Nuclear Instrume~ts and Methods in Pmb Res~arch B73 11993) 447-473 Nonh-Holland

IcL Data

Report

On the theory and simulation of multiple elastic scattering of electrons

J.M. Fernhndez-Vareaa, R. Mayola, J. Barob and F. Salvat a

a F d m & Fhicn (ECM), Uniwsimt de Bnrceiona and Societat Catalana de Fisica (IEC), Diagonal 617. 08028 Barcelona, Swirr Sewis Cient$~eTkttics. Uniwnitai & Barcelonq M#ti i FronquPs s/n, 08028 Barcelona, Spain

Received 21 August 1992

MuMple elastic mttcring of dWns in matter is a d y d cm the basis of a m m t e single seattcring differential a m s obtniaed fmm mdal wavt d d a k We giw a brief derivation of Molikre'a mulriple scat!& theory that

darifies its physid conmt mi poinu out its limitati- In particular, k is shown that v r r mean free paths calculated fmm the Molikrc single mtmimg cross section differ sisnifamitly fmn tbc vdua obtahcd from panid wave calculations. We present a mixed simulatiw a@r&n that owrwmes of the limi~lions of the mmmlly available wn&n& Monte Carlo eodcs. This algorithm taka advantage of the k t that most of the c d h i a a cxperiewpd by a m e r g y electron a l ~ a g i v ~ n p a t h t c ~ m w R i a . ~ ~ ~ h k l e s s a ~ d s m a l l v a l u t ~.Tbtglobdeffectdthtse a o f t ~ o n s i . -bed bywiwamultipkseattuhgappmhatio~ Had d i s h & with & ! l c ~ ~ n a ~ l u a a t b a n l r . occur in a moderately matt number and arc -bed pa in detailed simulatio~ This mixtsdgorithm csn bE applied to any mgk scattering differential toa as d o n , it hda to the wrmct spatial distributions and if oompletcly avoids pmbkm related to boundary um& Moreover, when the single scat- law mddw Molikrc's tbmry i s adopted, tk algwithm a n bt formulated in. a comprctdy analytical way.

Monte Carlo simulation is used as an e ff~cient method to solve electron (and positron) transpon probiems [ 1,2]. The existing simulation algorithms can be classified into twdjdifferent kinds, namely "detaBedn simulations and "condensed" simulations. Detailed shu~ations,~wbere all the cobions experienced by an electron are simuIated in chronologica1 succession, are fwble when the average number of collisions per track is not too large (say up to a few hundred). A p m from the inhereat sta- tistical uncertainties, detailed simulation is exact, i.e. it yields the same results as the rigorous solution of the transport equation. Experimental situations amenable la detailed simulation are those involving eitber electron m u m with low initial kinetic energies (say up to about 100 keV ) or special geometries such as electron beams impinging on thin foils. For larger initial energies, and thick geometries, the average number of wflisions experienced by an electron until it is effectively stopped b m e s very large, and detailed simulation becomes very inefficient. The hgh-energy simulation coda currently available [2-41 bave recourse to approximate multiple scattering theories, which allow the simulation of the glow effect of the collisions that occur in a track segment of a given length. Each tmck is simulated as a moderately small number of connected *stepsm of a specified length that is much larger than the mean free path between real collisions, so that a Iarge number of collisions takes place along each step. The step length is either internally determined by the simulation code or specified by the user. The net displacement, energy loss and change of direction of the electron after traveUing a certain step are evaluated from the multiple scattering theories. The accuracy of these condensed simulation methods is thus limited by the approximations introduced ia, the adopted multiple scattering theories.

C ~ ~ A W to: F. Salmt, Facultat dc Fisica (ECM). Univcmitat de Barcdolls, Dhgmd 647, E48028 Barodolur, Spain.

0168-583X/93/SMO600 @ 1993 - MET Seieace PoWisbers B.V. All ryhu rwemd

Angula~ deflections of rhe electron tracks are mainly due to elastic scattering with nuclei xmnd by the surrounding electron cloud. As regards Monk Carlo simulation, elastic scattering and energy toss pmxsa can be considered separately. This is strictly correct fot detailed simulation, and i: is approximately valid for mudeased simulations provided we consider only short steps swh that ,he fractional energy loss in each step is small compared with the kinetic energy at the beginning of he step. OHrlng to this fact, elastic W a r deflections in condensed Monte W o simulations can be described by using a multiple scattering theory that neglects tbe energy loss in each single step, i.e. a purely elastic multiple scattering theory . For the sake of simplicity, we shall limit our considerations to this b d of theories. Comctions to account for the energy loss dong each step can be introduced by using e.g. the continuous lowing down approximation as described by Lewis [S 1 and Berger [ 1 1.

The usual practice in condensed Monte Carlo simulations is to use the theories due to M o l i t ~ [6,7], Goudsmit and Saundemn [8,9 1 and Lewis 15 1. The first two of these theories give ouly fie angular &stribution after a given path length without any information about the spatial distribution. The approach of Lewis [5] also allows the calculation of the first moments of the spatial distribution. The main uncertainties in condensed simulations originate h m the lack of detailed knowledge on the probability distribution of the spatial displacement. Each sirnubtion code incorporates a well- defined aIgorithm to dekmine the spatial displacement of the electron at the end of each step. These algorithms are not exact and, thefore, simulated spatial distributions, and other related quantities, are influend by the value of the selected step length in a rat her unpredictable way [ 101. The reliability of the simulation results must therefore be checked tbrough a detailed study of their dependence on the selected step length. Generally, it happens that when reducing the step length the results converge to the c o r n value but the computation time increases rapidly (roughly in propopion to the invase of the step length).

Bielajew and Rogers [ I 1 ] have developed the algorithm PRESTA, based on the multiple scattering theory of Molihe, which appfoximately accounts for the differences between the actual longitudinal displacement and !he true path Iength of a step. f t also accounts for the lateral displacement in the step and includes a boundary crossing strategy, which ensures that electron tracks are properly simulated in the vicinity of interfaces. The work of these authors represents an effect ivc improvement of the previous condensed simulation procedures. In particular, the dependen= of the simulation results on the adopted step length is largely reduced, However, the use of the Molik t w r y is open to question [12] and, moreover, only the mean values of the longitudinal and transverse displawments in each step are considered, i.e. straggling in the spatial disphccment after each step is neglected. A simpler procedure 10 mt the transverse and longitudinal displacements was outlined by Berger [ I ] and described in detail by Seltzer (31.

The aim of tbe present work is to review briefly the most habitual multiple scattering theories, to discuss their reliability when used in condensed simulations and to present a "mixed" simulation algorithm that overcomes most of the step-length dependence of the condensed Monte Carlo codes currently available, Mixed simulation algorithms take advantage of the fact that most of the collisions experienced by a htgh-energy eiedron along a given path length are "soft ", i.e. they produce very small angular deflections. The global effect of these soft collisions can be d d b e d by using a "continuous scattering approximationn, in tbe same spirit as the habitual continuous slowing down approximation for inelastic scattering. The modmtely small number of "hard" collisions, with large scattering angles, can then be simulated in a detailed way. Mixed simulation algorithms have been previously proposed by Reirner and Uefting [ 13) and by Andreo and Bmhme 14 j. The one described here can be applied to any single scattering law, does nut require any preselected step length, yields the correct spat ia! distributions a d does not pose any problem with boundary crossing. Furthennore, when us& a single scattering differential crms section of the Wentzel type 1 1 5 1 (is. with the same analytical form as the one underlying Moliire's theory), our simulation algorithm can be formulated through a few very simple anaiytical expressions.

The paper is structured as follows. In section 2, reIevant multiple scattering theories are briefly reviewed. We offer a simple derivation of the Molikre theory that serves to point out its physical

J.M. Fm&n&z-Yaw a al. / S i m u l ~ h drnultide elmron scattering

ened ' n w 3 it is at the 3f the an be i.e. a

itions iumd : [ I ] . oliere ly the ition. tio on. F on weU- hese : i h , bdity =e on verge Verse

rries, ition odes ;ions maU UOUS

~tiun

Osed n be met sising ti& & a

content and limitations. Tbe new s ~ U I U ~ ~ ~ Q P algorithm is described in section 3. In section 4, this algorithm is applied to the Wenwe1 cross section (151 to yield a simulation procedure that, in spite of its simplicity, is more accurate than the currently available algorithms based on Molikre'a theory. Useful mathematical information is given in the appendices.

4

2. Multiple s d h h g Wsries

We consider electrons (or positrons ) with kinetic energy E moving in a hypothetical idmite h b mogeneous medium, with N scattering centres per unit volume, in which they experience oniy elastic collisions. We assume that the single mitering differential cross section (DCS) per unit solid angle da Q )/di2 depends only on the polar scattering angle X , i.e. it is axiaIly symmetric about the incident d i m i o a TBis assumption is satifid as bng as the scattering centres are spherically symmetrical atoms or randomly oriented molecul~~ (1 61. Moreover, interference effects resulting from coherent scattering by several centres are assumed LO be negligible, As a consequence, 'the theory is applicable only to amorphous materials and, with some care, to plycrystalline solids. For the sake of simplicity, we limit our considerations 10 single-element materials, the generalization to compounds is straight- forward. Notice that the number of scattering centers per unit volume is given by N = NAp/Aw, where NA is Avogadro's number, p is the mass density of the material and A, is thg atomic weight.

A class of Monte Carlo simulation codes uses the multiple scattering qeory of Molihe. [ 6 ] - see also refs. [7,17]. This thmry is based on certain assumptions about ~f single scattering DCS and incorporates matbematical approximations that render the fmal disuibuen fully analytical. The particular DCS underlying Molitre's theory is based on the cikonal approxietion [ 1 8,191, wbicb is a h@-energy, small-angle approximation and neglects spin effects. Nigam et al. [20] reformulated f he theory on the basis of the DCS obtained from the second Born approximation and a scattering field of the Wentael type [ 1 51, i-e. an exponentially screened Coulomb field. It is well known that the second Born approximation fails for intermediate and Mgh atomic numbers [2 1 1, even for an unscreened Coulomb field 122 1. Moreover, the actual screened field may depart appreciably from the Wentzel one (and also from the Thomas-Fmi field adopted by Molikre ) . 1

More accurate simulation procedures are based on the multiple scattering)heory of Goudsmit and Saunderson [ a ] and Lewis [ 5 ] , which allows the calculation of the exact anmar distribution due to multiple elastic scattering after a given path lengtb by means of an expansion & Legendre polynomials. This theory does not assume any particular form of the single scattering DCS, and is essentially exact, i.e. errors in the computed md tiple scattering distributions can always be tra& back to inaccuracies in the adopted DCS.

2.1. Single scarrering diflerential cross s~ctiotts

The most reliable DCSs available to date are obtained from reiativistic (Dirac ) partial wave analysis using a r d s t i c scattering field [23,21], which can be obtained from self-consistent Hartree-Fock atomic calculations. Detailed simulations bawd on these DCSs have been shorn to yield mul?s in good agreement with experimental data for kinetic energies up to - 100 keV [ 13 1 . Partial wave calculations for m n e d fields are feasible for energies up to -20 MeV (2 1 ] (see also ref. (241 ). For higher energies, the Mott DC5 for a point unscreened nucleus 1 1 6 1, suitably corrected to account for smniw effects [3,25 ] and for nuclear size effects [25], is accurate enough for simulation purposes. The DCSs used in the present work have been calcuhted by using the pmedure described in ref:

I21 1. The adopted scattering field is the analytical approximation (sum of three Yukawa tern) to the self-consistent Dirac-Hartdlater f d d of ref. 1261, where parameters for elements with atomic number Z from 1 to 92 are given. This analytical field is accurate enough for simulation purposes and al1ows the we of the WKB and Born approximations to compute the majority of phase shifts. Only phase shifts of partial waves with low angular momen turn need to be computed from the numerid

J.M. F d - V a r m ei a!. / Sirnubtion of tnorlripk electron xaftering

solution of the Dirac radial wave equation [27]. This procedure allows the calculation of mliable DCSs for electrons and pitsons with kinetic energies from - 1 keV up to -20 MeV. Hereafter, th.! DCSs computed in this way will be referred to as PWA-DCSs. The mean free patb rl between elastic mUisions and the single scattering angular distribution f, { X r

is given by

where

is the total c m section. Notice that the probability of having a polar scattering angle between x and x + dx in a single collision is given by 2n f1 (1) sin^ dx. For our purposes, it is useful to write f 1 ( ~ ) in the form of a Lcgendre series:

where f i are the L.egeodre polynomials and

- I

The quantities

will be referred to as the transport coefficients. Notice that Fo = 1 and Go = 0. Moreover, the value of FI decreases with I due to the faster oscillations of 4 (cosx ) and, hence, GI tends to unity when I goes to infinity.

The elastic scattering within a given medium is completely characterized by the transport mean free paths I I defmed by

In particular, we have

i l - ( oos~ ) and d"'x)d(cos~)= A , A;' = N2n ( 1 -msx)

The quantity "1 - a x * can be adopted as a measwe of the angular deflection in single dastic collisions. It is then apparent that the inverse of the first transport mean free path gives the average angular deflection per unit path length. By analogy with the "stopping power", which is defined as the mean energy loss per unit path length, A; ' is sometimes d s d the "scattering power" [28 1.

The mean free path and the first and second transport mean free paths computed from the PWA- DCSs are given in fa 1 for AI and Au, as representatives of bw and high atomic number elements.

able , the

and I(X

astic :rage s the

WA- eats.

ENERGY (eV) ENERGY (eV) F - 1. EIastic mean fm path, I , and fm a d aeeond wauqmrt mean free paths, d l and A2 or electrons m e r e d by Al and Au atoms as functions of tke kinelie taergy of the projectile, cornputad witb Ibe PWA- A d & W in ?he text. The

+ symbols indicptc partial wave rcsultn fmm rtf. [23], f 1 L

The values obtained from the total and transport sections computed by et al. (23 1 for E 5 256 kcV, using a different scatte- field, are also included far comparison purposes.

2.2. The theoria of Goudsmit and Saarnderson [8J and h i s 131

Assume that an electron starts off from a certain positiod, which we se!ect as the origin of our reference frame, moving in the direction of the z-axis. kt f ( r , d ; s l denote the probability density of finding the electron at the position r = ( x , y , z), moving in the direction &en by the unit vector d after having traveIled a path length s. The diffusion equation for this probleh is [ 5 ]

when 1 - cos-I (d . d I ) is the scattering angle corresponding to the angular deflection dl - d . This equation has to be solved under the boundary condition f (r, d ; 0) = ( 1 /s)d l x ) 6 ( 1 - ms 8 ), where 8 is the polar angle of the direction d . By expanding f (r, d ; s ) in sp hetical harmonics, Lewis 15 ] obtained general expressions for the angular distribution, and for the first moments of the spatial distribution after a given path length s. The angular distribution is given by

where A/ is the ith transport mean free path defined by eq. (6 ) . It is worth noticing that FGs (B; s)dfJ gives the probability of having a find direction in the solid angle element d I l around a direction defined by the polar angle 8. Evidently, the distribution given by eq. ( 10) is symmetrid about the z-axis, i.e. independent of the azimuthal angle of the final direction.

The result given by eq. ( 1 0 ) coincides with the distribution obtained by Goudsmit and sunderson (81 in a more intuitive way, which we sketch here to make its physical meaning clearer. Using the Legendre expansion given by eq. (3 ) and the persistence property of the kgendre poIynomiala (see ref. [16], p. 4701, the angular distribution after exactly n collisions is found to be

The pmbbdity distribution of the number n of collisions after a path length s is Poissonian 74th mean s/L, i.e.

Therefore, the angular distribution after a path length s can be obtained as

which coincides with expression 1 1 0). From the orthogonality of the Legendre polynomials, it follows that

In particular we have

(a 8)- = exp C-$/Al ), and

The theory of Lewis [ S ] is superior to that of Goudsmit and Saundenon [8] , since it also yields analytical formulae for the first moments of the spatial distribution and the correlation function of z and cos 8. Neglecting energy losses, the rtsul ts explicitly given in Lewis' paper simplify to

( z M s ~ ) ~ ~ z / z c o s ~ / ( r , i ; S ) d(COse)&

s

=exp( - s /~ , ) / 11 + 2exp( - l /A2) ] cxp( - t /A l )d f .

0

The quantities (15)-(19) am completely determined by the values oi l l and AZ. The Goudsmit and Saundersun expansion ( 10) and the results ( 1 5 )- ( 1 9) are exact. To compute

these quantities fur a given single scattering DCS, which usually is available only in numerical fom, we have to evaluate the transport coeficients GI as defined in eq. ( 5 ) . Nevertheless, for path lengths which are not too long, the convergence of the series ( 10) is rather slow [ 1 ] and a large number of terms are needed, Due to the fast oscillations of the Legendre polynomials, the numerical calculation of the integrals in eq. (4) for large I is a very deli a t e task. The usual practice to avoid this difficulty consists in replacing the exact single scattering distribution fi ( X ) by suitable analytical approximations

t ,23 ] that allow the easy evaluation of the transpon coefficients Gl by means of recurrence relations. However, we have not been able to fmd any analytical form able to reproduce the PWA-DCS to an

F I Z Wributions of 15.7 MeV ekctnms transmitted through Au foils of the indicated mass thicknesses. The continwus cumm are G o u d s r n i t S B u n ~ distributions calculated with the PWA-Dm, #be &bed curves are the dis- vibutions computed from Mdiht's formula tcq. (53) with the parameters given by eqs. (59 ) and (60)). The are

experimental d t s from H a u m et at. 1291-

accuracy better than .Y 5%. Therefore, we have computed the transport coepcients fmm the "exact" PWA-DCS by numerical integration. The integrals

have been mlculated by means of a 2&point Gauss formula complemented with an adaptive bipartition procedure that allows the control of integration errors, which have been kept below 0.00 1 %. In these calcuIatisns, the single scattering distribution (1 ) has been handled by means of spline interpolation from tabulated values in a grid dense enough to give a negligible interpolation error. Owing to the length of the calculation, only the fmt 300 coefflcien have been computed and, consequently, we will only consider large enough path lengths s such that the Goudsmit-Saunderson series effectively converges with this number of terms.

Angular distributions of 15.7 MeV electrons transmitted through gold foils are shown in fig. 2. Theoretical distributions have been calculated from the Goudsmit-Saundcrsoo theory with our P WA- DCSs. The agreement between our results and the experimental data of Hanson et al. [29] is seen to be satisfactory, although the theoretical distributions are slightly narrower. This is to be expected since , in the calculations, we are neglecting inelastic scattering by the atomic electrons, which tends to widen the multiple scattering angular distribution. For a h igh2 element such as gold, the relative effect of ineiastic collisions is small (of the order of 1 / Z [ 171 1.

mpute form, :nglhs ber of llation flculty ations ~tions. , to an

J. M. F A V m er al. / Simuhtion cfmdtiple electron was~m~ng

2.3. The Wenrzel mdel

The simple scattering model due to Wentzel [ 15 I has been repeatedly used in connection with multiple scattering theory [ 5 1. Here we describe it briefly because it will be useful to derive the multifle scattering theory of Moliere [6] in a way that makes the physical content and mathematical a m c y of this theory evident. Moreover, the analytical simplicity of the Wentzel DCS makes it particlilarly suited as the basis of fast simulation produres [30,3 1 1. In particular, the general simulation algoriQm described in sect ion 3 can be formulated in a completely analytical way when a DCS of the Wenuel type is adopted (see section 4).

The Wentzcl approach for describing elastic scattering of particles with charge Z'e ( Z ' = - 1 and + 1 for electrons and positrons, respectively) by atoms of atomic number Z is based on the simplified scattering potential

V ( r ) = (22'e2/r ) ex#(-r /R) , (213 where the exponential factor tries to reproduce the effect of screening. The screening radius R may be estimated from the Thomas-Fed model of the atom, which yields

R 2: 0.885 2 - l l 3 Q, (22 ) where is the Bohr radius. However, it is more expedient to determine R so as to obtain agreement with more accurate elastic scattering cross sections; this was the procedure adopted by MolZre [ 6 ] (see below). The DCS is obtained from the fmt Born approximation, which gibs

where p is the momentum and j? is the velocity of the scattered particle in units of the speed of light c. The screening parameter A is given by

The corresponding total cross section is

and the single scattering angular distribution is

The transport coefficients defined in eq. ( 5 ) are given by (see appendix A)

G : ~ ' = 1 -I[Ql-,(l + 2 A ) - ( 1 + 2 A ) Q ( l + 2 A ) ] . The explicit expmsions for the first two transport coefficients are

G : ~ ' = ~ A [ ( l + A)ln(( l + A ) / A ) - 1 1 .

G : ~ ) = ~ A ( I + A ) [ ( I + 2A)ln((l + A ) / A ) - 2 1 . (29 ) Certainly, the Wentzel DCS, eq. (23), with the weening parameter A given by q, (24), is not

very accurate and leads to erroneous moments of the multiple scattering distributions. Improved distributions can be obtained by using a screening parameter determined in such a way that the first transport mean free path calculated from the Wentzd DCS,

coincides with the value A, determined from the PWA-DCS. The value of the screening parameter is obtained as the root of the equation

J.M. Fm&&z+Varcrr al. / Sidnfim of mulfipie dmmn mttering

bn with 1ultipIe xuracy .cularly .orithm Veatzel

- 1 and ~plified

(21 may be

(233

)f light

129) is not )roved .e fmt

which may be easily solved, c.g. by Newton-Raphwn's method The Goudsmit-Saunderson distribu- tion for such a corrected Wentzel model, with the proper d u e of i2 will have its m a n equal to that ofthe "exact" distribution (i.e. the one obtained from tbe PWA-D@S) but higher order moments may still be in error (cf. eq. ( 1 6 ) ) .

In his onginal paper, Molihe [ 6 ] , and also Bethe 171, used a Wentzel model with an energy- and Z-dependent ~creenin~'~arameter that can be expressed as

where a = 1 / 137 is the fine-structure constant. This screening parameter agrees with eq. (24) except for the last factor, which amounts for corrections to the Born approximation. Moliire derived eq. (32) by fitting the DCS obtained from the eikonal approximation with the Thomas-Fermi atomic field.

The eikonal approximation leads to a DCS that is very accurate for high energies and small scattering angles [32 J *I and reduces to the Rutberford DCS

for large scattering angles [ 18 1 (irrespective of tbe adopted screening model). As the Wentzel DCS departs from the Rutherford DCS only for small X , it is clear that, by ushg a convenient value of the screening parameter, we can make the Wtntzd DCS almost identical $th the eikonal P C 3 for aIl angles. Thus, the uoptimum" screening parameter can be determined linambiguously by solving eq. (3 1 ) with the first transport mean free path computed from the eikod DCS. This procedure is essentially equivalent to the orre followed by Malike to derive his "screening angle". Therefore, the screening parameter determined in this way should not differ sidmntly fmm the vduc given by eq. (32). Hereafter, the Wentzel model with Molih's screening parameter will be r e f d to as the WM model.

The adequacy of the WM model to describe the actual scattering process is thus detemined by the combined effect of two different approximations, namely (1 ) the use of the Thomas-Fcrmi atomic field and (2) the eikonal approximation. The statistical Thomas-Fermi fiep gives a reasonably good representation of tbe actual scattering field except for the elements of low atomic number. The deficien- cies of the statistical model of the atom can be largely avoided by using a mdre realistic self-nsistent field, for instance the analytical Dirac-Hartree-SIater field [26] adopttd;in the calcuIa$on of our FWA-DCSs. In any case, the eikonal DCS for such a realistic field could s t d be closely approximated by the Wentzel DCS with a convenient value of the screening parameter. The essential limitations of the Wentztl model come from the Rutherford-like behaviour of the DCS for large scattering angles, which is correct only when spin and finite nuclear size effects are negligible. However, it happens that both types of effects modify the single scattering cross section for large angles [24,25] with the result that the eikonal DCS, obtained from a realistic scatterhg field, may differ considerably from the actual DCS at large angles. As a consequence, the WM model is otlly adequate for describing nnall-mgIe elastic collisions of highenergy electrons. The practical consequences of the enonsous behaviour of the WM-DCS at large angles will be analyzed hlow.

As regards Monte C d o simulation it is important to make sure that the simulated distributions have at least the correct values of the quantities given by eqs. ( 1 5 )- ( 1 9 1. This can be accomplished by simply using an approximate analytical D@S with the proper values of l and Az, i-e. those obtained from tbe PWA-DCS. An obvious candidate is the Wen tzel DCS, which now we write in the form

*l The accwacy af a e eikoad DCj for 1 = Q and cntrgia above -- 100 k V is surpringly gcml. We have found thpl the relative difhmca bmween thc DCSa for f o m d scattering d a l a d from the eilconal appmhatioa and fmm the panial wave method, with tiu same weened fidd, arc lesa than 0.1%.

J.M. Fernrindez- Varea el a/. / Simulation of multiple electron scarrering

Up to this point, we have only considered the possibility of varying the value of the screening parameter since we intended to keep the (incorrect) Rutherford large angle behaviour of the Wentzel DCS. Far our present purposes, we can consider A and dW) as adjustable parameters and determine them so that

(W) (W) lWUw) and 1 / A 2 = G , /A , 1 / R , = G, /L (35)

where E I / The screening parameter A is given by the root of the equation

and = I G : ~ ' . This Wenlzel model (hereafter referred to as the W2 model) leads to a DCS w h c h has the same "average" value as the PWA-DCS in the interval of angles that effectively contribute to I ; ' , eq. ( 7 ) , and A T ' , eq. ( 8 1. As the integrands in these eqs. vanish for x = 0, the Wentzel DCS and the PWA-DCS may differ considerably for very small angles, but these differences only affect the shape of the multiple scattering distributions for short path lengths (see below). Significant differences between the W2-DCS and the PWA-DCS appear also at large scattering angles, but there the DCSs take values that, for high-energy electrons, are exceedingly small.

Ln practice, when condensed Monte Car10 simulation is needed, i.e. for high-energy electrons, the screening parameter A is small. In this case, the evaluation of the transport coefficients (27,) can be simplified by using their limiting form derived in appendix A (eq. (A. I U ) ) ,

G : ~ ' = I ( 1 + 1 ) A [ l n ( ( l + A ) / A ) - 2 8 ( 1 ) + 1 ] + 0(14~'), where

I

a(/) = ( l / m ) . (38 1 m = l

Eq. (37) was fmt derived by Goudsmit and Saunderson [8 1, and rederived by Bethe [7], using tedious integration methods.

2.4. The Molikre theory

The Molikre theory 161, as reformulated by Bethe [7], is based on the assumption of a single scattering law that has the form (23) for small scattering angles and goes over to the Rutherford law, eq. (33), for large scattering angles. Indeed, the Wentzel DCS, q. (231, fdfds this last assumption and, hence, it will lead to the Molibre multiple scattering distribution if the screening parameter A ( ~ ) given by eq. (32) is adopted.

Assuming temporanly that a single scattering DCS of the form (34) is a good approximation, we will derive the Molihe multiple scattering distribution on the basis of the results of the previous subsection. Our derivation follows the same steps as that of Bethe [7], but it is cansiderably shorter and emphasizes the fact that the MoliGre theory is essentially the multiple scattering theory of the Wentzel model. A single scattering DCS having a form different from the Wentzel PCS will lead to muItip1e scattering distributions that differ from the Molikre one (see e.g. ref. [20] 1. We start from the approximate transport coefficients given in eq. (371. First we replace @(I ) by

the approximation 16 1

where y = 0.5772 is Eukr's constant, and keep only the first two terms, thus obtaining the Molike approximation for the multiple scattering coefficients;

c o r poin will eq- ( I = I becc 00, i that mod T

is ac

and Bess

whe

and,

In particular wc have

ciM' = Z A [ln ( ( 1 + A ) / A ) - 0.9651, ameter 3. For lem so

GM' = 6 A [In ( (1 + A ) / A ) - 1.9871 . Comparing these results with eqs. (28) and (291, we see that the approximations introduoed up to this point are not serious, provided A 8: I . However, from eq. (37) it is clear that the approximation GY' wil l fail for 1 values such that 126 - 1. Actually, when I increases from 0 to m, the right-hand side of eq. (40) first increases from 0 up to a maximum value - { 1 + A ) exp -2y ), which is reached when I = i,, - ( 1 + A-I )'I2 exp (- y 1, and for larger values of I it demasa monotonically and eventually becomes negative. As the 'exact" transport ~eificients G/'), eq. (27), tend to unity when 1 goes to oc, it is clear that we should limit the use of the approximation given by eq. (40) to path lengths s that are large enough to make sum that the Goudsmit-Saundemn series, eq. (lo), for the Wentzel model, converges with l a than I,, terms.

The mmainhg task is to avoid the summation of the Goudsmit-Saunderson series, eq. ( 10). This is accomplished by in tmducing the following approximation, due to Molikre [ 6 ] ,

1

which tribute

the :reaces DCSs

ns, the can be and replacing the summation in the GoudsmitSaundcrson series by an inte(jral over 1. Here, lo is the

Bessel function of the first kind. Assuming s 3 A, we have

ediow

slngte d law, iption r A'M'

Introducing the pameten S S

x ~ E ~ ~ A , b ~ l n [ ~ ( l + ~ ) ] + l - Z y

we wn write

p ( v ) = f x : ( y Z - f ) [b-h(hS2)],

In, we :vious horter of the ead to and, changing to the variable u = ~ y ,

TO facilitate the evaIuation of tbis integral, Molitre set

b = B - l n B , w = u ~ ' f ~ = ~ ~ ' / ~ ~ ,

so that he muld write

JY F d h - V a r m al. / Simulation qfmulripk electron scurrerIng

w ( .' )I2 / w c x p B (8-ln(w2/4))] Faa(B;s )= - - 2~ sin B X,ZB

0

Now, we note that eB/ B = @ S / A ( ~ ) , the average number of collisions in the path length 3. In practice B takes d u e s much larger than unity (e.g. B = 3.6 for S / A ( ~ ) = 10, which is a rather s d path length) and hence, for the values of w that effectively contribute to the integral in eq. (501, we can write

w2 I w2 I 1 oZ 2

~W[ah(~2/4/a)]=1+BTln~d/4)+--(-ln(w2/4)), 2! B2 4 and

exp [(~:/16) ( B - ln(02/4))] -. exp(x~~/16). With ali this, we obtain

where 8 E 8/ &,B 'la), and

This is exactly the same distribution obtained by Moliire 161 and Bethe 171. The first rerm in the series (53) is the Gaussian distribution

f (8) = 2 exp(-8'). ( 5 6 ) The functions /(I)(d) and f ("(19) have kcn tabulated by Beth 171. In the kumeriul clkulatioos reported below, these functions have been calculated by cubic spline interpolation from Bethe's tables (71 for 19 < 6. For 8 > 6, we use the expmsions

which are accurate to within a few tenths of a percent. With the screening parameter given by eq. (321, and assuming it to !x much less than unity, we

fmd

which coincide exactly with the parameters used by Molitre in his original paper 161. Therefore, the o m form of the M o l i h theory is aothmg more than an andycical approximation to the exact Goudsmit-Saunderson distribution for the WM model.

The mathematical approximations introduced in the derivation of the Moliire distribution, eq. (53 ), put certain limits on its range of validity. Firstly, the biting form of the transport coefficients

J.M. Ftmrfnda-V#m pr d. / S i d i o n cf mubipie d m mter ing

I . the xact

eq- ents

Scattering ongle (dsg)

Fig. 3. Muhi& mtterhg dktribu- of I MeV eltarom after travcbg a length s (cxpmdh units of the first transport man Frcc pa& 11;"' given in table 1) in At. Tbe continuous cum art Goudsmit-Saunderson +butions for the WM-DCS ( A = AtM) = 1.4 x 10-J). The Wed curves rqmscnt the Molibe dimibution, q. (53), fdthis WM-DCS, I.e. with the

parameters @vea by eqs. (59 ) and (60). Thc mmgc n u m k of dbions n = in each cast is a h indicated.

given by eq. (40) is valid only when the screening parameter A is small. This limits the application of the theory to high energies, far which the first transport mean free path is much larger than the mean free path. Wndly , we must have b > 1 {otherwise, the parameter B, see eq. (49 ), is xlo t defined ) . This meansthat the pathlengths must be larger than about 4A(W) (see eq. (46)), i.e. thescattering mustbe at least plural. Actually, a restriction of this sort was to be expected from the very beginning since, when s - A(W), the Goudsmit-Saunderson series is slowly convergent and contri@tiona from terms with I > I,, (see the discussion after q. (42)) may not be negligble. As G : ~ ' ij not adequate for these high order terms, the whole theory fads when s - A(W). Finally, the approximation given by eq. (43 ) is very accurate for small mgh, it remains valid for intermediate angles and bn&s down for yalues of 0 near 1 80° where the factor (81 sin 0) diverges. This divergence h not important when r c A ', since then the angular distribution is strongly peaked in the forward direction and the only effect of the diwrgmcc is a very narrow peak in the backward direction with a n-ble area. Undesirable effects of this divergence become prominent when s - liW). Under these circumstances, the distribution (53) shows a conspicuous jmk in the backward direction and therefore differs appreciably from the "exact" angular distribution, which tends to the isotropic distribution when s % 1,. To avoid this anomalous behaviour, we should limit to path lengths such that the Gaussian part of the Molitre distribution, which is given by eq. (56), has a width less than 1 rad [ 7 ] or, equivalently, such that x ~ B 5 1. In condusion, the Molitre distribution, eq. (531, give a g d approximation to the Goudsmit- Saundetson distribution for the Wentzel model when the conditions

A ( ~ ) q (liw' and a(W) < s < A$' (61)

are simultaneously fulfilled. This is exempIilied in f* 3 where the Goudsmit-Saunderson distributions for 1 MeV electrons in aluminium f Z = 13) computed from the WM-DCs are compared with the distributions obtained from MoliWs formula, q. (53), for four different path lengths.

Bethe (71 used the angular distributions of 1 5.7 MeV electrons in gold measured by Hanson et d. [ 29 J to illustrate the reliability of the original Malike theory (using the screening parameter given