Range of electrons and positrons

NUCLEAR

INSTRUMENTS

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I34 (I976) I 2 5 - 1 2 7 ;

© NORTH-HOLLAND

PUBLISHING

CO.

RANGE OF E L E C T R O N S A N D P O S I T R O N S T. M U K O Y A M A

Institute for Chemical Research and Radioisotope Research Center, Kyoto University, Kyoto, Japan Received 13 J a n u a r y 1976 T h e range o f electrons a n d positrons in detector materials has been calculated by the Wilson theory. T h e results obtained are c o m p a r e d with the values estimated f r o m the c o n t i n u o u s - s l o w i n g - d o w n model. A correction factor which depends only on the energy is presented. It is concluded that the corrected Wilson theory yields good a p p r o x i m a t e values for ranges o f electrons a n d positrons in the energy region above 70 keV.

In the Monte Carlo calculations of the detector response for g a m m a rays, it is often very important to trace the secondary electrons and positrons, produced from the interaction of the photon with the detector materials. These particles are followed until they escape froln the detector or are absorbed in it, and the energy released in the detector is estimated. The most rigorous method for this purpose is the electron Monte Carlo method1). However, this method is tedious and requires much computer time. Various approximate methods have been used to estimate the slowing down of electrons and positrons in the detector2'3). In the calculations of cascade showers, Wilson derived an analytic expression for the range of highenergy electrons4). Because of its simplicity, this formula has often been used to determine the mean path length and energy loss of electrons and positrons in the Monte Carlo programs for the gamma-ray detector 5 7). The photo-peak efficiencies calculated from these programs are in good agreement with the experimental results. The purpose of the present paper is to test the validity of the Wilson model by comparing the electron range with that calculated from the continuous-slowing-down approximation, and to develop a correction formula which can yield the ranges within a reasonable error limit over a wide energy range. Starting from the formula for radiation loss and including ionization loss, Wilson obtained a simple expression for the mean range of high-energy electrons. 3-'his range is shown to be given in radiation lengths by R = ln2 [1 + E / ( E c ln2)],

(1)

where E is the incident energy of the electron and E¢ represents the critical energy in the same units as E. The critical energy is defined as that energy at which the ionization loss due to collision with atomic electrons is equal to the radiative energy loss. This

energy is approximated by the expression E¢ = 800/(Z + 1.2) MeV.

(2)

Eq. (2) has been verified by the calculations of Berger and Seltzer8). The radiation length is the mean path length over which the electron has its energy reduced by a factor of e. Recently, Knasel 9) reported a formula to predict this length for elements Z~> 6 with accuracy of 1%: X0 = (l+l.4xl0-sZ x

2) x

A/E4~r 2 N Z ( Z + I) l n G ] ,

(3)

where G = 183/Z ~, r 0 is the classical electron radius, N is the Avogadro number, c¢is the fine structure constant, and Z and A are the atomic number and weight of the material, respectively. We have calculated the values of the mean range of electrons for three detector materials (silicon, germanium and sodium-iodide) in the energy region from 50 keV to 100 MeV. In the case of sodium iodide, the radiation length and the critical energy were estimated by the use of the average values of Z and A of the constituent elements, weighted in proportion to the fraction by weight of each element. The values obtained are compared with other theoretical values. The most precise values so far reported are those from the continuous-slowing-down approximationS'l°'11). Nelms l°) calculated the ranges for electrons and positrons. However, he included only the ionization loss by the collision of the atomic electrons. Taking into account the energy loss by bremsstrahlung, Berger and Seltzer s) evaluated the ranges of electrons for various materials and for energies between 10keV and 1000MeV. Recently, Pages et al. 11) published the most complete tables on the ranges of electrons in matter. Their values are in good agreement with those of Berger and Seltzer.

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x. MUKOYAMA

C o m p a r i s o n o f the present results with the results of Pages et al. 11) is shown in figs. 1-3. It is clear that the W i l s o n theory leads to larger values than the contin u o u s - s l o w i n g - d o w n m o d e l in the low-energy region b u t to smaller ones in the high-energy region. In order to m a k e the present values agree with those o f Pages et al., a correction factor is introduced and the range f o r m u l a is expressed as

R'

(4)

RF(E).

=

Here R is the range calculated f r o m eq. (1) a n d the

iO a

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i0-1

10 -2

10 2

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I ENERGY

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(MeV)

°

_

S i / , , ¢ ~ /

Fig. 3. Range of electrons in sodium-iodide crystal plotted against the energy. The open circles represent the continuousslowing-down model (ref. 11). The dashed curve shows the Wilson theory (ref. 4) and the solid curve indicates the corrected Wilson theory.

_

E D to

tm

OJ (.9 iO.i Z tic 10-2 / / t / /

correction factor is given by

F(E) = 1 . 5 - 1 . 3 e x p ( - 2 E ) , io-3

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i

I0 ENERGY (MeV)

0.1

I00

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Fig. 1. Range of electrons in silicon plotted against the energy. The open circles represent the continuous-slowing-down model (ref. 11). The dashed curve shows the Wilson theory (ref. 4) and the solid curve indicates the corrected Wilson theory. I°2

l

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0.1

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ENERGY (MeV)

1 I00

Fig. 2. Range of electrons in germanium plotted against the energy. The open circles represent the continuous-slowing-down model (ref. 11). The dashed curve shows the Wilson theory (ref. 4) and the solid curve indicates the corrected Wilson theory.

(5)

where E is the initial energy o f the electrons in MeV. The electron ranges calculated according to eq. (4) are also plotted in figs. 1-3. It can be seen from the figures that the agreement between the corrected W i l s o n theory a n d the c o n t i n u o u s - s l o w i n g - d o w n m o d e l is quite satisfactory. The difference between ranges o b t a i n e d f r o m these two models does not exceed 10% in the energy region larger t h a n 70 keV for silicon, 60 keV for g e r m a n i u m a n d 55 keV for sodium iodide, respectively. It should be noted that the difference in the ranges between electrons and positrons is less than 10%, according to the calculations of Berger a n d Seltzer8). This means that we can use eq. (4) as a f o r m u l a for positrons. W e conclude, therefore, that the corrected Wilson f o r m u l a can be used with a reasonable degree o f accuracy for r a p i d estimation o f the ranges o f electrons and positrons in the detector materials in the energy region between 70 keV a n d 100 MeV. W i t h the use o f the present formula, M o n t e Carlo calculations o f the response o f g a m m a - r a y detectors can be performed quickly with accuracy. The a u t h o r wishes to thank Prof. S. Shimizu for many helpful discussions.

R A N G E OF E L E C T R O N S A N D P O S I T R O N S

References i) M. J. Berger, S. M. Seltzer, S. E. Chappell, J. C. Humphreys and J. M. Motz, Nucl. Instr. and Meth. 69 (1969) 181. 2) G. Gagerro, Nucl. Instr. and Meth. 94 (1971) 481. a) M. Belluscio, R. De Leo, A. Pantaleo and A. Vox, Nucl. Instr. and Meth. 118 (1974) 533. 4) R. R. Wilson, Phys. Rev. 79 (1950) 204; 84 0951) 100. 5) W. F. Miller and W. J. Snow, A N L 6318 (1961). 6) M. Giannini, P. R. Oliva and M. C. Ramorino, Nucl. Instr.

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and Meth. 81 (1970) 104. 7) T. Mukoyama, Bull. Inst. Chem. Res., Kyoto Univ., 53 (1975) 49. a) M . J . Berger and S. M. Seltzer, N A S A - S P Publication 3012 (1964). 9) T. M. Knasel, Nucl. Instr. and Meth. 83 (1970) 217. lo) A. Nelms, NBS Circular 577 0956). 11) L. Pages, E. Bertel, H. Joffre and L. Sklavenities, At. Data 4 (1972) 1.