Multiple-Scattering corrections in ‘spherical’ and ‘ring’ geometry

J. Nucl. Energy II, 1959, Vol. 9. pp. 169 to 177

Pergamon Press Ltd.. London.

hinted

in Northern Ireland

MULTIPLE-SCATTERING CORRECTIONS IN ‘SPHERICAL’ AND ‘RING’ GEOMETRY* V. F. TURCHIN (Received 5 March 1957)

Abstract-A method is given for making a correction for multiple scattering in experiments to measure the angular distribution of elastically scattered neutrons, when the average path length of the neutron in the specimen is comparable with, but not sudstantially greater than, the scattering mean free path. In the case of isotropic nuclear scattering the probabilities of multiple scattering are evaluated for a sphere and for ring specimens of circular and rectangular cross-section; double scattering by direct reduction of the appropriate integrals and higher order processes by approximate methods. The anisotropic.scattering of neutrons of severa MeV is dealt with by representing the cross-section a(O) = ~~(6) + a,(O) as the sum of o,(O), a forward peak, and a,(B) a more or less isotropic remainder term. Nuclear scattering events are then divided into two types according as to which partial crosssection is involved, and double-scattering processes correspondingly divided into four classes. The probabilities of all the four last-mentioned classes are calculated from the results of the theory for isotropic scattering. Higher multiplicities of scattering are treated in the same way.

IN experiments on the angular distribution

of elastically scattered neutrons, the low counting rates often make it necessary to use a scattering specimen of thickness comparable to the mean free path of the neutrons. The recorded intensities then contain an appreciable contribution from neutrons that have undergone more than one collision. However, the specimen is never so large that three and fourfold scattering events are obtrusive, for if it were, the results would be virtually useless. A quantitative allowance for multiple scattering therefore calls for an accurate calculation of double processes, together with an estimate of the frequency of processes of higher order. Expressions for these quantities are derived in this paper. Consider a scattering specimen exposed to a unit neutron flux. Let X,(19,4) dQ be the number of neutrons undergoing single scattering into an element da of solid angle about the direction 0, 4. Let &(O, 4) dQ be the same quantity for double scattering and &JO, I$) dQ = &(O, 4) da + X4(0, 4) dQ + . . . the sum of the numbers of neutrons involved in higher order processes. If o(O) be the differential cross-section for elastic scattering, o the total cross-section and n the number of scattering nuclei cm- 3, then using the geometrical entities of Fig. 1

and

x2(e, $) =/vJn3u(elj0(e2)

exp [-na(rl

+ rJ] exp (L2tur12) dV, dV2.

(2)

We shall be interested mainly in arrangements with ‘ring’ geometry, as illustrated in Fig. 2. A typical experimental procedure would be to measure the ratio N/N,,, * Translated from Atomnaya Energiya4, 244 (1958). 169

V. F. TURCHIN

170

FIG. l.-Typical

neutron paths in the specimen.

where N is the counting rate due to scattered neutrons with the screen in place, and N,, is the counting rate observed with screen and scatterer removed. If the detection sensitivity is isotropic, this ratio is related to our macroscopic cross-sections by N -= N,

RR;

1”

2

2 (E

(3)

1 + xc, + U.

Scatterer

Fig. 2.-A

CORRECTION

scattering experiment with ‘ring’ geometry.

IN THE CASE

OF ISOTROPIC

SCATTERING

We calculate first the probabilities of single and double scattering in an isotropic specimen of volume V having a constant o(O) = c,j4~r, with o, the total elastic scattering cross-section. Equation (1) may be put in the form (4) where M, 4) = (l/V)

exp

[-n4rl

+

r&l dv;

(3

s

tl is the mean coefficient of penetration, defined as the probability, averaged over the volume of the scatterer, that the neutron will escape absorption and a second scattering. The equation for double scattering, equation (2), contains a factor exp [-nc(rl + r&l. This quantity is everywhere positive; its maximum value is

Multiple-scattering

corrections

in ‘spherical’ and ‘ring’ geometry

171

unity, while the minimum is not very small in as much as r1 $ r2 does not materially exceed one mean free path. Evidently we may write

where & = -

1

exp (-ncr12)

V2

tz(O,9) =

&

r122

dV, dV,;

(7)

exp[-n4rl + r2>ldVl dV2;

t@, 4) here is the coefficient of penetration for double scattering. According to (8), tz may be represented as the product of two volume integrals, since r1 depends only on the co-ordinates of the volume element dVI and r2 only on the co-ordinates of the volume element dV2. Spherical geometry

For a scatterer in the form of a sphere of radius R, (8) gives

with p the radius of the sphere in units of the neutron mean free path I, i.e. p = R/l = Rno. In equation (7), if the exponent is to be expanded, the formula rn = (n + 3-l) div (r?)

leads to 1 y

(-1)“.

9 .2”-+1

P12=RZ~=O(k+2)!(k+4)!Pl=~

Ring geometry

Consider a specimen in the form of a ring of radius R, whose radial cross-section a circle of radius R (Fig. 3). It will be assumed that R, is much greater than both R and the neutron mean free path 1. The neutron may be supposed to suffer its first .collision within the volume of a cylindrical section of indefinitely small thickness dh, and its second in some other part of the ring. For the purposes of calculating the contribution of such events to &, the ring may be straightened out and replaced by an infinite cylinder of the same cross-section; for with R, > 1, the volume regions in Fig. 3 that are not common to both ring and cylinder evidently play no more than a is

dh

FIG. 3.-A

scattering ring and the cylinder obtained by straightening it out.

172

V. F. TURCHIN

very small part. The problem therefore reduces to calculating the number d&.(O) dQ of neutrons that are scattered once in the small disk-shaped volume of thickness dh, and then a second time elsewhere in the cylinder in such a way as to emerge perpendicularly to its axis within a solid angle dSl about a line making an angle of 0 with the initial incident direction. The same discussion may be applied to a ring of arbitrary cross-section, rectangular or otherwise. Then for the whole ring, obviously Z2(f3) = fy

(11)

2TR,.

In the present instance, equations (6-8) do not apply as they refer to a scatterer of finite dimensions. However, their ‘two-dimensional analogues’ are easily found to be :-

dW) - dh

t2(e) =

= ( z)2,(e)

~_~d~~sexpiI~~“)d~~d~2,

$ exp(-4) dS,)(is (S

exp (--r2/l) dS2 .

>

(12) (13)

The integration is carried out over the cross-sectional area S of the ring. t,(O) depends on the geometry of the experiment, and can be derived without special difficulty for each particular case : for a ring of circular cross-section it proves to be independent of 0, and is I,=

[

l-$4+’

R 2(_)2-~(!)3+~(~)4--...]2.

(14)

T

The integral in (12) has been computed to within ~1 per cent by an approximate method, valid so long as S is not very large. It is found that t2(0)Z3[aS’+ F/lS’3’2 - YS’~],

(1%

in which 5” = S/12, and a, p and y are coefficients whose values are given in Table 1 TABLEl.-VALUES OF a, p AND y Rangeof values of S’

0 0.07 0.3 0.8


a

j

0 0 0 0.59

I I

p 1.56 1.43 1.30 1.09

i

7

’ 6.480

;

3.760 2.452 1648

for various ranges of S’. F is a factor depending on the shape of the cross-section of the scattering ring. For a circular cross-section, F = 32&/3 ; for a rectangular shape with sides a and b

F = -f{(b/a)*Pn tan iy + !Ztan (in - 3y)l + (a/b)tb tan Lb - SY)f 4 tan yl},

(16) with y = tan-l (b/a). F is plotted as a function of y in Fig. 4.

Multiple-scattering

corrections

in ‘spherical’ and ‘ring’ geometry

173

Correction for triple and higher order scattering

The macroscopic cross-section for single scattering is given accurately by equation (2) in terms of an easily evaluated integral; X&3) has been calculated approximately on the basis of simplifying assumptions. A comparatively rough estimate will be sufficient for C,(O). Consider a scattering sphere of radius R and suppose that there is no absorption, so that gS = G. We calculate the number of neutrons which pass unimpeded through

FIG. 4.--F(y) as a function of y = tan-’ (b/a) for a ring of rectangular cross-sectional

shape having sides a and b.

the scatterer and subtract it from the total number of incident neutrons. This gives the total number of scattered neutrons, and hence X, + Ze + X;,, where X1 = J&(0) dL2 and so on. On further subtraction of EC,and C, we have C,, which is found to take the form +37

(J”-

1.12 (J4+

. . ..

(17)

Proceeding to the case of an absorbing sphere, the situation is modified insofar as 0, < g. Accordingly we write Xi, & and X3 as follows: X1 = 4~ i

(S

(:ti(B)sin 6 d6

)(

nV%4~ , )

(18)

(equation (20) is derived in a similar fashion to equation (19)). If the absorption is now reduced progressively to zero, while cr is maintained constant, we arrive at the corresponding magnitudes E\“), E&O)and JZi”) applicable to a non-absorbing material. These quantities differ from C,, X2 and X3 only in having cS in place of (r. And

(21) (22)

174

V. F. TURCHIN

Recause & is a preponderant

part of C,, it will be permissible to put fs8X2’ 2, c,=;Q$*

(23)

Equation (23) is the solution for the higher order scattering term appropriate to an absorbing sphere. In an experiment with more complicated geometry it is considerably more difficult to make an estimate by the above method. However, the following procedure may be adopted instead. The formula (23) giving &/& for an absorbing sphere may be regarded as a function of Z-J&. We now assume that the functional relationship of &JXs to C,/Ic, is preserved, even though the geometry of the arrangement be altered in an arbitrary way. This assumption is, actually, the principal source of error in the ultimate formula for Xc,, in as much’as Xc, for a sphere was calculated quite accurately. If f0 is the function describing the dependence of Cc)/&‘) on Xp)/C$s), in a non-absorbing sample, then (23) and (21) give (24) and using (17) it may be shown that f&C) = 0*93x + 1*2x2 + . . . ANISOTROPIC

(25)

SCATTERING

We have so far assumed that o(0) is’s constant. In passing to a more generally applicable case it will now appear that the results obtained above are of value in deriving the multiple scattering corrections under conditions of wider validity. A typical angular distribution observed with elastically scattered neutrons at a few MeV consists of a sharp forward peak, covering the range of angles out to 19,+ 2040”, complemented by a more or less isotropic distribution at 19> 13,. In this situation, a@) may be formally represented as the sum of two terms, a#) + o;(e), the first describing the peak alone and going to zero at 8 = Bo, and the second a term having no specially pronounced variation throughout the interval 0 < 8 < r. We may correspondingly divide, in the imagination, the various neutron scattering events into two groups appropriate to the two cross-sections. The first group comprises exclusively small-angle scattering below t?,, while the second group covers large angle scattering together with a fraction of the scattering at 8 < Bo. Double scattering processes may now be divided into four classes according to which group, 1 or 2, the successive collisions belong to. Thus one can have different double scattering events represented by the symbols (1 + l), (2 + 2), (1 + 2) and (2 + 1). Neutrons in class (1 + 1) have scattered only on the cross-section cr,(O); their number will be estimated below, together with a discussion of the mixed classes (1 + 2) and (2 + 1). The number of neutrons in class (2 + 2) is determined by the formulae of the preceding section. We replace 02(f3)by an isotropic function o(0) = 1 0s -=_ u,(e) dQ, and write for the total cross-section c = j[o#) + a,(8)] dR + ci, 4r 47r s ui representing the inelastic processes. Since o,(B) has no markedly preferred directions, double scattering on rs, may certainly be regarded as isotropic. However, by

Multiple-scattering corrections in ‘spherical’and ‘ring’ geometry

175

averaging o,(8) over angle we do make a certain very slight error in the total number of neutrons in class (2 + 2), though it would be difficult to estimate this error quantitatively. To determine the numbers of neutrons in classes (1 _1-2) and (2 + 1) we shall replace a@) by a delta-function. The integral of this cross-section over solid angle is to remain unchanged. Such a substitution distorts the result for the distribution of doubly-scattered neutrons, which now takes on the form of a,(O) exactly; the true form is a distribution obtained by smearing out o,(O) over a range of angles up to 0,. We shall ignore this distortion on the grounds that it is only a second-order correction to the actual neutron intensity observed. A second source of error in the approximation stems from the change in the average path traversed by the neutron in the specimen. The real track of a doubly-scattered neutron is typically a zigzag line having two bends; this is now replaced by a line with one bend only. However, the lost corner of the zigzag normally represents only a small change in direction, so that the average path length must change only slightly, with a consequently small effect on the scattering correction. Scattering on the cross-section o@) is accordingly replaced by forward scattering through zero angle; but, of course, this is equivalent to no scattering at all. Thus in our approximation the processes (1 + 2) and (2 f 1) reduce simply to single scattering events of group two, associated with a total cross-section (r = j”+(O) dQ + ui. They may be calculated by the straightforward procedure for group two single scattering, but with the term Jo,(O) d!J removed from the total cross-section. It will be clear that similar processes such as (1 J- 1 J- 2) (2 $ 1 -t I), (1 f 2 + 1 + l), and so on are automatically taken into account at the same time. Returning to the consideration of class (1 + l), we consider first the example of a scatterer in the form of a flat section of thickness a and area S. According to equation (2)

Now o(O) falls off quickly with 0, so that values of the integrand at small 19,and 0, make a heavily preponderant contribution to the integral. If the cosines of these angles be replaced by unity it is not difficult to derive cg

+l)(e) = 9 e -nua s f q,,,(e>,

(26)

4,+,,(o) = s ~1(e)~1te2)d~

(27)

where

is the angular distribution of the doubly-scattered neutrons. A specimen that is not in the form of a plate can be dealt with by a fairly simple extension, inasmuch as the scattering in class (1 f 1) is largely in the forward direction. The area represented by the sample is to be broken up into elementary areas dS, each having its own a; and then

with

tizs

1

a&=!. s

s’

(2) = $1~2 ds.

176

V. F. TURCHIN

The principal source of error in (28) is, in fact, to be ascribed to the expressions (29). Strictly speaking, 1 and 2 in (28) should stand for the true mean and mean square path lengths traversed by the neutron; however, (29) makes no allowance for the deviation of the neutron track from a transverse straight line. If it were thought desirable, a more accurate ZZL1il)could be arrived at by introducing properly computed values for d and 2; the same refinement could be employed to improve equation (26). I

I

0.2

0

\

0.2

0.4

06

\ 0.0

PO

x

FIG. L-The

function Z,1+1j(x).

We determme Jo+l, (0) explicitly for the case:

In an approximation in which cos 8 =i=1 - &ez, this a@) has a linear dependence on cos 0, and thus roughly corresponds to what is encountered experimentally. For e,+(27)

can be shown to give

J,,,,,

w = P2b2( 1 - $) z(l+l)(g-)9

(30)

in which ZU,,, (x) = (; - 4x2) cos-1x + ;X (1 + $2

A graph

- +/m;

-&f 1.

of zo+1, (x) is reproduced in Fig. 5.

Use of the formulae in practice In the preceding section, the problem of determining the single and double scattering intensities for a given a(O) has been solved. However, in practice one is presented with the inverse problem; a scattering distribution is given, and it is required to Cnd the true o(0). The solution may be found by an iterative process, starting with some trial function. As a zero-order approximation, some function a,(O) is adopted, having been made as close as possible to the expected solution. This function is then divided into a#) and o,(8) and the probability of double-scattering calculated as

Multiple-scattering

corrections

in ‘spherical’ and ‘ring’ geometry

111

shown above. A correction to the scattering distribution is thereby obtained, which can be applied to the results to give a first-order approximation s(0). The procedure may be repeated until successive approximations sufficiently nearly coincide. An actual example may be quoted in which the zero-order approximation was taken to be the observed effective cross-section, unmodified except for an increase of 20 per cent. The second order approximation proved to be in adequate agreement with the first.

12