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TECHNICAL NOTE

489

Tables of Response Functions

For Silicon Electron Detectors

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

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NBS TECHNICAL NOTE 489

ISSUED AUGUST 1969

Nat. Bur. Stand. (U.S.), Tech. Note 489, 62 pages (Aug. 1969)

CODEN: NBTNA

Tables of Response Functions for

Silicon Electron Detectors

M. J. Berger, S. M. Seltzer, S. E. Chappell,

J. C. Humphreys, and J. W. Motz

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

This work was supported by

The National Aeronautics and Space Administration

under Contract R-80

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C, 20402 - Price 60 cents
Abstract

Calculated response functions are presented which describe

the pulse-height distributions produced by monoenergetic electrons incident on silicon detectors. It is assumed that the

direction of incidence is perpendicular, and that the detectors

are wide enough so that lateral leakage of energy is insignificant. The validity of the calculations has been confirmed

experimentally for incident energies between 0.25 MeV and 1.0

MeV and detectors with thicknesses between 0.06l mm and 3.0 mm.

The calculated response functions cover a wider range of conditions, including fourteen energies between 0.15 MeV and 5*0 MeV

and ten detector thicknesses between 0.05 and 10.0 mm. At

energies ^ 1 MeV, response functions are given not only for bare

detectors but also for an arrangement in which the detectors

are shielded on the transmission side by another detector operating in anti-coincidence.

Key words: Electron; energy absorption; electron backscattering; electron transmission; detector

response function; silicon detectors.

11
Preface

This report is an expanded version of a paper (published in Nuclear

Instruments and Methods 6°^, 181-193 (1969)) on the response of silicon

detectors to fast electrons. In that paper it was shown that transport

calculations by the Monte Carlo method yield response functions that are

in good agreement with experimental results. The new material in the

present report consists of nineteen pages of tables listing energy

deposition distributions D(T ,T) obtained in Monte Carlo calculations.

These distributions, discussed in Section 2, relate the kinetic energy

T of the incident electron to the amount T of energy deposited in the

detector, taking into account the transmission and reflection of energy

carried by electrons and secondary bremsstrahlung. Tables of D(T ,T)

are given for fourteen values of T between 0.15 MeV and 5»0 MeV and ten

o detector thicknesses between o 05 mm and 10,0 mm. These results can

easily be converted to response functions by folding in the experimentally determined noise distribution (intrinsic resolution) according to the

procedure given in the Appendix.

in
Contents

Page

Preface --------------------- ±±±

1. Introduction ---------------- \

2. The Response Function ------------ 2

2.1. Energy Deposition Distribution - - - - 2

2.2. Noise Distribution ---------- 3

3. Monte Carlo Calculation of the Energy

Deposition Distribution ----------- 5

3.1. Assumed Conditions ---------- 5

3.2. Sampling Procedures --------- 3

3.3« Utilization of Monte Carlo Histories - 7

h. Experimental Method ------------- 8

k-.l. Electron Source ----------- 8

k.2. Detectors -------------- 9

lj-,3. Amplification and Recording of

Pulses ---------------- 9

K.k, Background Correction -------- 10

5. Results ------------------- 11

5.1. Comparison of Calculated and Measured

Response Functions ---------- 11

5.2. Scaling 13

5.3. Backscattering, Transmission and

Absorption Coefficients ------- ll+-

5.^. Detection Efficiency --------- l6

iv
Contents (Continued)

5.5* Anticoincidence Arrangements ------- 16

5.6. Tables of Energy Deposition Distributions - 17

Appendix ----------------------- 18

References ---------------------- 19

Tables -------- _-_-____-_____ 21

Figures ----------------------- 50

1. Introduction

The use of silicon detectors for radiation measurements has become

increasingly important in recent years. The interpretation of such

measurements requires knowledge of the response of the detectors to

various types of radiation. The purpose of this paper is to provide

information about the response to electrons.

Data on the response of silicon detectors to electrons have been

reported by McKenzie and Ewan [l] , West, Burns, Wang and Ziemba [2],

Charoenkwan [3], Rester and Rainwater [4-], Huhn and Schneider [5] 5 Aitken, Emerson and Zulliger [6], Planskoy [7j. and Waldschmidt and

Wittig [8]. In the present investigation, extensive new data have been

obtained experimentally as well as by a Monte Carlo calculation. The

results provide information required for the application of silicon

detectors to the counting of electrons and the measurement of electron

spectra. They also confirm the accuracy of the Monte Carlo method used

as a tool for predicting the penetration and diffusion of electrons in

extended media.

Experimental response functions and corresponding Monte Carlo

results have been obtained for detectors with six thicknesses (0.061,

0.105, 0.191, 0.530, 1.0 and 3o0 mm) at four electron energies (0.25,

0.5, 0.75 5 and 1.0 MeV)[9,10]. Monte Carlo response functions have also

been calculated at fifteen energies between 0.15 and 5*0 MeV for

detectors with ten different thicknesses between 0.05 and 10.0 mm, and

are presented in Table 9 of the present report.

Figures in brackets indicate the literature references at the end of

this paper.
2. The Response Function

The pulse-height distribution H(h), observed with a given silicon

detector, is related to the actual electron spectrum S(T ) by the

integral equation

00 H(h) = |

S(T

o )R(To ,h)dTo

. (1)

The kernel R(T ,h) is called the response function of the detector, and

R(T ,h)dh is equal to the probability that an electron incident on the

detector with kinetic energy T will give rise to a pulse with a height

between h and h+dh.

It is useful to express the response function as the combination of

an energy deposition distribution D(T ,T) and a noise distribution

G(T,h) , by writing

T

o R(T ,h) = f D(T T) G(T,h)dT. (2) O' Hi O

In this expression D(T ,T)dT is the probability that an electron incident

with kinetic energy T will deposit an amount of energy between T and

T+dT in the detector, and G(T,h)dh is the probability that the deposition

of energy T will give rise to a pulse with a height between h and h+dh.

2.1 Energy Deposition Distribution

It is convenient to express this distribution as the sum of two

terms,J D(T

Q

,T) = De

(T

q

,T) + P

A

6(T-T

o

), (3)

The response function depends also on other parameters such as the

detector thickness and the direction of the incident electron. In

the present work, the direction of incidence is perpendicular to the

detector surface. For data with oblique incidence, see Ref. 8 C 2
where D (T ,T) is the probability density associated with the escape

of some energy from the detector, P is the probability of total absorption," and 6(T-T ) is a delta function. The calculation of D(T ,T) was

o o 7

done by a Monte Carlo method which is described in Section 3»

2.2 Noise Distribution

The noise distribution, or intrinsic resolution, can be adequately

represented by a Gaussian,

G(T,h) =-T=f- exp [- (h-h ) 2 /262 ] . (4)

P

The most probable pulse height, h , is a linear function of T. In this

Section and in the Appendix we shall assume, without loss of generality,

that the pulse-height scale is such that h is expressed in energy units

and is equal to T. The square of the full width at half maximum of this

distribution is

W^ = 8(4n2)d2

. (5)

One can resolve W

2 into the sum of three terms, W? , wf , and Wf , which

take into account, respectively, (l) fluctuations in the number of holeelectron pairs produced, (2) fluctuations of losses occurring during

charge collection, and (3) leakage current fluctuations and amplifier

noise.

In the present experiment, carried out at room temperature, Wf was found to have values ranging from 13 to 25 keV for different

detectors, and to be independent of the incident electron energy.

Therefore, the predominant contribution to W

2 was w| which is the only

term expected to be energy independent. The first term is relatively

3
small and is given by the equation

W? =8(£n2) FeT, (6)

where e is the average energy required to produce a hole- electron pair,

and F is the Fano [ll] factor which accounts for the fluctuations of the

number of pairs produced during the absorption of energy T. The experimental and theoretical information about these parameters has recently

been reviewed by Deshpande [12] and Klein [ 13] • The latter author

recommends the values e = 3*6 eV and F = 0.05 and thus obtains the

value Wi = 1 keV when T = 1.0 MeV. The magnitude of the term W^ can be

gr&atly reduced by increasing the bias voltage applied to the detector,

P P Our data indicate that W2 was small compared to W3 .

By combining Eqs. (2-4-) 5 one obtains the response function in the

form

K(T

o ,h)=R

E

(T

o

,h) + J^e-(

h- T

o>

2 /2c!\ (7)

where

T

R

E

(T

o'

h) = ~jk~a tf " dT D

E

(I

o'

T) exp ["(h-DVZa2

]. (8)

The second term on the right-hand side of Eq. (7) is usually said to

represent the absorption peak . ^(T ?h) may be called the escape

component of the response function. The numerical evaluation of the

integral in Eq. (8) is described in the Appendix for the case where a

histogram representing D (T ,T) is available from a Monte Carlo

E o calculation.

A
3. Monte Carlo Calculation of the Energy Deposition Distribution

3.1 Assumed Conditions

A narrowly collimated monoenergetic beam of electrons was assumed

to be incident perpendicularly on a silicon detector. For the purpose

of the calculation, the detector was considered to be a plane- parallel

target with lateral dimensions large enough so that the leakage of

radiation through the side walls was insignificant. (This condition

was also fulfilled in our experimental arrangement.) A one-dimensional

treatment of the transport equation was therefore adequate.

Most of our calculations were done for a single detector (case (i)

in Fig. l). In addition, results were obtained for two special anticoincidence arrangements (cases (ii) and (iii) in Fig. l) . by means of

which the escape component of the response function can be substantially

reduced.

3.2 Sampling Procedures

The following processes were taken into account: elastic scattering

of electrons by atoms, inelastic scattering by atomic electrons and the

production of knock- on electrons, the production of secondary bremsstrahlung photons, and the scattering and absorption of these photons. A

large number of collisions was sampled until the incident electron and

the secondary electrons and photons were either absorbed in the detector

or emerged from it.

The sampling procedures have been described elsewhere [14-, 15] and

will be reviewed only briefly here. For the purpose of sampling, each

electron track was divided into many short sections, chosen so that
the electrons lost, on the average, a fixed fraction (~ 2%) of their

energy per section. The actual energy loss in each section, caused by

many successive collisions with atomic electrons, was sampled from the

Landau [ 16] energy- loss distribution (with the Blunck-Leisegang [ 17]

correction to take into account binding effects). Each track section

was further divide into four subsections which, in our Monte Carlo

model, were assumed to be rectilinear. The electrons were allowed to

change their direction at the end of each subsection. This change of

direction was accomplished by sampling an angular multiple- scattering

deflection from the Goudsmit-Saunderson [18] distribution, and compounding it with the original direction at the beginning of the subsection. The Goudsmit-Saunderson multiple- scattering angular distribution was evaluated with the Mott [19] elastic scattering cross section

and a screening correction at small angles derived from the theory of

Moliere [20], Angular deflections associated with inelastic electronelectron collisions were taken into account approximately by replacing

the factor Z in the elastic scattering cross section by Z(Z+l),

where Z is the atomic number of the medium. In each subsection, the

following kinds of events were allowed to take place with the appropriate

frequency: (a) the production of knock-on electrons, sampled according

to the Miller [21] cross section; (b) the production of bremsstrahlung

photons, sampled according to a cross section based on the Bethe-Heitler

theory with empirical modifications suggested in a review article by

Koch and Motz [22], The histories of the bremsstrahlung photons were in

turn followed by conventional random sampling [23] through successive

Compton scatterings until the photons either emerged from the detector

6
or were absorbed photo- electrically or in a pair- production event. The

histories of all secondary Compton electrons, photo-electrons, or

electron- positron pairs were followed in the same manner as those of the

primary electrons.

3.3 Utilization of Monte Carlo Histories

For each source energy, the histories of 10 primary electrons,

and the histories of all descendant secondary electrons and photons

were sampled. This set of histories was analyzed to determine the

energy deposition distribution for many detectors with different thicknesses.

The sampled Monte Carlo histories were grouped into cascades. Each

cascade consists of the history of a single primary electron and the

histories of all of its descendant particles Let f denote the

fraction of the incident energy T that escapes through the transmission

boundary of the detector, as the result of being carried away by primary

or secondary particles. Similarly, let f be the corresponding fraction

of T escaping through the reflection boundary From each cascade,

information was extracted pertinent to the three detector arrangements

shown in Fig. 1;

(a) All cascades, regardless of the value of f IK. and f , were

allowed to contribute to the estimate of energy deposition for case (i).

The amount of energy left in the detector by the cascade was

T (l-f -f )o This energy was sorted into one of 50 energy bins covering

the region ° from T to 0.

o

(b) Cascades with f = were allowed to contribute to the estimate

7
of the energy deposition distribution for case (ii). The amount of

energy deposited in this case was T (l-f ).

(c) The relative frequency of cascades with f = f = provided

T R an estimate of the probability of total absorption, P .

4« Experimental Method

4.1 Electron Source

The primary source of electrons was a d.c. accelerator which was

stable to within ±0.1 percent in high voltage. The absolute energy of

the electrons was known to ± 2 percent. In order to prevent pulse pileup, a relatively small electron current was required. This requirement

-9 -7 was met by scattering the primary beam, which varied from 10 to 10

/pcm /pcm thick

film of collodion. As is shown in Fig. 2, the scattered electrons were

carefully collimated at an angle of approximately 30 degrees with

respect to the incident beam, and the electrons emerging from the collimator were moving in a direction perpendicular to the face of the

detector. The scattering chamber for this arrangement has been described

elsewhere [9]«

The scattered beam from the gold foil consisted mainly of elastically- scattered electrons, but also contained some inelasticallyscattered electrons of lower energy. The ratio of the inelastic to

the elastic scattering cross section is an increasing function of the

scattering angle. In order to minimize the relative amount of inelastical ly- scattered electrons in the beam, it was therefore desirable to
make the scattering angle as small as possible. The angle of 30 degrees

was the smallest that could conveniently be arranged with our experimental configuration. We have estimated that for primary energies

between 0.25 and 1.0 MeV the amount of inelastically- scattered electrons

in the beam at 30 degrees was of the order of one percent. The effect

of this on the response function measurement is discussed in Section $.1.

4-. 2 Detectors

Six silicon detectors which included surface- barrier , diffused, and

lithium-drifted types were employed at room temperature. Some of the

physical properties of the detectors are given in Table 1. For each

detector the magnitude of the reverse bias applied was such that practically the entire thickness was fully depleted. The dead layers at the

surfaces were negligibly thin, except for the lithium-drifted detectors

which had dead layers approximately 4-0 |~im thick on their exit surfaces.

According to the specifications supplied by the manufacturers, the

detectors were uniform in thickness to 1 or 2 percent over the entire

sensitive area. The area exposed to the electron beam was limited by

2 collimation to a central region of 4 TM , which was much smaller than

the sensitive area.

4-. 3 Amplification and Recording of Pulses

The charge collected from the detector was fed into the input of a

charge- sensitive preamplifier in which a pulse was formed. This pulse

was further amplified and stored in 2^6 channels of a pulse-height

analyzer. A rate of 4-00 to 800 pulses per second was maintained during

each experimental run. The minimum height for pulse detection was
established by a discrimination Level (cut-off), h , which was set at

the input of the analyzer. The values of h adopted for the various

detectors are given in Table 1.

The integral linearity of the entire electronic system was checked

with a pulser and found to be within ± 1 percent. The gain of the

system was kept approximately the same for all detectors and source

energies. By noting the position of the absorption peak of the response

function at various source energies, the linearity of the amplification

system was confirmed, and the proportionality constant relating pulse

height (channel number) and energy was established With the gain

setting employed, each channel had a width of 7.3 keV

.

4o4 Background Correction

The background radiation incident on the detector was found to consist of bremsstrahlung photons produced in the beam trap and other parts

of the apparatus, and a "dark current" of low-energy electrons originating from components of the accelerator tube other than the filament.

The background correction was obtained with two supplementary measurements: one with the scattering foil removed to determine the bremsstrahlung background, and another with the accelerator filament cold and the

scattering foil in place to determine the "dark current" background.

The results showed that the bremsstrahlung contribution to the

A check of the energy output of the d.c. accelerator was performed by

matching the channel number y_s energy relationship for the electrons

from the accelerator to that obtained for conversion electrons from

o3 Hg(0.196 and o 264- MeV) , 137 Cs(0.626 and O.656 MeV) , and

2o7 Bi(0o972 and I.O44. MeV) . This was done with a 3-mm thick detector,

keeping the gain of the electronic system constant.

10
background was considerably larger than that of the "dark current", and

increased with detector thickness. At primary energies of 0.5 MeV and

lower, the entire background correction amounted to less than 1 percent.

At higher energies, particularly for thick detectors (l and 3 mm), the

number of pulses in the background distribution amounted to 10 to 2$

percent of those in the principal distribution. The distribution of

background pulses was concentrated in the vicinity of the cut-off value

h , decreased strongly with increasing h, and vanished for h approximately equal to one- half the maximum pulse height.

5. Results

5.1 Comparison of Calculated and Measured Response Functions

Response functions for various combinations of electron energy and

detector thickness are plotted in Figs. 3a-d versus the relative pulse

height, h/h . The calculated results were obtained by folding the

sampled energy deposition distribution into a noise distribution according to the procedure outlined in the Appendix. The width Wf (see Eq. 5)

of the noise distribution, which was determined for each detector by

fitting a Gaussian distribution to the experimental absorption peak,

was found to be independent of the electron energy for each detector

and is given in Table 1. Each experimental response function shown is

5 based on a sample of ~ 10 counts. The electron current incident on

the silicon detectors was not monitored. Nevertheless, it was possible

The quantity h is the most probable pulse height which is associated

with the deposition of energy T in the detector. When pulse heights

are expressed r in energy units, then h = T . ° " o o

11
to normalize the experimental response functions absolutely, and

independently of the Monte Carlo calculations, by requiring that the

00 total integral _ R(T ,h)dh be equal to unity. Actually, the pulse-height

o distributions were obtained onlyJ above the cut-off value h . However,

c

7

in most cases the response function rapidly became small with decreasing

h, so that the additional contribution to the integral from pulses

below h was negligible. In a few exceptional cases, namely, for the

0.06l-mm detector at all T and for the 0.105-mm detector at

o T 2: 0.50 MeV , the normalization was performed by setting the area under

00 the experimental response function down to the cut-off, L R(T ,h)dh,

c equal to the detection efficiency T) obtained in the Monte Carlo calculations (see Eq. 9 of Section 5.4-)*

The agreement between calculated and experimental response functions

is generally good.1 For certain cases with energies of 0.25 and 0.50

MeV in Figs, 3a- d the experimental peak of the escape component occurs

at a somewhat smaller relative pulse height than the corresponding calculated peak. It appears that in these cases the most probable energy

loss of the electrons transmitted through the detector is somewhat

smaller than predicted We have been unable to explain the magnitude

and pattern of these discrepancies in terms of experimental errors or

1 In some of the experimental response functions, a small peak can be seen which is produced by inelastically scattered electrons from the

gold foil (see Section J+ 9 l) . This peak appears at a relative pulse

height h/h ~ 0.6 as predicted by the Miller formula for electrono electron scattering at 30

12
statistical errors in the Monte Carlo calculations. The Monte Carlo

model does not take into account "channeling effects", presumably caused

by diffraction ; of the type found by various authors [26-29] in electron

penetration experiments in crystalline media It is possible that this

omission is responsible for the discrepancies,,

5o2 Scaling

The extent to which a detector reflects or transmits electrons

depends primarily on the ratio of the detector thickness, z, to the

mean range,

2 r , of the incident electrons. A list of range values for

silicon is given in Table 2. When the ration z/r is small, the absorption peak is either absent or quite small, and there is a large escape

component associated with the transmission of electrons. As z/r

increases, the absorption peak becomes more prominent, and the escape

component is associated mainly with the reflection of electrons The

The random numbers needed for the Monte Carlo calculations were obtained with a multiplicative congruential generator ,, It has recently

been shown by Marsaglia [25] that such generators have a defect which

makes them unsuitable for certain Monte Carlo problems „ The statistical

analysis of different runs made with our computer program, and the

comparison of results obtained with different congruential generators

used singly or in combination, produced no evidence indicating a lack

of randomness which would tend to bias the results In particular, it

appears very unlikely that the observed discrepancies between calculation and experiment are due to a deficiency of the random numbers usedo

p The mean range is calculated from the expression

r = .; ° dT/(_ m , o «J ' \ dx/ '

dE where — is the mean energy loss per unit pathlength (stopping power)

due to collision losses and bremsstrahlung.

13
shape of the response function, for a given value of z/r . is very insensitive to the value of the incident energy T . This can be seen by

inspection of Figs. 3a- d and is brought out even more clearly by two

experimental examples in Fig. 4«

5.3 Backscattering, Transmission and Absorption Coefficients

The response function calculations also give values of various

reflection and transmission coefficients for plane- parallel silicon targets. These coefficients may be defined for primary (incident) electrons,

secondary electrons (delta rays, photoelectrons, Compton and pair electrons), and bremsstrahlung. They are in each case equal to the average

number of particles per incident electron that emerge from the entrance

(backscatter) surface or the exit (transmission) surface. Coefficients

for 1-mm and 3-mm thick detectors are shown in Table 3» It can be seen

that at high energies, because of the escape of secondary particles, the

probability of total absorption of an entire electron- photon cascade is

significantly smaller than the probability of absorbing the primary

electron.

The probability of total absorption, P , for a large number of

different detector thicknesses and incident energies is given in Table 4»

The estimated uncertainty in these results is ± one unit in the last

figure Q The statistical error is given by ,^/P (l-P )/l0 , where 10 is

the size of the Monte Carlo sample The systematic errors include the

effect of round-off, approximations in the Monte Carlo model, and uncertainties in the cross sections, and are believed to be of the same

order of magnitude of the statistical error. u
Values of the fraction of events without total energy absorption.

(l-P ), reported by various authors are compared in Table 5« The

present results (experimental and calculated) and Planskoy's results [7]

are in good agreement. They show a moderate rise with increasing T

which is to be expected because of the increased escape of secondary

particles. By contrast, the reflection coefficient shows a decrease

with increasing T The values given by Charoenkwan [3]j Rester and

Rainwater [4] 5 and Huhn and Schneider [5] are higher than the present

results at all energies, and those of Waldschmidt and Wittig [8] at some

energies. The experimental problems involved in the determination of

(l-P ) have been carefully investigated by Planskoy who found that,

depending on the detector and collimator geometry , spuriously high values

can easily be obtained in the following circumstances: (a) when the

incident electrons can lose some of their energy before reaching the

detector, as the result of inelastic scattering in the collimator, and

(b) when the electrons can escape with some energy through the side

walls of the detector. In the present experiment, precautions were

taken to make these effects negligibly small.

Table 5 also gives comparisons of the fraction of electrons backscatteredo The calculated values are in good agreement with the experimental result of Rester and Rainwater [4-] at 1 MeV, but are one- third

It was necessary, however, to lower the measured values of (l-P ) by ~ O0OO5, in order to discount the presence of inelastical ly

scattered electrons in the incident beam from the gold scattering

foil.

15
smaller than the coefficients calculated by Waldschmidt and Wittig [8]

at 0.3 and 0.6 MeV. The predictions of our Monte Carlo program have also

been compared with many other related experiments. For example, the

fraction of electrons backscattered from aluminum at 0.6 MeV has been

calculated to be 0.105 ± 0.003, which is in good agreement with the experimental values of Glazunov and Guglya [30] (0.094)5 Cohen and Koral

[31] (0.104 3o"ooP> and Miller and Hendricks [32] (0.109_q 001) •

5.4- Detection Efficiency

In practice, only those pulses are recorded that are greater than

some chosen threshold value, h , in order to reduce the pulses produced

by noise and background radiation. The detection efficiency, T)

, may

then differ significantly from unity, and can be calculated as an integral over the response function, i.e.,

CO \ = J

R(T ,h c )dh. (9)

h

c

It is not practical to make a complete tabulation of the efficiency

because it depends on a large number of parameters (e.g., incident

energy, detector thickness, threshold, width of noise distribution). In

Tables 6a- b, sample results are given which indicate the dependence on

the various parameters.

5.5 Anticoincidence Arrangements

The escape component of the response function can be reduced through

the use of an arrangement in which a detector operating in anticoincidence is placed on the transmission side of the principal detector

16
(case (ii) in Fig Q l) . The extent of the reduction is demonstrated in

Fig. 5 for incident energies between 2 and 5 MeV and a 3-mm detector

The efficiency T) of this arrangement, i.e., the fraction of events in

which an incident electron gives rise to a recorded pulse, is shown in

Table 7 for various incident energies and detector thicknesses. In case

(iii), with the silicon detector surrounded on both sides with detectors

operating in anticoincidence . the efficiency T| is, of course, equal to

the probability of complete absorption, P . It is also of interest to

consider the ratio P./Tl which represents the fraction of the area of

A

7 a

r

the response function that is under the absorption peak. As can be

seen in Table 8, the values of this ratio, for incident energies greater

than 1 MeV, are considerably greater in case (ii) than in case (i).

5.6 Tables of Energy Deposition Distributions

The Monte Carlo calculations provide estimates of the energy deposition distribution in the form of histograms. The escape component of

the distribution in histogram form is given by

T

D

E < T

o> = ( T

n+ 1 " V'M"

n+1VT

o> T> dI

> "*W.» ("»

n

where T and T ,, are the boundaries of the n+1 ' st energy interval. The

n n+1 bJ

histograms are given in Table 9 for all of the incident energies T

and detector thicknesses z listed in Table A. For T ^ 1 MeV, results

o ' are also given for the anticoincidence arrangement (ii), with an

auxiliary detector operating in anticoincidence on the transmission side.

17
Appendix

Convolution of the Energy Deposition and Noise Distributions

Given the energy deposition distribution in histogram form as

listed in Table 9> and knowing the experimental value of the standard

deviation d of the noise distribution, one can easily determine the

corresponding response function. The histogram of the escape component

of the response function may be defined, in analogy to Eq. ( 10) , as ^V = V,)"1/

1*1 VTD

,h)dh. (11)

h

n The energy deposition and response function histograms are then related

by a noise matrix G, i.e.

R^^D^C , (12) E !— E nm ' m

where

G = (h,-h T

1 [^dTi*

1"-1- 1 -^-

exp {-(h-T)V2^2

} (13) mn n+1 ny T

! h

J Jzff a m n

The evaluation of the double integral in Eq. (13) gives the result that

- <V2 f / h

n" T

' m+l \ / V^

nn " ^WV l ^

2

' * ^ V2

. n+1 nH-lA / rrt-1 m\"; . , )* F

V 6J2 JJ (U)

6/2 / ^ d/2 /-

1 -x

2 where F(x) = x erf (x) + ~y= e

2

X

-t2 and erf (x) = -j= e dt is the error function,

18
References

[l] J. M. McKenzie and G. T. Ewan, IRE Trans. Nucl. Sci. NS-8 , No. J. (1961) 50.

[2] H. I. West ; Jr., F. P. Burns, L. Wang, and F. P. Ziemba, Rev. Sci.

Instr. j£ (1962 380.

[3] P. Charoenkwan, Nucl. Instr. and Meth. ^

(1965) 93.

[4] D. H. Rester and W. J. Rainwater, Jr., NucL. Instr. and Meth. ^J. (1966) 51

o

[5] E. A. Hiihn and H. Schneider, Kerntechnik % (1967) 397.

[6] D. W. Aitken, D. W. Emerson, and H. R Zulliger, IEEE Trans.

Nucl. Sci NS-15 ? No. J. ( I968) 4-56.

[7] B. Planskoy, Nucl. Instr. and Meth. 61 ( 1968) 285.

[8] M. Waldschmidt and S. Wittig, Nucl. Instr. and Meth. 64 (1968)189.

[9] S. E. Chappell, J. C. Humphreys, J. W. Motz, M. J. Berger, and

S M. Seltzer, IEEE Trans. Nucl. Sci. NS-15 , No. J ( 1968) 359.

[10] M. J. Berger, S. M. Seltzer, S. E. Chappell, J. C. Humphreys,

and J. W. Motz, Nucl. Instr and Meth. 6g (1969) 181.

[11] U. Fano, Phys. Rev. 72 (194-7) 26.

[12] R Y. Deshpande, Nucl. Instr. and Meth. jS7 (1967) 125.

[13] C A. Klein, IEEE Trans. Nucl. Sci. NS-15 , No. j} ( 1968) 214.

[14] M. d. Berger, in Methods of Computational Physics , Vol. 1 , Ed.

B. Adler, S. Fernbach, and M. Rotenberg, Academic Press,

New York (1963) p. 135.

[15] M. J. Berger and S. M. Seltzer, NBS Reports 9836 and 9837 (1968)

unpublished.

[16] L. Landau, J. Phys. (U.S.R.R.) 8 (1944) 201.

[17] 0. Blunck and S. Leisegang, Z. Phys. 1_28 (1950) 500.

[18] S. Goudsmit and J. L. Saunderson, Phys Rev. _£7 (1940) 23.

[19] No F. Mott, Proco Roy. Soc. A124 ( 1929) 475.

19
[20] G. Moliere, Z Naturforsch 2a (194-7) 133; J3a (194-8) 78.

[21] C. Miller, Ann Physik U*. (1932) 53L

[22] H. W. Koch and J. W. Motz, Rev. Mod. Phys. 31 (1959) 920.

[23] U. Fano, L V. Spencer and M. J. Berger, in Encyclopedia of Physics , Vol. 38/2 , Ed. S. Fliigge, Springer-Verlag, Berlin (1959) p.660.

[24] M. J. Berger and S M. Seltzer, NASA Report SP-3036 ( 1966)

.

[25] G. Marsaglia, Proc. Nat. Acad. Sci. 60 (September 1968) 5.

[26] B. Domeij, Nucl. Instr. and Meth. J38 (1965) 207.

[27] E. Uggerh^j, Phys. Letters 22 ( 1966) 382

[28] E. Uggerh^j and J. U. Anderson, Can. J. Phys. ^6 ( I968) 54-3.

[29] W. L. Brown, W. M. Augustyniak, and L. G Feldman, Bull. Am.

Phys. Soc. JJ, No. ^ (1968) 4-03.

[30] P. Ya. Glazunov and V. G. Guglya, Soviet Phys. - Doklady 159

(1964) 632.

[31] A. J. Cohen and K. F. Koral, NASA Report TND-2782 (1965).

[32] Wo E. Miller and H. D. Hendricks, NASA Report TND-^363 (1968).

20
Table 1. Characteristics of the Silicon Detectors

Type

I

Detector

Thickness

(mm)

1

Sensitive

Area

(mm^)

FWHM, Wf (keV)

| Low-energy

Cutoff, h

c (keV)

Surface barrier 0.06l 50 12.8 35

ti it 0.105 50 11.2 20

it it 0.191 50 15A 25

Diffused 0.530 50 17.6 35

Lithium drifted

l 1.0 80 15.8 30

i ti ii

1 3-0

__ _

80

>

23.1 ko

1

-* At room temperature

21
Table 2. Mean range of electrons in silicon-xi ; Energy

(MeV)

r(mm)

5.00 12.7

ij-.oo 10.3

3.00 7-68

2.00 5.02

1.50 3.67

1.00 2.29

0.80 1.7k

0.60 1.20

0.50 O.936

o.Uo O.687

0.30 0A5-1

0.25 0.3^1

0.20 0.2^2

0.15 0.152

1

*Calculated by multiplying the values (in

g/cm2

) in Ref. 2k by 10/p, where

p = 2.33 g/cm3 is the density of silicon

22
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25
Table 6. Detection Efficiency, T\ , as a Function of Various Parameters

(a) Dependence on Electron Energy T , Cut-Off Energy h , and Detector Thickness z

T (MeV) h (keV)

c

^d

W

f (keV) 20

100

200

300

J4OO

z (mm) o.5 1.0 2.0

1 0.5 1.0 2.0

14.0 0.98

0.17

0.05

0.02

0.98

0.97

0.68

0.20

0.99

0.99

0.98

0.98

0.95

0.19

0.05

0.02

0.98

0.96

0.67

0.22

0.99

0.99

0.98

0.98

50

100

150

200

z (mm) 0.3 0.5 1.0 0.3 0.5 1.0

2.0 0.98

0.33

0.09

0.05

0.98

0.98

0.62

0.21

0.99

0.99

0.99

0.98

O.9I4 0.36

0.10

0.05

0.99

0.96

0.62

0.23

0.99

0.99

0.99

0.98

20

itO 60

80

z (mm) 0.1 0.2 0.3 0.1 0.2 0.3

1.0 0.9I4 0.17

0.06

0.03

0.99

O.98

0.59

0.23

0.99

0.99

0.98

0.89

0.83

0.29

0.08

0.03

0.99

0.92

0.60

0.27

0.99

0.99

0.97

0.85

10

20

30

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z (mm) 0.05 0.1 0.2 0.05 0.1 0.2

o.5 0.9I1 0.27

0.09

0.05

0.99

0.98

0.77

0.38

1.00

1.00

0.99

0.99

0.80

0.14I4 0.17

0.07

0.98

0.91

0.71

0.1*5

1.00

1.00

0.99

0.99

10

20

30

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z (mm) 0.05 0.1 0.2 0.05 0.1 0.2

0.25 0.99

0.80

0.3U

0.19

1.00

1.00

0.99

O.9I4

1.00

1.00

1.00

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0.93

0.71

0.1*2 0.23

1.00

0.99

0.97

0.91

1.00

1.00

1.00

0.99

26
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Carlo method. The results are given in histogram form. The first

two columns on the left give the limits of the energy intervals,

and the other columns give the quantity Mn

' (T ) defined by Eq. ( 10)

The numbers in the bottom row, labelled "TOTAl^ABSORPTION" , are identical with the values of P given in Table 4» The numbers following the symbols (+) or (-) indicate powers of ten.

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Case (i): Single detector; Case (ii): Detector 2 in anticoincidence

with detector 1; Case (iii): Detectors 2 and 3 in anticoincidence with

detector 1„

50
/

PRIMARY

BEAM

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SCATTERED

BEAM

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a = 2.25 mm

b

s 3.25 mm

DETECTOR (SI ) Diam. 7.98 and 10.09mm

Fig. 2. Experimental arrangement of scattering foil, collimator, and

silicon detector

51
T =0.25 MeV

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0.4 0.8 RELATIVE PULSE HEIGHT, f

1.2

Fig. 3a. Response functions of silicon detectors for various thick- nesses z. T = 0.25 MeV. The points are experimental and the solid

lines are calculated.

52
T =0.50 MeV

RELATIVE PULSE HEIGHT,

JJ

Fig. 3b. Response functions of silicon detectors for various thick- nesses z. T = 0o50 MeV. The points are experimental and the solid

lines are calculated.

53
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r i1 1 1 0.4 0.8 RELATIVE PULSE HEIGHT. £

Fig. 3c Response functions of silicon detectors for various thick- nesses z. T = 0.75 MeV. The points are experimental and the solid

lines are calculated.

54
T = 1.00 MeV

10':-

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Fig. 3d* Response functions of silicon detectors for various thick- nesses z T = 1.00 MeV. The points are experimental and the solid

lines are calculated.

55
cr

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Fig. 4« Comparison of experimental response functions for cases in

which the incident energy and detector thickness are varied such

that the scaled thickness, z/r . is constant

56
10' z = 3mm, Wf =20 keV

E—I r Z To " 2 MeV

RELATIVE PULSE HEIGHT, -£

Fig. 5o Effect of an anticoincidence arrangement on calculated response

functions. Case (i): Single detector; Case (ii): Single detector

with an auxiliary detector operating in anticoincidence.

57

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