Rinsuke ITO and Tatsuo TABATA

rssN 0285-8797 RADIATION CENTER OF OSAKA PRXFECTIIRE TECMTICAL REPORT 8

SEI"IIEMPIRICAL CODE

EDMULT

FOR DEPTH-_DOSE DISTRIBUTIONS OF ELECTRONS

IN

MULT]LAYER SLAB ABSORBERS:

REVISIONS AND APPLICATIONS

Rinsuke ITO and Tatsuo

TABATA

November l987

klil ik i

i

-

bk4'J #.!{'f

,,A

ti) r

3'"fii

RAD]ATION CENTER OF OSAKA PREFECTURE SHTNKE-CHO,

SAKAI, OSAKA 593,

JAPAN

ABSTRACT

The model and an outline of a code to compute depth-dose

distributions produced by plane-paral1e1 electron beams normally incident on roultilayer slab absorbers are described. The algorithm of the code 1s based on a simple model of electron penetration across the interface, and uses ernpirical equations for transmission and backscattering. It is applicable to incident energles from about 0.1 to 20 MeV and to absorbers consisting of one to three layers of materials with atomic numhers .from about 5.6 (polystyrene) to 82. Revisions and appJ.ications of the code are described. Listings of a revised versj-on af the F0RTMN code are given" Application programs for 6 persoftal computer have been developed to tabulate and plot depth-dose distributions and to calculate absorbed-dose coefficients. The present report supersedes Radiation Center of 0sals.a Pref ecture Tecl-rnical Report No . 1, ( I98l. ) and ibid . No . 3 ( 1983) , Appenclix II.

- r_11 -

* E vel$t{6 ir=f-d-

rF s>E+

6-=+.ffiffi:=

-

€>i#E*R

itffi F- E f) IVr u r_ -r EL

L=

{rffie. aHE* o+ffi€+#H*{ - #EffiIEtFi:*EcclF.t Lt: k E aWtrffiE }fi A EfHf 6 fi*) €t= - F i:= t ra, 4 olffiL.ffiffi tffi"(6 . i O : - F fiT tVJ t) X,A l*. €+4€rFtriBi&irHlf6ffi.HrrH4tL L $e L. frffitt*frffif;lEf,f?Bffi t{HHL{t,t5" LCI=-Ft*. 719fl44,{-*l 0.1"-20 Hey. HT#€*! 5.6 t,-11 Ur+vy) - SZ aWHa 1-3 Eli.6&6&rEtr[lf.fL(i6H"E6o r-]so& iTt[6H{:=I}(}S,\. dffiTffiz r- }- jv . e-Fot| 7},Az-.f" iBEffiE.lyfra ffit

BfliE*. 6

-?ffi,)ffiHyn/7J^tffi#L/:" dsffi6$f*. /#ffitr+E #i:[Y-(ftr,b.,o)achb

"

-l-v-

No.1

L.

IEI

uo.3

z$t

CONTENTS

Page CODE

I. II.

SI]MMARY

1

INTRODUCTION

.

OUTLINE OF THE

3 MODEL

4

1. Definitj-ons and Assumptions 2. The Model

II]. IV.

OUTLINE OF THE

CODE

V.

1-7

.

t7

.

t7

APPLICATIONS

20

1. General ... 2. Absorbed Dose Coefficient and Application Programs for a Personal Computer 3. Energy Resolution of Sampling Calorimeters VI. ILLUSTMTIVE RESULTS 1. Depth-Dose Distribution 2. Absorbed-Dose Coefficlent VII. LISTINGS OF THE CODE AND THE

20 22

25 34 DOCUMENTS FOR THE

35

1. Listings of EDMULT 2. Documents for the Applicatj,on programs APPENDIX 1. AN ALGORITHM FOR ELECTRON DEPTH_DOSE II.

20

25

APPLICATION PROGRAMS ...

APPENDIX

6

L2

REVIS]ONS

1. The lst Revision 2. The 2nd Revlsion

4

DISTRIBUTIONS IN MULTILAYER SLAB ABSORBERS ADDITIONAL FIGURES TO COMPARE THE ALGORITHM WITH EXPERIMENTAL AND MONTE-CARLO RESULTS

REFERENCES

35 58

62

/) B5

-v-

CODE SI]UMARY*

A. Title of

Code

a Code for Evaluating Electron Depth--Dose Dlstributlons in Multilayer Slab Absorbers. The present version supersedes the code listed in Radiat. Center Osaka Prefect. Tech. Rep. 3. EDMULT stands for energy deposition in multilayer absorbers. EDMULT:

B. Auxiliary Routlnes None.

C. Coding

Language

FORTRAN 77.

D. Nature of Problem

So1ved

computes deptb-dose distributlons produced by plane-para11e1 electron beams normally incident on single-, twoEDMULT

or three-1ayer slab absorbers. The information produced by EDMULT are differential and integral energy-deposition by an electron beam as a function of the depth 1n the absorber. E" Method of Solution The energy deposited. per incident electron per unit depth in the homogeneous semiinfinj-te absorber is computed by the

algorithm originally developed by Kobetich and Katz. The depth-dose distribution in the finite single-layer '* Adapted f rom Ref .

1.

-1-

absorber and that in the multilayer absorber are computed on the basis of a simple model of electron penetration across the interface. In the mode1, the following two assumptions are made: (1) The distribution is given by the sum of contributions from schematic segments of paths, each segment representing the forward or backward passage I the backward passage is considered to take into account the effect of backscattering from each layer. (2) An equivalence rule is applicable to the depttr-dose

distribution. The rule states that the distribution in the second layer of the multilayer absorber remains a,pproximately the same when both the residual energy and the half-va1ue angle of multiple scattering at the interface remain unchanged under the replacement of the material of the first 1ayer, its thickness and the incident energy of the electrons. F. Restrictions or Limitations The model is applicable to the incident energies of electrons from about 0.I to 20 MeV and to absorbers consisting of slabs of atomic numbers from about 5.6 (effective atomic-number of polystyrene) to 82. The thickness of a single-1ayer absorber and that of the second layer in a two-layer absorber can be either finite or semiinfinite. The last layer in a three-1ayer absorber must be of semiinfinite thickness. G. Typical Running Time The coniputation time required to evaluate a depth--dose profile with EDI'IuLT was estimated to be about 11400 to LlZo of the time required to compute the history of 1000 electrons wj-th the typical Monte Carlo code (data for the computation time of CCC-107/ETMN (Ref . 2) and SAND (Ref . 3) was used in the estimation

)

.

-2*

I.

INTRODUCTION*

To use electron beams efficiently in various applications, knowledge of depth-dose distributions in absorbers of multila.ver

configuratj-ons is frequently important. Experimental5-7 and eomputational3'8-10 methods to obtain this knowledge have been developed. Application of these methods, however, is timeconsuming and expensive. Therefore, it would be helpful to have a si-mp1e method of computing depth-dose distributions in multilayer absorbers. We have developed a number of empirical formulas related to penetration of electrons. Using these formulas, we have refinedll the algorithm given by Kobetich and Katzl2 for the depth--dose distributions of plane-para11e1 electron beams normally incident on multilayer slab absorbers. An outline of the latter algorithm has been described for two-1ayer absorbers.l3t In the present report, additional explanation is made of the fu11 algorithm including the case of three-1ayer absorbers. Then we give an outline of a FORTMN code for the algorlthm, revision and applications of the code, illustrative results of an application program and listings of the code and the application program. This report supersedes Ref. 4 and Appendix II. of Ref. 14.

* Adapted from Ref. 4. t Reference 13 is reproduced in Appendix I of this report.

-3-

II.

OUTLINE OF THE

MODET

1. Definitions and AssumPtions The following definitions are given for the symbols used in

this report. T0 = incident kinetic energy of electrons. x = depth in the absorber. d(x) = dose per unit fluence at depth x in the conciiti.ons treated here.

d51v(xi

r' m)

;.i.,i:":::J:i;::.:.::.-:1.:JTl:":#l;"' impi.nging on the surf ace.

Consider the absorber configuration as shown in Fig. 1. Region 0 is empty space. i'laterials and thicknesses of regions from l to 3 are as follows:

1: material m1 , thickness xtr1. Region 2: rnaterial m2, thickness x12. Region 3: material rnq, semlinfinite thickness" Regi"on

A plane-paraL1e1 beam of monoenergetic electrolls is normally lncident on the interface between reglons 0 and tr. The present algorlthm gives the doses d(x) given try these electrons in regions from 1 to 3 as a function of depth x.

-4-

REGION O

ELECTRON BEAM

REGION

REGION 2

REGION 3

ml

m2

m3

sro

seo

sso

I

sr Q

"21 c

Stz

'22

sts

-23 c

( stq

'Lr

"3

szq

;

szs

ss:

'Lr**Lz

DEPTH ------->

Fig. 1. Absorber configuration and schematic paths of electrons considered in the algorithm. Virtual paths required for correction purposes are i ncl uded

-5-

I

2. The Model For the single-1ayer absorber of semiinfj-nite thickness consisting of material m1, the dose d(x) to be calculated is simply given by dg1p1(x; Tg, m1). The function dgly is computed by a semiempirical algorithm as uiinus the derivative of the energy deposited per incident electron at the depths ix. This energy is approximated by the product of tire empirical formulas for the number transmission coefficient and for the residual energy of electrons passing through tl-re slab of thickness x. Compensation for the effects of the approximation is taken into account. The energy deposition via radiative process is neglected. Further details of the algorithm for dgly are described in Ref. 11. Minor modifications to the original algorithm are given in Appendix I. The dose d(x) in the single-1ayer absorber with finite thickness or in the two- or three-1ayer absorber is computed by repeated application of a basic model of electron penetration across an interface. In this computation, dg1p1 is used repeatedly. The basic model treats the penetration of electrons from layer 1 to layer 2, i.e. the two-1ayer problem. The application of this model to the three-1ayer problem j-s as follows: In the passage through the interface between regions I and 2r layer I is considered to consist of material m1 , and layer 2, of material m2. In the passage through the interface between regions 2 and 3, layer I is considered to consist of m2, and layer 2, of m3. Let us consider the basic two-1ayer problem with layers I and 2 consisting of m1 and m2, respectively. We use dglM(xi T0, m1) as a first approximation to d(x) for layer l. This corresponds to consider a virtual presence of m1 in the region of Layer 2. We must apply a correction for the effect of the

-6-

difference in the contribution of electrons backscattered from the actual and virtual layers. The correction is given by the function dgly having appropriate arguments and multiplied by a normalization factor. In computing the terms other than the main term for the first layer, an equivalence rule is used. The rule can be stated as follows: The depth--dose distributlon in the second layer of the multilayer absorber remains approximately the same when both the residual energy and the half-value angle of multiple scattering at the boundary remaj-n unchanged under the replacement of the material of the first 1ayer, its thickness and the incident energy of the electrons. To apply this rule to the main term of d(x) in the second 1ayer, for example, the equivalent thickness x1' of the first layer consisting of rn2 and the equivalent incident energy Tr are determined. Then the main term for the second layer is given by dg111(x-xl+x1 r, T!, m2) rnultiplied by a normalj-zation factor, where x1 is the thickness of the actual first 1ayer. The normalization factor to each term is determined so that the integral of the term might be matched to the residual energy of the relevant component of the electron f1ux. The recipe useci in the present algorithm to calculate the energy deposition caused through the aforementioned processes of

penetration across an interface is given in Appendix I. The processes of transmission and backscattering considered in evaluating the main and correctj-on terms of d(x) have been described there by using schematic paths in a diagram. A similar diagram of the schematic paths* for the three* In Fig. 1 of Appendix I , solid lines are used to represent actual paths, and dashed lines , for virtual paths. In Fig. I of the present report, (continued on the margin of the next page) -7 -

iayer absorber is given in Fig. 1. The terms contributing to d(x) from all the segments of the paths are listed in Tatrle i' where each term is represented by the symbol for the Symbol"s f,or the segments are normal"ization factor applied to it. denoted as S1i, where the subscript i refers to the region, and j to the order of the correction related. The symbols for the normalization f,actors are denoted as cij[]r where the subscripts 1 and j corresponci to those of si3, and the subscripts k and 1 are rised to discriminate backscatterers assumeci. For example, the rows for segment Sr1 in Table I mean the following. The contributions from S11 consist of two terms) one being negative and the other positive. The first term- is a correction to the main term for the absence nf rirateriaL ml in region 2. It is equal to minus the contribr-ltion due to virtual presence of m1 in this region (shown by the symbol 2-(mt) of the backscatterer). The second term is a correction for the actual presence of ni2 i.n region 2 (the backscatteret 2-m2'1 . The rows for segment S12 ilean the foliowing. The first .lerm frorir Sl2 is a correction to the first ierni frcm 51i for the absenee of m1 in region 0 (the backscatterer 0-(m1)). Tl-ie second term from S12 is a similar correction to the seccnci term froni Sit. The corr:espondence to the terms fr:om Sll is indicated by tire symbols for the fi-rst backscatterers (2-(mi) and Z-mz). The meaning of the other rows can al.so be traceri simiiarly. !"or the tirree-layer absorber r.ve have thus nine terms to expless rl(x) in region 1, eieveu in region 2 ana five in ::egion 3. For the two-layer absorber, only the first five terms are required ia (continued frr:nr the margin of the previous page) tiowever, all [he paths are drawn by solid lj-nesl when p1ura1 lines differ c-.n1y in the material assumed of the backscatterer' these are represented by a si-ng1e 1ine.

-B-

Table I. Terms contributing to d(x) from the segments of the schematic paths considered are listed by using symbols for the normalization factors to the terms. In the last column, the backscatterers assumed for the related path are given. For example, the expression 2-(m1) means that the backscatterer assumed to be in region 2 is material m1, and the parentheses enclosing the symbol for the material indicate that the presence of this materiaL is virtual.

Normallzation factor Segment to contributing terms

Sro stt srz

q. ^ oI

3

sr+

Backscatterer assumed

clooo (=1)

-cllol

2- (m1 )

c1102

2-*2

cizol

2-(m1), 0-(mr )

-ctzaz

2-m2, 0-(mi

-L I 3 0 I cI3o2

3- (m2 ) 3-m3

3-(m2), 0-(m1)

c1+oi -c1402

szo

c2ooo

szr

c2loo

)

3-m3

,

0- (m1 )

2-m2, 1-*l

-9-

Table I (Continued). Backscat

Segment

Normalization factor to contributing terms

szz

-czzoo

2-m2t 0-(m1)

Sza

-cz3oI

3- (m2 )

sz+

3-*3

c24Il

3-(m2), 1-(mz) 3-(m2), 1-*i

c2422

3-m3, 1-(m2) 3-m3, 1-*t

cz5oI

3-(mz), 0-(nt)

-C

c r, o

LALL

r

3-m3, 0-(m1 )

-c2so2

sao

caooo

S:t

catoo

saz

-ca2oI

3-m3, 2-*Z 3-m3, 1-(m2) 3-m3, 1-*1

Cecnc

Ssg

assumed

c23oz

-c24l2

Szs

terer

3-m3, 0-(mt

-ca3oo

-10-

)

region l, and the first three in region 2. rn addition to the normalizatj,on factors, equivalent energies, equivalent depths and effective depths, used as the arguments in dSIM, must be evaluated. Definitions of these parameters (we call them path parameters) and methods for evaluating them are described in Appendix I.

-

11

-

III.

OUTLINE OF THE

CODE

The calculations outlined in Sec. IIl

is carried out by a part, the subprogram

function subprogram EDliUl,T. In its first evaluates the path palameters by using subprograms REXTAU, TRANS' suBRl, SUBR2 and sUBR3. In its second part, EDMULT evaluates d(x) by using a function subprogram FUNC1. The subprogram directly ca11ed by EDIIULT in turn use a total of twenty other su-bprograms. A11 the subprograins can be classified into the following five categories: ) !1ain functions: FUNCl , FUNC2. (2) Subprograms to evaluate path parameters: (

I

SUBR1, SUBR2,

SUBR3.

(3) subprograms to evaluate the depth and energy paranleters: tsACK, SEQ, TRANS.

(4) Subprograms to evaluate quantities related to penetrat.ion of electrons: DTBYT' EDEP0S' ETABN, ETABNI, FSATUR, MSCATT, REXTAU, TAURES, THETAH,

(5) 0perational subprograms:

XDIF, XTHH'

ETATN,

XTRN.

ARYPUT' NCPUT, TRPUT, XTPPUT,

ZAPUT.

A brief description of the subprograms i-s given by the cornment statements of the code listed in Sec. VII. For the convenience of the use for separate problems related to penetration of electrons, the paranleters that can be evaluated by the subprograms of category 4 are listed in Table 1I. The empirical and approximate relations develcped by the present authors and used in these subprograms have been described in

-t2-

Table II. General purpose subprograms on penetration of electrons contained in the listing of EDMULT. I,leanings of the abbreviations used j-n the description cf parameters are as foLlows. SLT: slab target, SIT: semiinfinite target, ni: normal incidence, ii: isotropic incidence. Entries in the column of the subprograms required are included as the entries of the subprogram in this tab1e. Parameter

of Subprograms subprogram required

Ref.

I'SATUR

none

t5

DTBYT

none

i6

Name

Symbol Description Backscattering

fs

Saturation factor fclr the backscattering coefficient (SLT; ni)

ATg/Tg Average energy-loss fraction of backscattered electrons (SIT; ni) nBi

Number-backscattering coefficient (SIT; ii)

ETABNI

none

i3

lBn

Number-backscattering coefficient (SIT; ni)

ETABN

none

t7

-

13

-

Table II (Continued). Parameter Subprograms of subprogram required

Naure

SymboI Description

Ref.

Absorpti-on

dE

EDEPOS

Energy deposition

(SIT; ni)

DTBYT, ETABN,

ll 13;

IAURES

x1

i

XTRN

REXTAU

1

Extrapolated range (SIT; SLT; ni)

REXTAU

none

18,

Average residualenergy of electrons at a given depth

TAURES

Depth for a give

residual energy (SIT; ni) Ru"

t

134

none

11,

l9

(Srr; ni) Transmi-ssion

nT

Number-transmlssion coefficient (SLT; ni)

-L4-

ETATN

none

20

Table II (Continued). Parameter

of Subprograms subprogram required

Ref.

I{SCATTb

none

15

XDIF

none

15

XTHH

XDIF,

21,

Inl1 lAri

:/

MSCATT

21

Name

Symbol Description Multiple scattering B

Parameter in MoliEre's

theory of multiple scattering (SLT) xD

Depth of cornplete

diffusion (Sff; ni) xH

Thickness for a given half-value angle of

multiple scattering (SLT; ni) 0g

Half-value angle of multiple scattering (SLT; ni)

THETAH

22

a Partial modificati-on is described. b Another parameter Xq is also evaluated.

-

15

-

, prevlous

1 t I ? I C-rn )aJ Lv papers. :rtrJ

The code EDI4ULT can be used to evaluate d(x) and its i-ntegral over x for incident energies Tp=Q"1-20 Mev in a singie-"

two- or three-1ayer siab absorbers. The thickness of a singielayer absorber and that of the second layer in a two-la;rer absor-ber can -he either finite or semiinfinite" The iast iayer in a three-layer absorber mrist be of semiinfinite thickness. The absorbers can consist of materials of atomic numbers from about 5.6 (polystyrene) to 82. For mixtures or compounds, the atomic number Z ana the atomic weight A are rep1aced by (1)

Leff = F,LriLj. i

Aef

f =

Zef.f (Z/ A) ef

f-L

(2)

,

respective1y, where

(zIA)eff = Ltizilti

(3)

,

r

f; as tire fraction by weight of the constituent element with the atomic number 21 and the atomic weight A1.

and

-16-

IV.

REVISIONS

1. The First Revision The first

revision of the F0RTMN code for EDMULT given in Ref. 4 was made in 1983, and a revised code was given in Ref. L4. The main change in this revision was niade in the expression for an interpolatj,on factor f. This factor appears in the expression (Uq.(5) in Appenciix I) for the backscattering coefficient of the Civergent electron flux after passing the layer of a gi.ren il-rickness x1. The o1d expression f or f is given by Eq. (6) in Appendix I. TI-re new expression is simpler, and is given by

f = exp(-x1/rxp)

(4)

Here the symbol r denote a constant independent of incident energy and target material, and xp is the diffusion depth defined as the depth where the::ms deflection of transmitted electrons reaches saturation. The value of c was determined as

r = 2.5

(s)

2. The Second Revlsion The change of the code from the one given in Ref. 14 to the

one given in this report constitutes the second revislon. main points of this revision are as follows:

The

(1) Segments S31 and 532 of the schematic paths shown in

-17*

Fig. I of Ref. 4 have been found unnecessary, and have 0.3 MeV. - - lTo mc2 MeV zcz for ro