Neutron flux measurements by resonance activation of foils

NUCLEAR

"NS'IRUMENTS

AND

METHODS

23 (1963)

29--35;

NORTH-HOLLAND

PUBLISHING

CO.

NEUTRON FLUX MEASUREMENTS BY RESONANCE ACTIVATION OF FOILS A. M. J U D D

U K . A . E . A . , Dounreay Experimental Reactor Establishment, Thurso, Caithness, Scotland R e c e i v e d 27 N o v e m b e r 1962

Various a s s e s s m e n t s of flux depression a n d self-protection b y u foil i r r a d i a t e d in a diffusing m e d i u m are c o m p a r e d . F r o m a consideration of self-protection for resonance acttvation, an

e v a l u a t i o n of t h e " s a n d w i c h " technique of m e a s u r i n g n e u t r o n flux a t resonance energies is m a d e .

1. Introduction

resonances, the energy distribution of flux m a y be estimated. This article seeks to collect some of the more imp o r t a n t theoretical results on the a c t i v a t i o n of foils. F r o m these, a n e v a l u a t i o n of the " s a n d w i c h " technique of exploiting self-protection is made. (This is b a s e d on the work of Ehretl).)

N e u t r o n flux m e a s u r e m e n t s are often m a d e b y m e a n s of foils. T h e foil is placed in t h e position of interest, i r r a d i a t e d for a k n o w n time, a n d the q u a n t i t y of some radioactive nuclide produced is assessed b y c o u n t i n g decay p- or 7- radiation. T h u s a n e s t i m a t e of t h e reaction r a t e for t h e p r o d u c t i o n of this nuclide is obtained. B y using materials w i t h cross sections which v a r y w i t h energy i n a distinctive m a n n e r , a n e s t i m a t e of t h e s h a p e of the flux s p e c t r u m c a n sometimes be obtained. F a s t t h r e s h o l d reactions can be used, b u t i t is difficult to o b t a i n more t h a n a crude picture of t h e s p e c t r u m b y t h i s means. This is due to t h e similarity of t h e shapes of m o s t t h r e s h o l d reaction cross section curves, a n d t h e closeness of t h e i r effective t h r e s h o l d energies. Moreover, this only provides i n f o r m a t i o n a b o u t t h e MeV energy range, since t h e r e are few reactions w i t h a t h r e s h o l d less t h a n 1 MeV. The other distinctive features of crosssection curves are t h e energies of t h e resonances of greatest a m p l i t u d e a n d these m a y be more useful in e x a m i n i n g spectra. I n finding reaction rates from foil a c t i v a t i o n results, corrections h a v e to be m a d e for "flux depression" b y t h e foil, a n d " s e l f - p r o t e c t i o n " w i t h i n it. (These t e r m s are defined below.) The selfprotection effect m a y be used to assess the activation due to a d o m i n a t i n g resonance i n t h e activation cross section. The n e u t r o n flux a t the energy of t h e resonance m a y t h u s be found, a n d b y employing a series of materials with suitable p r o m i n e n t

2. Symbols T h e following symbols are used t h r o u g h o u t this article : p = reaction rate per unit area of a foil ; P,cs = reaction r a t e due to resonance neutrons; ¢(E) = n e u t r o n flux per u n i t energy; (¢(E) d E = flux of n e u t r o n s w i t h energies i n the range E to E + dE); T(E) = thickness of foil i n m e a n free paths, = t 27,(E) for a purely a b s o r b i n g material, where t = foil thickness; 27a(E) = macroscopic absorption cross section, = N a n ( E ) , where N = n u m b e r of a t o m s of foil m a t e r i a l per u n i t volume ; ~a(E) = microscopicabsorptioncross section; ~(T) = m e a n p r o b a b i l i t y for a n isotropic flux t h a t a n e u t r o n i n c i d e n t on the foil will be a b s o r b e d ; FD = flux depression factor; Fs = a[2~ = self-protection factor; 1) G. E h r e t , A t o m p r a x i s 7 (1961} 393. 29

30

A.M.

I(M)

= mean log energy decrement for elastic scattering in the diffusing medium surrounding the foil; = mean log energy decrement for foil material; = energy of resonance t i n foil material ; = total width of this resonance; = t Xa(Eoi ) for a pure absorder; = thickness of foil in mean free paths at the peak of the resonance; = resonance activation function;

¢1

= $(Eoi) ;

Co,C1)

= reaction rates in respectively the outer and inner foils of a sandwich.

~v Eo i F~ Ms

3. Foil Activation

If we have a foil so small that it does not distort the flux distribution in an appreciable manner, and if it is embedded in the diffusing medium, we have

p = f

E

z(E)$(E)dE.

(1)

In practice, the result of an activation experiment has to be corrected for two effects to obtain this ideal reaction rate. These are: a. The lowering of the flux near the foil due to the introduction of a neutron sink into the diffusing medium. We shall call this flux depression. b. The shielding of the centre of the foil by the outside layers of the foil. We shall call this self-

protection. These effects have been dealt with by several authors. We shall discuss their results, and in particular apply them to resonance detectors and " s a n d w i c h " foils. 4. Flux Depression and Self-Protection

Bothe2), Skyrme3), Ritchie and Eldiidge4), and Dalton and OsborneS), have calculated the reaction rate for thermal neutrons in a thin foil with high absorption cross-section. A comparison of these calculations is given in fig. 1. Bothe evaluates ~(~), assuming the foil to be of E a (x) = ½ { ( 1 - - x ) e - x + xSE5 (x)}, and E x (x) = - - E l (--x). This Ei notation is used in some of the literature. These iunctions are t a b u l a t e d b y J a b n k e and Erode6).

JUDD

large extent in its plane compared with its thickness. In this case: =

1 -

2E3(,),

(z)

where E3(~)t is a particular case of the exponential integral defined by e-~tdt E.(x) = $1 tn

[~

If we define a flux depression factor F v and a self-protection factor Fs, such that equation (1) becomes

p = [ ,FDFs¢ dE,

(3)

d E

where p is the observed reaction rate and ~ the undisturbed flux, and F s represents the self-shielding effect if there were no depression (i.e. if $ were the actual incident flux), then Bothe's results m a y be expressed in the form : F s = ct/2x

(4)

and I[FD = 1 + ½cx(3RL/22t, (R + L) - 1) forR >> 2t, (5)

I/F v = 1 + 0.34~R/3t, for R ~

~tr,

(6)

where 2tr and L are the transport mean free path and the diffusion length in the surrounding medium, and R is the radius of the foil, which is taken to be circular. Eq. (5) results from a diffusion theory approach, where the foil is represented by a sphere, and the boundary condition between this and the surrounding medium is expressed by means of an albedo derived from ~, and eq. (6) from an evaluation of the probability of a neutron passing through the sphere two or more times. Both are in the form due to TittleT), which is an empirical modification of Bothe's expression. Tittle recommends that eq. (5) should not be used for 5) W. Bothe, Z. Phys. 120 (1943) 437; AEC-tr-1691. a) T. H. R. Skyrme, Reduction in neutron density caused by an absorbing disc. M.S. 91 (1943) (Reprinted 196 l). 4) R. H. Ritchie a n d H. B. Eldridge, Nucl. Sci. and E n g i neering $ (1960) 300. 5) G. R. Dalton a n d R. K. Osborne, Nucl. Sci. and Engineering 9 (1961) 198. 6) E. J a h n k e and F. E m d e , Tables of Functions, ( D o v e r Publications, New York, 1954). 7) C. W. Tittle, Nucleonics 8 (1951) 5; 9 (1951) 60.

NEUTRON

FLUX MEASUREMENTS

o.

BY R E S O N A N C E

ACTIVATION

OF F O I L S

31

I

°.7o

ooo'

ooa

ooa FOIL

o'

o;o,

THICKNESS

tp

o o.

o;o

o o.

o'oos

o o.

ooo.

INCHES

Fig. 1. Comparison of calculation of flux depression and self protection, a. Reduction in flux in a 1.5 c m radius gold foil in water.

b. Reduction in flux in a 1.25 c m radius i n d i u m foil in graphite - Dalton a n d Osborne . . . . . Skyrme - - - Bothe ........ Ritchie and Eldridge.

R < 22tr, a n d eq. (6) should n o t be used for highly a b s o r b i n g diffusing m e d i u m . F o r the case of a t h i n foil (z <~ 1), we h a v e

R < 0.4L. I n general, Skyrme's results are valid f o r z ~ l a n d t / R ~ 2. Ritchie a n d Eldridge use a variational m e t h o d to solve the t r a n s p o r t equation, a n d o b t a i n results:

a = 2z + terms of order z 2 log z ,

(7)

a n d F s ~ 1, the e x p e c t e d result for a weakly absorbing foil. F o r z >> 1, ct ~ 1 a n d F s ~ l[2z. Skyrme uses a more sophisticated a p p r o a c h involving t r a n s p o r t theory, a n d his results m a y be w r i t t e n in t h e following form (after Ritchie a n d Eldridgea)). F s = ct/2z as betore 1~El)

=

I +

½c~(D t -

D'I) ,

(8)

F s = o¢/2z,

(10)

1/F D = 1 + ½o~g,,(y, z) ,

where y = X d X,. These r e s u l t s a p p l y f o r a n i n f i n i t e foil (R ~ ~ ) , a n d a p p r o x i m a t e to Skyrme's results for z *~ 1, a n d to B o t h e ' s for z > 4. The values of gv are presented graphically. If B o t h e ' s a n d Skyrme's results are expressed in a similar form, B o t h e ' s value of F D m a y be w r i t t e n I / F o = 1 + ½ctgB, where ga = 3RL/22tr(R + L) -

where

D1 =

2(Ztz - L-2) [½ - ~(2R/L)] L (Y,sZt Zs-- Ztz+ L -2)

Similarly, S k y r m e ' s results are expressed b y : 1/FD = 1 + ½ctgs,

and

~b(z) = 7t

o (1

-- /2)~-e-*'

-

where

gs=Di-D'l

dt.

(L, 27s a n d 2 t are the diffusion length, a n d scattering a n d t o t a l cross-sections in the diffusing medium). D~ is a still more complicated expression, b u t Skyrme points out t h a t it is less t h a n 5 % of D~, a n d does not e v a l u a t e it. The values of D z are presented graphically. F o r 27t - 27s ~ 2, (the n o r m a l diffusion t h e o r y condition), a n d 2R ~ L, we h a v e D1 "~ l4 ~R z~ '- (

1 ,

~6/t ~-)

(9)

a n d this is accurate to w i t h i n 5 % for L 2 t > 5, a n d

Since gv only applies in the case of an infinite foil, Ritchie a n d Eldridge suggest t h a t for a foil of finite radius, we should write 1/Fr, = 1 + ½~g,

(11)

g = gs(R) gs(R gv = oo) "

(12)

where

B o t h e ' s a n d Skyrme's results h a v e been compared with experiment. Tittle 7) (using i n d i u m a n d dysprosium in hydrogeneous media), K l e m a a n d Ritchie s) (indium a n d gold in graphite), F i t c h a n d 8) E. D. K l e m a and R. H. Ritchie, Phys. Rev. 87 (1952) 167.

32

A.M. JUDD

D r u m m o n d 9) (indium in water), and Gallagher t°) (indium in graphite), find good agreement with Bothe's t h e o r y and Gallagher prefers Bothe to Skyrme. Thompson11), using indium in graphite, finds better agreement with Skyrme, but seems to have omitted to apply the F s correction in Bothe's theory. Dalton and Osborne 5) have probably given the most accurate treatment of the subject. They transform the transport equation into an integral equation, and integrate numerically. They do not distinguish between flux-depression and self-protection, and their results are presented as a series of g~.aphs. T h e y compare well with experiment (fig. 2) EXPERIMENTS

FDFS

[~

~] GALL AGNER 0 KLEMA

o

AND

A- I.gcm RITCHIE

.Q.- ,.9 c rn.

0.7 - Az2cm 0

0002 EOII-

0.004 THICKNESS,

0.006

O.OOS

INCHES.

Fig. 2. Comparison of calculation, by the method of Dalton and Osborne, with experiment. (Figure reproduced from Nucl. Sci. and Engineering 9(1961) 202).

but the authors point out that the margins of error on results of most experiments are so wi(,e as to minimise their validity in confirming theoretical predictions. Much more accurate experiments must be performed before the results can be used to criticise the various theoretical approaches On this basis, it seems reasonable to suggest that for v e r y accurate work, Dalton and Osborne's m e t h o d should be used: this would probably necessitate the performance of a new set of calculations for each set of experiments done. For ordinary work, h o w e v e r the simplest results, those of Bothe, seem most useful, since they are expressed in a 9) S. H. Fitch and J E. Drummond, Nuclear detector perturbations. LRL95 (1954). 10) T. L. Gallagher, Nucl. Sci. and Engineering 3 (1958) 110 zl) M. W. Thompson, Some effects of the self-absorption of fl-rays and neutrons in neutron detecting foils. A E R E R P / R (1954) 1549.

simple analytic form, and the errors involved are less than most experimental errors In addition, the Fs correction, which Is common to all three simpler approaches is usually the most important. 5. Resonance Detectors

Flux depression is important when we are considering activation by thermal neutrons, since a thermal neutron that is absorbed m passing through the foil is not available to diffuse back into i~. This gives rise to a flux depression within a distanc~ from the foil comparable with the neutron mean free path in the diffusing medium. The situation is different when we are considering resonance activation. For a cross-section that consists only of a single narrow resonance (F ,~ lEo), in the slowing doom part of the spectrum, there is no flux depression effect. A neutron absorbed in the resonance as it passed through the foil would not have been available for absorption on passing through a second time since to do so, it must suffer at least one scattering collision, which would remove it from the resonance energy range. The flux at, and below, an energy of about Eo(l - ¢) will be perturbed, but not the flux at Eo itself (Fitch and D r u m m o n d 7) point out that there is a finite probability of a neutron suffering a large number of small angle scatterings and returning with small total energy decrement, but this probability is itself small. These authors also point out that the depression effect of one resonance on another at lower energy is small provided the first resonance is narrow, and they are separated by 3 or 4~E o. Thompson's assessment of resonance depression 1°) appears to be in error). In practice, we usually find that z(E) is small except at a resonance energy. At the resonance energy, there is a small flux depression effect due to the small, but non-zero, value of z at energies Eo(1 + ~) and above. Thus, as an approximation, it seems best to calculate FD(E) from ,(E) except for a resonance energy, where FD should be calculated from z at an energy immediately above the resonance. (Often, we have FD--~ 1 except at thermal energies). We now consider the reaction rate in a foil for which the reaction cross section consists of a single

NEUTRON

FLUX

MEASUREMENTS

BY

absorpt';on resonance. (The effect of a super-imposed scattering resonance is considered below.) F r o m equation (3) we have P,e; =

RESONANCE

ACTIVATION

OF

FOILS

33

10£ / /

z.x...v

/

//

FDFslrd? d E . 0

/

We assume F D is constant, and~b is constant over the resonance range (E o 4- F). Then, using equation (4), #,e, = ½FD~b

/o

•

e dE,

/

//

(13)

0 /

where e -- 1 - 2Ea(z ), w = t2;,, and we use the Breit-Wigner expression for resonance absorption cross section a.(E) = ~ 2 g ( ~ 9 ) ½

(E -

F'F7 ~ o ) 2 + (½r)""

DOl

[1 -- 2E 3 (M/(1 + q2))] dr/, -oo

where q = T 2 (E - Eo)

and m = t Y,. . . .

= t N . 4n]~ g F.F~/F 2 .

(The lower limit of the integral should be - 2 E o / F , but it is simpler and nearly correct to

write - oo, since the integrand is negligible for large l ~ [ ) . We can write this result in the form P,,, = ¼FD¢hrI(M),

(15)

where I(x) =

S /

(14)

We note that the expression (13) for Pre, is no longer linear in t. We now approximate by saying the fractional change in energy over the resonance width is small, ( £ 4` Eo) , and so we write E o / E = 1. Then Or,, = J;Fa

J

/ f // /

[1 - 2E3(x/(l + y2))] @ .

Ol

,r

lO

Fig. 3. I (x). In practice, this latter approximation is useful for x>6. 6. Scattering Resonances For light elements (Z < 40), an absorption resonance is usually accompanied by a scattering resonance of larger amplitude. If a resonance neutron is scattered in the foil, it will be degraded in energy, and if ~FEo > F, it will probably be scattered out of the resonance. Thus we m a y say that whatever reaction a resonance neutron undergoes it disappears as a resonance neutron, and that a fraction a d ( a , + a.) of the disappearances result in the production of an active nuclide. Now a s = n~Io2 g

F2

(16)

(E - Eo) 2 + (½F)2

and so, ignoring variation in E / E o as before,

*.I(..

+ ,,,) =

r, lr

-ot~

I ( x ) has been evaluated numerically, and is shown

and equation (15) is modified to p,,, = ¼ F D C r r l ( M ) ,

in fig. 3. For

x 4. 1,

I(x) ~ 2 n x .

For

x ~ 1,

I(x) ~ { ~ / ~ ~ 4.73 ~/~

where we now have M = t Zt=.x Xt =., = N . 4n~

g F./F .

(17)

34

A.M.

JUDD

Typically the foils used are a b o u t 0.005" thick: i.e. t = 0.013 cm. I n general, density ~ 10 a n d a t o m i c weight ~ 100. T h u s t Z ~ 0.001a, where a is in barns. So as a rough guide, we m a y say t h a t if a is less t h a n 100 barns, t ~ < 0.1, a n d the foil m a y be regarded as " t h i n " . F o r most of t h e detector materials we are interested in, a < 100 b a r n s except for resonances. The only exception is dysprosium at t h e r m a l e n e r g i e s I g n o r i n g this, we m a y confine ourselves to foils t h a t are ' t h i n " , except at reson a n c e energies. I n this case, we can rewrite e q u a t i o n (3) in the form

Co -

p = (

d

eDvq~ dE + Z FD, P,e,,, non--res

(18)

i

where the integral is t a k e n over all energies for which the foil is " t h i n " , a n d the sum over all " h i g h " resonances (for which the foil is n o t " t h i n " ) . FD depends on t, b u t only weakly. T h u s the in tegral in eq. (18) represents the p a r t of p t h a t varies linearly w i t h t, while the sum represents the nonlinear part.

7. Sandwich Foils I n m a n y cases, the resonance region is d o m i n a t e d b y one particularly h i g h resonance. If the p a r t of p t h a t is non-linear in t consists m a i n l y of a c t i v a t i o n due to this resonance (i.e. the foil is " t h i n " at all energies except the resonance), the use of " s a n d wich foils" is a simple m e a n s of isolating the nonlinear part, a n d thus of m e a s u r i n g the flux a t the resonance energy. This m e t h o d is described in detail b y E h r e t l ) . We consider irradiation of a sandwich consisting of three similar foils clamped together• If p(t) is the reaction rate per unit area of a foil of thickness t, then C o = ½p(t) - {p(2t) + ½p(3t), C, = p(2t) - p(t), • C O - C 1 = ½ E3p(t) - 3p(2t) + p(3t)i] .

(19)

If we s u b s t i t u t e the values of p from eq. (18), the factors FD, FD, are the same for outer a n d inner foils, a n d so the integral term, being linear in t, vanishes from Co - C1, a n d we are left with

3I(2M,)

C, = ~ Z Fo,(3t)dPf ~,(3I(M,) i

(20)

+ I(3M3),

where the sum is t a k e n over all " h i g h " resonances. If there is one v e r y high resonance t h a t d o m i n a t e s over all others, we can regard Co - C1 as being due to this one resonance only. A l t h o u g h in t h e o r y we can eliminate the contrib u t i o n of non-resonance n e u t r o n s b y this m e t h o d , in practice if this c o n t r i b u t i o n is large Co - Ct is the small difference between two large numbers. F r e q u e n t l y t h e non-resonance c o n t r i b u t i o n is due to a large flux of t h e r m a l neutrons, a n d this can be eliminated b y performing the irradiation u n d e r c a d m i u m . B y this means, E h r e t obtains values of Co/C t in the range 1.1-1.5 i r r a d i a t i n g gold, m a n ganese, indium, t u n g s t e n a n d l a n t h a n u m in a l I E spectrum. He shows t h a t , for these materials, 9 5 - 1 0 0 % of C o - C 1 is due to n e u t r o n s a t the energy of the single d o m i n a t i n g resonance.

8. Applications A useful application of this t e c h n i q u e is in shielding work. Reactor shielding calculations (for instance as described b y A v e r y et a/.12)), n o r m a l l y utilise a small n u m b e r of large n e u t r o n groups. Typically, six groups are used to cover the range from 2 MeV to t h e r m a l energies. W i t h such wide groups, the problem of weighting cross-sections w i t h a flux d i s t r i b u t i o n w i t h i n the group is of importance. Thus a n e x p e r i m e n t a l m e t h o d of determ i n i n g the flux s p e c t r u m in a reactor shield is of interest. The use of sandwich foils seems to provide a simple m e t h o d of obtaining information on n e u t r o n

Substance

Indium Gold Tungsten Lanthanum Manganese

E n e r g y of high resonance 1.46 4.9 18.8 73.5 337

eV eV eV eV eV

H a l f - l i f e of active nuclide I

54 2.7 24 40 2.56

m d h h h

IS) A. F. Avery, et al., M e th o d s of c a l c u l a t i o n for use in th e design of shields Ior p o w e r reactors. A . E . R . E . - R 3216 (1960).

NEUTRON

FLUX

MEASUREMENTS

BY

spectra in the range 1 eV - 1 keV. E h r e t suggests the use of five substances as follows. The l a n t h a n u m can be replaced b y cobalt, with a resonance energy of 132 eV, which E h r e t dismisses because of its long half-life (5.28 years). The use of these detector materials should give an e s t i m a t e of the spectrum over the eV range.

RESONANCE

ACTIVATION

OF

FOILS

35

In such an application, in order t( compare measurements made with different detector materials, we must make absolute, rather than relative, determinations of flux. In this case, the various correction factors due to local flux disturbances, as described in the first part of this paper, are of importance,