energy loss of 24.8-mev electrons by interacting with thick materials

Energy loss of 24.8MeV electrons by interacting with thick materialsTomohisa Mikado and Takio Tomimasu Citation: Journal of Applied Physics 54, 3677 (1983); doi: 10.1063/1.332601 View online: http://dx.doi.org/10.1063/1.332601 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/54/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Monte Carlo scattering calculations of 24.8MeV electrons through C, Al, Cu, and Pb foils J. Appl. Phys. 60, 2944 (1986); 10.1063/1.337083 Energy spectra of electrons backscattered from thick layers of C, Al, Cu, and Pb bombarded with 15.0 and 24.8MeV electrons J. Appl. Phys. 56, 634 (1984); 10.1063/1.333992 Emissionangle dependence of energy spectra of 24.8MeV electrons passing through thick materials J. Appl. Phys. 47, 3948 (1976); 10.1063/1.323215 Thick Target Bremsstrahlung Produced by Electron Bombardment of Targets of Be, Sn, and Au in the EnergyRange 0.2–2.8 MeV J. Appl. Phys. 41, 2682 (1970); 10.1063/1.1659282 Electron Energy Straggling Measurements for Thick Targets of Beryllium, Aluminum, and Gold at 4.0 and 8.0MeV J. Appl. Phys. 41, 678 (1970); 10.1063/1.1658732

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Energy loss of 24.8-MeV electrons by interacting with thick materials8)

Tomohisa Mikado and Takio Tomimasu Electrotechnical Laboratory, Sakura-Mura, Niihari-Gun, Ibaraki 305, Japan

(Received 25 October 1982; accepted for publication 15 March 1983)

Energy spectra of24.8-Me V electrons after emerging from C, AI, Cu, and Pb were measured at 0°, 5°,15°,30°,60°,120°, and 150°. The material thickness ranged from -1 to -3.6 glcm2

• Total energy spectra, defined as 21T f~ I(E,() )dE sin ()d(), were composed from these observed energy spectra. Average energy loss was calculated from the total energy spectrum for each material. By plotting the experimental data including the one obtained by Ziegler, an empirical equation estimating the average energy loss per unit path length was obtained. The most probable energy losses at 0° for C and Pb agree well with theoretical predictions, while those for Al and Cu are generally larger than calculated values by -6-7%. Discrepancy between the experimental and calculated values of the full width of/(E,OO) at half-maximum due to electron straggling becomes smaller as Z increases.

PACS numbers: 29.70.Gn, 29.30.Fq

I. INTRODUCTION

For the purpose of obtaining precise knowledge on mechanisms through which the energy of electrons is transferred to a material, we should know accurately the dependence of the energy spectrum of electrons inside the material upon the depth of the material and upon the angle with respect to the incident direction. For the past several years, the present group has performed systematic measurements of energy spectra of 24.8-Me V electrons after passing through and backscattering from C, AI, Cu, and Pb. The energy spectra of electrons emerging from the materials observed at the straightforward direction have been shown previously. I The comparison and minute discussion of the most probable energy loss Ep and the full width of the energy distribution at half-maximum r due to straggling of electrons with the available theoretical predictions are also given in Ref. 1.

The energy spectra of24.8-MeV electrons after passing through 2. 31-g/ cm2 Pb measured at emission angles () of 0°, 5°, 15°, 30°, and 60° have been shown in a previous paper2 as a representative. Also pointed out there are (i) for most cases studied, the minimum point (named a valley) of the spectrum was found to lie below - 5 Me V; and (ii) the ratio of the electron intensity at the peak to that at the valley can be not only a useful parameter to express the spectrum shape, but also a good measure of angular distributions of multiplescattering electrons up to 15°.

In order to estimate how much energy of primary electrons is lost when they interact with a material, we have to know a total energy spectrum which is defined in Sec. III A of the emerging electrons. Therefore, the present group has measured the energy spectra of 24.8-MeV electrons also at 120° and 150° as well as at forward directions mentioned above. In the present paper, a whole set of the energy spectra observed at seven directions for eight target materials (two thicknesses for each substance) are shown in Sec. III B. The thickness dependence of the average energy loss calculated from the total energy spectra, which in turn have been composed from the above-mentioned energy spectra, is discussed

-) Work supported by the Japanese Atomic Energy Bureau.

in Sec. III C. In Sec. III D and Sec. III E, Ep and r obtained at 0° are compared with other authors' data.

II. EXPERIMENTAL INSTRUMENTATION AND PROCEDURE

A. General description

The electron source employed in the present work was the 4O-MeV linear electron accelerator which used to be installed at the old site of the Electrotechnical Laboratory. The experimental procedure was the same as that described previously.2 Therefore, only some important parameters are listed below.

Incident electron energy: 24.8 MeV; Energy spread of incident electrons: 0.83% (206 keY); Repetition rate and pulse duration of incident electron beam: 50 pulses/sec and 2 flsec; Electron current at the target position: -4-6 X 10-7 A; Dimensions of the electron beam at the target position: -5 mm (width) X -2 mm (height).

The shape of the target materials used in the present work was a disk with a diameter of 30 mm for C (reactorgrade graphite) and a rectangle whose dimensions were 40 mm in width and 30 mm in height for AI, Cu, and Pb. The mass thicknesses of the target materials examined in the present work are given in Table I. When electrons traverse a

TABLE I. Mass thicknesses t and path lengths, t A and tN' corrected according to the Yang theory of the target materials examined in the present work.

t tA tN Element (glcm2

) (glcm2) (glcm2

)

C 1.72 1.73 1.72 3.42 3.47 3.44

Al 1.67 1.69 1.68 3.34 3.42 3.37

Cu 1.79 1.84 1.81 3.57 3.75 3.63

Pb 1.04 1.07 1.05 2.31 2.46 2.36

3677 J. Appl. Phys. 54 (7), July 1983 0021-8979/83/073677-11$02.40 © 1983 American Institute of Physics 3677


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material, the path length of the electrons is slightly longer than the physical thickness t of the material due to multiple scattering of the electrons in the material. 3.4 The third and fourth column of Table I gives t A and t N which correspond to the case I and case II, respectively, defined by Yang.3

A transmission ionization chamber was used to monitor the electron current incident upon a target material; the bremsstrahlung rays produced in a target material penetrated the monitor ionization chamber which charged a polystyrene film capacitor whose voltage was measured by the use of a vibrating reed electrometer. The ionization-chamber response to unit incident electron fluence has been calibrated for each of the target materials studied in the present work. Therefore, the electron fluence incident upon the material could be evaluated just by measurement of the integrated electronic charges across the capacitor.

B. Detectors

A thermoluminescence dosimeter (TLD) served as an electron detector for the present work. The host crystal of the thermoluminescence (TL) phosphor is magnesium silicate, and a small amount of terbium is doped as an activator (MgzSi04 :Tb).s About 30 mg of granulated MgzSi04 :Tb phosphor is encapsulated in a thin-walled container made of Pyrex glass whose dimensions are 2 mm in outer diameter, 10 mm in effective length, and -0.1 mm in thickness. The response of the MgzSi04 :Tb TLD to high-energy electrons has been investigated by the present group.6 The principal conclusions drawn from Ref. 6 are

(i) The TL output of the MgzSi04 :Tb TLD per unit electron fluence is energy independent for D.5-27.5-MeV electrons within an experimental error of about 3%;

(ii) The response of the MgzSi04 :Tb TLD to high-energy electrons is proportional to incident electron fluence up to - I X 109 electrons/ cm Z and is expressed as

1= 3.71 X 10-8 t,b 10, (1)

where I is the TL output measured by the use of a TLD readerS in units of R 6OCO equivalence and t,b is the incident electron fluence in units of electrons/cm2

;

(iii) The supralinearity takes place when ¢ exceeds - I X 109 electrons/cm2 and is expressed as a straight line on a fully logarithmic paper and the following simple power law holds for 1 X 109 S t,b S 5 X 1011:

1= 5.93 X 1O- 1Ot,b 120. (2)

In measuring energy spectra of electrons emerging from a material, the present authors adjusted the exposure time of a TLD to energy-analyzed electrons aiming for the TLD that would not exhibit the supralinear response. When it was anticipated that the overexposure of the TLD could hardly be avoided by any means, an energy spectrum was measured with a Faraday cup (FC) overlapping several data points with those obtained by the use of MgzSi04 :Tb TLD's and a whole spectrum was obtained by combining these two partial spectra. The Fe consisted of a reactor-grade graphite core whose dimensions were 80 mm (width) X 80 mm (height) X 100 mm (depth). The front face of the graphite core was a flat surface but it was not necessary to make any cor-

3678 J. Appl. Phys .. Vol. 54. No.7. July 1983

rections due to electron backscattering because the total backscattering coefficient of graphite is very small in the energy region above -15 MeV, where the FC was used to measure the energy spectrum, as shown graphically by Tabata. 7 The method of measuring the energy-analyzed electrons collected by the FC was similar to that employed in the measurement of electronic charges induced in the monitor ionization chamber. When an energy spectrum of emerging electrons was measured with the FC, a Cu-C composite collimator was placed between the exit window of the magnetic spectrometer and the FC. The thickness of Cu was 20 mm and that of C was 50 mm. The collimator had a slot of which dimensions were 20 mm in width and 16 mm in height. The height of the slot was determined to correspond the spatial distribution of the incident electrons, whose energy spread was 0.83% at 24.8 MeV, at the exit window of the spectrometer. I

For the purpose of measurement of background signals due to stray radiations, the following method was tried: When the energy spectra were measured using MgzSi04 :Tb TLD's, additional TLD's were placed where they would not be exposed to the energy-analyzed electrons directly; when the FC was used, a beam stopper, identical to the aforementioned collimator except for a slot, was inserted between the exit window of the spectrometer and the Fe.

C. Data processing

The TL output of the Mg2Si04 :Tb TLD's exposed to the energy-analyzed electrons were measured by the use of the TLD reader which is commercially available from the TLD manufacturer.s The maximum temperature of the heater pan on which a TLD was placed was about 300 ·C. 6

•8

After the completion of the readout of the whole set of the Mg2Si04 :Tb TLD's for one run, the contribution of the background radiation measured with the extra TLD's was subtracted and the net TL emitted per unit incident electron fiuence was calculated.

In the case of measurement of the energy spectra with the FC, the background radiation measured with the beam stopper per unit response of the monitor ionization chamber was subtracted from the total output obtained with the collimator per unit monitor response.

Finally, for either the measurement with the TLD's or that with the FC, the net number of outgoing electrons per unit number of incident electrons was corrected to constantenergy intervals because the relative momentum resolution of the magnetic spectrometer is constant.

III. RESULTS AND DISCUSSION

A. Definition of parameters

We will now define some parameters to be used below.

F(E )dE = 21T IT f(E,e )dE sin ede (3)

is the total energy spectrum, wheref(E,e) is the number of emerging electrons per unit incident electron fiuence having energy between E and E + dE observed at e.

T. Mikado and T. Tomimasu 3678


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tAV = Eo _ (EO F(E)EdE/1(EO F(E)dE JE~2 JE~2

(4)

is the average energy loss, where Eo is the incident electron energy ( 24.8 MeV for the present work). In the study of electron interactions with matter, the more energetic electron after scattering is generally treated as the primary electron because of the indistinguishability between the incident and scattered electrons.4 Therefore, the minimum energy of the primary electrons emerging from the material is E0I2

instead ofO as Eq. (4) suggests.

tp = Eo - Ep (5)

is the most probable energy loss at 0, and Ep is the electron energy where the energy distribution of the outgoing electrons measured at 0 reaches its maximum.

r = (T~bs - r~)1/2 (6)

is the full width at half-maximum of the energy distribution at 0,1.9 where robs is the energy spread of the outgoing electrons observed at 0 and ro is that of the incident electron beam (=206 keV for the present work).

B. Observed energy spectra

The energy spectra of 24. 8-MeV electrons after emerging from C, AI, Cu, and Pb measured at 0 = 0·,5·, 15·,30·, 60·, 120·, and 150· are shown in Figs. 1-4. In these figures, f(E,O ) is illustrated as smooth curve rather than individual data points for the sake of avoiding confusion, and typical statistical errors are shown by vertical bars unless they are smaller than the size of data points. In each figure, (a) and (b) show the variation of energy spectra with target thickness. Because the target thicknesses ofC (Z = 6), Al (Z = 13), and Cu (Z = 29) are nearly comparable to each other, Figs. 1-3 also show the atomic-number dependence of the energy spectra of the outgoing electrons.

C. Average energy loss

It is obviously impossible to measure the total energy spectrum at one time because of the finite acceptance solid angle of any type of spectrometers. Therefore, when we wish to know the total energy spectrum of electrons emerging from a material, we have to integratef(E,O ) over 0 and composeF (E) as Eq. (3) suggests. In the present work, the integration was made numerically according to the Simpson 1/3 rule with f(E,O) being given by means of interpolation or extrapolation to those obtained at 0 = 0·, 5·, IS·, 30·, 60·, 120·, and 150·. The composed total energy spectra for four substances listed in Table I are illustrated in Figs. 5-8.

There are a few authors10-

12 who report the experimental values of t AV as far as the present group is aware. It seems that those authors define the average energy loss, tav' as

LEo ~IEO tav = Eo - f(E,O·)EdE f(E,O·)dE, E~2 E~2

(7)

rather than as Eq. (4). It might be correct to approximate tAV

with tav for a very thin material when a great majority of electrons penetrating it can be detected by the use of a spectrometer situated at 0 = D·. However, when the material

3679 J. Appl. Phys., Vol. 54, No.7, July 1983

Ii) I-

Z

(a)

10-4 1.72-gkm2 C Eo. 24.8 .... V

1""0 c 206 k~V

o 5 10

120"

15 ELECTRON ENERGY (M~V)

30"

10-4 ,----,------r---..----...,.----,--, (b)

3.42 -g/cm2 C

Eo" 24.8 M~V 1""0. 206k~V

::::> 10-7

ai

30"

a: ~

60"

120"

• 150"

1 0-12.~ __ __L __ ........l ___ _'__ __ ..I_ __ -..IL...J

o 5 10 15 20 25 ELECTRON ENERGY (M~V)

FIG. 1. Angular dependence of energy spectra of24.8-MeV electrons after emerging from C of (a) t = 1.72 glcm2 and (b) 3.42 glcrn2

•

thickness is increased, a proportion of electrons scattered to directions other than D· becomes greater and the above approximation becomes more inappropriate. This can be concluded by observation of Figs. 5-8 in whichf(E,O·) are shown for comparison purposes.



136.165.238.131 On: Sun, 21 Dec 2014 12:27:10

Iil I-

Z

10-J ..------____,r----__.__---.-----.-----,.----, (0)

1.67-glcm2 AI Eo = 24.8 MeV (70 = 206 keV

:::> 10-6

<Ii It: S

1 0-11L-___ L-__ ---L ___ .L.--__ ----L __ ...I.-~

o 5 10 15 20 25 ELECTRON ENERGY (MeV)

10~r----r----.-----~--~---~ (b)

3.34-g/cm2 AI Eo = 24.8 MeV (70 • 206 keV

1O'.'2L ___ l...-__ ..L ___ L-__ .....L __ ----:~

o 5 10 15 20 25 ELECTRON ENERGY (MeV)

FIG. 2. Angular dependence of energy spectra of 24. 8-MeV electrons after emerging from Al of (a) t = 1.67 glcm2 and (b) 3.34 glcm2.

In Fig. 9, the experimental values of X,E AV It A are plotted against X, EoI(ZtA ). where X, is the radiation length of the substance considered and is defined by 13


10~~--____,r----__.__----r_--_r----~ (0)

10-11

1. 79-glcm2 Cu Eo. 24.8 MeV (70. 208 keV

ISO·

(f

5°

60"

120·

10-12.~------:~--_':_----'-------L---.JJ1...J o 5 10 15 20 25

ELECTRON ENERGY (MeV)

10-',......----r-----.----.----.----,.-,

Oil ... z :::> . 10-7

m It: S

~ 10-8

... !il -'

"" ::>-t 10-9

iii i ... !ilH' iil

(b)

3.57-glcm2 Cu Eo. 24.8 MeV ~.206keV

10-12L ___ ---'L-__ --L.. ___ L-__ .....L ___ J.LJ

o 5 10 15 20 25 IELECTAOfI ENERGY (Mey)

FIG. 3. Angular dependence of energy spectra of 24.8-MeV electrons afrer emerging from eu of (a) t = 1.79 g/cm2 and (b) 3.57 g/cm2

.

Xr = (1 + I.4X 1O-sZ2)A

I[ 4aNA Z(Z + 1)"; In(lS3/Z 1/3)]. (S)



136.165.238.131 On: Sun, 21 Dec 2014 12:27:10

In .... z ::;)

(a)

1 O-n

o

1. 04 -glcm2 Pb Eo. 24.8 .... V

1"0. 206 k.V

-"'-... --___ 120"

5 10 15 20 ELECTRON ENERGY ( .... V)

25

10-4r-------~----_,------~------_r------~ (b)

2.31-g/cm2 Pb 10-5 Eo. 24.8 .... V

1"0. 2 0 6 k.V

iii 1 0- 7 ~:--..= ....... -a: ~

10~-------L------L-----~------~----~~ 10 15 o 5

ELECTRON ENERGY (1.1. V)

FIG. 4. Angular dependence of energy spectra of24.8-MeV electrons after emerging from Pb of (a) t = 1.04 g!cm2 and (b) 2.31 g/cm2.

here a is the fine structure constant, 'e is the classical radius of an electron, NA is the Avogadro number, and A andZ are, respectively, the atomic weight and atomic number. The experimental data obtained by Ziegler is also shown \4 because


o

toIat.: C

Eo: 24.8 "'.V roo 206 kl!V

/ /

\ // \ ........... /"

/ /

/

/ /

/ /

/ /

/

I I

/ /

/

5 10 15 20 ELECTRON ENERGY ( .... V)

1.72

25

FIG. 5. Composed total energy spectra for C bombarded with 24.8-MeV electrons. Figures attached to curves are tA in g/cm2 units. Dashed curves are/(E,OO) which are normalized so that peak heights agree with those of corresponding F (E ).

o

IoIat.: AI Eo: 24.8 toI.v rei: 206 k.V

5 10 15 20 ELECTRON ENERGY (toI.V)

1.67

I I \ I I \ \ I \ I I \ I I

25

FIG. 6. Composed total energy spectra for AI bombarded with 24.8-MeV electrons. Figures attached to curves are tA in g/cm2 units. Dashed curves are/(E,OO) which are normalized so that peak heights agree with those of corresponding F(E).



136.165.238.131 On: Sun, 21 Dec 2014 12:27:10

o

l4at.: Cu

Eo: 24.8 I4.V TO:206hV

./ ./

/"

5

./

?' ./

./

10 15 ElECTRON ENERGY (I4.V)

1.79

20 25

FIG. 7. Composed total energy spectra for Cu bombarded with 24.8-MeV electrons. Figures attached to curves are fA in g/cm2 units. Dashed curves are /(E,aO) which are normalized so that peak heights agree with those of corresponding F (E).

z o a: .... u W ...J W

=: 10-7

l:

Vi z li1 .... u w iil

a

Mat.: Pb Eo: 24.8 M@V

IQ:206keV

2.31

5 10 15 20 ELECTRON ENERGY (M@V)

1.04

I I 1 I I I I I I I I I I I I

25

FIG. 8. Composed total energy spectra for Pb bombarded with 24.S-MeV electrons. Figures attached to curves are fA in g/cm2 units. Dashed curves are/(E,aO) which are normalized so that peak heights agree with those of corresponding F (E).


• tA (glcm2) Mat.

• 1.0 0 1.73 C

• 3.47 Il. 1. 69

3.42 Al ...

0 1.84

• 3.75 Cu

V 1. 07 Pb • 2.46

10°L-LL~~~-L-L~~~ __ ~LL~~~~-L~~~ 100 101

102 103

X,Eo/(ZIA)

FIG. 9. Experimental values of X, cAV/fA against X, EoI(ZtA ) for Eo = 24.8 and 32 MeV. Solid line is the best-fit curve to the data.

Eo (32 MeV) is much closer to that of the present work among Refs. 10-12. The solid line is the best-fit curve to the nine data points and is expressed as

(9)

The above empirical equation can be used to estimate € AV ItA of materials bombarded with electrons of Eo;:::; 20-30 MeV.

D. Most probable energy loss

The well-known Landau formula to predict €p of electrons penetrating to () = O· is expressed as 15

€LS = (As t N I,82)[ Bs + 0.891 + 21n(plme c) + In(As tN I,82) _,82 - 0] MeV, (10)

where tN is the material thickness corrected after Yang,3 As and B s are parameters specific for each substance considered, 16 0 is the density-effect correction,16,17 and other symbols have their usual meaning. The original formula derived by Landau does not include 0. Further inadequacies of the original Landau formula are (i) the resonance-energy transfers are treated with insufficient accuracy and (ii) the radiative energy loss is neglected. These two inadequacies are resolved, respectively, by Blunck and Leisegang'8 and by Blunck and Westphal. 19 The calculated values of €p appearing in the following discussion are what include all the corrections mentioned above.

The experimental values of €p obtained at () = O· in the present work are compared with calculated €p and experimentally obtained €p by previous authors I ,10--12,20--30 in Figs. 10-13 for C, AI, Cu, and Pb.31

As can be seen in Fig. 10, the experimental values of €p

for C appear to support the theoretical predictions very well for Eo> several MeV, The discrepancies between the experimental and calculated values are smaller than - 6% at any thicknesses shown in Fig. 10.

In the case of AI, the experimental values are generally larger than the calculated values by -6% for tN ~ 0.6 glcm2

except those obtained by Ziegler '2 and the group at the Naval Postgraduate School, Monterey. 29.32 Although Hall et al. 22 give the experimental values very close to the calculated



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c

ones, the results observed experimentally in the present work, by Breuer,2S and by Tomimasu et al.' are larger than the calculated Ep by about 10%-15% for materials thinner than -0.6 glcm2. On the other hand, the discrepancy between the experimental and theoretical values for the thickest material is only -5%.

Observed values of Ep for eu are systematically larger than calculated values by up to - 7% for the thicknesses ranging from -0.5 to -5 glcm2 for Eo~5 MeV except

Al

'> 100

... ~

(1

\iI

10-1

Ie!>

3683 J. Appl. Phys .. Vol. 54. No.7. July 1983

FIG. 10. Experimental values of €p for C as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); ~ Paul and Reich, Ref. 10 (4.7); \l Hudson, Ref. 21 (ISO); .... Ziegler, Ref. 12 (32); T Theissen and Gudden, Ref. 28 (53); 0 Tomimasu et al., Ref. I (24.8). Solid line shows the calculated €p for 24.S-MeV electrons in C. Difference between the calculated €p for Eo = 4.7 MeV and that for Eo = 24.8 MeV is negligibly small.

those given by the group at the University of Illinois20.22 as may be seen in Fig. 12. Tomimasu et al.' give the experimental results larger than the calculated values at ~0.44 and -7.3 glcm2 by -12%, while the Monterey group29 reports the experimental results larger than the predicted values by -13% even at -0.71 and -5.8 glcm2.

A very few groups have performed the measurements of Ep for Pb as far as the present authors are aware. The experimental values obtained in the present work and by Tomi-

FIG. 11. Experimental values of €p for Al as a function of (N' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); 6. Goldwasser et al., Ref. 20 (15.7); \l Kageyama and Nishimura, Ref. 11 (1.414); .... Hudson, Ref. 21 (150); T Ziegler, Ref. 12 (32);. Hall et al., Ref. 22 (7.7 and 11.7); 0 Grishaev et al., Ref. 23 (18); • Knop et al., Ref. 24 (1.06); <l Breuer, Ref. 25 (53.61);'" Rester and Rainwater, Ref. 27 (1.0); I> Monterey group, Ref. 29 (53.57,74.63, and 96.93); ~ Davies and Jennings, Ref. 30 (28.14); 0 Tornimasu et al., Ref. 1 (24.8). Solid and broken lines show, respectively, the calculated €p for 24.8- and 1.0-MeV electrons inA\,



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Cu

10° > II

~

0-W

10-1

10-1

masu et al. 1 seem to support very well the predicted values at least for the thicknesses ranging from -0.3 to -4 glcm2, whilst those obtained by the Monterey group29 are systematically larger than the calculated values as shown in Fig. 13. The result obtained with 4.7-MeY electrons lO agrees very well with the calculated values.

In Figs. 11-13, the experimental values for Eo~ 1 Mey ll •24,26.27 are also shown for comparison purposes; for

Pb

o

:; 10° II

~

0-W

10-1


FIG. 12. Experimental values of Ep for Cu as a function of t,. The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); L, Goldwasser el al., Ref. 20 (15.7); 'V Kageyama and Nishimura, Ref. 11(1.414);.& Halletal., Ref. 22 (7.7,11.7,16.1, and 19.9); .. Hall et al., Ref. 22 (3.9); D Knop et al., Ref. 24 (1.06); • van Camp and Vanhuyse, Ref. 26 (2.65);. Monterey group, Ref. 29 (52.8, 74.8, and 94.3); o Tomimasu el al., Ref. I (24.8). Solid, dash-dot, and broken lines show, respectively, the calculated Ep for 24.8-, 2.65-, and 1.06-MeV electrons in Cu.

almost all the cases the experimental values are much smaller than the calculated ones. This may be due to the fact that the Yang theory overestimates4

,26 the increment in path length especially for low-energy electrons.

E. Energy distribution due to straggling

Landau has derived a simple formula which relates the material thickness and the full width of the energy distribu-

FIG. 13. Experimental values of Ep for Pb as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); L, Paul and Reich, Ref. 10 (4.7); 'V Paul and Reich, Ref. 10 (2.8); .& Kageyama and Nishimura, Ref. 11 (1.414); .. Monterey group, Ref. 29 (53.9, 74.7, and 91.7); 0 Tomimasu et al., Ref. I (24.8). Solid and broken lines show, respectively, the calculated Ep for 24.8- and 1.414-MeV electrons in Pb. Difference between the calculated Ep for Eo = 2.8 or 4.7 MeV and that for Eo = 1.414 MeV is negligibly small especially for IN

~ 0.08 g/cm2•



136.165.238.131 On: Sun, 21 Dec 2014 12:27:10

c

L

tN (g/cm2 )

tion at half-maximum r L of electrons after passing through the material of R cm asl5

r L = 3.98aR MeV,

where

a = 21rNA epf(mev2A ) MeV fcm,

(11)

(12)

N A is the Avogadro number, and all the other symbols have their usual meaning. The Landau distribution has been modified 18 taking into account the resonance transfer of en-

L

lrf


FIG. 14. Experimental values of r for C as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8);!::,. Hudson, Ref. 21 (150); 'V Ziegler, Ref. 12 (32); ... Theissen and Gudden, Ref. 28 (53); 0 Tomimasu et al., Ref. I (24.8). Solid, dash-dot, and broken lines show, respectively, rBW, r BL , and r L for 24.8-MeV elec· trons in C.

ergy during distant collisions. Besides this, Blunck and Westphal have included the additional broadening due to the emission of bremsstrahlung rays. 19 In the following discussion, these two modified widths will be denoted as r BL

and r BW ' respectively.

The experimental values of rmeasured at e = O· for C, AI, Cu, and Pb obtained in the present work and by previous authors l •IG-12.20.21.24-26.28-30.33 are compared with r

L, r

BU

and r BW in Figs. 14-17.

FIG. IS. Experimental values of r for AI as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); l'::,. Goldwasseretal., Ref. 20 (15.7); 'V Kageyama etal., Ref. 33 (1.414); ... Hudson, Ref. 21 (150); ... Ziegler, Ref. 12 (32); 0 Knop et al., Ref. 24 (1.06);. Breuer, Ref. 25 (53.61); 0 Monterey group, Ref. 29 (53.57, 74.63, and 96.93):. Davies and Jennings, Ref. 30 (28.14); 0 Tomimasu et al., Ref. I (24.8). Solid, dash·dot, and broken lines show, respectively, r BW '

r BU and r L for 24.8- and 1.06-MeV electrons in AI.



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FIG. 16. Experimental values of r for Cu as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); 6 Goldwasseretal., Ref. 20(15.7); 'V Kageyamaetal., Ref. 33 (1.414)' lit. Knop et al., Ref. 24 (1.06); .... van Camp and Vanhuyse, Ref. 26 (3.66); 0 Monterey group, Ref. 29 (52.8, 74.8, and 94.3); o Tomimasu et al., Ref. I (24.8). Solid, dash-dot, and broken lines show, respectively, row, r OL ' and r L for 24.8- and 1.06-MeV electrons in Cu.

Despite that the experimental values of Ep for C appear to agree with the calculated values, the experimental values of r do not seem to support the predictions by theory as shown in Fig. 14. The experimentally obtained r for Care generally larger than r BW by - 15 to - 85 % except those reported by Ziegler l2 and by Theissen and Gudden28 for t N :S 1 g/cm2 that agree fairly well with r BW'

A large number of groups have measured r for AI. The result obtained in the present work with 1.67-g/cm2 AI, that by Goldwasser et al. 20 with 0.863-g/cm2 AI, and that by Davies and Jennings30 with 0.674-g/cm2 Al show a good agreement with theory. But almost all other results are systematically larger than rBW by up to -45% for tN ranging from -0.2 to -5 g/cm2.

Goldwasser et al.,20 similarly to the case of AI, have obtained a slightly smaller value than r BW for Cu as shown in Fig. 16. Excluding this, all the results for Eo ~ 5 Me V are larger than rBW by -15 to -25% for tN:S 2.5 g/cm2 although the Monterey group29 reports much larger values even in the range of t N mentioned above.

There are only a few groups that have investigated r experimentally for Pb as is the case of Ep for Pb. Among the four elements studied in the present work, Pb seems to support best the theoretical predictions at least 0.4 :S t N :S 1.5 g/ cm2 as shown in Fig. 17. The Monterey group gives r much larger than r BW but they claim, same as the case for Al or Cu, that the inclusion of ro into calculation results in a good


Pb

L

FIG. 17. Experimental values of r for Pb as a function of tN' The data sources are listed below with Eo in MeV units in parentheses .• present work (24.8); 6 Kageyama etal., Ref. 33 (1.414); 'V Monterey group, Ref. 29 (53.9,74.7, and 91.7); o Tomimasu etal., Ref. I (24.8). Solid, dash-dot, and broken lines show, respectively, row, rou and r L for 24.8-MeV electrons in Pb. Also shown are r OL and r L for 1.414-MeV electrons in Pb.

agreement between experiment and theory.29.34 In order to calculate rBW for 1.414-MeV electrons in Pb, one has to extraordinarily extrapolate the curves given by Blunck and WestphaI. I9 Therefore, the rBW curve for 1.414 MeV is omitted from Fig. 17.

IV. SUMMARY

By integrating energy spectra observed at e,f(E,e )dE, over e, the total energy spectra were composed for 24.8-MeV electrons emerging from thick materials. The total energy spectra are compared tof(E,OO)dE in Figs. 5-8 for the sake of demonstrating that it is inappropriate to evaluate the average energy loss merely fromf(E,OO)dE.

An empirical universal relation has been obtained as Eq. (9), which may apply to estimate EAV/tA for electrons of - 20-30-MeV region.

The experimental values of Ep for C agree well with the theoretical predictions. 19 For Al and Cu, the experimentally obtained Ep are generally larger than the calculated values by -6-7%. In the case of Pb, the experimental values at 0.3:S tN :S 4 glcm2 seem to support the theory.

In the case of r, the discrepancy between the experimental and calculated values becomes smaller as Z increases; -20% for C, -17% for AI, -15% for Cu, and -10% for Pb at tN::::: 1 g/cm2.

It is desirable to improve the Z dependence of theory predicting, especially, r.



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ACKNOWLEDGMENTS

The authors wish to express their thanks to Dr. E. Teranishi for his continuous encouragements. They are greatly indebted to Dr. T. Yamazaki, Mr. M. Chiwaki, Miss Y. Yamamoto, Mr. K. Hayashi, and Mr. T. Tsujimura for their valuable help during the present work.

IT. Tomimasu, T. Mikado, Y. Tsuchiya, S. Sugiyama, and M. Chiwaki, Bull. Electrotech. Lab. 35, 92 (1971).

2T. Mikado, T. Tomimasu, and T. Yamazaki, J. Appl. Phys. 47, 3948 (1976).

3c. N. Yang, Phys. Rev. 84, 599 (1951). 4R. D. Birkhoff, Handbuch der Physik, edited by S. Fliigge (Springer, Berlin, 1958), Vol. 34.

'Kyokko TLD 1200 System. Manufactured by Kasei Optonix, Ltd., Odawara, Kanagawa, Japan.

"T. Mikado, T. Tomimasu, T. Yamazaki, and M. Chiwaki, Bull. Electrotech. Lab. 42, 29 (1978); Nucl. Instrum. Methods 157, \09 (1978).

7T. Tabata, Phys. Rev. 162, 336 (1967). "T. Yamazaki, T. Tomimasu, T. Mikado, and M. Chiwaki, Bull. Electrotech. Lab. 42, 44 (1978); J. Appl. Phys. 49, 4929 (1978).

9T. Tomimasu, T. Mikado, and T. Yamazaki, J. Phys. Soc. Jpn. 37, 286 (1974); Phys. Rev. B 10, 2669 (1974).

lOW. Paul and H. Reich, Z. Phys. 127,429 (1950). "s. Kageyama and K. Nishimura, J. Phys. Soc. Jpn. 7,292 (1952). 12B. Ziegler, Z. Phys. lSI, 556 (1958). 13See, for example, T. M. Knase1, N ucl. Instrum. Methods 83, 217 (1970). 14Ziegler mentions nothing about material thickness except for Be. There-

fore, the present authors treat that he has performed the measurements using 1-g/cm2 materials. In addition to this, the corrected thicknesses, tA


and tN' are also treated as 1 g/cm2. "L. Landau, J. Phys. USSR 8,201 (1944). 16See, for example, R. M. Sternheimer, Phys. Rev. 91, 256 (1953). 17R. M. Sternheimer and R. F. Peierls, Phys. Rev. B 3,3681 (1971). 1"0. Blunck and S. Leisegang, Z. Phys. 128, 500 (1950). 190. Blunck and K. Westphal, Z. Phys. 130,641 (1951). 2°E. L. Goldwasser, F. E. Mills, and A. O. Hanson, Phys. Rev. 88, 1137

(1952). 21A. M. Hudson, Phys. Rev. 105,1 (1957). 22H. E. Hall, A. O. Hanson, and D. Jamnik, Phys. Rev. 115,633 (1959). 231. A. Grishaev, A. N. Fisun, A. S. Litvinenko, V. M. Grizhko, B. I. Shra-

menko, and I. N. Onishchenko, Sov. Phys. JETP 37(10), \031 (1960). 24G. Knop, A. Minten, and B. Nellen, Z. Phys. 165, 533 (1961). 2'H. Breuer, Z. Phys. 180, 209 (1964). 26K. J. van Camp and V. J. Vanhuyse, Phys. Lett. 19, 504 (1965). 27D. H. Rester and W. J. Rainwater, Jr., J. Appl. Phys. 37,1793 (1966). 2"H. Theissen and F. Gudden, Z. Phys. 191, 395 (1966). 29F. A. Bumiller, F. R. Buskirk, J. N. Dyer, and R. D. Miller, Z. Phys. 223,

415 (1969); F. A. Bumiller, F. R. Buskirk, and J. N. Dyer, ibid. 234,185 (1970); F. R. Buskirk, J. N. Dyer, X. K. Maruyama, and K. E. Woehler, ibid. 271, 69 (1974).

30G. R. Davies and R. E. Jennings, J. Phys. B 3, 804 (1970). 31 A slight increase in Ep caused by ro has been eliminated by the similar

method described in Ref. 9 for the present values. 32These exceptions should not be stressed too much because Ziegler reports

Ep/t rather than Ep and the calculated values of Ep are not proportional to tN' while the primary electron beam used by the Monterey group has already been scattered by a ~0.02-g/cm2 AI foil.

33S. Kageyama, K. Nishimura, and Y. Onai, J. Phys. Soc. Jpn. 8, 682 (1953). 34J. L. Matthews, D. J. S. Findlay, and R. O. Owens [Nucl. Instrum. Meth

ods 180, 573 (1981).] recently comment that they do not obtain the same results that the Monterey group has reported even though they have followed the calculation procedure stated in Ref. 29.



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energy loss of 24.8-mev electrons by interacting with thick materials

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