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Multiple scattering correction for inelastic scattering from cylindrical targets
NUCLEAR
INSTRUMENTS
AND
METHODS 80 (197 O) I 8 7 - 1 9 1 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
MULTIPLE SCATTERING CORRECTION FOR INELASTIC SCATTERING F R O M CYLINDRICAL TARGETS C.A. E N G E L B R E C H T
Atomic Energy Board, Pelindaba, South Africa Received 15 August 1969 Simple expressions are developed for the corrections for flux attenuation and multiple scattering of neutrons from cylindrical targets.
1. Introduction
The accurate extraction of neutron cross sections from experimental results obtained with thick targets is possible only after application of corrections for flux attenuation and multiple scattering. Although programs such as M A G G I E 1) are now available for the calculation of every conceivable correction including even effects such as inhomogeneity and nonmonochromaticity of the incident beam, the nonavailability of a suitable computer, the excessive demands in computer time, or simply the sheer amount of human labour required in any case, make it desirable to have a simple formula available by means of which at least some of these corrections may be estimated with the minimum of trouble. The purpose of this note is to satisfy this demand. To obtain such a simple formula, one must, of course, simplify the general problem somewhat. The scattering of neutrons from a hollow cylindrical target will be considered and the incident beam will be assumed to be monoenergetic, homogeneous, unidirectional, and normal to the axis of the cylinder. Only the corrections which are necessary for the extraction of inelastic scattering cross sections will be considered. Recoil corrections will be ignored and the angular distribution for elastic scattering will be taken into account in a very primitive way only. The inelastic scattering will be assumed to be isotropic, which also makes it reasonable to ignore elastic scattering occurring after the production of an inelastically scattered neutron: there should be no nett change in the number of inelastic neutrons emerging in the solid angle subtended by the detector. Removal of inelastic neutrons due to further inelastic scattering or other absorptive processes will be taken into account in an approximate way. 2. Formulation of the problem
The target is taken to be a hollow cylinder with inner and outer radii RI and R 2 and half height H. 187
The total macroscopic cross section Z is composed of an elastic, an inelastic, and an absorptive part: (1)
Z = Eel-{-,~in-~-Zabs
where Z~b includes inelasticscattering to all levels other than the level or levels excited with cross section Si.. The total yield of elastically scattered, inelastically scattered or absorbed neutrons is denoted W and is simply the sum of these three component parts: W = Xo + Yo + Zo •
(2)
Here X o denotes the yield of neutrons whose first interaction is an elastic scattering and similarly Yo for primary inelastics and Zo for primary absorbed neutrons (if the absorption is real, Z 0 is of course not a "yield" in the usual sense). These two decompositions are in the same ratio Xo_ Eel
Y o _ Zo _ W. Zln
Zab
(3)
E
If the yields are measured per unit target volume and per unit incident flux, this ratio is the flux attenuation coefficient So = w / z ,
(4)
which allows for the fact that the nuclei on the shadow side of the target do not experience the full incident flux. A related quantity is the transmission coefficient, the probability of a neutron being transmitted through the target without any interaction, averaged over the part of the incident beam hitting the target: Qo = l - e o .
(5)
The flux attenuation coefficient is simply a positiondependent transmission coefficient averaged over the volume of the target. The fraction Po of incident neutrons which do interact, are decomposed according to (2). Hence, the fraction which are scattered elastically is simply ~Po
188
C. A. ENGELBRECHT
where ¢ - Z~,IZ,
(6)
whereas the primary inelastic fraction is t/Po where t1 - Z~,IZ.
(7)
For the elastic fraction one may again define a transmission coefficient Q1 = l - P 1 .
(8)
The fraction P~ which interact, once again consists of three components W 1 = XI'4-YI-I-Z1
(9)
in the same ratio as (2). Thus the fraction of incident neutrons scattered inelastically after a single elastic scattering is given by t/P~ ~ Po while ~PI ~Po are scattered elastically for a second time. In the same way a fraction t/P 2 of these are scattered inelastically. Thus the total inelastic yield including multiple scattering to all orders is given by
Y,=
The first line expresses the underlying basic approximation of splitting attenuation, multiple scattering, and attenuation into three independent processes. In actual fact the overall correction factor is not separable in this way since each of the three processes gives rise to a positional dependence which again affects the next process. Numerically the approximation is quite good, however, as may be ascertained by comparing (12) with results of Monte-Carlo calculations. 3. Flux attenuation and transmission
Since the incident beam is normal to the axis of the target, it is easy to show that
= Yo(1 +P~ ~ + P 2 ~P1 ~ + P a ~P2 ~PI~+ ""). (10) Any preferred position or direction in the target disappears rapidly with successive scatterings so that it is quite a reasonable approximation to replace P2, P3 .... in (10) by P1. The error which this introduces is made even smaller by the fact that in practical examples the P's will always be small. The infinite series can then be summed with the result Y0 = Y,(1-P1 3).
(11)
To deduce the inelastic cross section Z~in from the experimental yield Yt one requires, in addition to the multiple scattering correction ( 1 - P I ~) and the flux attenuation So of the incident beam, also the attenuation of the outgoing beam of inelastically scattered neutrons. As was mentioned in the introduction, the effects of further elastic scatterings may be ignored. To a good approximation one may therefore simply use a further flux attenuation coefficient Sf calculated in terms of the non-elastic cross section Z--Zel corresponding to the energy of inelastic neutrons instead of the total dross section Z at the incident energy. Combining this with (3), (4) and (11), one finally obtains Z~n = Y t S i - ' ( 1 - P 1 ¢ ) S o '
= Y,[(1-P~ ~)/(SoSf)].
(12)
(13)
S O = G2(2ZR2, R1/R2),
(14)
where the functions G, are defined as
o.(~, ~)
G. (c~, O) +
d x [ g . { ~ ~/(1 - ~2 x ~) -
=
"fo
-
~/(1-x2)}-
r. n=0
Qo = G I ( 2 Z R 2 , R1/R2),
0.(~,/(1-~2x2)}],
(15)
while the functions g, are simply gl(x) = e-x,
(16)
gz(X) = x - l ( 1 - e - X ) .
(17)
When ? vanishes, one obtains the corresponding quantities for a solid cylinder G 1(c~,0) =
dx e -~4(~ -x2),
(18)
0
fl
Gz(~,0)= o dx
[1 --e -'4(1-x2)]
~x/(l_xZ )
(19)
It is of interest also to exhibit the forms these quantities would have had if the beam had been incident along rather than perpendicular to the target axis: Qh = g ~(2 ZH),
(20)
Sh = gz(2~,H).
(21)
These functions were in fact used in the derivation of (15). Although at least the function Gx(~, 0) may be expressed in terms of Bessel functions, an easily calculable numerical form is desired for the present
189
MULTIPLE SCATTERING CORRECTION FOR INELASTIC SCATTERING
purposes. To first order in ~ the following relations hold: G,(c~,7) = G,(c~ [ 1 - 7 2 J , 0).
(22)
In the limit when y tends to zero, they are clearly satisfied identically for all cc Numerical evaluation of the various integrals indicate that these relationships also hold to a high degree of accuracy when y tends to unity. For intermediate values of y the two sides of eq. (22) differ by less than 4% when ~ = 3 and less than 12% when ~ = 5 for the case n = 1. When n = 2, the error does not reach 4% until c~ = 9 and even for ~ = 20 the error is still less than 7%. Since actual applications will always be limited to small values of ~, the problem has thus been reduced to the determination of G,(c~, 0). The lowest order expansion and asymptotic forms of G1 (~, 0) are easily shown to be Gj(~,0) ~ 1 - ¼ n ~ , 1/~ 2,
~ ~ 1;
(23)
~ ~ 1.
(24)
Hence, G 1 m a y be approximated by a function of form F1 (~) =
1+A~
1+
(A+¼rc)o~+Bo~2+A~ 3
and the values of A and B varied to obtain the best possible fit to values of (18) obtained by numerical integration. The resulting approximation is F1 (~) =
1 + 0.346 c~ 1 + 1.145 ~ + 0.386 ~2 + 0.346 ~3 •
(25)
Similarly, G 2(0~, 0) has a lowest order expansion and
asymptotic form Gu (~, 0) ~ 1 -- ~ n~,
"~ ½n/a,
ct ~ 1 ;
(26)
~ ~ 1;
(27)
I + 0.242 ~ +0.052 ~ 2 1 + 0.742 ~ + 0.255 c~2 + 0.052 ~3
As was shown in section 2, the multiple scattering correction is determined by P1 = 1 - Q~. The calculation of Q~ differ from that of Qo in two respects. In the first place the neutron may originate anywhere in the target rather than enter it at the outer surface. This provides it with a nature (particularly the asymptotic behaviour in the limit of strong absorption) which is more like that of the flux attenuation S than that of the transmission Q. The starting point in the target is of course determined by the probability distribution for the previous interaction but the separability approximation discussed in section 2 implies that all volume elements should have the same weight. The second difference from Qo is that the direction of motion of the neutron is no longer that of the incident beam (i.e. normal to the target axis) but is determined by the angular distribution of elastic scattering. This angular distribution will be discussed later. For the present all directions will be taken to be equally likely. Even this simplified problem cannot be handled analytically. Instead, the cases of propagation along and normal to the axis will be considered separately, and QI ultimately be assumed to be some combination of the two. For propagation along the axis Qx is in fact identical to the flux attenuation factor Sh(e q. 21) which was approximated by eq. (29). For propagation normal to the axis, with all directions equally likely, one obtains Sr = G4(2ZR2, RI/R2),
G4(e,~) = (28)
Although S~ (eq. 21) has the analytic form given in eg. (17), the requirement of uniformity would suggest that it also be approximated by ratio of polynomials. The function F 3 (~) =
4. Multiple scattering
(35)
where the function G 4 is given by the double integral
and may be approximated by 1+0.139 c~+0.016 c~2 F2(~) = 1+0.535ct+0.109ct2+0.010c~ 3.
does this to within less than about one half of a percent for all a. In all of these approximation formulae, only the first order terms (23 and 26) may be kept if ~ is small enough but it will be left to the user to decide on the accuracy he requires.
(29)
;;;o dx
do
2x exp [ - e ( l l - 1 2 ) ] . n(l _~2) (36)
11 = ½(1--x2sin2(p) "~- ½ x c o s q ~ , 12 =
(yZ-x2sinZq~)~H[qg-n-sin-~(y/x)],
(37) (38)
where H(t) is the heaviside function (unity for positive argument and zero otherwise). For solid cylinders
190
C. A. ENGELBRECHT
(7 = 0) this reduces to G4(e,0) =
dx
dtp(2x[n)e -'t' .
(39)
Once again calculations confirm relation (22) for n = 4. This function has lowest order expansion and asymptotic form G4(~,0 ) ~
l -- ~C~/rC,
~.. 2,,/2/a,
~ "~ 1;
(40)
~ >> 1;
(41)
and may be approximated by F,(~) =
1 - 0.305 c¢+ 0.029 ~c2
(42)
R 3H Q = - - S h + - - S r
R+3H
(43)
was found to give excellent agreement. Here R - R2(1--R~/R~). The effect of anisotropic elastic scattering, represented by the expression a(O) = [ae,/(4n)] [1 + ~ to, P,(cos 0)] ,
(44)
was also investigated in the Monte Carlo calculations. To reduce the number of parameters, the forwardbackward asymmetry parameter a(O°)-- °'( 180° ) O"( 0 ° ) -[--O"( 1 8 0 ° )
=
to1 +to3 +to5 + " "
This is reminiscent of the transport cross section •tr = Sel(l"l-½tol)q-Sin--~'ZVab
Having obtained sufficiently accurate representations of the functions S h and St, we may now attempt to express Q~ in terms of them. To obtain the value of Q I for various combinations of cross sections and geometrical parameters, a Monte Carlo program was written for the investigation of life histories of individual neutrons in the target, At first the values of S, and Sr given by eqs. (21) and (35) were corrected for directions other than along and normal to the axis and sophisticated combinations allowing for the relative solid angles and for escape along the hole through the target were tried as representations for Q~. These attempts were not very successful. Instead, the following simple combination of the uncorrected Sh and S,
o9 =
(46)
Z0 = Sel ( 1 - - ~- to) q-Sinq---Yab .
1 +0.120c~-0.092 ~2 +0.010 c~3 "
R+3H
bility of deflecting the neutron from its direction of motion (thus preventing it from escaping) is smaller than in the isotropic case. To lowest order one may think of the elastic scattering as consisting of an isotropic part plus a part in the forward direction. For the purpose of Q~ only the isotropic part is effective. Thus the effective elastic scattering cross section for this effect may be obtained by subtracting a part proportional to the forward-backward asymmetry. It was found that very good agreement is obtained by replacing Z in the calculation of (43) by
(45)
1 -k- ~ 2 q- to4-{- •. •
was introduced before comparing the Monte Carlo results with expression (43) for isotropic scattering. If elastic scattering is preferentially forward, the proba-
,
(47)
which is, however, enhanced by forward scattering. 5. S u m m a r y and conclusions
For maximum accessibility, the results will be summarized briefly. The geometrical parameters of the target and the components of the cross section were defined in the first sentence of section 2. We also define Z o = X - :} to Z~,,
~'1 = "Y'-f'el
(48)
(at outgoing energy!),
(49) (50)
= se~/~,,
(5l)
R = Rz(I-R~/R2),
where the asymmetry parameter w has been given by eq. (45) in terms of the angular distribution (44). If the inelastic scattering cross section without corrections for flux attenuation or multiple scattering (but containing all other corrections such as detector sensitivity etc.) is written as ai,(unc), the corrected cross section is given by aln(COrr) 1-~(1-Q) O'in ( u n c )
(52)
S O SI
The flux attenuation factors are So = F2 (2 S R ) ,
(53)
$1 -- F2(2Z1 R),
radial ,1
= F 3(2.rt H),
axial, ]
(
(54)
MULTIPLE
SCATTERING
CORRECTION
where the two parts of eq. (54) refer to the detector being in the radial or axial direction from the target. The multiple scattering correction is given by Q = R F 3 (2 2~o H) + 3 H F 4 ( 2 ~o R) R+3H
(55)
and the functions F,(~) are given by (28), (29) and (42). If the target contains, besides the nucleide under consideration (with cross sections S0, '~el, '~i,, '~ab), number fractions flk of nucleides number k (with cross sections S k etc.), the same procedure may be followed provided the following cross sections are used: flo
= 1- S
fl~,
k=l
= t oZ° + Eel = S k=O
k= 1
k=O
flk~kel,
Sl. =/~oZf, Lab
=
~--~el--~in,
+... tO
k=O
S
k=0
:o,
+.
To avoid confusion with the macroscopic cross sections Z, the symbol S is used here to indicate summation.
FOR
INELASTIC
SCATTERING
191
Apart from the purely numerical approximations made in the representation of these functions, the following approximations in principle were made. Firstly, the multiple scattering correction was treated as independent of the positional and directional distribution resulting from the flux attenuation process. Secondly the interaction probabilities P1, P2, P3 . . . . in eq. (10) were taken to be equal. Finally, the values of Q obtained from Monte Carlo calculations were related to S h and S r in a purely empirical way. To evaluate the corrections, one already needs the cross sections. The procedure to be followed, is therefore to use the uncorrected cross sections for the calculation of the corrections to lowest order. The resulting corrected cross sections may then be used for a better calculation of the corrections. If the corrections are small, as they should be in acceptable experimental set-ups, this procedure should converge rapidly. The correction to the integrated elastic cross section may also be obtained by adding the multiple scattering corrections to all inelastic levels. Although this algorithm lacks much of the sophistication available in certain computer codes, its simplicity of application seems to merit its wider distribution. The author is indebted to D. Reitmann for acquainting him with the problem, to E. Barnard for performing a number of comparisons with M A G G I E , and to W. Cilli6 and A. le Roux for doing most of the computer programming. Reference 1) j.B. Parker, J.H. Towl¢, D. Sams, W.B. Gilboy, A. D. Purnell and H. J. Stevens, Nucl. Instr. and Meth. 30 (1964) 77.
INSTRUMENTS
AND
METHODS 80 (197 O) I 8 7 - 1 9 1 ;
©
NORTH-HOLLAND
PUBLISHING
CO.
MULTIPLE SCATTERING CORRECTION FOR INELASTIC SCATTERING F R O M CYLINDRICAL TARGETS C.A. E N G E L B R E C H T
Atomic Energy Board, Pelindaba, South Africa Received 15 August 1969 Simple expressions are developed for the corrections for flux attenuation and multiple scattering of neutrons from cylindrical targets.
1. Introduction
The accurate extraction of neutron cross sections from experimental results obtained with thick targets is possible only after application of corrections for flux attenuation and multiple scattering. Although programs such as M A G G I E 1) are now available for the calculation of every conceivable correction including even effects such as inhomogeneity and nonmonochromaticity of the incident beam, the nonavailability of a suitable computer, the excessive demands in computer time, or simply the sheer amount of human labour required in any case, make it desirable to have a simple formula available by means of which at least some of these corrections may be estimated with the minimum of trouble. The purpose of this note is to satisfy this demand. To obtain such a simple formula, one must, of course, simplify the general problem somewhat. The scattering of neutrons from a hollow cylindrical target will be considered and the incident beam will be assumed to be monoenergetic, homogeneous, unidirectional, and normal to the axis of the cylinder. Only the corrections which are necessary for the extraction of inelastic scattering cross sections will be considered. Recoil corrections will be ignored and the angular distribution for elastic scattering will be taken into account in a very primitive way only. The inelastic scattering will be assumed to be isotropic, which also makes it reasonable to ignore elastic scattering occurring after the production of an inelastically scattered neutron: there should be no nett change in the number of inelastic neutrons emerging in the solid angle subtended by the detector. Removal of inelastic neutrons due to further inelastic scattering or other absorptive processes will be taken into account in an approximate way. 2. Formulation of the problem
The target is taken to be a hollow cylinder with inner and outer radii RI and R 2 and half height H. 187
The total macroscopic cross section Z is composed of an elastic, an inelastic, and an absorptive part: (1)
Z = Eel-{-,~in-~-Zabs
where Z~b includes inelasticscattering to all levels other than the level or levels excited with cross section Si.. The total yield of elastically scattered, inelastically scattered or absorbed neutrons is denoted W and is simply the sum of these three component parts: W = Xo + Yo + Zo •
(2)
Here X o denotes the yield of neutrons whose first interaction is an elastic scattering and similarly Yo for primary inelastics and Zo for primary absorbed neutrons (if the absorption is real, Z 0 is of course not a "yield" in the usual sense). These two decompositions are in the same ratio Xo_ Eel
Y o _ Zo _ W. Zln
Zab
(3)
E
If the yields are measured per unit target volume and per unit incident flux, this ratio is the flux attenuation coefficient So = w / z ,
(4)
which allows for the fact that the nuclei on the shadow side of the target do not experience the full incident flux. A related quantity is the transmission coefficient, the probability of a neutron being transmitted through the target without any interaction, averaged over the part of the incident beam hitting the target: Qo = l - e o .
(5)
The flux attenuation coefficient is simply a positiondependent transmission coefficient averaged over the volume of the target. The fraction Po of incident neutrons which do interact, are decomposed according to (2). Hence, the fraction which are scattered elastically is simply ~Po
188
C. A. ENGELBRECHT
where ¢ - Z~,IZ,
(6)
whereas the primary inelastic fraction is t/Po where t1 - Z~,IZ.
(7)
For the elastic fraction one may again define a transmission coefficient Q1 = l - P 1 .
(8)
The fraction P~ which interact, once again consists of three components W 1 = XI'4-YI-I-Z1
(9)
in the same ratio as (2). Thus the fraction of incident neutrons scattered inelastically after a single elastic scattering is given by t/P~ ~ Po while ~PI ~Po are scattered elastically for a second time. In the same way a fraction t/P 2 of these are scattered inelastically. Thus the total inelastic yield including multiple scattering to all orders is given by
Y,=
The first line expresses the underlying basic approximation of splitting attenuation, multiple scattering, and attenuation into three independent processes. In actual fact the overall correction factor is not separable in this way since each of the three processes gives rise to a positional dependence which again affects the next process. Numerically the approximation is quite good, however, as may be ascertained by comparing (12) with results of Monte-Carlo calculations. 3. Flux attenuation and transmission
Since the incident beam is normal to the axis of the target, it is easy to show that
= Yo(1 +P~ ~ + P 2 ~P1 ~ + P a ~P2 ~PI~+ ""). (10) Any preferred position or direction in the target disappears rapidly with successive scatterings so that it is quite a reasonable approximation to replace P2, P3 .... in (10) by P1. The error which this introduces is made even smaller by the fact that in practical examples the P's will always be small. The infinite series can then be summed with the result Y0 = Y,(1-P1 3).
(11)
To deduce the inelastic cross section Z~in from the experimental yield Yt one requires, in addition to the multiple scattering correction ( 1 - P I ~) and the flux attenuation So of the incident beam, also the attenuation of the outgoing beam of inelastically scattered neutrons. As was mentioned in the introduction, the effects of further elastic scatterings may be ignored. To a good approximation one may therefore simply use a further flux attenuation coefficient Sf calculated in terms of the non-elastic cross section Z--Zel corresponding to the energy of inelastic neutrons instead of the total dross section Z at the incident energy. Combining this with (3), (4) and (11), one finally obtains Z~n = Y t S i - ' ( 1 - P 1 ¢ ) S o '
= Y,[(1-P~ ~)/(SoSf)].
(12)
(13)
S O = G2(2ZR2, R1/R2),
(14)
where the functions G, are defined as
o.(~, ~)
G. (c~, O) +
d x [ g . { ~ ~/(1 - ~2 x ~) -
=
"fo
-
~/(1-x2)}-
r. n=0
Qo = G I ( 2 Z R 2 , R1/R2),
0.(~,/(1-~2x2)}],
(15)
while the functions g, are simply gl(x) = e-x,
(16)
gz(X) = x - l ( 1 - e - X ) .
(17)
When ? vanishes, one obtains the corresponding quantities for a solid cylinder G 1(c~,0) =
dx e -~4(~ -x2),
(18)
0
fl
Gz(~,0)= o dx
[1 --e -'4(1-x2)]
~x/(l_xZ )
(19)
It is of interest also to exhibit the forms these quantities would have had if the beam had been incident along rather than perpendicular to the target axis: Qh = g ~(2 ZH),
(20)
Sh = gz(2~,H).
(21)
These functions were in fact used in the derivation of (15). Although at least the function Gx(~, 0) may be expressed in terms of Bessel functions, an easily calculable numerical form is desired for the present
189
MULTIPLE SCATTERING CORRECTION FOR INELASTIC SCATTERING
purposes. To first order in ~ the following relations hold: G,(c~,7) = G,(c~ [ 1 - 7 2 J , 0).
(22)
In the limit when y tends to zero, they are clearly satisfied identically for all cc Numerical evaluation of the various integrals indicate that these relationships also hold to a high degree of accuracy when y tends to unity. For intermediate values of y the two sides of eq. (22) differ by less than 4% when ~ = 3 and less than 12% when ~ = 5 for the case n = 1. When n = 2, the error does not reach 4% until c~ = 9 and even for ~ = 20 the error is still less than 7%. Since actual applications will always be limited to small values of ~, the problem has thus been reduced to the determination of G,(c~, 0). The lowest order expansion and asymptotic forms of G1 (~, 0) are easily shown to be Gj(~,0) ~ 1 - ¼ n ~ , 1/~ 2,
~ ~ 1;
(23)
~ ~ 1.
(24)
Hence, G 1 m a y be approximated by a function of form F1 (~) =
1+A~
1+
(A+¼rc)o~+Bo~2+A~ 3
and the values of A and B varied to obtain the best possible fit to values of (18) obtained by numerical integration. The resulting approximation is F1 (~) =
1 + 0.346 c~ 1 + 1.145 ~ + 0.386 ~2 + 0.346 ~3 •
(25)
Similarly, G 2(0~, 0) has a lowest order expansion and
asymptotic form Gu (~, 0) ~ 1 -- ~ n~,
"~ ½n/a,
ct ~ 1 ;
(26)
~ ~ 1;
(27)
I + 0.242 ~ +0.052 ~ 2 1 + 0.742 ~ + 0.255 c~2 + 0.052 ~3
As was shown in section 2, the multiple scattering correction is determined by P1 = 1 - Q~. The calculation of Q~ differ from that of Qo in two respects. In the first place the neutron may originate anywhere in the target rather than enter it at the outer surface. This provides it with a nature (particularly the asymptotic behaviour in the limit of strong absorption) which is more like that of the flux attenuation S than that of the transmission Q. The starting point in the target is of course determined by the probability distribution for the previous interaction but the separability approximation discussed in section 2 implies that all volume elements should have the same weight. The second difference from Qo is that the direction of motion of the neutron is no longer that of the incident beam (i.e. normal to the target axis) but is determined by the angular distribution of elastic scattering. This angular distribution will be discussed later. For the present all directions will be taken to be equally likely. Even this simplified problem cannot be handled analytically. Instead, the cases of propagation along and normal to the axis will be considered separately, and QI ultimately be assumed to be some combination of the two. For propagation along the axis Qx is in fact identical to the flux attenuation factor Sh(e q. 21) which was approximated by eq. (29). For propagation normal to the axis, with all directions equally likely, one obtains Sr = G4(2ZR2, RI/R2),
G4(e,~) = (28)
Although S~ (eq. 21) has the analytic form given in eg. (17), the requirement of uniformity would suggest that it also be approximated by ratio of polynomials. The function F 3 (~) =
4. Multiple scattering
(35)
where the function G 4 is given by the double integral
and may be approximated by 1+0.139 c~+0.016 c~2 F2(~) = 1+0.535ct+0.109ct2+0.010c~ 3.
does this to within less than about one half of a percent for all a. In all of these approximation formulae, only the first order terms (23 and 26) may be kept if ~ is small enough but it will be left to the user to decide on the accuracy he requires.
(29)
;;;o dx
do
2x exp [ - e ( l l - 1 2 ) ] . n(l _~2) (36)
11 = ½(1--x2sin2(p) "~- ½ x c o s q ~ , 12 =
(yZ-x2sinZq~)~H[qg-n-sin-~(y/x)],
(37) (38)
where H(t) is the heaviside function (unity for positive argument and zero otherwise). For solid cylinders
190
C. A. ENGELBRECHT
(7 = 0) this reduces to G4(e,0) =
dx
dtp(2x[n)e -'t' .
(39)
Once again calculations confirm relation (22) for n = 4. This function has lowest order expansion and asymptotic form G4(~,0 ) ~
l -- ~C~/rC,
~.. 2,,/2/a,
~ "~ 1;
(40)
~ >> 1;
(41)
and may be approximated by F,(~) =
1 - 0.305 c¢+ 0.029 ~c2
(42)
R 3H Q = - - S h + - - S r
R+3H
(43)
was found to give excellent agreement. Here R - R2(1--R~/R~). The effect of anisotropic elastic scattering, represented by the expression a(O) = [ae,/(4n)] [1 + ~ to, P,(cos 0)] ,
(44)
was also investigated in the Monte Carlo calculations. To reduce the number of parameters, the forwardbackward asymmetry parameter a(O°)-- °'( 180° ) O"( 0 ° ) -[--O"( 1 8 0 ° )
=
to1 +to3 +to5 + " "
This is reminiscent of the transport cross section •tr = Sel(l"l-½tol)q-Sin--~'ZVab
Having obtained sufficiently accurate representations of the functions S h and St, we may now attempt to express Q~ in terms of them. To obtain the value of Q I for various combinations of cross sections and geometrical parameters, a Monte Carlo program was written for the investigation of life histories of individual neutrons in the target, At first the values of S, and Sr given by eqs. (21) and (35) were corrected for directions other than along and normal to the axis and sophisticated combinations allowing for the relative solid angles and for escape along the hole through the target were tried as representations for Q~. These attempts were not very successful. Instead, the following simple combination of the uncorrected Sh and S,
o9 =
(46)
Z0 = Sel ( 1 - - ~- to) q-Sinq---Yab .
1 +0.120c~-0.092 ~2 +0.010 c~3 "
R+3H
bility of deflecting the neutron from its direction of motion (thus preventing it from escaping) is smaller than in the isotropic case. To lowest order one may think of the elastic scattering as consisting of an isotropic part plus a part in the forward direction. For the purpose of Q~ only the isotropic part is effective. Thus the effective elastic scattering cross section for this effect may be obtained by subtracting a part proportional to the forward-backward asymmetry. It was found that very good agreement is obtained by replacing Z in the calculation of (43) by
(45)
1 -k- ~ 2 q- to4-{- •. •
was introduced before comparing the Monte Carlo results with expression (43) for isotropic scattering. If elastic scattering is preferentially forward, the proba-
,
(47)
which is, however, enhanced by forward scattering. 5. S u m m a r y and conclusions
For maximum accessibility, the results will be summarized briefly. The geometrical parameters of the target and the components of the cross section were defined in the first sentence of section 2. We also define Z o = X - :} to Z~,,
~'1 = "Y'-f'el
(48)
(at outgoing energy!),
(49) (50)
= se~/~,,
(5l)
R = Rz(I-R~/R2),
where the asymmetry parameter w has been given by eq. (45) in terms of the angular distribution (44). If the inelastic scattering cross section without corrections for flux attenuation or multiple scattering (but containing all other corrections such as detector sensitivity etc.) is written as ai,(unc), the corrected cross section is given by aln(COrr) 1-~(1-Q) O'in ( u n c )
(52)
S O SI
The flux attenuation factors are So = F2 (2 S R ) ,
(53)
$1 -- F2(2Z1 R),
radial ,1
= F 3(2.rt H),
axial, ]
(
(54)
MULTIPLE
SCATTERING
CORRECTION
where the two parts of eq. (54) refer to the detector being in the radial or axial direction from the target. The multiple scattering correction is given by Q = R F 3 (2 2~o H) + 3 H F 4 ( 2 ~o R) R+3H
(55)
and the functions F,(~) are given by (28), (29) and (42). If the target contains, besides the nucleide under consideration (with cross sections S0, '~el, '~i,, '~ab), number fractions flk of nucleides number k (with cross sections S k etc.), the same procedure may be followed provided the following cross sections are used: flo
= 1- S
fl~,
k=l
= t oZ° + Eel = S k=O
k= 1
k=O
flk~kel,
Sl. =/~oZf, Lab
=
~--~el--~in,
+... tO
k=O
S
k=0
:o,
+.
To avoid confusion with the macroscopic cross sections Z, the symbol S is used here to indicate summation.
FOR
INELASTIC
SCATTERING
191
Apart from the purely numerical approximations made in the representation of these functions, the following approximations in principle were made. Firstly, the multiple scattering correction was treated as independent of the positional and directional distribution resulting from the flux attenuation process. Secondly the interaction probabilities P1, P2, P3 . . . . in eq. (10) were taken to be equal. Finally, the values of Q obtained from Monte Carlo calculations were related to S h and S r in a purely empirical way. To evaluate the corrections, one already needs the cross sections. The procedure to be followed, is therefore to use the uncorrected cross sections for the calculation of the corrections to lowest order. The resulting corrected cross sections may then be used for a better calculation of the corrections. If the corrections are small, as they should be in acceptable experimental set-ups, this procedure should converge rapidly. The correction to the integrated elastic cross section may also be obtained by adding the multiple scattering corrections to all inelastic levels. Although this algorithm lacks much of the sophistication available in certain computer codes, its simplicity of application seems to merit its wider distribution. The author is indebted to D. Reitmann for acquainting him with the problem, to E. Barnard for performing a number of comparisons with M A G G I E , and to W. Cilli6 and A. le Roux for doing most of the computer programming. Reference 1) j.B. Parker, J.H. Towl¢, D. Sams, W.B. Gilboy, A. D. Purnell and H. J. Stevens, Nucl. Instr. and Meth. 30 (1964) 77.