Finite-angle multiple scattering

Nuclear Instruments and Methods in Physics Research B21 (1987) 1-7 North-Holland, Amsterdam

1

FINITE-ANGLE MULTIPLE SCATI'ERING K.B. W I N T E R B O N Atomic Energy of Canada Limited Research Company, Chalk River Nuclear Laboratories Chalk River, Ontario, Canada KOJ 1JO

Received 24 June 1986 and in revised form 6 October 1986

The sn0all-angleBothe formule for multiple scattering is derived from the formally-exact Goudsmit-Saunderson (GS) formula, and the lowest-order correction terms are examined. In the GS formulation, the length variable is path length, rather than projected depth; the lowest-order corrections from path length to depth are also derived. The more formal part of the calculation is independent of the scattering cross section. Power cross sections are introduced to obtain more specific analytical results, and finally the formulae are evaluated for the m = 1/2 power cross section. The corrections are small for reasonably small multiple-scattering widths when projectile and target masses are about equal, but can become large if the projectile is much heavier than the target atoms.

1. Introduction

Within the usual paradigm of multiple scattering of swift ions, certain assumptions are customary. The particles undergoing multiple scattering ("projectiles") scatter from various scattering centres, or "target atoms". The problem is assumed to be linear: that is, the projectiles do not scatter from one another, nor from target atoms that are still in motion from having been struck by other atoms in the beam. The target atoms are distributed randomly and homogeneously in space. Whatever motion they may have is negligible relative to the motion of the projectiles. The scattering is purely binary, and is azimuthally symmetric. Energy losses are usually neglected, or equivalently scattering cross sections are assumed to be independent of projectile energy. This assumption has occasionally been relaxed, but that is not the subject of this paper. Within this paradigm, angles may be treated exactly, or a small-angle approximation may be made. If angles are treated exactly, as was done by Goudsmit and Saunderson [1] (GS) and more recently by Scanlon and Castell [2], one obtains the multiple-scattering (ms) distribution as an infinite sum (for example a Legendrepolynomial sum) in which the coefficients are best obtained by recursion - that is, there may be no explicit form for the coefficients taken singly - and the number of terms needed in the sum to represent the ms distribution can be of the order of several hundred or even a few thousand, with the number increasing as the target thickness decreases *. In the small-angle approxima* This is the conventional view of the GS calculation, but I suspect the difficulties may be overcome. Legendre coeffi-

tion, commonly associated with the name of Bothe [3] (see also ref. [4]), the ms distribution is represented by an integral which may easily be computed. The price paid is that the distribution applies only in those cases where the ms distribution is confined to angles small enough that angles, sines, and tangents are mutually indistinguishable, and cosines may be indistinguishable from unity. In the GS treatment, the distance variable must be interpreted as the projectile path length, rather than a projected depth variable; in the Bothe treatment, the two are indistinguishable. The GS theory may be rewritten in terms of projected length, at the cost of making it even more difficult to compute anything: whereas in the original GS theory, each term in the sum is the solution of a differential equation, which solution may in principle easily be written down by inspection, in the modified GS theory this infinite number of independent trivial differential equations becomes an equal number of coupled differential equations. Fortunately, the coupling is one way, so that the coefficient of P0 is. unity, the coefficient of/'1 is readily written down, that of/'2 depends on the P1 solution, and so forth. In this paper, we begin with the GS equation in its original path-length formulation and obtain the Bothe equation by a limiting process, retaining the lowest-order correction terms. The path-length correction to the Bothe equation is also obtained. The latter derivation may be cients are equivalent to moments of a distribution, so good non-negative approximations to the ms distribution should be obtainable from a relatively small number of the GS Legendre coefficients by moment techniques. However that is another project and will not be addressed further here.

K.B. Winterbon / Finite-anglemultiple scattering instructive in that it uses Barnes-integral representations of sums to handle apparently-divergent integrals more properly. Ne~.t the normalization of the ms distribution with corrections is obtained, and finally analytical results for the m = 1 / 2 power cross section are given. The notation of Abramowitz and Stegun [5] is used wherever applicable, and formulae therein are quoted as "AS 9.1.71" for example.

the isotropy, reduces to a Fourier-Bessel transform. Using the small-angle approximation, and then interpreting a as extending over the entire plane,

F(x, a)= fd/,~o(ka)P(x, k)

(6a)

P( x, k)=fda,',So(/,a)F(x, a)

(6b)

and we find [4] 2. The basic equations

/7(x, k) = (2~r) -1 exp(-Nxo(k)),

We start with eq. (39) of ref. [4], and we shall largely follow the notation of that source: the ms distribution F(x, a) is given by

o(k) = f d , / c ( , )

OF(x, a) ax

(1 - J0 (k*))

and

V( x, a) = (2~)-l f

N f dC K( ¢ )[ F( x, a)

(7a)

dkkJo(ka) exp(- Nxo( k ) ).

(la)

(7b)

where x is the particle path length, N is the number density of target atoms, and K(#0 is the cross section for scattering through an angle ~b. No small-angle approximations have yet been made. From (la) we may obtain either the GS or the Bothe formulation. The corresponding equation with x the projected depth rather than path length is

Again o(k)=0 for k = 0 , so the normalization of F is preserved. Going beyond the path-length approximation in the GS equation, i.e. solving (lb), we find instead of (3) the more complicated relation

- F ( x , a + ~b)],

- cos a

aG(x,a) ax

Nrdq'g'q,'ra'x,J ~ )t t

-G(x, a +,)], (lb) where G(x, a)= F(x, a ) / c o s a and F is still the desired ms distribution. To obtain the GS equation, write in (la)

½f d'q Pt(*l)F(x, a),

(7 = cos a)

(2a)

- ~

=/INo~,

(3)

At x = 0, F = 8(a), so ft(O) = (4vr) -1, and ~us = exp( - Nxo t)/(4 ~r)

(4)

whence

F(x,

a) =

and

It is easy to verify from (8) that the normalization, f0 or gx, is independent of x. The path-length correction to the Bothe equation is done perturbatively, and we leave it until later.

3. From G S to Bothe

with 0, = f d , K ( ¢ ) [ 1 - V,(cos ¢b)].

ft( x )

(8a)

(2b)

and we have

dr1

g; = dgt/dx, /,= [(l+ X)g,+x + tg,_~]/(2t+ 1).

SO

f,(x)

(8)

where gt is defined analogously to ft,

a)

F(x, a) = • ( 2 1 + 1)ft(x)Vt(cos a),

- [(1+ 1)g;+ 1 + lg;_t] = (21+ 1)Uotg t,

I E ( I + ½) exp(-Uxot)Vt(cos a).

(5)

The heart of the limiting process of deriving the Bothe equation is the relation between Legendre polynomials and Bessel functions. It is well known (see for example AS 9.1.71) that for large l and small a,

Vt(cos a) - J0(ta);

(9)

however we require a slightly different relation. We use instead the expansion [6] Pt(cos a) = ( a / s i n a) 1/2 £ a,(a)k-'J~(ka),

(10)

i--O

We note from (3) that %-=0, so f0(x) = (4~r) -1, and the normalization of F is preserved. To obtain the Bothe equation from eq. (la), we use a two-dimensional Fourier transform, which, because of

where we have written k for 1+ ½. The ai(a ) are elementary, and in particular ao=l,

al=2-3(cot a-a-t).

(11)

K.B. Winterbon / Finite-angle multiple scattering Expanding the prefactor and a~, we get el(cos a) = J 0 ( k e t ) 1 + - ~

where a is a screening radius, f ( ~ ) a screening function, and *1= t sin 0/2, with 0 the center of mass scattering angle and t Lindhard's energy function [8],

- 2-~Jt(ket)

+ higher-order terms.

(12)

We shall also need the Poisson sum formula to transform the GS sum of eq. (5) to the Bothe integral, (6a). Writing (5) as F ( x , a) = E~-0g(l), we rename the g( I) as q~(2~'(I +

½)) =

3

e-

M2E a ( M l + M 2 ) ZlZ2e2.

The laboratory angle is given by

cos ~ = (1 - ~(1 + ~ , ) r n 2 / d ) ( 1

- ~¢/2)-1/2,

(16)

where ~ = 4 M 1 M 2 / ( M 1 + M2) 2 and g = M j M x and so, writing ~ for rlegT/t 2 ,

q~(2~rl + ~r),

so our sum is q~2= .~2 +

~4.

(16a)

12p, 2 /-0

n-O

Then from (12) we get

From ref. [7], pp. 76-7, we have

1 - el(cos q,) --, 1 - 4 ( k ~ )

~ ~ ( 2 r r n + or) 0 1 2~r

f. i~g

+ --241,

d r O ( r ) e_i~ ,

(17)

=

d k ¢b(2~rk)

e -2~ri"k

O~

= f~dk

1 + -i~

OO

q~(2~rk) + E ' i'f~ d k q~(2~rk)e

2~r b,k

(13) where the prime on the last sum indicates that the r = 0 term is omitted. The first term is the lowest-order approximant, and the sum terms are corrections. Now ¢b(2vr/) = g ( I - ½), and

g(t) = ~(1+

-~) Pl(cos

,~)

It is only in this last term that we can see the shape of the ms distribution depending on the projectile-target mass ratio. We now specify a power cross section, fO/) = ;t~1"2'' in eq. (15) and integrate to get

Nxo(k)=Nxcta2k

(MI+M2)e

22"+1T(1 + m )

×{1-~~2[~-~m2(l-m)

e - Nxol,

+ i~(4,,,

- 1)

,

(18)

with o, = fd~,,(,I0(1 - el(cos ~,)), so, using (12) and neglecting correction terms,

which we shall write as u k 2 " ( 1 - vk-2), where the v term is a correction. Then the Bothe ms distribution is

• (2,~k) = ~1a o

F(x, ~) = ~1 £~dk~o(ka) e -".2",

( h a ) e -NX'¢k),

with

(14)

o(k) = f d , ~(,)(1 - J 0 ( k , ) ) so that we have indeed obtained the Bothe eqs. (6) and (7).

4. The correction terms

[4] n"

F ( x , a) =

-1

(-f)" r2(mn + 1) sin ~tmn

22 n~l

(15)

n!

(19a)

,~,~,~2 . - 0

To evaluate the correction terms, it is necessary to specify a cross section, so we consider the corrections to the cross section first. The cross section may be written

d ~ x(q~) = d o = ~ra2 d-~-~f ( ~ ) ,

and we know [4] that by expanding * either the exponential or the Bessel function, we get, respectively,

(n!) 2

.9b

* The integrals obtained by expanding the exponential are of course divergent, but the combined sum and integral, and others occurring subsequently, can be justified by replacing the sum by its Barnes-integral representation. Such a procedure is carried out later in this paper where it is necessary that it be done explicitly.

K.B. Winterbon / Finite-anglemultiplescattering wher~ ~ = u(2//a) 2m and ~"= a//2u -1/2m, so ~ = ~-2m. The normalization integral in the GS formalism is 2~r f~' d e . s i n a - ( . - . ) , and that in the Bothe forrealism is 2~r f ~ d a . a . ( - . - ) . Hence we can expect that a Bothe formalism with correction terms will also have a differeat normalization, with the first corrections to the normalization being O ( a 3) at small a. On the other hand we can see from (19a) that F(x, a) = O ( a - 2 - 2 " ) at large a which, for small m, is only just normalizable. The corrections to the normalization must therefore at each stage make the normalization integral more convergent, not less, and so in the lowest order of correction we include an extra factor (1 + a2/p2) -1 in the normalization integral. With this norm, corrections must be o(1) at infinity, a n d ' s o correction terms which are O(1) or larger must be of higher order. The correction from the v term in the cross section is

corrections

We now evaluate the path-length correction in the Bothe equation, starting from (lb). Replace cos a on the left by 1 - a2/2, for the moment, and make the usual Fourier-Bessel transform to get

~d +~x~ J/'dotot J ° ( k ~ ) T a2c ( X ' ~) ---ax =

d(x, k)N,,(k).

(23)

Let (Go, G0) be a solution of the usual Bothe equation, G0(x, k ) = (2¢t) -x e x p ( - N x o ( k ) ) , and let G1 be the solution of (23) with G in the integral replaced by Go. Then

~. (-O~r2(~(,,+l))sin~-m(,,+l)

°~

(2¢r) 2 . - 0

5. Path-length

SO

Ao=2-~rfdkk2"-lJo(ka) e-"k~" =

expansion of the hypergeometric function. The first is manifestly O(1), and so is of higher order. These are all the correction terms arising in the path-length formalism. Corrections to this formalism are discussed in the next section.

n!

(20a)

Gl(x, k) = (2~r)-lA(x) exp(-Nxo(k)),

(23a)

where (4~m)

n_ 0

(.!)2

m

"

A ( x )= 1 - ½foXdyeNV°tk)fo°°daa3Jo( ka)

At large a, Ao = O ( a - 2 " ) , so it is normalizable at this level. Now consider the correction arising from the other substitutions (12). This is in two parts; the first, the J0 term, is trivial. The second is 1 2,r fai f r o d k Y l ( k a ) e - U : "

A jl

-U

1

48¢r2

-

F(mn)F(mn+l)sin~rmn.

X fo°°dppJo(pa)No(p) e -Ny°(e).

(23b)

For convergence, we put in the extra factor we shall require for the norm, (1 + aE/p2) -1, but it will turn out that we do not need the value of p2, if we assert that we are working to lowest order in 0-2; the mere existence of the factor prevents a divergence that would at least formally occur.

o

(21) This is O(1) at large a, so must be a higher-order term. Next consider the corrections from the Poisson sum formula, which may be reduced to the form S PLANE

1 *~ Av" = 2-~fo d k k J ° ( k a ) e -~k,-~os (sin)(2~rvk), v

=

1,2,

1

JO

• • •

~, ( - u ) "

= ~ .-o-

;,-~

F(2mn+2)

1

2 3

Cos

(2~,,)~,..+2 (s~°)(~'")

x F ,,,,,+1, m.+3/2;1; (2'~) 2 .

(22)

Now a/2 ~i, should be very small for all reasonable values of a, and so Ap, may be considered as a sequence of approximations, one for each term in the

Fig. 1. Contour of integration for the Barnes-integral representation of Bessel functions. The poles of F ( - s ) , at the nonnegative integers, are marked.

K.B. Winterbon / Finite-anglemultiple scattering m2~

In the power-law approximation, we get U

A_l=_-~f

-

)'. 12¢r2 n - o

01 dt e,tk:" f °° daa3J°( ka)

-o

2)

.-0

(28b)

(25)

where the contour $' (see fig. 1) runs from - i o o to +ioo, passing to the left of all the poles of F(-s), and can be closed to the left or to the fight. With this substitution, we get

( drY(- - r ) ( 1 / / 4 )

r

There is a further correction term, in transforming from G to F. Recalling F = G cos a, we see that this correction, A x2, is simply - a 2 / 2 times the uncorrected Bothe term.

6. Normalization

F(r+ 1)

fdsr(-sl(k2/4)"°°daa '+2"+2"

× f = d p p 2"+t+2r e -"iv''

(26)

~0

The p integral is 1 F(R)

(n!)2

× [~(nlm) 2.- ½(n/m) + ~].

1 [ dsr(-s) J°(Pa) = 2-~i J~er ( s + 1) ( p 2 a 2 / 4 ) s '

I

(28a)

(24)

We write the two Jo's as Barnes integrals, e.g.

- 2f0 Idte"tk2"21~tiJ,

( n + 2 ) ( 2 n + 3)

n!

×F2( m( n + 1)) sin ~m( n + 1)

× fo°°dpp 2" +tSo (pa) e- "'v2".

A -1=

(

In the transformation from the GS formalism to the Bothe formalism, as we argued above, the normalization integral changes from 2¢tfda sin a to 2vrfdaa, so we expect the lowest-order corrections to the Bothe equation to require a norm containing terms to third order in ct, and there should be no second-order terms. Hence the general third-order norm is

whereR= r+m+l

2m (ut) R'

m

2,

provided Re r > - 1 - m; that is, provided that the r contour ~ crosses the real axis between - 1 - m and 0. Similarly, the a integral is ¢/.04 + 2s + 2 r

for - 2 < R e ( r + s )

2sincr (s + r ) '

A n expansion of the I + a2/O 2 denominator for small a is an expansion in negative powers of 0; thus we want the first pole to the left of the integration-variable range in the a integral. That is, s + r = - 2 , whence

2u fl A-l=---mJoate

utk:" 1 t d r ( r + l ) 2 F ( R ) 2~ri J~¢ ! k2,+4(ut) a

"

Closing the contour to the left to encircle the poles of the F function, we get

A - 1 = 2uk2m-2m2foldt e utk2~ X ~

(-utk2m)" n!

( n + 1)2

n--0

= 2rn2uk2"-2(1 - 3uk 2" + (uk2")2/3),

(27)

o2fd..(: + =21-*

In the O -o oo limit this reduces to the Bothe norm. For the Bothe term alone, we have N

< -1.

2

fd..(1 + . % 2 ) - ' =

d~

F(x,

d.~

I

21r02jo ~°/" (02 + a 2)

=2.o2fo= 2

~)

fdkkSo(k~)e_UXO,k,

oo

= O fo d k k K ° ( k o ) e-uk2" = ~ (--q')" F2(mn + 1), n! 0

(29)

where ~b= U(2/o)2m; we see that for O ~ oo we get just N = 1, as desired. A series in ascending powers of P can be derived from (29) through its Barnes-integral representation, b u t it is more complicated, because all the left-hand poles are double poles. For the first correction, ~o, the norm is

No=vuO2fo°° daa fo~d kk2m_iJo(ka)e_.k2,. + o2) = vuO2fo~°d kk2m-lKo(kO) e-~k 2"

SO A X l = -2m2U -

2¢t

fdkk2,,-1Jo(ka) e-,k2"

× [1 - ½uk2m+ (uk2m)2/3].

=

: ( m ( . + 1)). 4

0

(3o)

"

The a so correction requires replacing Jo in N by

K.B. Winterbon / Finite-angle multiple scattering (1 + 32/12)

Jo. Now

so all integrals can be evaluated in closed form. The solution to the Bothe equation is known to be

1 + 32/12 = ( ot2 + p2)/12 + 1 - p2/12,

1

F(x,a) =

SO

u

2~r (u 2 + a2)3/2"

(34)

p2

p2 + ( 1 N+Njo = -~ _ -~]N.

(31)

The A jl and Z~p, corrections were seen to be of higher order. The path-length corrections are as follows. The first part of the norm is

The value at a = 0 is (2¢tu2) -1, and the half-angle is at/E= 2 ¢ ~ - - 1 . Recall that angles have not been scaled, so a is in radians. The correction terms are

1

Ao = 2~r (u2

Nxl = 2m~p2f (p2daa +.~)

vu

+ 0t2)l/2

where v

( i - 2~)

'

(35)

8~ 2

f dkk2"-~:o( k a ) 1 u (u4+u232+634) 2~r 12 (u2 + 32)5/2

Axl Xe-=k2" [1 _ ~uk 2m + u2k 4m/3]

1

(36)

a2U

= 2m2up2fdkk2m-lKo(kf) x e - ~ * ' ' [1 -

m2t['P212y"

½uk2"+=2k4m/3]

(~. t)" (n + 2)(2n

and the norms are p2 N= [X-l] i,/2 _ p2

(38)

(32a)

No = vp2 X

(39)

(32b)

Nx~=

+ 3)

XF2(m(n+l)). Using the same arguments as for (31), we have p2

Nx2=-~(N-1)

and N x = N x t + N x 2 .

Then the normalization for the corrected distribution is given by finding p such that

N + Njo + No + Nx= I.

~bP2~-~~v)nF(m(n+ l))2 "

0,

(40)

(u2_p2)2

-p

u> p

(41a)

p>u

(41b)

(34) U

>

]

u

X = ~ u2 - p2

Now N < 1 for p < oo, so the second sum must be positive. The leading term of this sum is positive only if m 2 + 30

u4

where

-~-- o

[ m2(2n 2 + 2n + 1) + 30].

(,,:_u'p~)~ ) x

6(u2-02)

3u 2 + 2 ( u 2 _ p2) +

(33)

Combining these terms, we have I + N+

1

(35)

i.e., there is a range of mass ratios for which the first-order finite-angle corrections to the Bothe formula are not normalizable. This is because the correction terms fall off more slowly at large angles than the Bothe term, and for this range of angles the correction terms are sufficiently negative at large angles to have a negative norm. It is believed that this lack of normalizability will nor preclude the use of these corrections at angles at which the corrected distribution remains positive.

~p2 _ u 2

arctan( ~ - u 2 u

)

and in the limiting case of u 2 = p2, N = ½

(38a)

N o = vp 2

(39a)

37p2 Nx = 180"

(40a)

We recall that N + Njo = p2/12 + (1 - p2/12) N and Nx2 = p2/2(N - 1).

8. D i s c u s s i o n 7. Analytical evaluation for m = ½

For the special value m = ½, the awkward exponential e x p ( - u k 2m) reduces to a simple exponential in k

It has been proposed by Marwick and Sigmund [9] and may be verified by calculation, that power-crosssection ms distributions are good approximations to the distributions calculated with more complicated cross

K.B. Winterbon / Finite-angle multiple scattering

~.o~

I

I

_

should apply to all target thicknesses, although the numerical results for the m = 1 / 2 power law are merely for illustration. The multiplier v in the o correction term is the only quantity, other than scaling parameters, which depends on projectile-target mass ratios. Fig. 2 shows results for equal masses (short dashes), for a half-angle of 30 ° and for U ions on He (long dashes, half-angle of 3°), the case for which v has its maximum practical value with a monatomic target. It is clear from this figure that heavy-projectile cases can have large finite-angle corrections. The path-length corrections are believed to be new. I thank G.E. Lee-Whiting for discussion.

References

o

0

I

I

I

2

3

i/2

Fig. 2. Finite-angle corrections to m =-~ multiple-scattering distribution. The solid curve is the Bothe small-angle distribution. The short-dashed curve (labeled 30) is for projectile and target masses equal. The target corresponds to a half width of 30 ° in the small-angle limit, and the calculated half width of the corrected curve is 27.5 °. The long-dash curve (labeled 3) is for a projectile 59.5 times as heavy as the target atoms. The target thickness corresponds to a half width of 3° in the small-angle limit, and the calculated half-width of the corrected curve is 4.86 ° . The curves are normalized to unit peak height. sections, provided the " p o w e r " of the cross section is properly chosen according to the thickness of the target. Thus the present results for general power cross sections

[1] S. Goudsmit and J.L. Saunderson, Phys. Rev. 57 (1940) 24; ibid. 58 (1940) 36. [2] P.J. Scanlon and B. Castell, Z. Phys. A317 (1984) 127. [3] W. Bothe, Z. Phys. 5 (1921) 63. [4] P. Sigmund and K.B. Winterbon, Nucl. Instr. and Meth. 119 (1974) 541; ibid. 125 (1975) 491. [5] M. Abramowitz and I.A. Stegtm, Handbook of Mathematical Functions (US Govt. Printing Office, Washington, 1964). [6] G. SzegS, Proc. Lond. Math. Soc. 36 (1933) 427, quoted in Erdelyi et al. Higher Transcendental Functions, vol. 2, (McGraw-HiU, New York, 1953) §7.8. [7] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, (Interscience, New York, 1953). [8] J. Lindhard, V. Nielsen and M. Scharff, K. Dan. Vidensk. Selsk. Mat. Fys. Med. 36 (1968) no. 10. [9] A. Marwick and P. Sigmund, Nucl. Instr. and Meth. 126 (1975) 317.