A method to measure the thermal neutron scattering properties of a sample by neutron wave propagation
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158 Nuclear Instruments and Methods in Physics Research A253 (1986) 158-162 North-Holland, Amsterdam
A METHOD TO MEASURE THE THERMAL NEUTRON SCATTERING PROPERT IES OF A SAMPLE BY NEUTRON WAVE PROPAGATION
Urszu la WOZNICKA
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
The determanation of the neutronic properties of geologacal samples gives valuable reformation winch supplements that which can be obtained in other ways . 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. , Kreft et al. , Tittle and Crawford  and by Hams et al. . 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 . For a recent revmw of the subject see WoSmcka .
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 ) 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. Basic theory
Diffusion of thermal neutrons from a harmonically modulated source in an infimte medium can be de-
scribed by one-group diffusion theory  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(~) ~ dx 2 D ~(x )=0. (4)
For x>0 and a boundary condition hmx~0J (x )= S0/2 we have the solution:
~0(x) = SoL--,,/L 2~ e ' (5)
L 2 = D//~a (6)
defines the diffusion length L of the medium. For the wave part 8~(x) we get:
d218~(x) ] za +~' / ~ a , (x ) =0 (7)
and for stmilar conditions as descnbed above we have
U Wo~mcka / Thermal neutron scattermgproperttes 159
8SL,~ SO(x) = ~exp( -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 wntten"
1/L~ = a + ifl, (10)
0/2 ~ ~a "~--~ + 2--o~O ffo2z:~2 + to 2 (11)
f12 2DZ a 1 2 2 to2 = - + 2v--- -~/v ~a + " (12) The neutron flux
(x , t )= SO2DL e-X/L + -~--~eSS . . . . e (~'-~), (13)
is therefore propagated as an attenuated wave. To obtain useful information on the neutron trans-
port properties of a medium from neutron wave propa- gation experiments one usually observes the dispersion of the neutron waves. From this one can determine the inverse diffusion length. As seen from eqs. (10-12) this is a complex function of the modulat ion frequency to. A simple investigation using the diffusion approximation shows that the criteria for materials to be suitable for such experiments are: sufficiently large diffusion length L of the neutrons and small absorption cross section ~'a = Do/L2" Tills is the reason why only materials like graptute, beryll ium and heavy water have been investi- gated and are of future interest in wave experiments, see table 1.
Special effects occur when the neutron wave propa- gates across the boundary between two adjacent media. At the interface there is one component which is re- flected, whereas another is transmitted. The solution of the diffusion equation must include terms of the inci- dent, reflected and transmitted components. The magni- tude of the reflected wave depends upon the properties of the medium reflecting it. By observing the dis- turbance which one medium causes on the neutron wave in an adjacent second medium, it may be possible
to obtain some information on the dlffUston properties of the first medium . However, the interface effect through reflection is negligible after a few mean free paths 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 utlhzing the wave propagation properties. The sample under investigation is placed between two layers of moderator. The plane modulated source is situated in the point x = 0 on the outer surface of the moderator. The geometry is shown in fig. 1. The modulated neutron flux travels through the first, second and third medium and is reflected by the two interfaces at x = d and x = d + a The attenuation and phase shift of the trans- mitted wave (propagating through medium 3), depend upon the neutron properties of both media.
The calculations to be described show that for a certain value of the frequency to* the phase shift of the neutron wave in medium 3 IS the same as for the homogeneous moderator 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 elaborate an experimental method for measure- ment 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
160 U Wokmeka / Thermal neutron scattermg properttes
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)
hm ~3(x) = 0 x ~ o o
l (x )=~2(x) fo rx=dandx=d+a
d~a(x) d~2(x) DI dx D2 dx
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 exp( -x /L ,o l ) + ni l exp(x / t , ,1 ) (18)
~t~2(X) =A22 exp( -x /L ,~2) + B22 exp(x/L ,o2) (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:
801(x , t) = [exp( -x /L~) + B~ exp( (x -2d) /L~x) ]
xe ' ' ' t (21)
~2(X , l) = [A 2 exp( - (x - d) /L ,o2)
+s2 exp( (x -d ) /L ,~2) ] e ''~' (22) t) = A 3 exp( - (x - a) /L ,~l) e ''', (23) ~,1,3 (x,
D1 1 (K I /K2) exp(2a/L,~l) + 1 B, = 2 - - - 1 (24)
L"I r l (K2 /K1) 2 exp(2a /L , .4 ) - 1
D1 Kz exp(2a/L,. ,2) exp( -d /L~l ) & = 2- - (25)
L'~I K2 (K2 /K , ) 2 exp(2a/L,~l) - 1
D1 1 exp( -d /L ,~ l ) B 2 = 2 - - (26)
L,ol K1 (KE/K1) 2 exp(2a /Ld)_ 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 exp( -xa l ) exp(i(tot - f l lx + O)) (30)
where p exp( -xa 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).
The relation between the phase shift (O - f l lx) 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 muated below the curve for the heterogeneous system for low frequencies (before the antersection point) be-
0 i . [ . . . . i . . . . ~ . . . . i . . . . ,
t~ -3 ' '
I ~ *
-4 ...... I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 500 1000 1500 2000 2500
FREQUENCY [ 1 /sec ]
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
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 .~
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 modulation 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 At= (2~r/~)/20 =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)
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.
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.
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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