158 Nuclear Instruments and Methods in Physics Research A253 (1986) 158-162 North-Holland, Amsterdam

A M E T H O D T O M E A S U R E T H E T H E R M A L N E U T R O N S C A T T E R I N G P R O P E R T I E S

O F A S A M P L E B Y N E U T R O N W A V E P R O P A G A T I O N

U r s z u l a W O Z N I C K A

lnsntute of Nuclear Physws, 31-342 Krakbw, uL Radztkowsklego 152, Poland

Received 18 June 1986

For the determination of the thermal neutron diffusion coefficient for samples of limited size, it is proposed to use the neutron wave or pulse propagation method. A thin slab of the material under investigation is then placed between two blocks of a medium with known properties Numerical calculations have been performed on such a system. It is found that, for a certain frequency and some spemfied condiuons, the phase shift in the outer medium will be the same as if the sample had the same properties as the outer medium This fact may be used for a measurement of the diffusion coefficient of the sample

1. Introduction

The determanation of the neutronic properties of geologacal samples gives valuable reformation winch supplements that which can be obtained in other ways [1]. One difficulty is that the samples frequently are of hmlted size. For the determination of the thermal neu- tron absorption cross section several methods are avail- able. Recent work has been descnbed by Czubek et al. [2], Kreft et al. [3], Tittle and Crawford [4] and by H a m s et al. [5]. A knowledge of the thermal neutron diffusion coefficient will gave information on the scattenng propertxes of the sample and help in the interpretation of the neutron well log prospecting in nuclear geophysxcs. However, the diffusmn coefficient is more difficult to measure

In this paper it ~s proposed to use the neutron pulse of wave propagation method for the deterrmnation of the diffusmn coefficient. This method is well known, and especially dunng the 1960 much work was done with it. Many papers were presented at the Gainesville conference [6]. For a recent revmw of the subject see WoSmcka [7].

The same pnnclples govern the propagation of a neutron pulse and a neutron wave, since a neutron pulse can be considered to be built up of waves of different frequencies (Moore [8]) Therefore, we treat only the wave propagatmn in detail However, the results are to a large extent possible to apply to the propagation of neutron pulse, too

2. Bas ic theory

Diffusion of thermal neutrons from a harmonically modulated source in an infimte medium can be de-

scribed by one-group diffusion theory [9] to obtaan a physical background of the phenomenon. The space and Ume distribution of the neutrons in a medmm in which there is a sinusoidally modulated source is ob- tained from the time dependent equation'

1 3~(x , t) Da20 (x , t) Z . ¢ ( x , t ) . (1)

v at ax 2

The plane modulated source is assumed to be situated in x = 0 and have the total strength:

S = S o q- ~S e '~at. (2 )

We seek a solution in the form of the sum of the stationary and modulated parts of the neutron flux:

O ( x , t ) = Oo(X) + 8 ~ ( x ) e ''°'. (3)

For the stationary part ~0 (x ) we have:

d~o(~) ~ d x 2 D ~ ° ( x ) = 0 . (4)

For x > 0 and a boundary condition h m x ~ 0 J ( x ) = S0/2 we have the solution:

~ 0 ( x ) = SoL--, , /L 2 ~ e ' (5)

where

L 2 = D/ /~a (6)

defines the diffusion length L of the medium. For the wave part 8¢~(x) we get:

d 2 1 8 ~ ( x ) ] z a + ~ ' ° / ° ~ a , ( x ) = 0 (7)

d x 2

and for stmilar conditions as descnbed above we have

U Wo~mcka / Thermal neutron scattermgproperttes 159

the solution:

8SL,~ S O ( x ) = ~ e x p ( - x / L ~ ) . (8)

Here, L,o is a complex diffusion length defined by '

1 ~:~a + i oJ/v 1 i to - + ( 9 )

L~ D L 2 vD

The complex inverse diffusion length used in the follow- ing can be wnt ten"

1/L~ = a + ifl, (10)

where:

0/2 ~ ~ a "~--~ + 2--o~O ffo2z:~2 + to 2 (11)

f12 2DZ a 1 2 2 to2 = - + 2v - - - -~ /v ~ a + " (12)

The neu t ron flux

¢ ( x , t ) = SO2DL e-X/L + -~--~eSS . . . . e (~ ' -~) , (13)

is therefore p ropaga ted as an a t tenuated wave. To obta in useful in format ion on the neu t ron trans-

por t propert ies of a medium from neu t ron wave propa- gat ion experiments one usually observes the dispersion of the neu t ron waves. F rom this one can determine the inverse diffusion length. As seen from eqs. (10-12) this is a complex funct ion of the modula t ion frequency to. A simple invest igation using the diffusion approx imat ion shows that the criteria for materials to be suitable for such exper iments are: sufficiently large diffusion length L of the neut rons and small absorp t ion cross section ~'a = Do/L2" Tills is the reason why only materials like graptute, beryl l ium and heavy water have been investi- gated and are of future interest in wave experiments, see table 1.

Special effects occur when the neu t ron wave propa- gates across the bounda ry between two adjacent media. At the interface there is one componen t which is re- flected, whereas ano ther is t ransmit ted. The solution of the diffusion equat ion must include terms of the inci- dent , reflected and t ransmi t ted components . The magni- tude of the reflected wave depends upon the propert ies of the med ium reflecting it. By observing the dis- tu rbance which one medium causes on the neu t ron wave in an adjacent second medium, it may be possible

to obta in some informat ion on the dlffUston propert ies of the first medium [10]. However, the interface effect th rough reflection is negligible after a few mean free pa ths and decreases with the frequency, which makes this idea difficult to realize experimentally.

3. Proposed method of measurement

In the present paper we suggest a new method for measurement of the diffusion parameters of a sample by ut lhzing the wave propaga t ion properties. The sample under investigation is placed between two layers of moderator . The plane modula ted source is si tuated in the point x = 0 on the outer surface of the moderator . The geometry is shown in fig. 1. The modula ted neu t ron flux travels through the first, second and third medium and is reflected by the two interfaces at x = d and x = d + a The a t tenua t ion and phase shift of the trans- mi t ted wave (propagat ing through medium 3), depend upon the neu t ron propert ies of bo th media.

The calculat ions to be described show that for a cer tain value of the frequency to* the phase shift of the neu t ron wave in med ium 3 IS the same as for the homogeneous modera tor with ZaI and D I as parame- ters. Also it has been found that a dependence between the frequency to* and the diffusion coefficient of the sample D E exists (fig. 3). It may be recognised as a first step to e laborate an experimental method for measure- men t of the diffusion coefficient for relatively small samples.

"F/.ooE.,,,oR/I \ \ g/ ooE.A,o. / / , / '///IsAMPLE //'''''' '///,

, , ILIA; /z.,, D,////

7/////. 0 d d*a x

Fig. 1 Calculation geometry. The sample 2 is placed between two layers of moderator 1 and 3 which extend over 0 _< x _< d

and d + a _ < x < + o o .

Table 1 Thermal neutron properties of different media

Graphite Heavy Berylhum Lead Light water water

L [cm] 50 100 22 13 2 8 v.~. [ l / s ] 80 20 230 1260 4900

160 U Wokmeka / Thermal neutron scattermg properttes

4. Calculations

The calculatmns have been based on the following derivations. The diffusion equation (1) is solved for the geometry presented in fig. 1 with the boundary and interface conditions:

x~okhm(-D'd~l(x))~x YS (14)

(15)

(16)

hm ~ 3 ( x ) = 0 x ~ o o

• l ( x ) = ~ 2 ( x ) f o r x = d a n d x = d + a

d ~ a ( x ) d ~ 2 ( x ) DI d x D2 d x

for x = d and x = d + a. (17)

We seek for the modulated part of the flux the general solution of the form:

~l~l(X ) = A l l e x p ( - x / L , o l ) + n i l e x p ( x / t , , 1 ) (18)

~t~2(X) =A22 e x p ( - x / L , ~ 2 ) + B22 e x p ( x / L , o 2 ) (19)

~3 ( x ) = A 33 exp ( - x/L,~ 1 ), (20)

where 8 ~ . (x ) denotes the modulated part of the neu- tron flux an medium 1, 2 and 3, respectively The terms with A . . coefficients represent the transmitted parts of the fluxes and the terms with B . . coefficients the re- flected ones According to the boundary condltmns (15-18) and with assumption Atl = 1 we have:

8 0 1 ( x , t ) = [ e x p ( - x / L ~ ) + B~ e x p ( ( x - 2 d ) / L ~ x ) ]

x e ' ' ' t (21)

~ 2 ( X , l ) = [A 2 exp( - ( x - d ) /L ,o2 )

+s2 e x p ( ( x - d ) / L , ~ 2 ) ] e ''~' (22) t ) = A 3 exp( - ( x - a ) /L ,~ l ) e ''°', (23) ~,1,3 (x,

where

D1 1 ( K I / K 2 ) exp(2a /L ,~ l ) + 1 B, = 2 - - - 1 (24)

L"I r l ( K 2 / K 1 ) 2 e x p ( 2 a / L , . 4 ) - 1

D1 Kz exp(2a/L, . ,2) e x p ( - d / L ~ l ) & = 2 - - (25)

L'~I K2 ( K 2 / K , ) 2 exp(2a /L ,~ l ) - 1

D1 1 e x p ( - d / L , ~ l ) B 2 = 2 - - (26)

L,ol K1 ( K E / K 1 ) 2 e x p ( 2 a / L d ) _ 1

exp(a /L '~2) (27) = 4(r - r ? ) / q exp(2a/L 2) - IC

K 1 = DE/L,,,2 - DI/L,.,1 (28)

K 2 = D2/L,o 2 + D1/L,o 1 . (29)

The modulated part of the transmitted flux 8~3(x , t) can be represented as a function:

8 ~ 3 ( x , t ) = p e x p ( - x a l ) exp(i( tot - f l lx + O)) (30)

where p e x p ( - x a l ) is the attenuation factor of the neutron wave m medium 3 and (8 - f l lX) is the corre- sponding phase shift. The values a I and fll are de- ternuned by eqs. (11,12).

5. Results

The relation between the phase shift (O - f l l x ) of the transmitted wave ~3(x, t) and the frequency ~0 for &fferent parameters of the medium 2 has been found numerically. The results are presented in fig. 2. The relation between the phase shift fll and the frequency ~0 for the homogeneous medium with the neutron parame- ters ~al and D 1 is drawn as a dotted line in this figure, too The curve for the homogeneous system crosses the other curves, which means that the phase shift for both cases is the same in this point. The intersection of the curves can be explained on the basis of the phase shift determlnatmn equation (12). The phase shift (see table 2) increases when the absorption cross section decreases or when the diffusion coefficient decreases The depen- dence upon the cross section is strong for low frequen- cies and weak for high frequencies. For high frequencies the phase shift increases strongly when the chffusion coefficient decreases and increases weakly when the absorption cross section decreases.

Now, we can explmn the behaviour of the curves m fig 2. The curve for the homogeneous moderator is m u a t e d below the curve for the heterogeneous system for low frequencies (before the antersection point) be-

0 i . ¸ [ . . . . i . . . . ~ . . . . i . . . . ,

" O

1

L-2

t~ - 3 ' '

I ~ *

- 4 ...... I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 500 1000 1500 2000 2500

FREQUENCY [ 1 / s e c ]

Fig 2 The relation between the phase shaft (0 - f l i x ) of the transmitted flux ~3(x, t) and the frequency ~0 for 3 different values of the dlffusxon coefficmnt D 2 of the sample Graphate is used as medmm 1 and 3 Xal = 0 00036 cm- 1, D1 = 0 9 cm The parameters of the sample ~a2 = 0 01 cm- 1 D2 = 0 9 cm (curve 1), D 2 = 0 27 cm (curve 2), D 2 = 0 1 cm (curve 3) The dotted line is for homogeneous graphite medmm The thick- nesses are for medium 1 d = 20 cm, for medium 2 a = 10 cm The calculations were made at the point x = 40 cm m medium

3

U Wokmcka / Thermalneutron scattermgproperttes 161

Table 2 The dependence of the phase shift fl for an infinite homoge- neous medium upon the frequency with decrease of the neu- tron parameters , ~ and D

Increase of Low High phase shift frequency frequency fl(to, "~a, D)

Strong X a ~ D ,: Weak D .: Z' a .~

cause ~'al << ~X'a2" Beyond the f requency ~o* the curve for the modera to r is s i tuated above the curve which is calculated for the heterogeneous system because D 2 < D i The intersect ion poin t moves towards higher fre- quencies when the diffusion coefficient D E of the sam- ple approaches tha t of the moderator .

The dependence between the f requency ~0" for the intersect ion po in t and the diffusion coefficient of the sample D2 has been calculated for different values of the absorp t ion cross section Xa2 and for different mod- era tors and geometrical d imensions A n example is shown in fig. 3. Graph i t e is used as a modera to r and three curves are presented for different values of the absorp t ion cross section ~ a 2 of the sample. The diffu- s ion coefficient of the u n k n o w n sample may be found exper imental ly if the frequency ~0" is measured and the absorp t ion cross sect ion of the sample and the proper- ties of modera to r are known.

10 4

r~ E O \

>-

210 3

C) b]

b_

- ' F " I ' ' " ' " 1 ' ' " ' " 1 " '

/

3 \ /

/

/ / / / /

' ' " 1 ' ' " ' " 1 ' " Y " I ' "

/ / /

/ /

i 1 0 2 ] , J , , l , , ) , i , , , i i , , , l , i ) , , , , , J , l , , , , ) , , , ) , , , i , J , , ) , , J )

.1 .2 .3 .4 .5 .8

1]IFFUSION COEFFICIENT I] 2 [cm]

Fig. 3. The relation between the diffusion coefficient of the sample D 2 and the frequency to* for 3 different values of the absorption cross sccuon of the sample. Curve 1. Xa2 = 0 005 cm- l ; curve 2:"~32 = 0.01 cm-I ; curve 3. "~32 = 0.018 cm -1

Moderator - graphite, d~menslons as m fig 2

The possibility of a measurement from the poin t of view of neu t ron yield has also been investigated. We assumed a sinusoldally modula ted source (eq. (2)) with total s t rength S O = 10 6 n / s and with the max imum dep th of modula t ion 8S = S o. For this source we found the solution for the t ransmi t ted flux ~ l (X , t ) and q~3(x, t) f rom eqs. (2, 5, 18, 20) with assumpt ion (14) which determines the new value for the cons tant Ali .

The transverse dimensions of the investigated system were assumed to be b and c, respectively. The effective diffusion length L and the complex diffusion length Lo, are then modified by the transverse buckl ing of the system according to

1 / L 2 = 1/L2,,mf + ( ~ ' / b ) 2 + ( o r / c ) 2. (31)

a

b

S

Cn/s]

iO s

n/at,

3008

2 0 0 0

1000

X=O Cm

2 4 B 8 10

TIME Cms]

2 4 B 8 I0

TIME [ms]

C

rl/ ~t

15

10

5

x = 8 5 cm

. . . , . . . . , . . . . , . . . . , . . . . , . . . . , . . . . ) . . . . , . . . , . . . . , . . . . , . . . . ,

0 2 4 B 8 10 TIME [ms]

Fig. 4 The effect of modulation of the neutron flux in medmm 1 (curve b) and in medium 3 (curve c) for the sinusoidally modulated source The neutron source ymld So = 8S = 106 n / s (curve a). The frequency to = 500 s-1 gives the period T =

2*r/to = 12.6 ms.

162 U Wokntcka / Thermal neutron scattermg properttes

This IS the consequence of an assumption of zero neu- tron flux at the transverse boundaries

The neutron flux was calculated m fixed points of the medium 1 and 2 as a function of time using the described assumptions. An example is shown in fig 4 The frequency of the source modulat ion is equal to

= 500 s 1. The system (fig. 2) consists of a graphite moderator and a sample with the neutron parameters ~a2 = 0.001 cm - I and D 2 =0 .1 cm. The thickness of the medium i is equal to d = 50 cm, the thickness of the sample a = 20 cm and the transverse sizes b and c of the system are equal to 100 cm each. The number of counts is observed in the points x = 10 cm (medium 1) and x = 85 cm (medium 3) during the period of time A t = ( 2 ~ r / ~ ) / 2 0 = 0 . 6 ms. The ratio of the maximum and minimum numbers of counts is about a factor of two in the point x = 85 cm (i.e. 15 cm after the sample)

6. Discussion

good results. In the range of frequencies and sizes wtuch are interesting from the experimental point of view the spectrum and transport theory effects must also be taken Into account.

Acknowledgements

I want to acknowledge Prof. Nils GiSran SjiSstrand, head of the Department of Reactor Physics, for giving me the opportumty to perform this work and for his help and valuable assistance in the preparahon of this paper.

This work was performed while I was a guest at Department of Reactor Physics, Chalmers University of Technology, G~Steborg. The stay there was made possi- ble through a grant from the Wilhelm and Martina Lundgren Foundation. The work was also supported by the Swedish Natural Soence Research Council.

This is only a rough estimation because of the simple diffusion approximation used The transverse dimen- sions for the measured system were assumed to be large to simplify the calculations In a realistic experiment smaller dimensions wall be expected The interpretation of experiments performed on prismatic assemblies can- not be made without taking care of the leakage through lateral boundaries. This is often done in the sense of asymptotic reactor theory by assuming a real transverse buckhng which is added to the infinite medium complex inverse diffusion length (eq. (31)) to obtain the finite medium value. In fact, the buckling must be a function of frequency A way of taking this into account is to introduce a complex transverse buckling [11,12]. The introduction of the complex transverse buckling may affect the values of diffusion parameters deduced from the neutron wave experiment with increasing frequency and with decreasing size of the system under investiga- tion. The example shown above has been performed in that region of dimensions and frequencies where the corrections are not necessary.

7. Conclusions

This first investigation of the problem shows that the wave propagation method can give the diffusion coeffi- cient from a hmited sample s~ze. If pulse propagation is used an ordinary pulsed neutron generator is a suitable neutron source. The detection and analysing system is relatively simple. There are some disadvantages of the method. A careful experimental design is needed for

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