The Collected Works of Tatsuo Tabata

INSTITUTE FOR DATA EVALUATION AND ANALYSIS TECHNICAL REPORT No. 9

The Collected Works of Tatsuo Tabata Volume 5 Interactions of Electrons with Matter in Bulk (3), 1975–1983

Edited with Commentary by Tatsuo Tabata January 11, 2018 Last Modified May 8, 2018

INSTITUTE FOR DATA EVALUATION AND ANALYSIS SAKAI, OSAKA, JAPAN

Institute for Data Evaluation and Analysis Technical Reports (IDEA-TR) are issued irregularly and available as PDF files only. IDEA is a virtual institute established in 1999 by Tatsuo Tabata. IDEA-TR 9 The Collected Works of Tatsuo Tabata Volume 5: Interactions of Electrons with Matter in Bulk (3), 1975–1983 / Edited with Commentary by Tatsuo Tabata. Copyright © 2018 by Tatsuo Tabata.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

Papers 27. A Generalized Empirical Equation for the Transmission Coefficient of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

28. An Empirical Relation for the Transmission Coefficient of Electrons under Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

29. An Improved Interpolation Formula for the Parameter B in Molière’s Theory of Multiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

30. Interpolation Formulas for Quantities Related to Radiative Energy-Loss of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

31. Approximation to cos γ Appearing in the Formula for the Coulomb Scattering of Relativistic Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

32. Approximations to Landau’s Distribution Functions for the Ionization Energy Loss of Fast Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

33. An Algorithm for Electron Depth–Dose Distributions in Multilayer Slab Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

34. Review of the Work at the Radiation Center of Osaka Prefecture on the Passage of Electrons through Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

i

Preface The present volume contains eight papers, published by Tatsuo Tabata and his coworkers from 1975 to 1983, in the form of the post-print re-edited by the use of LATEX. The studies described belong to the category of interactions of electrons with the matter in bulk (also called the passage of electrons through matter) and were made at the Radiation Center of Osaka Prefecture. Paper 34 at the end of this volume gives a brief review of the papers published by Tabata and his coworkers on this topic from 1967 to 1981. Each paper, except for Paper 31, includes a commentary by the present editor at its end. The editor has also corrected minor errors found in the published versions of the papers.

ii

Paper published in Nuclear Instruments & Methods, Vol. 127, Issue 3, 15 August 1975, Pages 429–434 (doi:10.1016/S0029-554X(75)80016-4) Copyright © 1975 by Elsevier B.V.

A Generalized Empirical Equation for the Transmission Coefficient of Electrons Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

(Received 5 May 1975) An empirical equation for the transmission coefficient of monoenergetic electrons normally incident on the absorber has been formulated by modifying the expression proposed by Rao. It utilizes a semiempirical equation recently developed for the extrapolated range of electrons, and the dependence of a single remaining parameter upon the incident energy T0 and the atomic number Z of the absorber has been determined by using a total of 79 experimental transmission curves for T0 = 8 keV– 30 MeV and for Z = 4–82. The rms absolute error of the equation is about 0.03 in most cases.

1. Introduction In the measurement and utilization of electrons, information is frequently required about the transmission coefficient ηT for the slab absorber of a given thickness; it is defined as the ratio of the number of transmitted electrons to the number of incident electrons. In general, ηT for monoenergetic electrons is a function of the absorber thickness x, the incident kinetic energy T0 of the electrons, the angle of incidence θ, and the parameters featuring the absorber material. The case of θ = 0◦ (perpendicular incidence) is of general importance, and considerations are restricted to this case in the present paper. Empirical equations for ηT are useful not only for facilitating interpolation or extrapolation of experimental data in particular cases but also for providing a set of input data for basic calculations of radiation effects and detection. Such equations have been proposed and investigated by many authors1−8 ), but the equations formulated are valid in rather narrow regions of T0 (see table 1). The present paper describes a new equation which covers all the regions of validity of the existing equations. An account of a preliminary stage of this work was given previously9 ). 2. Sources of data Experimental determinations of ηT are commonly made with two different methods. In the first method, we measure the charge transmitted, for example, with a Faraday cup, regarding both the primary and the secondary electron as the transmitted electron. In 1

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons Table 1 Previous empirical equations for ηT . Author Makhova Rao Dupouy et al. Mar Ebert et al. a

Region of incident energy T0 considered (MeV) 0.001–0.027 0.01–3 0.05–1.2 0.4–10 4–12

Ref. 1 7 5 6 8

Dependence on absorber material has not been incorporated in the equation.

the second method, we use counters and record associated electrons emerging together from the absorber as a single pulse. The number of pulses counted in the second method is close to the number of transmitted primary electrons, a very small difference being caused by the case in which only the secondaries enter the counter. In order to formulate the present empirical equation, transmission curves (the curves of ηT as a function of the thickness x) obtained with the second method have been used preferentially among published data.∗ One reason of this preference is that the definition of the transmitted electron as the transmitted primary electron is expected to serve more applications considering from large difference of energies possessed by primary and secondary electrons. The other reason is: For thin absorbers, the result of the first method frequently shows values of ηT greater than unity reflecting net emission of electrons from the absorber, and much complication of the expression is caused to reproduce this trend. The sources of data used are listed in table 2 together with experimental conditions. Table 2 Sources of data used to formulate the present empirical equation. The energy region, the absorber atomic numbers, and the detector of each experiment are also shown. Author Viatskin, Makhova Seligerb Nagai et al.c Nakai et al.c Matsuda et al.c Nakai et al.c Nakai et al.b Harder, Poschetb a b c

T0 (MeV) 0.0015–0.018 0.159–0.96 0.2–0.4 0.6–0.8 0.6–2 1–2 1.5, 2 4.17–30

Z 14, 29, 32, 83 13, 47, 50, 82 22 4, 13, 29, 47 22 4, 13, 29, 47 6 6, 13, 29, 82

Detector Charge collector 2π counter Solid-state detector Charge collector Charge collector Charge collector Charge collector Proportional counter

Ref. 13 14 15 16 17 18, 19 20 21

Only the data for Z = 32 cited in flg. 1 of ref. 1 have been used. Values have been read off from the graphs in the reference. Numerical data have been obtained from the first-named author through private communication. ∗

Lack of measurements in the energy region T0 . 2 MeV has made a consistent choice impossible. The inconsistency, however, is not so important, because the difference between results of the two methods are rather small for T0 . 1 MeV (refs. 10, 11).

2

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons 3. Formulation 3.1. Dependence Upon x As far as the dependence upon the thickness x is concerned the previous empirical equations can be classified into two types. One of them has been developed by Makhov1 ), and has been used by most of the previous authors2−6,8 ); it can be written as: ηT = exp (−γxβ ),

(1)

where γ and β are constants for a given absorber and energy T0 . By using the extrapolated range Rex , defined as the point where the tangent at the steepest point on the almost straight descending portion of the transmission curve meets the x-axis, eq. (1) can be rewritten as8 ): ηT = exp [−α(x/Rex )β ], (2) where:

α = (1 − 1/β)1−β .

(3)

The other type of equation has been proposed by Rao7 ): ηT = [1 + exp (−µx0 )]/{1 + exp [µ(x − x0 )]},

(4)

µ = n/Rex , x0 = n0 Rex ;

(5) (6)

where:

n and n0 are constants for a given absorber and T0 . While Rao has given independent expressions for the three parameters n, n0 and Rex as functions of the atomic number Z and the atomic weight A of the absorber, the relation: n = 2/(1 − n0 ),

(7)

should be satisfied in order that the values of Rex used might coincide with those determined from eq. (4), according to the aforementioned definition9 ). By using eq. (7) and putting: s0 = 2n0 /(1 − n0 ), (8) eq. (4) can be rewritten as: ηT = [1 + exp (−s0 )]/{1 + exp [(s0 + 2)(x/Rex ) − s0 ]}.

(9)

To make a selection between eqs. (2) and (9), we have compared the goodness of fit of eq. (2) to the experimental data with that of eq. (9). For Rex , a semiempirical equation developed by Tabata et al.12 ) (see appendix) has been used† , and the optimum value of a single remaining parameter in each equation, i.e., β in eq. (2) and s0 in eq. (9), has been sought by the method of least squares for each of a total of 79 transmission curves. The rms absolute error δ was minimized in the fit. Values of δ, averaged over a given region †

Although this equation well reproduces experimental values of Rex for Z & 6, appreciable deviations occur for Z = 4. Since these deviations are almost constant in fraction, correction has been applied to the equation by multiplying it by a factor of 0.852 in the case of Z = 4.

3

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons Table 3 Average values of the rms absolute error δ for the Makhov-type expression [eq. (2)], the Rao-type expression [eq. (9)] and the final expression obtained [eq. (9’)]. The symbol n denotes the number of experimental transmission curves considered. Z 4

Average value of δ × 102 eq. (2) eq. (9) eq. (90 ) 4.0 3.6 4.1

T0 (MeV) 0.6–1.4

n 5

1.5, 2 5.05–30

2 4

2.9 4.4

2.6 5.4

3.3 6.2

13

0.159–0.336 0.6–2 4.17–27.24

3 9 5

1.7 2.4 2.0

1.5 2.3 2.2

2.9 2.6 3.0

22

0.2, 0.3 0.6–2

2 8

4.7 2.5

3.7 1.7

4.5 1.9

29

0.6–2 5.05–30

8 6

1.7 3.7

1.3 2.8

2.5 3.7

32

0.008–0.02

4

5.2

4.6

4.9

47

0.159–0.336 0.6–2

3 8

7.6 5.2

5.6 3.5

5.6 3.6

50

0.18–0.362

3

5.6

3.5

3.5

82

0.159–0.96 5.05–30

4 5

4.3 3.4

3.0 3.1

3.0 3.3

79

3.6

2.9

3.4

6

Overall

0.008–30

of T0 for each elemental absorber, are shown in columns 4 and 5 of table 3. They are smaller for eq. (9) than for eq. (2), except only the cases of the highest energy region for Z = 6 and 13, so that we have decided to use eq. (9) in the present formulation. 3.2. Dependence Upon T0 and Z In fig. 1, the optimum value of s0 for each transmission curve is plotted as a function of T0 in logarithmic scales. Although the points show a considerable scatter, S-shaped curves are seen to be formed, suggesting that the relation between s0 and T0 may be expressed by an equation of the form: s0 = a1 exp [−a2 /(1 + a3 τ0 a4 )],

(10)

where ai (i = 1, 2, 3, 4) is a constant for a given absorber, and τ0 = T0 /mc2 ; mc2 is the rest energy of the electron. In order to find the dependence of ai on Z, the values of ai for representative elements have been determined by the least-squares fit with eq. (10) to the values of s0 . From an analysis of the results, the following facts have been found: The parameters a1 and a2 are well expressed by power functions of Z; a3 and a4 can be regarded as constants

4

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons

Fig. 1. Parameter s0 as a function of the incident energy T0 . Curves show the empirical relation, eq. (10); various symbols show the values obtained through the least-squares fit with eq. (9) to the individual experimental data13−21 ) for ηT . 4 4 Be, N 6 C, 13 Al, 22 Ti, 29 Cu, 32 Ge, O 47 Ag, H 50 Sn, × 82 Pb.

independent of the absorber. Thus we write: a1 = b1 /Z b2 , b4

a2 = b 3 Z , a3 = b 5 , a4 = b 6 ,

(11) (12) (13) (14)

where the symbols bi (i = 1, 2, . . . , 6) denote constants. Values of bi have been sought through the least-squares fit with eq. (10), in which eqs. (11)–(14) are substituted, to the total set of values of s0 . Table 4 Values of the constants bi . Errors attached are those of least-squares fit. i 1 2 3 4 5 6

bi 10.63 ± 0.86 0.232 ± 0.027 0.220 ± 0.041 0.463 ± 0.046 0.042 ± 0.018 1.86 ± 0.29

4. Results and discussion Values of the constants bi determined are presented in table 4. The relation given by eq. (10) with these values of bi is compared in fig. 1 with the values of s0 determined for the individual transmission curves. While the rms deviation of eq. (10) from the data points of s0 is as large as 12%, the rms absolute error δ of the final expression for ηT , i.e., eq. (9) in which eqs. (10)–(14), values of bi and the equation for Rex are substituted [this expression is called eq. (90 )], is about 0.03 in most cases as shown in the last column of table 3. In figs. 2–5 transmission curves given by eq. (90 ) are compared 5


Fig. 2. Comparison of the present empirical equation for ηT with Seliger’s experimental results14 ). The incident energy T0 is 0.159 MeV.

Fig. 3. Comparison of the present empirical equation for ηT with the experimental results of Nakai et al.18,19 ). The incident energy T0 is 1 MeV.

with representative experimental results. The abscissas of these figures represent the ratio x/R0 , where R0 is the range of the incident electrons computed in the continuous slowing-down approximation. Values of R0 used have been taken from the table of Berger and Seltzer22 ). The curve for the 0.159-MeV electrons incident on the silver absorber (fig. 2) shows an appreciable deviation from the experimental points; the value of δ in this case amounts to 8.8%, and is the largest of all the cases considered. For the other cases illustrated in the figures, agreement is moderately good. The experimental data used to determine the expression for s0 lies between T0 = 8 keV and 30 MeV, as shown in table 3. The lower limit to the region in which eq. (90 ) is generally valid is considered to be lower than 8 keV and to go down at least to about 1 keV because of the following reason: The dependence of s0 on T0 is rather weak for T0 . 0.5 MeV, so that the dependence of ηT on T0 in this region is governed principally by Rex , and the expression for Rex has been determined by using the experimental data in the region 0.3 keV–30 MeV 12 ). 6


Fig. 4. Comparison of the present empirical equation for ηT with the experimental results of Harder and Poschet21 ). The incident energy T0 is: 5 MeV for Al; 5.05 MeV for C, Cu, and Pb.

Fig. 5. Comparison of the present empirical equation for ηT with the experimental results of Harder and Poschet21 ). The incident energy T0 is: 27.24 MeV for Al; 30 MeV for C, Cu, and Pb.

Although a modified form of Rao’s expression has been chosen in the present formulation, the Makhov type expression, eq. (2), also gives a fairly good fit over the whole of the regions considered of Z and T0 as can be seen from column 4 of table 3. Neither of these expressions was derived from any physical model of electron transport. However, the Rao-type expression can roughly be regarded as a modification of the following solution to the transport equation23,24 ): ηT = 1/[1 + (3x/2λ)],

(15)

where λ, is the transport mean free path. Because of assumptions involved in deriving this equation [negligible energy loss and diffuse distribution of electrons24 )], it cannot directly be compared with the experimental data. A simple recipe to obtain from eq. (15) an expression which well reproduces the experimental data is to replace the term (3x/2λ) with an appropriate increasing function of x; the increase should be much faster than x for large x, because an effective value of λ varies approximately as (1 − x/Rex )k (k u1.5– 7

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons 1.6, ref. 25) on account of energy loss. Eq. (9) corresponds to the use of an exponential function for this replacement [the term exp (−s0 ) in the numerator of eq. (9), required to make ηT = 1 for x = 0, is rather small compared with unity in most cases]. References 1) 2)

3)

4) 5) 6)

7)

8)

9)

10 ) 11 ) 12 )

13 )

14 )

15 ) 16 ) 17 ) 18 )

19 ) 20 ) 21 ) 22 ) 23 )

24 )

25 )

A. F. Makhov, Fiz. Tverd. Tela 2 (1960) 2161 [English transl.: Soviet Phys.–Solid State 2 (1961) 1934]. B. N. S. Rao, Proc. Indian Acad. Sci. 51 (1960) 28. A. Ya. Vyatskin and A. N. Pilyankevich, Fiz. Tverd. Tela 4 (1962) 1038 [English transl.: Soviet Phys.–Solid State 4 (1962) 765]. A. Ya. Vyatskin, A. N. Pilyankevich and V. V. Trunev, Fiz. Tverd. Tela 6 (1964) 1563 [English transl.: Soviet Phys.–Solid State 6 (1964) 1230]. G. Dupouy, F. Perrier, P. Verdier and F. Arnal, Compt. Rend. 258 (1964) 3655; 260 (1965) 6055. B. W. Mar, Nucl. Sci. Eng. 24 (1966) 193. B. N. S. Rao, Nucl. Instr. and Meth. 44 (1966) 155. P. J. Ebert, A. F. Lauzon and E. M. Lent, Phys. Rev. 183 (1969) 422. T. Tabata and R. Ito, Ann. Rept. Rad. Center Osaka Prefect. 14 (1973) 27. M. J. Berger, NASA-CR-112838 (1970). S. M. Seltzer and M. J. Berger, Nucl. Instr. and Meth. 119 (1974) 157. T. Tabata, R. Ito and S. Okabe, Nucl. Instr. and Meth. 103 (1972) 85. A. Ia. Viatskin and A. F. Makhov, Zh. Tekhn. Fiz. 28 (1958) 740 [English transl.: Soviet Phys.–Tech. Phys. 3 (1960) 690]. H. H. Seliger, Phys. Rev. 100 (1955) 1029. A. Nagai, M. Hiro and K. Ishida, private communication [cited by Okabe et al., Oyo Buturi 9 (1974) 909]. Y. Nakai, K. Matsuda and T. Takagaki, private communication. K. Matsuda, T. Takagaki and Y. Nakai, Resumes for the 27th Annual Meeting of the Physical Society of Japan, No. 3 (1972) p. 5. Y. Nakai, H. Inomo, K. Matsuda, T. Osuga and K. Kimura, Ann. Rept Japan. Assoc. Rad. Res. Polym. 3 (1951) 11. Y. Nakai, K. Matsuda, T. Takagaki and K. Kimura, Ann. Rept Japan. Assoc. Rad. Res. Polym. 5 (1964) 7. Y. Nakai, K. Matsuda, T. Takagaki and K. Kimura, Ann. Rept Japan. Assoc. Rad. Res. Polym. 7 (1966) 5. D. Harder and G. Poschet, Phys. Letters 24B (1967) 519. M. J. Berger and S. M. Seltzer, NASA SP-3012 (1964). H. A. Bethe, M. E. Rose and L. P. Smith, Proc. Am. Phil. Soc. 78 (1938) 753. J. H. Jacob, J. Appl. Phys. 45 (1974) 467. H.-W. Thümmel, Z. Physik 179 (1964) 116.

Appendix The semiempirical equation for Rex developed by Tabata et al.12 ) is given by: Rex = c1 [(1/c2 ) ln (1 + c2 τ0 ) − c3 τ0 /(1 + c4 τ0 c5 )],

8

(16)

27. Generalized Empirical Equation for the Transmission Coefficient of Electrons where: c1 c2 c3 c4 c5

= 0.2335A/Z 1.209 g/cm2 , = 1.78 × 10−4 Z, = 0.9891 − 3.01 × 10−4 Z, = 1.468 − 1.180 × 10−2 Z, = 1.232/Z 0.109 .

(17) (18) (19) (20) (21)

Commentary Figures 1 and 5 of the published version include the following simple errors. Corrected figures are used in this volume. 1 For the abscissa of fig. 1, the label 10−2 is placed at a wrong place. 2 In fig. 5, ticks for the ordinate and the abscissa are missing. Using the equation in the present paper, Lazurik et al.1 derived an analytic expression for the “average depth of electron penetration.” They also showed that this parameter would be useful as the characteristic depth of energy and charge deposition in a target irradiated by electrons. [The terminology “average depth of electron penetration” is confusing with the “average penetration depth” (also called “projected range”), though the two definitions are rather close. The parameter of Lazurik et al. means the average of the deepest points of electron tracks in an semi-infinite target. These points are not necessarily the endpoints of electrons’ paths but often lie deeper than the endpoints because of the backward scattering of electrons in the medium.] An advanced empirical equation for the transmission curve was proposed later.2

1

V. Lazurik, V. Moskvin and T. Tabata, IEEE Trans. Nucl. Sci. 45 (1998) 626. T. Tabata and V. Moskvin, “Transmission coefficients and residual energies of electrons: PENELOPE results and empirical formulas.” Slides presented at The Third International Workshop on Electron and Photon Transport Theory Applied to Radiation Dose Calculation, Indianapolis, Indiana, August 8–12, 1999. DOI: 10.13140/RG.2.1.4733.6482. 2

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Paper published in Nuclear Instruments & Methods, Vol. 136, Issue 3, 1 August 1976, Pages 533–536 (doi:10.1016/0029-554X(76)90377-3) Copyright © 1976 by Elsevier B.V

An Empirical Relation for the Transmission Coefficient of Electrons under Oblique Incidence Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

(Received 26 April 1976) An empirical relation for the transmission coefficient of electrons impinging on aluminum absorbers is given as a function of absorber thickness x, incident energy T0 , and angle of incidence θ. It has been formulated by incorporating the dependence upon θ in the empirical equation for the case of normal incidence reported previously by the present authors. Numerical constants in the relation have been determined through least-squares fit to the data for T0 = 0.5–10 MeV generated by the Monte Carlo code of Berger and Seltzer. The rms absolute deviation, evaluated over θ = 0◦ –75◦ , of the relation from the Monte Carlo data is about 0.03 in the entire energy region considered.

1. Introduction We have recently reported an empirical equation for the number transmission coefficient of electrons normally incident on the slab absorber1 ). For application purposes, the knowledge of the average transmission coefficient of electrons constituting a flux with an angular spread is also important; such an average would be evaluated most conveniently by the use of an empirical equation for the transmission coefficient in which the dependence upon angle of incidence θ is incorporated. Using the Monte Carlo code of Berger and Seltzer2 ), Watts and Burrell3 ) have obtained the transmission coefficients of 0.5–10 MeV electrons incident on aluminum slabs with θ = 0◦ , 30◦ , 45◦ , 60◦ , 75◦ , and 89.9◦ . From an analysis of these data, Shreve and Lonergan4 ) have found the following relation∗ : ηT (x; T0 , Z, θ) = ηT (x; T0 , Z, 0◦ ) cosα θ,

(1)

where α is given for Z = 13 by α = 0.41 + 3.7(x/R0 )2 + [0.14(x/R0 ) − 0.039]T0 . ∗

(2)

The expression in ref. 4 contains typographical errors. We have been informed of the correct expression by D. C. Shreve.

10

28. Transmission Coefficient of Electrons under Oblique Incidence Here ηT (x; T0 , Z, θ) is the transmission coefficient of electrons impinging on the slab of thickness x and of atomic number Z with incident kinetic energy T0 (expressed in units of MeV) and with obliquity θ; R0 is the continuous slowing-down approximation (CSDA) range of the incident electrons. When this expression is used with an adequate expression for ηT (x; T0 , 13, 0◦ ), it well reproduces the Monte Carlo data for ηT (x; T0 , 13, θ) in the region of thickness x/R0 & 0.2. However, it shows an unreasonable behavior† near x/R0 = 0, giving too low values for larger θ. The present paper describes a different approach of formulating an empirical relation for ηT (x; T0 , Z, θ) as well as the result obtained by fitting the new relation to the same Monte Carlo data that were used by Shreve and Lonergan. 2. Formulation 2.1. Dependence upon θ In the empirical equation for ηT (x; T0 , Z, 0◦ ) reported in our previous paper1 ), we have used two parameters: s0 and Rex . Both of them are functions of T0 and Z, and Rex has the physical implication of the extrapolated projected-range (usually called simply extrapolated range) of the incident electrons. We consider here that these parameters are also functions of θ, and use the same form of equation as was used for ηT (x; T0 , Z, 0◦ ): ηT (x; T0 , Z, θ) = {1 + exp[−s0 (θ; T0 , Z)]} .

1 + exp{[s0 (θ; T0 , Z) + 2][x/Rex (θ; T0 , Z)] − s0 (θ; T0 , Z)} , (3)

where Rex (θ; T0 , Z) has the implication of a generalized extrapolated range of the electrons incident with obliquity θ. By analyzing the results of fitting eq. (3) to the Monte Carlo data, s0 (θ; T0 , 13) and Rex (θ; T0 , 13) for 0 5 θ 5 60◦ and for 0.5 5 T0 5 10 MeV have been found to be well expressed by power functions of cos θ, so that we write: s0 (θ; T0 , Z) = a1 cosa2 θ, Rex (θ; T0 , Z) = Rex (0◦ ; T0 , Z) cosa3 θ,

(4) (5)

where ai (i = 1, 2, 3) is a constant for a given set of values of T0 and Z, and Rex (0◦ ; T0 , Z) is the conventional extrapolated range pertaining to the transmission curve for θ = 0◦ . For Rex (0◦ ; T0 , Z), we use the semiempirical equation of Tabata et al.5 ), and a set of optimum values of ai has been sought for each incident energy by the method of least squares. In this search eq. (3), in which eqs. (4) and (5) are substituted, has been fitted to a combined set of data for θ = 0◦ –75◦ ; only the data for 89.9◦ have been excluded because of too large deviation of the equation at this angle. 2.2. Dependence upon T0 In fig. 1, the optimum values of ai obtained are plotted as a function of T0 in logarithmic scales. From this plot both ln a2 and ln a3 are seen to be linear functions of ln T0 . Essentially the values of a1 should be equal to the values of s0 for normal incidence given †

While ηT (0; T0 , Z, θ) should be unity for any value of θ considering from the definition of the transmission coefficient, the value of ηT (0; T0 , 13, θ) given by the relation of Shreve and Lonergan is less than unity for θ > 0◦ , and decreases with increasing θ.

11

28. Transmission Coefficient of Electrons under Oblique Incidence

Fig. 1. Dependence of ai (i = 1, 2, 3) upon incident energy T0 . The dashed line shows the values of s0 for normal incidence given by the equation in the previous paper (ref. 1), and the solid lines represent the results of fit with eqs. (6)–(8).

by the equation in the previous paper1 ) (shown by a dashed line in fig. 1), because we have s0 (0◦ ; T0 , Z) = a1 from eq. (4). With this regard a discrepancy is seen in fig. 1, and it is due to the difference between definitions of the transmitted electrons: In the case of the Monte Carlo data used in the present work, energetic secondary electrons ejected from the absorber were counted in as the transmitted electrons, whereas they were excluded in most of the experimental data used in the previous paper. The dependence of a1 upon T0 , however, is quite similar to that given by the dashed curve. This fact suggests that the same equation as was used for s0 for normal incidence can possibly be used for a1 by readjusting numerical values of some of its constants. According to the inspections mentioned, we express ai by the following equations: a1 = b1 exp[−b2 /(1 + 0.042τ0 1.86 )],

(6)

a2 = b3 /τ0 b4 ,

(7)

b6

a3 = b5 /τ0 ,

(8)

where τ0 = T0 /mc2 , mc2 is the rest energy of the electron, and the symbols bj (j = 1, 2, 3, . . . , 6) denote constants for a given absorber material. The two numerical values in eq. (6) have been taken from the previous paper1 ), in which they were determined as constants independent of absorber material. Values of bj have been determined by the least-squares fit with eqs. (6)–(8) to the values of a1 –a3 , respectively. 3. Results and discussion The values of bj obtained are presented in table 1. The relations given by eqs. (6)– (8) with these values of bj are shown in fig. 1 (solid lines). The empirical relation for ηT (x; T0 , 13, θ) is thus given by eq. (3) in which eqs. (4)–(8) and the values of bj are substituted. The rms absolute deviation δ of this relation from the Monte Carlo data 12

28. Transmission Coefficient of Electrons under Oblique Incidence Table 1 Values of the constants bj . Errors attached are those of least-squares fit. j 1 2 3 4 5 6

bj 6.929 ± 0.055 0.652 ± 0.039 1.711 ± 0.071 0.086 ± 0.018 0.5237 ± 0.0069 0.0191 ± 0.0063

Table 2 Values of the rms absolute deviation δ of the present empirical relation from the Monte Carlo data. T0 (MeV) 0.5 1 2 3 4 5 6 10

δ 0.035 0.034 0.032 0.030 0.028 0.029 0.028 0.028

used has been evaluated at each energy over θ = 0◦ –75◦ , and the results are listed in table 2. The value of δ is about 0.03 for all the energies considered, indicating that the fit is generally satisfactory.

Fig. 2. Curves given by the present empirical relation for the transmission coefficient ηT of electrons incident with obliquity θ are compared with the Monte Carlo data. Absorber material is aluminum, and the incident energy is 0.5 MeV. The abscissa is in units of the CSDA range R0 of the incident electrons (0.2243 g/cm2 ).

13

28. Transmission Coefficient of Electrons under Oblique Incidence

Fig. 3. Curves given by the present empirical relation for the transmission coefficient ηT of electrons incident with obliquity θ are compared with the Monte Carlo data. Absorber material is aluminum, and the incident energy is 2 MeV. The abscissa is in units of the CSDA range R0 of the incident electrons (1.212 g/cm2 ).

Fig. 4. Curves given by the present empirical relation for the transmission coefficient ηT of electrons incident with obliquity θ are compared with the Monte Carlo data. Absorber material is aluminum, and the incident energy is 10 MeV. The abscissa is in units of the CSDA range R0 of the incident electrons (5.841 g/cm2 ).

Transmission curves given by the present relation are compared with the Monte Carlo data in figs. 2–4. Appreciable deviations are seen for the largest values of θ at lower energies (figs. 2 and 3). These deviations, however, are of little importance, because the component of the electron flux with θ & 75◦ has a minor contribution in most applications. Owing to the effect of energetic secondary electrons described in the previous section, the 14

28. Transmission Coefficient of Electrons under Oblique Incidence Monte Carlo data for ηT (x; T0 , 13, θ) at the smallest obliquities and at the highest energies are greater than unity in wide regions of thickness (see fig. 4), and the present relation cannot reproduce such a feature. When the contributions of secondaries are subtracted from the data, a better fit would be obtained. The functional form of the present relation is considered to be applicable also to materials other than aluminum. References 1)

T. Tabata and R. Ito, Nucl. Instr. and Meth. 127 (1975) 429. M. J. Berger and S. M. Seltzer, Computer Code Collection 107, Oak Ridge Radiation Shielding Information Center (1968). 3 ) J. W. Watts, Jr. and M. O. Burrell, NASA TN D-6385 (1971). 4 ) D. C. Shreve and J. A. Lonergan, NASA CR-SAI 71-559-LJ (1971). 5 ) T. Tabata, R. Ito and S. Okabe, Nucl. Instr. and Meth. 103 (1972) 85. 2)

Commentary In the published version, the following typo was found: In the footnote on page 533, “expresssion” should read “expression”. A slightly different approach was later proposed for the empirical equation for the transmission coefficient of electrons under oblique incidence (see footnote 2 on p. 9).

15

Paper published in Japanese Journal of Applied Physics, Vol. 15, Issue 8, August 1976, Pages 1583–1584 (doi:10.1143/JJAP.15.1583) Copyright © 1976 by the Japan Society of Applied Physics

An Improved Interpolation Formula for the Parameter B in Molière’s Theory of Multiple Scattering Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Shinke-cho, Sakai, Osaka

(Received May 17, 1976)

In utilizing beams of charged particles, rapid estimation is frequently required of their angular spreading caused by multiple Coulomb scattering in target materials. To evaluate the angular distribution or measures of the angular width from Molière’s theory of multiple scattering1) or from its modifications,2,3) one must know values of the parameter B defined by a transcendental equation. Although they can be computed to a desired accuracy within a rather small number of iterations, it is quite tedious to perform iterative methods unless a digital computer is available. Approximate values of B can be obtained through interpolation from tables,1,3,4) or by the use of an interpolation formula proposed by Scott.5) In this note, we show that this formula can be improved significantly by a simple modification. Table I. Values of B and deviations of interpolation formulae from the exact values. log10 Ω 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

B 3.360 4.886 6.290 7.635 8.944 10.230 11.498 12.753 13.997 15.233 16.462 17.685 18.903 20.117 21.326 22.533 23.736

16

Deviation (%) Scott present 11.19 0.13 2.90 −0.06 0.47 0.04 −0.32 0.07 −0.47 0.05 −0.36 0.02 −0.11 0.00 0.18 −0.03 0.50 −0.04 0.83 −0.04 1.15 −0.04 1.46 −0.03 1.75 −0.02 2.03 0.00 2.30 0.02 2.56 0.05 2.80 0.07

29. Parameter B in Molière’s Theory of Multiple Scattering The parameter B is defined as the larger of the two roots of the equation: B − ln B = ln(Ωe/γ 2 ),

(1)

where Ω is the mean number of collisions, e is the base of the natural logarithm, and γ is Euler’s constant. Scott’s formula is given by B = 1.153 + 2.583 log10 Ω.

(2)

Deviations of this formula from exact values are shown in column 3 of Table I. For the purpose of improving eq. (2), we introduce an additional term, and write as B = a1 + a2 log10 Ω − a3 /(log10 Ω + a4 ),

(3)

where the symbols ai (i = 1, 2, 3, 4) denote constants. Values of ai have been sought by least-squares relative-error calculation. The range of log10 Ω considered is from 1 to 9, and is the same as that given in Molière’s table,1) which covers well enough the region of values appearing usually. Values of B used in the fit have been obtained by iteration of eq. (1) within an error of 5×10−8 , and those rounded to the three places of decimals are presented in column 2 of Table I. The values of ai determined are given in Table II. As shown in column 4 of Table I, the deviation of the present formula is less than 0.1% except near log10 Ω=1. Recent increase in available energy of charged-particle accelerators is causing extension of the region of interest of log10 Ω toward the lower end of Table I, where the deviation of eq. (2) exceeds 1%. In this situation, the present formula will receive greater application. Table II. Values of constants ai (i = 1, 2, 3, 4) in the present formula. i 1 2 3 4

ai 2.600 2.3863 3.234 0.994

References G. Molière: Z. Naturforsch. 3a (1948) 78. B. P. Nigam, M. K. Sundaresan and T.-Y. Wu: Phys. Rev. 115 (1959) 491. J. B. Marion and B. A. Zimmerman: Nuclear Instrum. and Methods 51 (1967) 93. H. Bichsel: American Institute of Physics Handbook, 3rd edition, ed. D. E. Gray (McGrawHill, New York, 1972) Chap. 8, p. 142. 5) W. T. Scott: Phys. Rev. 85 (1952) 245.

1) 2) 3) 4)

Commentary In a later paper, the authors published a revised set of coefficients: a1 =2.53270, a2 =2.3910505, a3 =2.95031, and a4 =0.889936. These coefficients were obtained so as to minimize the maximum deviation (best approximation or minimax approximation), and the maximum error was reduced to 0.076%. See: T. Tabata and R. Ito, Ann. Rept. Radiat. Center Osaka Prefect. 20 (1979) 87. 17

Paper published in Nuclear Instruments & Methods, Vol. 146, Issue 2, 15 October 1977, Pages 435–438 (doi:10.1016/0029-554X(77)90730-3) Copyright © 1977 by Elsevier B.V

Interpolation Formulas for Quantities Related to Radiative Energy-Loss of Electrons Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

(Received 22 April 1977) An interpolation formula is given for the quantity Φrad /Φ that is proportional to the radiative energy-loss divided by the total energy of the incident electron. Errors caused by the formula have been checked for three sets of values of Φrad /Φ, which have been computed by Berger and Seltzer with different empirical corrections to reduce Born-approximation errors. Incident energies from 1 keV to 1000 MeV and atomic numbers of material from 1 to 92 have been considered. Values of six parameters in the formula have been determined by using Tchebyschev’s criterion of approximation, and the maximum error has been found to be less than 1.9% for the intermediate set with Aiginger–Rester correction as well as for the no-correction set. A table of parameters in the case of the Aiginger–Rester set is provided for 59 elements. An interpolation formula for the Aiginger–Rester correction factor is also given.

1. Introduction In treating the transport problems of fast electrons through thick layers of material, it is frequently necessary to evaluate the collision and radiative energy-losses of electrons for a number of energies along the course of slowing down. Computation of the radiative energy-loss from the cross section is laborious, so that it is desirable to develop an interpolation formula valid in a wide region of energies. The present paper describes a formula which accurately reproduces the most reliable tables now available1 ). Recent computations1−3 ) of the radiative energy-loss have been based on cross-section formulas recommended by Koch and Motz4 ), and an empirical correction factor fcorr has been applied to reduce the effect of Born-approximation error in these formulas. Since fcorr is separately required in computing the number of bremsstrahlung events that is differential in photon energy and angle, we have developed also an interpolation formula for this factor. 2. Definitions and source of data Tables of computed values of the radiative energy-loss have been published by Berger and Seltzer2 ) and by Pages et al.3 ); these authors have used the values of fcorr determined 18

30. Quantities Related to Radiative Energy-Loss of Electrons by Koch and Motz4 ) (Koch–Motz correction). Berger and Seltzer1 ) have estimated fcorr from the later experimental results of Aiginger5 ) and of Rester and Dance6 ) (Aiginger– Rester correction), and have given three sets of values of Φrad /Φ based on fcorr = 1 (no correction) and on the two different corrections mentioned. The quantities Φrad and Φ are respectively defined by Φrad = −(dE0 /dx)rad /N E0 Φ

2

= αre Z(Z + 1),

(1) (2)

where −(dE0 /dx)rad is the radiative energy-loss per unit path of an electron, N the number of atoms per unit volume of the material, E0 the total energy of the incident electron, α the fine structure constant, re the classical electron radius, and Z the atomic number of the material. Among the three sets of Φrad /Φ, the Koch–Motz set is essentially equivalent to the tables of refs. 2 and 3. The Aiginger–Rester set is intermediate between the other two, and has been extensively used in recent Monte Carlo calculations of Berger and Seltzer. We have made an interpolation formula for Φrad /Φ, and have checked its errors for the three sets. Incident energies from 1 keV to 1000 MeV and atomic numbers from 1 to 92 have been considered. A table of parameters in this formula is given for the case of the Aiginger–Rester set. In formulating the equation for fcorr , the Aiginger–Rester correction factor, whose values are also given in ref. 1, has been considered. 3. Formulation 3.1. Φrad /Φ When Φrad /Φ is plotted in the semilogarithmic scales as a function of kinetic energy T0 of the incident electron, an S-shape is formed in the energy region above about 0.3 MeV; below this energy, Φrad /Φ slightly increases with decreasing energy (see fig. 1). The following is one of possible equations to express these trends: Φrad /Φ = a1 /(1 + a2 τ0 −a3 ) + a4 /(1 + a5 τ0 a6 ),

(3)

where τ0 = T0 /mc2 , mc2 is the rest energy of the electron, and ai (i = 1, 2, . . . , 6) are constants for a given material. We have also tried to use rational functions or a combination of the first term of eq. (3) with a rational function, and have found that eq. (3) gives the best result judging from smallnesses of the error and the number of parameters used. 3.2. fcorr Values of fcorr in the case of Aiginger–Rester correction are plotted as a function of T0 by circles in fig. 2 (for Z 5 13, fcorr is always unity; for Z = 14, it is unity in the energy regions T0 5 0.02 MeV and T0 = 5 MeV). An inspection of this plot followed by preliminary computations of fit has revealed thal fcorr can be well expressed by the relation: fcorr = where

 1 + b 1,

1 (τ0

− b2 ) exp(−τ0 ), for Z = 14 and τ0 > b2 , for Z 5 13 or for Z = 14 and τ0 5 b2 , b1 = c1 (ln Z)2 + c2 ln Z + c3 , 19

(4)

(5)

30. Quantities Related to Radiative Energy-Loss of Electrons

Fig. 1. Dependence of Φrad /Φ upon kinetic energy T0 of the incident electron. Values plotted have been computed by Berger and Seltzer (ref. 1) with Aiginger– Rester correction. A spurious bump in the curve for Pb at about 0.8 MeV is due to the effect of using a patchwork of cross sections.

Fig. 2. Dependence of correction fcorr upon kinetic energy T0 of the incident electron. The circles represent the Aiginger–Rester correction factor estimated by Berger and Seltzer (rel. 1), and the lines represent the present interpolation formula, eq. (4).

and b2 and cj (j = 1, 2, 3) are constants independent of material. 4. Determination of the constants Values of the constants in eqs. (3)–(5) have been determined so as to satisfy the criterion of the best approximation in Tchebyschev’s sense. The problem can be phrased in the form: Find a = (a1 , a2 , . . . , ap ) to minimize the maximum value of 20

30. Quantities Related to Radiative Energy-Loss of Electrons

Table 1 Maximum values δmax of eq. (3) when Z is varied from 1 to 92 are shown for three sets of Φrad /Φ. Values of atomic number Z at which δmax reaches the maximum are also shown. Set of Φrad /Φ No correction Aiginger–Rester correction Koch–Motz correction

Maximum valueof δmax (%) 1.9 1.9 2.6

Z 2 2 9

Table 2 Values of parameters ai (i = 1, 2, . . . , 6) in the interpolation formula for Φrad /Φ. Values δmax of this formula and those of the interpolation formula for fcorr are also given. Element 1H 2 He 3 Li 4 Be 5B 6C 7N 8O 9F 10 Ne

a1 20.365 19.307 18.875 18.645 18.469 18.308 18.162 18.022 17.891 17.772

a2 6.912 7.542 7.177 6.621 6.106 5.689 5.366 5.111 4.906 4.738

a3 0.7318 0.7563 0.7542 0.7459 0.7363 0.7270 0.7187 0.7116 0.7056 0.7005

Φrad /Φ a4 a5 6.164 1.158 7.093 1.243 7.554 1.437 7.657 1.597 7.632 1.739 7.590 1.874 7.557 1.995 7.526 2.103 7.500 2.196 7.473 2.276

a6 0.5334 0.4045 0.4087 0.4531 0.5069 0.5581 0.6031 0.6429 0.6782 0.7097

δmax (%) 1.8 1.9 1.8 1.6 1.4 1.2 1.0 0.9 0.8 0.8

fcorr δmax (%) ... ... ... ... ... ... ... ... ... ...

11 Na

17.655 17.544 17.445 17.353 17.263 17.178 17.097 17.012 16.931 16.851

4.598 4.482 4.372 4.306 4.249 4.197 4.150 4.111 4.082 4.059

0.6961 0.6926 0.6884 0.6855 0.6831 0.6809 0.6789 0.6779 0.6772 0.6769

7.450 7.427 7.403 7.397 7.388 7.379 7.369 7.359 7.357 7.356

2.343 2.398 2.452 2.441 2.424 2.403 2.380 2.349 2.314 2.275

0.7381 0.7632 0.7869 0.7972 0.8066 0.8146 0.8219 0.8279 0.8312 0.8335

0.7 0.6 0.6 0.5 0.4 0.4 0.3 0.3 0.3 0.3

... ... ... 0.02 0.04 0.07 0.09 0.1 0.1 0.2

16.778 16.693 16.604 16.515 16.428 16.332 16.245 16.155 16.063 15.981

4.031 4.033 4.042 4.071 4.102 4.153 4.203 4.261 4.331 4.392

0.6759 0.6777 0.6801 0.6839 0.6876 0.6928 0.6973 0.7025 0.7081 0.7127

7.354 7.353 7.353 7.354 7.355 7.358 7.361 7.365 7.387 7.405

2.240 2.179 2.113 2.027 1.947 1.859 1.781 1.704 1.632 1.572

0.8348 0.8344 0.8323 0.8273 0.8219 0.8113 0.8016 0.7912 0.7785 0.7682

0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3

12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 A 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn

21

30. Quantities Related to Radiative Energy-Loss of Electrons Table 2 (Continued) Element 32 Ge 33 As 35 Br 36 Kr 38 Sr 40 Zr 42 Mo 47 Ag 48 Cd 49 In

a1 15.830 15.741 15.584 15.497 15.312 15.158 15.006 14.613 14.536 14.468

a2 4.501 4.564 4.704 4.792 5.003 5.157 5.314 5.821 5.928 6.019

a3 0.7201 0.7248 0.7335 0.7389 0.7514 0.7599 0.7680 0.7882 0.7919 0.7946

Φrad /Φ a4 a5 7.419 1.463 7.425 1.408 7.435 1.307 7.441 1.254 7.449 1.149 7.411 1.062 7.385 0.987 7.386 0.831 7.386 0.804 7.386 0.781

50 Sn

14.389 14.313 14.238 14.164 14.090 14.015 13.950 12.666 12.590 12.276

6.149 6.257 6.365 6.470 6.583 6.691 6.781 8.469 8.550 8.845

0.7990 0.8021 0.8050 0.8077 0.8102 0.8125 0.8141 0.8247 0.8239 0.8197

7.384 7.384 7.383 7.383 7.382 7.382 7.381 7.373 7.373 7.372

0.754 0.732 0.710 0.690 0.670 0.652 0.636 0.434 0.427 0.399

0.6332 0.6277 0.6224 0.6173 0.6116 0.6066 0.6023 0.5152 0.5103 0.4914

0.6 0.7 0.7 0.7 0.7 0.7 0.7 1.2 1.2 1.3

0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.8 0.8 0.9

12.202 12.126 12.039 11.961 11.884 11.807 11.641 11.342 11.176

8.890 8.953 9.035 9.094 9.150 9.203 9.320 9.424 9.503

0.8182 0.8168 0.8159 0.8143 0.8127 0.8111 0.8080 0.7995 0.7955

7.371 7.371 7.371 7.371 7.371 7.370 7.370 7.367 7.367

0.393 0.387 0.380 0.375 0.369 0.364 0.353 0.337 0.328

0.4877 0.4834 0.4786 0.4742 0.4700 0.4659 0.4575 0.4441 0.4364

1.3 1.3 1.4 1.4 1.4 1.4 1.5 1.6 1.6

0.9 0.9 0.9 0.9 0.9 0.9 0.9 1.0 1.0

51 Sb 52 Te 53 I 54 Xe 55 Cs 56 Ba 73 Ta 74 W 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 86 Rn 90 Th 92 U

a6 0.7520 0.7437 0.7273 0.7178 0.6983 0.6900 0.6819 0.6499 0.6439 0.6389

δmax (%) 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.6 0.6 0.6

fcorr δmax (%) 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6

|w(xi )[y(xi ) − F (a, xi )]|, i = 1, 2, . . . , n, where xi is the independent variable forming a discrete set, w(xi ) a weighting function, y(xi ) the dependent variable to be fitted, F (a, xi ) the fitting function, and n > p (the discrete T-problem). When x takes on continuous values in a given interval and w(x), y(x), and F (a, x) are continuous on the same interval, we have the continuous T-problem. In order to solve the T-problem. we have utilized the linear-programming technique7,8 ) together with the Osborne–Watson algorithm for the nonlinear case9 ). While this technique properly treats the discrete T-problem, the criterion can be made essentially equivalent to that of the continuous T-problem by exchanging part of the data in each iteration; the data removed are those nearest to the extremal points of the error curve, and the data which enter are those at the extremal points estimated by interpolation. When the independent variable is τ0 , i.e. except the case of fitting eq. (5) to values of b1 , this exchange has been performed by using quadratic interpolation of the error curve.

22

30. Quantities Related to Radiative Energy-Loss of Electrons We have put w(xi ) = 1/y(xi ) to minimize the maximum relative error δmax defined as δmax = max |[y(xi ) − F (a, xi )]/y(xi )|. i

(6)

5. Results and discussion 5.1. Φrad /Φ The value of δmax of eq. (3) varies as a function of Z, and its maximum values, i.e. the values of max [δmax (Z)] 15Z592

are shown in table 1 for the three sets of Φrad /Φ. Except in the case of the Koch–Motz set, the maximum error over the whole region of energy and atomic number considered is less than 1.9%. Even if any revision is made of the table of Φrad /Φ in future, eq. (5) would be well fitted also to the revised table considering these values of the maximum error and the rather large differences between Φrad /Φ values of Koch-Motz and no-correction sets. The accuracy of corrected cross-sections on which Φrad /Φ is based has been estimated by Koch and Motz4 ) to be ±20% up to 2 MeV, ±5% between 2 and 15 MeV, and ±3% above 15 MeV. Comparison of these accuracies with the maximum values of δmax shows that the increase of possible errors caused by the use of the present interpolation formula is unimportant. Values of ai (i = 1, 2, . . . , 6) and δmax in the case of the Aiginger– Rester set are given in table 2 for 59 elements. Since the dependence of ai upon Z is considerably weak, it is possible to obtain the values of ai for the desired element by quadratic interpolation from part of this table. An example of error curves is depicted in fig. 3, where typical characteristics of the best approximation can be seen. 5.2. fcorr Values of constants in eqs. (4) and (5) are given in table 3. Examples of curves given by eq. (4) are compared with data points in fig. 2. The values of δmax of eq. (4) are given in the last column of table 2, and are less than 1.0% in the entire region of Z considered.

Fig. 3. Relative error δ of eq. (3) is plotted as a function of kinetic energy T0 of the incident electron for the case of aluminum. The data fitted have been taken from the Aiginger–Rester set in ref. 1. The error δ takes on the maximum value δmax = 0.58% at seven (= the number of parameters +1) points of energy, exhibiting typical characteristics of the best approximation.

23

30. Quantities Related to Radiative Energy-Loss of Electrons Table 3 Values of constants in the interpolation formula for fcorr . Constant b2 c1 c2 c3

Value 0.039 0.060632 −0.15287 −0.00677

The authors wish to thank Y. Matsuda and A. Mizohata for their valuable aid in using the computer. References 1) 2) 3)

4) 5) 6) 7) 8)

9)

M. J. Berger and S. M. Seltzer, RSIC Code Package CCC-107 (1968). M. J. Berger and S. M. Seltzer, NASA SP-3012 (1964). L. Pages. E. Bertel. H. Joffre and L. Sklavenitis, At. Data 4 (1972) 1. H. W. Koch and J. W. Motz, Rev. Mod. Phys. 31 (1959) 920. H. Aiginger, Z. Physik 197 (1966) 8. D. H. Rester and W. E. Dance. NASA CR-759 (1967). J. E. Kelly Jr., J. Soc. Indust. Appl. Math. 6 (1959) 15. M. R. Osborne and G. A. Watson, Computer J. 10 (1967) 172. M. R. Osborne and G. A. Watson, Computer J. 12 (1969) 63.

Commentary The following typos were found in the published version. Page 437 437 437

Line 3rd of Abstract 3rd–4th of 2nd par. in §1 10th from bottom of right col.

Now reads whith empricial fcorr the . . .

Should read with empirical fcorr , the . . .

After “or” in the second condition line of eq. (4), “for” has been inserted in the present version to make it clear that the expression “Z = 14 and τ0 5 b2 ” represents a joint condition.

24

Paper published in Nuclear Science and Engineering, Vol. 65, Issue 2, 1978, Pages 414–415 (doi:10.13182/NSE78-A27168) Copyright © 1978 by Taylor & Francis

Approximation to cosγ Appearing in the Formula for the Coulomb Scattering of Relativistic Electrons Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, 704 Shinke-cho, Sakai, Osaka, Japan

(Received July 28, 1977; Accepted August 18, 1977)

ABSTRACT An approximate expression for the function cos γ, defined by the use of the gamma function of a complex argument, has been developed to economize the computations pertaining to the single- or the multiple-Coulomb scattering of relativistic electrons. The maximum absolute error of the expression is 2.2×10−6 .

In treating the problems related to the single- or multiple-Coulomb scattering of relativistic electrons. the approximate formula of Bartlett and Watson1 for the Mott scattering cross section at small scattering angles is frequently utilized.2−4 In the evaluation of this formula, it is necessary to know the values of cos γ, defined by 1 Γ( − iq)Γ(1 + iq) 2

cos γ = Re

1 Γ( + iq)Γ(1 − iq) 2

,

(1)

with q = αZ/β,

(2)

where α = fine-structure constant Z = atomic number of the medium β = electron velocity divided by the light velocity. The values of cos γ can be computed by the use of approximate expressions for the gamma function of a complex argument4,5 or can be obtained through interpolation of a table.6 To save the computer memory and computation time, however, it is desirable to develop a direct expression approximating cos γ, and such an expression is described here. We divide the region of q into two by a point qc and use different forms for the respective regions. Noting that cos γ is an even function of q, we employ a rational function of q 2 for 0 6 q < qc . In the region q > qc , we utilize the result of the expansion as a power series in 1/q, which has been derived as follows. Let us define G(z) by G(z) = Γ(z)/Γ(¯ z ), 25

(3)

31. Approximation to cos γ for the Coulomb Scattering of Relativistic Electrons where z = x + iy.

(4)

Here, x and y are real variables, and z¯ is the complex conjugate of z. Using the asymptotic expansion of ln Γ(z) (Stirling’s series), we obtain, under the condition |y| > |x|, ∞ X 1 (−1)n G(z) ' exp i 2y(ln |y| − 1) ± π x − + S (x) 2n−1 n 2 n=1 y

( "

#)

(5)

,

where the upper of the ± sign is to be used for y > 0 and the lower for y < 0. The function Sn (x) is given by Sn (x) =

n X 1 (−1)k 1 x2n − x2n−1 − Bk Hn+1−k (k)x2(n−k) , n(2n − 1) 2n − 1 k k=1

with Hn (k) =

M −1 X m=0

(6)



n for n 6 k (2k + n − m − 3)! , M = (2k − 2m − 1)!(2m)!(n − m − 1)! k for n > k.

In Eq. (6), the constants Bk are the Bernoulli numbers (B1 = 61 , B2 = these relations, we can write cos γ for q > 1 as

1 , 30

1 cos γ = Re G − iq G(1 + iq) 2 (∞ ) X (−1)n 1 ' sin Sn ( ) − Sn (1) . 2n−1 2 n=1 q

(7)

. . . ). Using

(8)

Applying the Taylor expansion, we obtain cos γ '

1 23 593 1 + + + + ... . 3 5 4q 128q 8192q 262 144q 7

(9)

To reduce the error near the point q = qc , we use the ninth-order term in 1/q with an adjustable coefficient. Thus, our expression is written in the form  3 P   1+ 

cos γ =  P 4  

n=1

n=1

an q 2n

1+

3 P n=1

cn /q 2n−1 + d/q 9

bn q 2n

for 0 6 q < qc

(10a)

for q > qc .

(10b)

Values of the coefficients cn (n = 1, 2, 3, 4) are as given in Eq. (9). The other coefficients, an , bn (n = 1, 2, 3), and d, as well as the value of qc have been adjusted so that Eq. (10) might satisfy the criterion of the best approximation in the Tchebyschev sense. We have taken aim at an accuracy not worse than the table of Berger and Seltzer (an absolute error of 5×10−6 ). The reference data used for solving the best-approximation problem have been generated within an error of 5 × 10−10 by a program adapted from that of Felder.4 The values determined for qc and the coefficients are given in Table I. The maximum absolute error of Eq. (10) with these numerical values is 2.2×10−6 . For q & 5, the error of Eq. (10b) is approximately expressed as 1.5/q 9 . Modifications of Eq. (10) for obtaining higher accuracies can easily be achieved on the basis of the present work; for large values of q, Eq. (8) can also be utilized without expanding the sine. 26

31. Approximation to cos γ for the Coulomb Scattering of Relativistic Electrons TABLE I Values of qc , an , bn , (n = 1, 2, 3), and d qc a1 a2 a3

Value 1.72 1.6750639 1.0484753 0.0544309

b1 b2 b3 d

Value 5.518470 6.471978 1.484431 0.004778

ACKNOWLEDGMENTS Our thanks are due to A. Mizohata and Y. Matsuda for their valuable aid in using the computer. Appreciation is also expressed to W. E. McBride of Western Illinois University for kind instruction on his algorithm for the nonlinear best approximation. REFERENCES 1 J.

H. BARTLETT, Jr. and R. E. WATSON, Phys. Rev. 56, 612 (1939). V. SPENCER, Phys. Rev. 98, 1507 (1955); see also. “Energy Dissipation by Fast Electrons,” NBS Monograph 1, National Bureau of Standards, Washington, D. C. (1959). 3 M. J. BERGER, in Methods in Computational Physics, Vol. 1, p. 135, B. ADLER, S. FERNBACH, and M. ROTENBERG, Eds., Academic Press, New York (1963). 4 R. M. FELDER, “MOGUS—A Code for Evaluating the Mott Scattering Cross Section and the Goudsmit-Saunderson Angular Multiple-Scattering Distribution for Use in Electron Transport Calculations,” BNL 50199 (T-549), Brookhaven National Laboratory (1969). 5 D. W. VANDE PUTTE, “Electron Energy-Loss and Multiple-Scattering Distributions Below 2 MeV,” PEL 235. Atomic Energy Board, Pelindaba, Republic of South Africa (1974). 6 M. J. BERGER and S. M. SELTZER, “ETRAN: Monte Carlo Code System for Electron and Photon Transport Through Slabs,” RSIC Code Package CCC-107, Radiation Shielding Information Center, Oak Ridge National Laboratory (1969). 2 L.

27

Paper published in Nuclear Instruments & Methods, Vol. 158, January 1979, Pages 521–523, (doi:10.1016/S0029-554X(79)95440-5) Copyright © 1979 by Elsevier B.V.

Approximations to Landau’s Distribution Functions for the Ionization Energy Loss of Fast Electrons Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Sakai, Osaka, Japan

(Received 19 June 1978) Approximations are given for the universal function φ(λ) appearing in Landau’s theory on the energy loss distribution of fast electrons by ionization and also for the function P −1 (r) inverse to the integral of φ(λ). Values of parameters have been determined with the criterion of the best approximation in the Tchebyschev sense. Two results are presented for φ(λ); the simpler one is accurate within an absolute error corresponding to 1.0% of the peak value of φ(λ) over the interval −2.9 ≤ λ < ∞, and the other, within a relative error of 1.4 × 10−3 over −3.275 ≤ λ ≤ 100. The approximation to P −1 (r) is accurate within a relative error of 9 × 10−4 over 0.001 ≤ r ≤ 0.999.

1. Introduction In estimating the spectral distribution of the energy loss of fast electrons by ionization, theories developed by Landau1 ) and by Blunck and Leisegang2 ) are commonly utilized; these theories give distributions using a universal function φ(λ) defined by1 ) 1 Z σ+i∞ exp(u ln u + λu)du. (1) 2πi σ−i∞ Sampling of energy losses in Monte Carlo calculations of electron transport is frequently made from a distribution given by the above theories. For this purpose the function P −1 (r) inverse to r = P (λ) is required, where P (λ) is the integral defined by φ(λ) =

P (λ) =

Z λ

φ(λ0 )dλ0 .

(2)

−∞

While Börsch-Supan3 ) has given a complete tabulation of φ(λ), the development of approximate expressions for φ(λ) and P −1 (r) is desirable to save computer memory and to make rapid evaluation in calculations pertaining to the energy loss distribution of electrons. In the present paper, we describe two approximations to φ(λ) with different ac−1 curacies and an approximation to P −1 (r), which are denoted as φIa (λ), φII a (λ), and Pa (r), respectively. 2. Definition and formulation We make here the following definitions: 28

32. Landau’s Distribution Functions for the Ionization Energy Loss • the error δf of fit = the maximum deviation (in magnitude) of the approximation from the data used as the value of the function; • the error δd of data = the upper bound on the deviation (in magnitude) of the data from the true value of the function; • the error δa of approximation = the upper bound on the deviation (in magnitude) of the approximation from the true value of the function. We have sought approximations with the criterion of the best approximation in the Tchebyschev sense. The problem of the best approximation is to find the values of parameters that minimize the value of δf for a given approximate expression. The outline of the method used for solving this problem has been described in a previous paper4 ). To make the formulation feasible, we consider a limited interval of the independent variable for each of the approximations, excluding the intervals on which the function to be approximated is practically unimportant. In formulating φIa (λ), importance has been attached to simplicity of the expression rather than accuracy, and δf and δa have been considered in the sense of the absolute −1 error. In the cases of φII a (λ) and Pa (r), we have aimed at achieving values of δf not greater than values of δd ; since the latter values are known in the sense of the relative error (see section 3), δf and δa have also been considered in this sense. Expressions tried are of the form of a rational function. In the cases where it has been found difficult to obtain a good approximation with a rational function of the original independent variable, transformation has been applied to the variable. To make the expression simpler, the interval considered for each of the approximations has been divided into two subintervals, and different expressions have been used for the subintervals. 3. Data and values of δd Values of Börsch-Supan’s table have been used as the data for the best approximation problem of φ(λ). The upper bound on the relative error of this table has been given3 ) as 7 × 10−4 , which is the value of δd in this case. The data used for P −1 (r) have been computed through numerical integration of φ(λ) by the use of Börsch-Supan’s values (λ ≤ 100) and those given by an asymptotic formula1,3 ) Table 1 Values of P −1 (r). The symbol E indicates powers of ten. r 0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.999

P −1 (r) −2.6292 −1.0922 −4.8279E−1 7.3855E−2 6.6472E−1 1.3557 2.2431 3.5260 5.7666 1.1644E+1 9.8498E+2

29

32. Landau’s Distribution Functions for the Ionization Energy Loss (λ > 100). Values of P −1 (r) thus obtained are shown in table 1 for several points of r. The relative error of the asymptotic formula is less than that of Börsch-Supan’s table over the interval where the formula has been used, and the relative error in the data caused by the process of numerical integration has been checked to be less than 5 × 10−6 . Therefore, the value of δd for P −1 (r) is about the same as that for φ(λ). 4. Results and remarks 4.1. φIa (λ) and φII a (λ) Expressions used for φIa (λ) and φII a (λ) can be written in a common form φa (λ):

φa (λ) =

 , k  X     ai ξ 2i   i=0 m  X

    

i=0

1+

,

ci ξ

i

1+

l X

!

bi ξ 2i , for λ1 ≤ λ < λp ,

i=1 n X

(3a)

!

di ξ

i

for λp ≤ λ ≤ λ2 ,

,

(3b)

i=1

where ξ = λ − λp ,

(4)

λp is the point at which φ(λ) reaches the maximum and is given by λp = −0.2225.

(5)

Respective values of k, l, m, n, λ1 , and λ2 for φIa (λ) and φII a (λ) are given in table 2 as well as those of ai , bi , ci , di , δf , and δa . As a measure of unimportance of φ(λ) outside the interval λ1 ≤ λ ≤ λ2 , the ratios φ(λi )/φ(λp ) (i = 1, 2) are shown in the last rows of table 2. The choice of the values of m and n has been based on the relation n = m + 2,

(6)

which is considered to be appropriate from the asymptotic behavior1,3 ) of φ(λ) for large positive λ. In the case of φIa (λ), the value of λ1 has been chosen to satisfy the relation φ(λ) < δa for λ < λ1 , which allows one to put φIa (λ) = 0 for λ < λ1 within the error δa . A preliminary result for φIa (λ) has shown that deviations of eqs. (3a) and (3b) from the data are of opposite signs at λ = λp and that the value of φIa (λ) jumps there by a magnitude greater than the value of δf attained. To avoid this situation, the value of c0 for φIa (λ) has been set equal to the value of a0 , and only the remaining parameters in eq. (3b) have been adjusted. The value of φ(λp ) is 0.18066, so that the value of δa for φIa (λ) corresponds to 1.0% of φ(λp ). In the case of φII a (λ), the value of δf is equal to that of δd , making the value of δa twice as large as the latter. 4.2. Pa−1 (r) Out of the entire interval 0 < r < 1, we consider the interval 0.001 < r < 0.999 and divide it into two subintervals by a point r = rc . The expressions used on the subintervals can be written in a common form: Pa−1 (r)

=

K X

,

Ai s

i

i=0

1+

L X i=1

30

!

Bi s

i

,

(7)

32. Landau’s Distribution Functions for the Ionization Energy Loss Table 2 Values of parameters and related quantities for φIa (λ) and φII a (λ). φIa (λ)

φII a (λ)

k l m n

1 2 0 2

λ1 λ2

−2.9 ∞

−3.275 100

a0 a1 a2 a3

0.179 04 −0.025 30 ... ...

0.180 −0.051 0.005 −0.000

787 844 018 163

5 50 404 851 7

b1 b2 b3 b4 b5

0.0482 0.1132 ... ... ...

−0.037 0.063 −0.008 0.001 0.000

727 593 391 548 007

5 4 53 11 653 3

c0 c1 c2

0.179 04 ... ...

0.180 536 0.178 142 0.012 577

d1 d2 d3 d4

0.0134 0.1355 ... ...

0.972 0.330 0.090 0.012

δf

1.7×10−3 (absolute)

7×10−4 (relative)

δa

1.8×10−3 (absolute)

1.4×10−3 (relative)

φ(λ1 )/φ(λ)p φ(λ2 )/φ(λ)p

7.17×10−3 0

4.12×10−4 5.95×10−4

3 5 2 4

05 00 31 359

where s = ln[r/(l − r)],

(8)

and each of K, L, Ai , and Bi takes on different values for the subintervals. The value of rc has been determined so as to balance the values of δf on the subintervals, and we have obtained rc = 0.811. (9) Values of K, L, Ai , Bi , δf , and δa are shown in table 3. The value of δf is less than that of δd in this case, so that the value of δa is greater than the latter by only a fraction of it. We are indebted to Prof. T. Doke of Waseda University for his useful comment. We also

31

32. Landau’s Distribution Functions for the Ionization Energy Loss Table 3 Values of parameters and related quantities for Pa−1 (r). K L

0.001 ≤ rc ≤ 3 3

A0 A1 A2 A3 A4

1.355 1.375 −0.099 0.027 ...

B1 B2 B3

−0.402 700 7 0.060 765 45 −0.005 040 17

821 008 755 415

6 2 9 7

rc ≤ r ≤ 0.999 4 2 1.757 0.827 0.861 −0.187 0.057

368 982 266 555 282

9 34 02 67 253

−0.202 162 17 0.010 800 195 ...

δf

1.6 × 10−4 (relative)

δa

9 × 10−4 (relative)

wish to thank A. Mizohata and Y. Matsuda for their valuable aid in using the computer. References 1)

L. Landau, J. Phys. USSR 8 (1944) 201. O. Blunck and S. Leisegang, Z. Physik 128 (1950) 500. 2 ) W. Börsch-Supan, J. Res. Natl. Bur. Standards 65B (1961) 245. 4 ) T. Tabata and R. Ito, Nucl. Instr. and Meth. 146 (1977) 435. 2)

Commentary The following errors were found in the published version. Page 522 522 522 523

Location Table 1, 2nd col., 3rd l. from bottom Table 1, 2nd col., 2nd l. from bottom Table 1, 2nd col., bottom l. Table 3, 1st col., 5th entry

32

Now reads 5.7985 1.1644−1 9.8501E−2 A3

Should read 5.7666 1.1644E+1 9.8498E+2 A2

Paper published in Japanese Journal of Applied Physics, Vol. 20, Issue 1, January 1981, Pages 249–258 (doi:10.1143/JJAP.20.249) Copyright © 1981 by the Japan Society of Applied Physics

An Algorithm for Electron Depth–Dose Distributions in Multilayer Slab Absorbers Tatsuo Tabata and Rinsuke Ito Radiation Center of Osaka Prefecture, Sakai, Osaka 593

(Received August 23, 1980; accepted for publication October 25, 1980) An algorithm to evaluate depth–dose distributions produced by plane-parallel electron beams normally incident on two- or three-layer slab absorbers has been developed. It is based on a simple model of electron penetration across the interface, and makes use of empirical equations previously formulated as well as ones newly developed. Distributions obtained by the algorithm have been compared with available experimental and Monte Carlo results for electrons of incident energies from 0.1 to 10 MeV, showing good agreement in most cases. The algorithm is considered to be valid for incident energies from 0.1 to 20 MeV and for absorbers consisting of slabs of atomic numbers from about 5.6 (polystyrene) to 82.

§1. Introduction When a layer of material m preceded, followed or sandwiched by a different material m0 is irradiated by electrons, the energy deposition profile in m is very different from that in the equivalent layer in a homogeneous absorber of m, mainly because of the effect of backscattering across the interface between the different materials. In order to achieve efficient utilization of electron beams in various applications, knowledge of depth–dose distributions in such multilayer configurations is often important. Whereas methods of Monte Carlo calculation1,2) and of experimental measurement3−5) for these distributions have been developed, it would also be helpful to have a simple method of computation. For depth–dose distributions in a semiinfinite medium consisting of a single material, semiempirical algorithms have been formulated by Kobetich and Katz6) and by Tabata and Ito.7) On the basis of repeated use of the algorithm of the latter authors, we have developed an algorithm for depth–dose distributions of plane-parallel electron beams normally incident on two- or three-layer slab absorbers.∗ Distributions obtained by the present algorithm show moderately good agreement with available experimental and Monte Carlo data, and the algorithm is considered to be valid for incident energies from 0.1 to 20 MeV for absorbers consisting of slabs of atomic numbers from about 5.6 (polystyrene) to 82. ∗

Part of this work was presented at the 6th International Congress of Radiation Research, 1979, Tokyo.

33

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers §2. Formulation 2.1 Definitions The following definitions are given for the symbols used in this Paper. T0 = incident kinetic energy of electrons. d(x) = dose per unit fluence at depth x under the conditions treated here. dSIM (x; T, m) = dose per unit fluence at depth x in a semiinfinite medium m due to a plane-parallel electron beam of incident energy T normally impinging on the surface. It is evaluated here by the algorithm of Tabata and Ito7) with the minor modifications given in Appendix A. 2.2 General For the sake of simplicity, an outline of the formulation of the algorithm is given here for the case of two-layer absorbers; details of the algorithm for two- or three-layer absorbers will be given elsewhere. Let us consider the configuration as shown in Fig. 1. Region 0 is empty space, region 1 is a layer of thickness xb consisting of material m1 , and region 2 is a layer of effectively semiinfinite thickness consisting of material m2 . A plane-parallel beam of monoenergetic electrons is normally incident on the interface between regions 0 and 1. We use a simple model of electron penetration across the interface. The features of the model can be explained by the schematic paths illustrated in Fig. 1. First, we consider the paths shown by solid lines. In region 1, segment s10 of the paths represents penetration of incident electrons, and s12 , penetration in the reverse direction of electrons backscattered from region 2. In region 2, s20 represents penetration of electrons transmitted through region 1, and s21 , penetration of electrons backscattered from region 1 after being backscattered from region 2. The contribution to the dose d(x) from each segment is evaluated by tracing the average behavior of electrons and using the function dSIM defined in the previous subsection. The use of dSIM makes it necessary to consider the other paths shown by dashed lines for correction purposes. They represent imaginary penetration processes of electrons backscattered from a layer of m1 supposedly placed in region 0 or 2. In the following, approximate expressions for the contributions to d(x) from segments s1i and s2j (i = 0, 1, . . . , 4; j = 0, 1, 2) are denoted as d1i (x) and d2j (x),

Fig. 1. Absorber configuration and schematic paths of electrons considered in the algorithm. Solid lines represent actual paths; dashed lines, imaginary paths of electrons backscattered from the layer of material m1 supposedly placed in region 0 or 2.

34

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers respectively, with appropriate signs. 2.3 Expressions for d(x) In region 1, the main contribution to d(x) is d10 (x), which comes from segment s10 . To express this by dSIM , we assume that region 2 is filled with material m1 instead of m2 . Then d10 (x) is given by dSIM (x; T0 , m1 ). In the next approximation, we subtract the contribution d11 (x) due to electrons backscattered from the semiinfinite layer of m1 assumed to be in region 2 (segment s11 ), and add instead the contribution d12 (x) from segment s12 . If we assume again that region 0 is filled with material m1 , both d11 (x) and d12 (x) can be expressed by dSIM with appropriate parameters and multiplied by a normalization factor. These terms are corrected in turn for the effect of m1 assumed to be in region 0. This correction is given by −d13 (x) and −d14 (x) (segments s13 and s14 ). Neglecting the effect of further backscattering processes, we have d(x) = dSIM (x; T0 , m1 ) − [d11 (x) − d13 (x)] + [d12 (x) − d14 (x)]

(x ≤ xb ).

(1)

In region 2, the main contribution is from d20 (x), and the term d21 (x) is added to include the effect of backscattering from m1 in region 1 (segments s20 and s21 ). The latter term is evaluated under the assumption that region 0 is filled with m1 . A correction to compensate for this assumption is made by subtracting the term d22 (x) (segment s22 ).† Neglecting further backscattering processes, we have d(x) = d20 (x) + d21 (x) − d22 (x)

(x ≥ xb ).

(2)

It is fairly simple to express d11 (x), d13 (x) and d14 (x) by using dSIM and normalization factors, because in these cases the electrons enter material m1 after passing through the same material supposedly placed in region 2 or 0. In the case of d12 (x) and all the three terms for region 2, on the other hand, the effect of the change of material across the interface lastly traversed by the relevant electrons should be taken into account. The method of calculating these terms is explained in the next subsection, taking the case of d20 (x). 2.4 Expressions for d20 (x) and similar terms To evaluate d20 (x) by the use of dSIM , we replace material m1 of the first layer by material m2 , at the same time replacing the incident energy T0 and the thickness xb of the first layer by an equivalent energy T0 0 and an equivalent thickness xb 0 , respectively (see Fig. 2). We then assume that the first approximation to d20 (x) is given by dSIM (x0 ; T0 0 , m2 ) multiplied by a normalization factor; here x0 is given by x 0 = x − xb + xb 0 .

(3)

The equivalence condition used to determine T0 0 and xb 0 is that the mean energy per residual electron and the angular width of multiple scattering, both evaluated for the electrons transmitted through the first layer, remain unchanged. The basic idea of this condition was originally proposed by Tanaka et al.‡ †

It is unnecessary to consider the continuation of the segment s13 into region 2, because the assumption under which s13 is considered necessary is irrelevant to the terms d20 (x) and d21 (x). ‡ R. Tanaka, K. Yotsumoto and Y. Nakamura: Preprints of the 32nd Autumn Meeting of the Japan Society of Applied Physics, Osaka, November, 1971.

35

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers

Fig. 2. Equivalent configuration and schematic paths considered in the evaluation of d20 (x) (lower part); upper part of the figure shows correspondence with the original configuration. Circled numbers indicate the order of backscattering across the plane x=xb 0 . The “1st and 2nd terms” are those of eq. (4).

The shape of dSIM (x0 ; T0 0 , m2 ) includes the effect of backscattering occurring within m2 across the depth xb 0 . However, the second- and higher-order backscattering processes between regions 1 and 2, i.e. the backscattering from region 1 to region 2 and further backscattering processes across the interface (see Fig. 2) do not in fact take place within m2 , but between m1 and m2 . Correction for the absence of these backscattering processes within m2 is made by using the depth–dose curve for the electrons that have experienced backscattering processes up to the second order together with a normalization factor; in this factor the reduction in the total energy of these electrons due to third order backscattering is taken into account. [The effect of the actual second order backscattering between m1 and m2 has been accounted for by the term d21 (x) in eq. (2).] Thus we have d20 (x) = c1 [dSIM (x0 ; T0 0 , m2 ) − c2 dSIM (x00 ; T0 0 , m2 )],

(4)

where c1 and c2 are normalization factors, and x00 is the effective depth traversed by the electrons to reach a depth x0 after the second order backscattering across a depth xb 0 (see Fig. 2). The expressions for d21 (x) and d22 (x) are similar to eq. (4) except that the second term in eq. (4) is neglected in the case of these dose components. 2.5 Evaluation of parameters To obtain detailed expressions for the dose components d1i (x) and d2j (x) (i = 1, 2, 3, 4; j = 0, 1, 2), methods of evaluating the normalization factors, the equivalent energies, the equivalent depths and the effective depths mentioned in the previous subsections should be specified. The normalization factor is given by the ratio D1 /D2 , where D1 is the residual energy per incident electron evaluated for the electrons that have reached the initial point of the relevant segment of the schematic path, e.g. the point P in Fig. 1 in the case of d11 (x); and D2 is the integral of the relevant dSIM over the depth parameter from the value corresponding to the same initial point to infinity. We evaluate D1 as the product of the residual number-fraction of electrons and the mean energy per residual electron. The effective depth is defined as the depth in a single semiinfinite medium at which 36

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers the mean energy per residual electron has fallen to the same value as the mean energy per residual electron after traveling along the relevant schematic path down to the point considered. The principle of evaluating the equivalent energy and depth was described in the previous subsection. The procedure for this evaluation is to solve nonlinear simultaneous equations, and an iterative method is used for this purpose. 2.6 Empirical and approximate formulas used The residual number-fraction of electrons is evaluated by the use of empirical formulas for the transmission and backscattering coefficients. Empirical formulas for these quantities in the case of normal incidence have been given previously.8,9) In the present algorithm, a formula is also required for the backscattering coefficient η of the divergent electron flux after passing the first layer of a given thickness, say x1 ; the backscatterer is the second layer consisting of the same material as the first,§ and is of semiinfinite thickness. We express η by the weighted average of the backscattering coefficients ηn and ηi for normal and isotropic incidence, respectively: η = f ηn + (1 − f )ηi .

(5)

Here f is a function which decreases from unity to zero with increasing x1 , and we assume that it is given by f = exp[−(x1 /κ1 xD )κ2 ]; (6) the symbols κ1 and κ2 denote constants independent of incident energy and target material, and xD is the diffusion depth defined as the depth where the rms deflection of transmitted electrons reaches saturation. Empirical formulas for xD and ηi have been developed for use here, and are given in ref. 10 and Appendix B of the present paper, respectively. The values of κ1 and κ2 were determined so as to minimize the deviation of the present algorithm from the experimental and Monte Carlo data of depth–dose distributions in two-layer slab absorbers reported by Eisen et al.3) The results are given by κ1 = 2.2, κ2 = 1.15.

(7) (8)

The mean energy per residual electron is evaluated by using an empirical formula7,11) for this quantity in the case of transmission or by using an empirical formula12) for the mean fractional energy loss in the case of backscattering. Although the latter formula was developed for normal incidence, it has been assumed to be approximately valid also for a divergent flux. In evaluating the angular width of multiple scattering required for the calculation of the equivalent energies and depths, the approximate expression of Hanson et al.13) for the width given by Molière’s theory14) has been used up to the depths where this theory is invalid; this is permissible here because it is not the angular width itself but the relative extent of multiple scattering in different media that matters. To obtain values of the parameter B appearing in the expression for the angular width, an approximate §

It suffices here to consider the case in which the first and second layers are of the same material, because we consider the equivalent layer as described in §2.3 when the materials are different.

37

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers Table I. Results of comparison of the present algorithm with the experimental and Monte Carlo data of Eisen et al. (ref. 3) for the case of two-layer absorbers. The incident energy of electrons is 2 MeV. In the column of the absorber, polystyrene is denoted by (C8 H8 )n . Thicknesses of absorbers are expressed as fractions of the continuous slowing-down approximation range R0 of the incident electrons in each material. Values of R0 are 1.21 g/cm2 for Al, 1.56 g/cm2 for Au, 1.00 g/cm2 for (C8 H8 )n , 1.34 g/cm2 for Cu, 1.48 g/cm2 for Sn (ref. 16). In the columns of the rms deviation δ, “Alg.–Exp.” means the deviation of the result of the algorithm from the experimental data, and “Alg.–MC”, the deviation from the Monte Carlo data. Values of the deviation “Exp.–MC” of the experimental data from the Monte Carlo results are also given in the last column. Case No. 1 2 3 4 5 6 7 8 9 10 11 12

Absorber Al–Au Al–Au Au–Al Au–Al (C8 H8 )n –Cu (C8 H8 )n –Cu Cu–(C8 H8 )n Cu–(C8 H8 )n (C8 H8 )n –Sn (C8 H8 )n –Sn Sn–(C8 H8 )n Sn–(C8 H8 )n Overall

Thickness (Fraction of R0 ) 0.15–∞ 0.33–∞ 0.11–∞ 0.22–∞ 0.21–∞ 0.41–∞ 0.14–∞ 0.27–∞ 0.21–∞ 0.42–∞ 0.12–∞ 0.25–∞

Alg.–Exp. 4.7 5.3 10.1 8.6 10.8 8.7 6.9 7.4 9.1 8.8 10.5 12.1 8.8

δ (%) Alg.–MC 4.4 4.8 5.5 8.5 5.5 7.2 8.5 7.5 7.4 5.5 13.3 11.8 7.9

Exp.–MC 6.1 5.9 10.3 9.2 8.3 8.3 12.7 5.6 5.9 8.2 11.7 7.9 8.6

formula developed by the present authors10) was used. §3. Results and Discussion The depth–dose distributions given by the present algorithm have been compared with the experimental and Monte Carlo results of Eisen et al.3)¶ for the case of two-layer absorbers, and the results are summarized in Table I. Eisen et al. reported the distributions caused by electrons of 2-MeV incident energy in absorbers of aluminum–gold, polystyrene– copper and polystyrene–tin; four cases were studied for each combination of materials by exchanging the first- and second-layer materials and by using thin and thick layers in front. Their Monte Carlo results were generated with the multilayer electron-transport code zebra developed by Berger and described by Buxton.15) In Table I, the thicknesses of the absorbers are expressed as fractions of the continuous slowing-down approximation range R0 of the incident electrons in each material, the values of R0 used being taken from the table of Berger and Seltzer.16) The last three columns of Table I give values of the rms relative deviation δ as a measure of agreement. It is defined for the case of the deviation of the algorithm from the experimental data (this is denoted as Alg.–Exp. in Table I), for example, by δ=

n X

wi {[d(xi ) −

dexp i ]

2 dexp i }

.

!1/2

n

.

i=1 ¶

Their results have been read off from non-reduced copies of drawings provided by Eisen.

38

(9)


Fig. 3. Depth–dose profile of 2-MeV electrons normally incident on an aluminum–gold slab absorber (case No. 1 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Fig. 4. Depth–dose profile for 2-MeV electrons normally incident on an aluminum–gold slab absorber (case No. 2 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Here n is the number of data points, wi is the weight for the ith experimental data point dexp i , and d(xi ) is the dose calculated by the algorithm for the depth xi . We use wi given by‖ , wi =

n X

2 (dexp i )

2 (dexp i ) .

(10)

i=1

To reduce the effect of the rather large scatter of points in the original experimental data, sets of data obtained by averaging two successive points were used in the evaluation of δ. As can be seen from Table I, the deviations of the algorithm from the experimental and Monte Carlo results are nearly the same as the deviation of the experimental data from the Monte Carlo results. The accuracy of the algorithm, therefore, can be said to be satisfactory for many applications. The depth–dose distributions for case Nos. 1–4, 5, 8, 10 and 11 in Table I are shown in Figs. 3–10, respectively. In the cases of Fig. 3, for example, Eisen et al.3) found that the gold backing increased the dose in the first layer by 50–60% compared with the dose received by the equivalent aluminum layer in a semiinfinite aluminum absorber; the present algorithm gives exactly the correct amount of this increase. It can be seen from ‖ The minimization of δ with wi given by eq. (10), used in the determination of the values of κ1 and κ2 , is equivalent to the minimization of the rms absolute deviation with equal weights for all the data points.

39


Fig. 5. Depth–dose profile for 2-MeV electrons normally incident on a gold–aluminum slab absorber (case No. 3 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Fig. 6. Depth–dose profile for 2-MeV electrons normally incident on a gold–aluminum slab absorber (case No. 4 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Fig. 7. Depth–dose profile for 2-MeV electrons normally incident on a polystyrene–copper slab absorber (case No. 5 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Figs. 3–10 that the algorithm reproduces moderately well the experimental and Monte Carlo depth–dose profiles for various combinations of materials and thicknesses. The results of comparison with some of the data generated by the Monte Carlo program sand1)∗∗ are summarized in Table II, and the depth–dose profiles for case Nos. 13 and ∗∗

M. J. Kniedler: Dissertation, Institute for Physical Science and Technology, University of Maryland, College Park, Maryland, 1968.

40


Fig. 8. Depth–dose profile for 2-MeV electrons normally incident on a copper–polystyrene slab absorber (case No. 8 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Fig. 9. Depth–dose profile for 2-MeV electrons normally incident on a polystyrene–tin slab absorber (case No. 10 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

Fig. 10. Depth–dose profile for 2-MeV electrons normally incident on a tin–polystyrene slab absorber (case No. 11 of Table I). Points, experimental data (ref. 3); histogram, Monte Carlo result (ref. 3); curves, present algorithm.

20 are illustrated in Figs. 11 and 12, respectively. The incident energy of the data ranges from 0.1 to 10 MeV. The physical model of sand, as well as that of zebra, assumes continuous slowing-down of electrons, and depth–dose profiles computed with this model show a faster decrease of the dose at large depths than actual profiles, especially in the case of high incident energies (see Fig. 12). Taking this fact into consideration, we can say that agreement is also satisfactory for the two-layer absorbers listed in Table II.

41

33. Electron Depth–Dose Distributions in Multilayer Slab Absorbers Table II. Results of comparison of the present algorithm with the Monte Carlo data of Kniedler and Silverman (ref. 1) for two-layer absorbers and with the experimental and Monte Carlo data of Lockwood et al. (ref .4) for three-layer absorbers. Values of R0 used as a unit of thickness are (ref. 16 for T0 =0.1, 0.5 and 10 MeV; ref. 4 for 1 MeV): T0 =0.1 MeV: 0.0140 g/cm2 for H2 O, 0.0211 g/cm2 for Fe; T0 =0.5 MeV: 0.174 g/cm2 for H2 O, 0.248 g/cm2 for Fe; T0 =10 MeV: 4.88 g/cm2 for H2 O, 5.93 glcm2 for Fe; T0 =1 MeV: 0.490 g/cm2 for C, 0.551 g/cm2 for Al, 0.772 g/cm2 for Au. Case No. 13 14 15 16 17 18 19 20 21 22 23

Absorber H2 O–Fe Fe–H2 O H2 O–Fe Fe–H2 O Fe–H2 O Fe–H2 O Fe–H2 O Fe–H2 O Fe–H2 O C–Au–C Al–Au–Al Overall

Thickness (Fraction of R0 ) 0.1–0.5 0.5–0.5 0.1–0.5 0.1–1.0 0.2–1.0 0.5–0.5 0.1–1.0 0.2–1.0 0.5–0.5 0.18–0.06–∞ 0.08–0.06–∞

T0 (MeV) 0.1 0.1 0.5 0.5 0.5 0.5 10 10 10 1 1

δ (%) Alg.–Exp Alg.–MC ... 9.0 ... 11.6 ... 4.8 ... 11.7 ... 10.7 ... 7.1 ... 11.7 ... 12.1 ... 11.8 16.1 9.9 8.5 7.8 12.9 10.1

Fig. 11. Depth–dose profile for 0.1-MeV electrons normally incident on a water–iron slab absorber (case No. 13 of Table II). Histogram, Monte Carlo result (ref. 1); curves, present algorithm.

The algorithm was checked for three-layer absorbers by using the experimental and Monte Carlo data reported by Lockwood et al.,4) and the results are shown in the last two rows of Table II. The depth–dose distributions for these cases are shown in Figs. 13 and 14. Their Monte Carlo data were generated by the code tiger.2) Agreement between the results of the algorithm and the data is poor for the first layer in the carbon–gold–carbon configuration, and the values of the dose calculated by the algorithm for the second goldlayer are systematically 10–20% higher than the data in both Fig. 13 and Fig. 14. In these cases, the high atomic number and the small thickness of the second layer make the correction terms for the first two layers very large, so that errors of approximation accumulate to a rather large fraction of the dose, showing the limitations of the present

42


Fig. 12. Depth–dose profile for 10-MeV electrons normally incident on an iron–water slab absorber (case No. 20 of Table II). Histogram, Monte Carlo result (ref. 1); curves, present algorithm.

treatment. Better agreement is expected for smaller atomic numbers and larger thicknesses of the second layer. As for the third layer, however, agreement is fairly good in both cases, making the application reliable to cases where only the dose in the third layer is important. An example of this is radiation processing with electrons of energies lower than about 600 keV; the effect of the accelerator output-window and a layer of air is not negligible for these electrons, and the dose in the sample should be evaluated by considering the three-layer configuration. The algorithm has not been checked for energies above 10 MeV because of lack of data for comparison. The function dSIM and other empirical relations used are valid at least up to 20 MeV, and the effect of backscattering across the interface between different materials decreases with increasing energy. Considering these facts as well as the results of the comparison described, it can be concluded that the present algorithm is valid for incident energies from 0.1 to 20 MeV and for two- or three-layer absorbers consisting of slabs of atomic numbers from about 5.6 (polystyrene) to 82. From the values of δ for the overall data given in Tables I and II, the rms deviation as defined by eq. (9) is expected to be around 8–10% for an arbitrary case satisfying the conditions of validity. The advantages of the present algorithm over the Monte Carlo method are its simpler computer program and shorter computation time. The algorithm consists of about 800

Fig. 13. Depth–dose profile for 1-MeV electrons normally incident on a carbon–gold–carbon slab absorber (case No. 22 of Table II). Points, experimental data (ref. 4); histogram, Monte Carlo result (ref. 4); curves, present algorithm.

43


Fig. 14. Depth–dose profile for 1-MeV electrons normally incident on an aluminum–gold–aluminum slab absorber (case No. 23 of Table II). Points, experimental data (ref. 4); histogram, Monte Carlo result (ref. 4); curves, present algorithm.

lines of fortran,†† this number being about one tenth of that of a typical Monte Carlo program, etran,17) for transport of fast electrons. The computation time required to evaluate a depth–dose profile with the algorithm has been estimated to be about 1/400 to 1/20 of the time required to compute the history of 1000 electrons with the Monte Carlo method (data for the computation time of etran and sand have been used). Acknowledgement The authors would like to express their appreciation to Dr. H. Eisen for sending us non-reduced copies of the drawings in ref. 3. They would also like to thank Professor J. Silverman for his encouragement and a gift of a copy of the dissertation by Dr. M. J. Kniedler. The authors wish to acknowledge S. M. Seltzer for providing us with his new computation results for the backscattering coefficient of electrons and Dr. H. W. Thümmel for sending us a copy of his manuscript on the diffusion depth of electrons. Thanks are also due to Professor D. Harder for his valuable discussion with one of the authors (T. T.) and also to A. Mizohata for his help in using the computer. Appendix A In this appendix, we describe minor modifications of the algorithm of ref. 7 for depth– dose distributions in a semiinfinite medium consisting of a single material. The modifications made are: (1) the expression for low atomic-number materials of the parameter a1 in the extrapolated-range formula and (2) the expression for β appearing in the formula for the fractional number of electrons deposited at depths ≥ x. In ref. 7, the parameter a1 is given by a1 = 0.2335A/Z 1.209 ,

(A.1)

where A and Z are the atomic weight and atomic number of the absorber, respectively. The modification of this parameter is that for Z