�ciency is chie¯y governed by theelectronic stopping power in the carbon foilthrough the secondary electron yield.

For analytical purposes it is convenient todescribe g in terms of an empirical formula. Pre-vious studies [27,30] found that for restrictedranges of light nuclei, the detection e�ciencyg � ��

g1g2

pfor a single time detector ®tted a law

based on Poisson probability for non-detection:

g � 1ÿ exp�ÿkKSe�; �5�where k is the probability that a single electronimpinging on the MCP triggers the CFD and K isthe proportionality constant between secondaryelectron yield and electronic stopping in the C-foil.

We found that our data, which spans a wider rangeof electronic stopping and particles, were not ®ttedwell by Eq. (5). This may be associated with theinvalidity of Poisson statistics when the probabilityof non-detection exceeds 10% and increase in MCPdetection e�ciency for coincident multiple-electronimpingement. By an empirical approach it wasfound that the stopping dependence of g for thedi�erent CFD thresholds could be ®tted well by aempirical polynomial, a sum of rational functionswhere the constants can be determined using asmall number of reference standards.

g � a0 � a1

Se

� a2

S2e

� a3

S3e

; �6�

where a0, a1, a2, and a3, are the ®tting parameterslisted in Table 2. Se is the electronic stoppingpower in carbon foil, which can be calculated byrecoil atomic number and energy [35]. The devia-tion for a ®xed MCP bias was found to be less than3% using the appropriate ®tting constants corre-sponding to the di�erent CFD threshold values.As can be seen from Figs. 9 and 10, Eq. (6) is asmooth monotonic function of electronic stoppingpower and may hence be used with con®dence todetermine the detection e�ciency for recoils ofintermediate elements.

Fig. 9. The detection e�ciency g of one time detector vs.

electronic stopping power in the carbon foil for all recoils and

the empirical ®t given by Eq. (6): (a) ÿ50 mV, (b) ÿ125 mV and

(c) ÿ250 mV CFD thresholds.

Fig. 10. The detection e�ciency of one time detector and the

empirical ®t given by Eq. (6) as a function of electronic stopping

power in the carbon foil for H, Li and B measured withÿ10 mV

CFD threshold.


3.5. Relation to the overall detection e�ciency

The relative telescope detection e�ciency gtotal

and the overall telescope detection e�ciency g0total

are simply related according to the equations

gtotal � g1g2 � g2 � a0 � a1

Se

� a2

S2e

� a3

S3e

� �2

; �7�

g0total � gtotal a1a2�1ÿ b�; �8�where a1 and a2 are the recoil optical transparencyof the two time detectors, respectively. The prod-uct a1a2 for the entire telescope, totally 6 grids, isabout 78% for the system in Uppsala and 72% inHelsinki. b is the fraction of recoils that are scat-tered out of the solid angle DX subtended by thedetector telescope. As a result of collisions withgrid wires and scattering in the carbon foils, not allthe recoils entering the time detectors can reachthe energy detector to be detected. Considering thedistance between the time detectors and the energydetector, the scattering caused by the carbon foilof the ®rst time detector has a signi®cant in¯uenceon the total detection e�ciency of the whole tele-scope system. The e�ect on the e�ciency g0total isdependent on the probability of the scattering,which depends on recoil species and energy. Thescattering in the carbon foil results in a dispersionof the recoil trajectories from the target. Thismeans that some recoils inside DX, that would hitboth of the time detectors in the case of straighttrajectories, will be scattered out of the DX. This,however, also means that some recoils outside DXwill be detected ± an e�ect which is detrimental tothe depth resolution. More details have been in-vestigated by Arstila et al. [18]. Monte Carlosimulations suggest that the contribution associ-ated with scattering in the carbon foil (4 lg cmÿ2

for the Uppsala geometry) can exceed 10% forheavy recoils. This e�ect is expected to be mostsigni®cant in the case of low velocity heavy recoilsin the thick carbon foils. The e�ect of grid scat-tering is more di�cult to estimate. A simple esti-mate of the upper limit based on the relation ofwire diameter to ion range suggests this contribu-tion will exceed 1%. The de¯ection of chargedparticles by the electrical ®elds in time detectorsmay also have a minor in¯uence to the detectione�ciency.

4. Conclusions

In a ToF-E telescope using two carbon-foil timedetectors, a single time detector detection e�-ciency is critically limited by the requirement thatthe yield of secondary electrons is su�cient toenable the resulting pulse to be registered.1. The relative detection e�ciency of a single time

detector when plotted against electronic stop-ping power in a carbon foil follows a singlesmooth curve that can be approximated by asimple empirical formula.

2. In the energy range of recoils measured in thiswork, the detection e�ciency g of a time detec-tor is correlated with the electronic stoppingpower in carbon foils which depends on the re-coil element and energy.

3. The detection e�ciency depends on the settingof the threshold of the constant fraction dis-criminators. Higher e�ciency is obtained forlower discriminator thresholds.

4. For very light recoils the detection e�ciency issensitive to MCP bias voltage and CFD thresh-old. Higher voltages need to be applied to theMCP to improve the detection e�ciency of hy-drogen. The threshold should be set as low as

Table 2

Fitted parameters for Eqs. (6) and (7) and Figs. 9 and 10

CFD threshold (mV) System a0 a1 a2 a3

ÿ50 Uppsala 1.012467 ÿ0.1706339 0.01414224 ÿ0.005638352

ÿ125 Uppsala 1.061439 ÿ0.8334188 0.41429333 ÿ0.089546537

ÿ250 Uppsala 1.105514 ÿ2.6543250 2.72345140 ÿ0.98655701

ÿ10 Helsinki 1.032903 ÿ0.0577570 0.00143073 ÿ0.000012647


possible, but high enough to avoid backgroundcounts.

5. Observed di�erences between the detection e�-ciency results obtained in Uppsala and in Hel-sinki can be understood to origin from thedi�erent thresholds, electronics and MCP biasvoltages used in the two measurement systems.

Acknowledgements

We are thankful to Prof. Water Loveland, LarsWesterberg, Elbert Jan van Veldhuizen for con-tributing to some of the measurements. Thanks arealso due to the sta� at the tandem acceleratorlaboratories in Uppsala and Helsinki for assistancewith ToF-ERD measurements. H.J. Whitlow andY. Zhang are grateful to the Royal PhysiographicSociety in Lund for travel support. We would liketo express our gratitude to the Wenner GrennFoundation for ®nancial support for I.F. Bubb.The work of the Lund authors has been carriedout under the auspices of the NFR-NUTEKNanometer Structure Consortium. The work ofthe Helsinki authors has been supported by theResearch Council for Natural Sciences and Engi-neering of the Academy of Finland.

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detection efficiency of time-of-flight energy elastic recoil detection analysis systems

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