Ann. nucl. Energy, Vol. 9, pp. 525 to 532, 1982 Printed in Great Britain
0306-4549/82/100525-08503.00/0 Pergamon Press Ltd
APPLICATION OF THE T H E O R Y OF R A N D O M MATRICES TO A REACTOR-NOISE PROBLEM F. C. DIFILIPPO* Department of Nuclear Engineering,The University of Tennessee,Knoxville,Tennessee,U.S.A.
(Received 14 April 1982) Abstract--The stochastic equation corresponding to a nuclear reactor with random distribution of materials is written in a matrix form, in this way the probability distribution of reactivitiesis derived as the probability distribution of the higher eigenvaluesof random matrices.The theoretical results are illustrated and validated with Monte Carlo simulations based on the diffusionequations and the matrix approach. INTRODUCTION There are neutron-reacting systems where a degree of randomness exists in the spatial distribution of materials. Fluctuations in composition appear, for example, as a consequence of the operational characteristic of boiling-water and pebble-bed reactors. Traditionally these types of systems have been approximated by homogenizing methods that use average values for the distribution of materials; this deterministic approach gives values of reactivities and flux distributions that do not agree with a stochastic treatment of the problem (Williams, 1974; Van Dam, 1977 ; Yamada et al., 1980a,b). According to Williams (1974) a knowledge of the statistical laws obeyed by the system is necessary in order to obtain an equivalent homogeneous system. In this context, a model has previously been developed by Van Dam (1977) which gives the first two moments of the reactivity distribution in an ensemble of reactors having a specified random distribution of macroscopic absorption and fission cross sections. An important result of this previous work is that the use of higher-order perturbation theory gives a positive value for the average reactivity as opposed to a value of zero when calculated by ordinary perturbation theory. A recent work, Yamada et al. (1980a), confirms this conclusion but these same authors, Yamada et al. (1980b), stress that the sign of the average reactivity depends on the fluctuations in the moderation process. In the present work we have recast the equations derived by Van Dam (1977) in a matrix form. In this way the distribution of reactivities can be described by the distribution of the highest eigenvalues of random matrices, with the result that mathematical techniques
developed for the study of the statistical laws of excited nuclei, Porter (1965), can be applied to the similar reactor-physics problem ; so the entire distribution of reactivities, instead of only the two first moments, can be calculated. Following a presentation of the theory, a numerical example is given, for which a Monte Carlo simulation of several ensembles of reactors is made.
THEORY In this section the stochastic equation related with the balance of neutrons is written in a matrix form, the distribution of the matrix elements is discussed and finally the probability distribution of eigenvalue (reactivities) is derived.
Matrix form of the stochastic equation In this section the nomenclature of Van Dam (1977) is followed. The space-dependent macroscopic absorption and fission cross section of one member of the ensemble of reactors are given by: 52a(r) : ~2aO+ ~ a ( r )
(1)
~f(r) = ~fo 4- ~f(r),
(2)
where it is assumed that Ea0 and Efo are independent of position. In equations (1) and (2) the incremented variables indicate the effect of the random dispersal of materials upon the absorption and fission macroscopic cross sections. Neglecting the effect of random dispersal upon the diffusion constant, the static equation in one-group diffusion theory can be written as OV2q~--(E~o + fiY.a)q~4
v(Efo +,SEf) ~b = 0, k
(3)
*Present address: Oak Ridge National Laboratory, Building 3500, P.O. Box X, Oak Ridge, TN 37830, U.S.A. where k is the multiplication constant given by the 525
526
F.C. DIFILIPPO
higher eigenvalue of equation (3). Expanding the flux in a basis given by the Helmholtz equation,
V2@, + B ~ , = 0,
of thickness a, in this case the eigenfunction ¢~ are given by
(4)
where B~ is the buckling of mode i and qJ~is zero at the extrapolated boundary ~ can be written as
ck = L alibi.
Oi(x) =
sin(i+ 1)~-; i = 0,1,2 ....
(12)
and from the definition of C u the variances 0.u are given by
(5)
i=0
Substituting equation (5) in equation (3) and multiplying scalarly by ~j Van Dam (1977) obtained
(PJ -- P)V~'f°aJ = ~i al jl [-f~"~a(r)- Y~~f(r)]l~i~//j
dr,
O~(x)O,(x)f~j(x)dx,
(13)
f°(x) = J0 (fZ(x)fE(y))~Oi(y)~bj(y) dy
(14)
(VZ~o)2 where
(6)
where pj = 1
Z~0+DB~
(71
VEfo
fZ(x) = vf£f(x) - fE.(x).
is the reactivity of mode j in the homogeneous system taken as reference, and P-
k-1 k
(8)
is the reactivity of one member in the ensemble of reactors ; in equation (6) the assumption has been made that p << 1, Yamada et al. (1980a) have discussed the implications of this approximation. Equation (6) can be written in the matrix form : [C]a = pa
(9)
where a is the column vector formed with the coefficients a~.The matrix elements of[C], Cu, are given by
Cij = Iij/(v~Zfo) + pifij,
(10)
where fu is the Kronecker symbol and I o are given by,
llj = jV [vf~f(r) -- 6Z.(r)]~b~#
and by definition
dr.
(11)
Due to the linear relation between l~j and the random variables fE~ and fEf, and applying the central limit theorem, C u is a random variable with Gaussian distribution. If the ensemble averages of rE. and fief are zero for all r, the mean values of C u are zero for nondiagonal elements and equal to p~ for diagonal elements.
To continue with the formulation it is necessary to postulate some form for the correlation function (fE(x)fE(y)). It will be assumed that the slab can be divided in N slices of thickness 2 (N2 = a) in such a way that fE is the same at all the points of a slice and that a complete lack of correlation exists between the macroscopic cross section of different slices ; thus for x belonging to slice nfo(x ) is given by (n+ 1)2
fo(x) = ((fY.)2)n
fdn2
~O,.(y)~j(y)dy
(16)
Substituting equation (16) into equation (13) the variances are given by
f"+ q,,(x)~j(x) dx ]2, (17)
0.'~ - ~
.-% LJo~
where the additional assumption that ((f]~)2) is independent of position was made. In practice the formalism based upon the infinite dimension matrices [C] is approximated by one based on matrices of finite dimension [I], if N satisfies the condition ( I + l ) x / N < < l equation (17) can be approximated by 0-2 -- ((fZ)2) ~, fa ffj2(X)[//2(x) (V'~7~fO)2 3o
dx.
(18)
After substitution of the eigenfunctions given by equation (12) into equation (18) the variances are given by
( ( f z ) 2)
0-~ -
a (V~-~fO)2
Variance of the matrix elements In this section an expression for the variance of the matrix elements is obtained in the case of a slab reactor
(15)
0.~
i~j
3 2 ((rE) 2) -
2 a (VZfo)2 '
(19)
(20)
Application of random matrices to reactor noise
where ct is the ratio between capture and fission cross sections in the fuel, NF and NM are the atomic densities for fuel and moderator, respectively. No correlation between 6NF and 6NM was assumed.
This is a remarkable result, it was shown that under certain conditions all the off-diagonal and diagonal elements have, respectively, a common value for their variances. These properties are important in the derivation of the probability distribution of reactivities. More exact values of a 2 can be obtained from equations (17) and (12), the results are: a~ -
<(6Z)2) ~_1 f 1 [n nnq L '~~ z . , / s i n ( i - j ) ~ (n + 1 ) - sin(/-j) q T | (VEf0)27z2 n=o (U--J)L iv 1~¢A
~' a~]- <(FZ)2> ~ .~o {.1
sin(i+j+2)~(n+ 1) -- sin(i + j + 2 ) ~ - J ~
((~E)2) - (e~)-- k ~ (efea) + k ~ (e~)
(VEfo)2
(23)
where ef = 6£f/Ef and ea = 6Za/Za" Assuming constancy of average microscopic cross sections the following results are obtained,
6NM
((~-M)
i :~j
1 . 1)N(n + 1 ) - sin nnl]2 2n(/-+l)[ sin 2(i+ 2(i+l)~-Jj~
Table 1 compares values given by equations (21) and (22) with the approximations given by equations (19) and (20) as function of N for the case of 2 x 2 matrices, for N = 20 the approximate formulae agree within 1% with the more exact ones. In order to calculate the variances it is necessary to relate <(fiE)2) with variations in material composition. From the definition given by equation (15),
2
nnll 2
n
1
(i+j+2)
k~
527
(24)
(21)
.
(22)
Probability distribution of reactivities In the previous sections it was shown that under certain assumptions the probability distribution of p, p(p), is given by the probability distribution of the higher eigenvalues of random matrices that have the following properties : (1) They are real and symmetric. (2) Their elements are Gaussian distributed. (3) The off-diagonal elements have mean values equal to zero. (4) The diagonal elements have mean values equal to Pi. (5) All the off-diagonal elements have the same variance, a2o. (6) All the diagonal elements have the same variance,
O'2D"
2
(7) Defining f12 = 2a2D/a2D yield f12 = 4/3.
/
,2~laNAi\/ (2s)
= k~-(l +oO/(aNF'~2~
\\N~j/'
(26)
The probability distribution of eigenvalues of random matrices are known for particular cases ; if the diagonal elements have also zero mean and if f12= 1 the probability distribution is given by the Wishart law (Wilks, 1943) a distribution well known for its application to the analysis of levels distribution in
Table 1. a2o, ao2, and a2~ as function of N*
.go
-~,
~g,
N
Exactt
Approx.~:
Exact
Approx.
Exact
Approx.
2 5 10 15 20
0.500 0.287 0.148 0.0995 0.0748
0.750 0.300 0.150 0.100 0.0750
0.500 0.257 0.144 0.0981 0.0742
0.750 0.300 0.150 0.100 0.0750
0.360 0.170 0.0960 0.0655 0.0495
0.500 0.200 0.100 0.0667 0.0500
* In units of (6Z2) (vZ~0)2"
Exact: equations (21) and (22). :~Approximate : equations (19) and (20).
528
F.C. DIFILIPPO
excited nuclei (Lynn, 1968). A relaxation of these conditions, namely diagonal elements with zero mean but f12 # 1, was studied by Porter and Rosenzweig (1960) in the case of 2 x 2 matrices. In what follows, p(p) is derived for matrices fulfilling conditions (1~(7) and with dimension two. The probability distribution for C u is Gaussian, i.e. G(Cij ) --
1
N//~ai j exp
(Cu-(Cu))21,
[
2a 2
j
(28)
and the highest eigenvalue is given by P-
2
+
-
-+C2a.
(29)
If Coo, Ca~ and p are elected as the independent variables the probability distribution for the new variables is P(Coo, Cla, p)
8Coa
=
2HECoo, Co a(P), C111W ,
(30)
where Co a(P) is the inverse of equation (29), the factor 2 appears because given Coo, Ca 1 and p two values ofC 0 a satisfy equation (29). Defining the auxiliary variables Coo - Ca 1 v- - 2 2
p(p) = 4 K e x p { - I 2 p 2 + 2 [pa' p + p2 ]~
JJ fo zl(z)
+ 2~o~ dz, (33) -L-[z2-2pz-lPdz ~o
l(z)
= x~-exp(-B2z2) L F(n+½)[-2B2]"
I(z)=j
=
~_~_~
K = 1/[(2rr)3/23o2oaoa]
dv
(34) (35)
LTJ z""fcz)
.:o
(38) where I, is the modified Bessel function of order n. Introducing equation (38) into equation (33), and rearranging terms, the following expression is derived for p(p),
P(P)=rcK'ff~aooexp{-[ pzx4~oo °+
(IpO+2P)2]' ° [Sao2 JJ
a I-2B 2q" r(n +~),,,
x
(39)
in equation (39) h.(p) is given by
aoo " + 1 fo '~ 9,(s, p) ds,
h.(p) =
(40)
where
g,(s, p)
=
s" +Xl,(ws)G(s -- So)
(41)
w = IPl[ / aoo
(42)
s o = (2p + Ipd)/4aoo.
(43)
In equation (41) G(s-So) is the Gauss function with variance one. In the limiting case B = 0, IPd = 0 equation (39) reduces to a form which can be derived from the Wishart distribution (Porter and Rosenzweig, 1960). As Fig. 1 suggests when So < 0 the integral in equation (40) tends to zero which implies that p(p) tend to zero when p < - IPd/2. In general the contributions to the integral in equation (40) are important for those
j~G
where
(37)
expanding exp(BZz 2 sin 2 0) in power series of z, I(z) is given by
(32)
and integrating equation (30) with respect to Coo and Ca x yields
x exp
exp(B2z 2 sin 2 0) x exp(Cz cos 0) dO
(31)
Coo+CH
Z --- p
l(z) = e x p ( - B 2 z 2)
(27)
where ( C u ) = pi6o; a reference reactor in the critical state will be assumed so Po = 0 and the p~ are negative numbers. The joint probability distribution for the elements of 2 x 2 random symmetrical matrices is
H(Coo, Co,,Cxa) = G(Coo)G(CoOG(Cla)
The fact that in the present c a s e B 2 ~ 0 and IPl] va 0 complicates the integration of equations (33) and (34). With the change of variable v = z cos 0 and the definition C = Ipd/ago equation (34) can be written as
~
( s - s o)
~
S
N*I I N (ws)
D
,
0
S
So
B2
1
1
Oo~o 2~o~a
(36)
Fig. 1. The function g,(s, p) is the product of the two functions shown in the figure.
Application of random matrices to reactor noise values of s around So, in this region the argument of the Bessel function is
PIPll IPll 2 WSo = ~2ao2 -~ 4ao2°
a~+l exp[sow+WZ]f ~ r~/ zw 2 j j _ ¢~o+w)/~ x exp[-q2](.4/2q+So+W) "+l/z dq.
F
+
W2~
fp
gmaxP(P) d p = A.
exp/sow + ~-1 ~x/2w L z d
x (So+w)"+'2 ~
exp(-q~) dn
(46)
or
h.(p)=m{So+W)"+l/Zexp
~/2zw
E w21 SoW+ 2- .
(47)
Due to the dominance of the Gaussian factor in equation (52), the contribution of the tails to the integral in equation (53) is assumed to be negligible so the following formal substitution in equation (53) can be made : A --+ 1 ; Pmin OO; Pmax -{-cX3,in this way K " = 1/(~/~%o ). The corresponding moments are given by
2)5"~1/2
--
~
even n
(p") = ~oo(n-- 1)!!
(54)
odd n
0532 +1 (pn> = ~ - ~ o o n[!
(55)
and the following values for the mean, the standard deviation, skewness (S) and kurtosis (K) are obtained
Substituting h.(p) given by equation (47) into equation (39) p(p) can be approximated by
P(P) = K'( P~\IPd+
(53)
mln
---+
h.(p) = ~
P
Due to the approximations made the distribution given by the previous equation is valid within the limit Pmi,~-lPll/2 and Pmax~
(45)
The Gaussian factor in the previous equation is centered at zero and has a variance equal to 0.5, then consistent with the two conditions stated above equation (45) can be approximated by o.n+ 1 O0
In the present case f12= 4/3 and with three digit accuracy E 1 = 1.63, ~"~2 : 1.35 then p(p) is approximated by
(44)
so an asymptotic approximation of the Bessel function can be used when p > - I P l l / 2 and for systems where aoZo<
h.(p)
529
(56)
o- = Ooo[1-(aoo 0.532/1P11)2]1/2
(57)
~fO'532tr°° "~3[ - " - (0"532tr°°.'~2]- 3/2 (58)
expl- 2~o]
s =
X ~ o ( 1 - ~ )l'~"F(n+½) l~/pp~ + ~-) 5 ", ~
(48)
where K' collects all the constant factors. The probability distribution is, according to equation (48), the product of a Gaussian term, centered at zero, and a function of p, so this function is relevant only for small values of p which justify a power-series expansion in terms ofp/lp 11truncated after the first term, the result is
)
L
i.ll / j
1 2( 0"532a°°~2 K=3
-
(0.532ao o~,~
[1-\(°.532oq7 /j
\~l--]
-\
~
]
(59) Equations (58) and (59) show that S > 0 and K <~ 3, these two parameters illustrate the departure from a Gaussian distribution.
p ( p ) = K " e x p [ - - 2 ~ o 2 o ] [ l + ( l + ~ - ) ~ 2 E 2 "P~ ]], (49) MONTE CARLO SIMULATION where K" collects all the constant factors, E 1 and Z2 are given by
(2n- 1)!!
Z1 =
n!
- ~
(50)
n=O
Z2 =
n! n=0
n ~--
.
(51)
The stochastic parameters that describe the ensembles of reactors in the Monte Carlo simulations are the same as those used by Van Dam (1977), in this way it is possible to compare results from diffusiontheory simulation with the present matrix approach. The data of the reference core is shown in Table 2 and the stochastic parameters that describe each ensemble
530
F . C . DIFIL1PPO I
Table 2. Data for the reference reactor Slab width Z, Zf v D Po Pl
I
I
i
I
I
I
I
I
1.2 ENSEMBLE 2
100 cm 0.1 cm -1 0.5 cm -1 2.01974 1 cm 0.0 -0.02932
1.0
_L__
Q8 0.6 (14
t
t
-I
ENSEMBLE 6
Table 3. Stochastic parameters for the ensemble of reactors Ensemble 2(crn) 1 2 3 4 5 6
4 4 4 4 10 10
<(6ZJZ~)2> <(6"~f/~f)2> 3.33E -- 3J" 0 8.33E - 4 0 8.33E-4 0
0 3.33E-3 0 8.33E- 4 0 8.33E-4
O-O0"
Q6 0.4
1.40E - 2 1.41E-2 7.00E- 3 7.07E- 3 1.11E-2 1.12E-2
02 ENSEMBLE 4
O.4 Q2 : f - - ~ 0
* From equation (20).
T 3.33E- 3 = 3.33 x 10
3
......
-i. . . . . .
I ....
i
I
l
i
I
I
I
I
t
2
5
4
5
6
7
8
9
10
I (ORDER OF THE MATRIX)
of reactors are s h o w n in Table 3. It was assumed, as did Van D a m (1977), t h a t the rectangular probability distributions for 6Ea a n d fief were w i t h o u t reference to any particular variation of composition. These distributions were symmetric a b o u t zero a n d were scaled to provide relative variations of +10~o (ensembles 1 a n d 2) a n d + 5 ~ (ensembles 3-6), for each ensemble; a sampling of 1000 reactors was run. T h e i n t e r c o m p a r i s o n between the m a t r i x simulation a n d the simulation based on the diffusion e q u a t i o n is s h o w n in Fig. 2 as function of the order of the matrix I ; there is a n asymptotic behavior, after a certain m i n i m u m value of the matrix order the two simulations agree. F o r small p e r t u r b a t i o n s simulations based o n low-order matrices give reasonable results, similar results were obtained for the cases not s h o w n in Fig. 2. The theory predicts a G a u s s i a n distribution for the matrix elements, in this respect Table 4 c o m p a r e s results from the simulations a n d the theory, a good agreement was found for the mean, s t a n d a r d deviation, skewness a n d kurtosis. Table 5 intercompares integral parameters from the probability distribution o b t a i n e d
Fig. 2. Average value of reactivities based on matrix and diffusion theory simulations. The continuous lines, from Van Dam (1977), are based on the numerical solution of the diffusion equation, the dashed lines are the ones providing 90~o confidence limits. The size of the symbols for the matrix simulations correspond to the 68.3~o confidence limits. in the previous section, e q u a t i o n (52), a n d simulation based on 2 x 2 matrices, the results agree within statistical errors. An i n t e r c o m p a r i s o n between histograms c o r r e s p o n d i n g to the p r o b a b i l i t y distribution, e q u a t i o n (52), a n d the 2 x 2 matrix simulations of ensemble 6 is s h o w n in Fig. 3 revealing g o o d consistency; the a s y m m e t r y of the probability distribution due to the linear term in e q u a t i o n (52) can be appreciated. As can be seen in Fig. 2 a n expansion o f t h e flux up to the second h a r m o n i c is not e n o u g h in cases where the p e r t u r b a t i o n is large, as a consequence the average reactivity deduced from e q u a t i o n (52) is good only for small p e r t u r b a t i o n s ; nevertheless the shape of the a p p r o x i m a t e d probability distribution compares fairly well with the distribution of eigenvalues from
Table 4. Monte Carlo simulation of matrix elements* Ensemble 1
Mean Standard deviation Skewness Kurtosis
_
Ensemble 4
Theory
Simulation
Theory
Simulation
0.0 0.0140 0.0 3.00
(3.32 _+4.53)E - 4 0.0143 +0.003 0.063 _+0.134 3.12 + 0.40
0.0 0.00707 0.0 3.00
(2.02 _+2.33)E - 4 0.00736 _+0.00152 - 0.0396 + 0.109 2.82 _+0.23
* Matrix element Coo ; errors correspond to 68.3~ confidence limits.
Application of r a n d o m matrices to reactor noise
531
44 "4E
40
+
56
++
-
+
32
+1+1+1+1+1+1
4-
28
--
+÷
--
÷ ~< 24 +
4-+
_
÷
+_
+
4-
+-
+
4-++
16 4-
.4--+
,,71
- -
-+-
-
zo z
+-
4-
-
~-+-
--
7
+1+1+1+1+1+1 -20
r.)
-~6
q2
- 0 . 8 -04
~11111
0.0 p (%)
O4
oe
12
t6
20
Fig. 3. Histogram corresponding to the 2 x 2 matrix simulation of ensemble 6. The crosses are the results of a sampling of 1000 reactors, the lines bracket the 68.3~o confidence limits based on the probability distribution obtained in this work, equation (52), and the finite sample. t'--I x
simulation using higher-order matrices. This is substantiated in the case of ensemble 2 in Fig. 4, where the central moments up to order four, from simulations based on matrices up to order 9, are compared with the theoretical values from the previous section. In this case, as Fig. 2 shows, an expansion up to the seventh harmonic is necessary to obtain the average reactivity, while Fig. 4 shows that a second-harmonic expansion is enough to calculate the shape-related parameters a, S, K. In the previous paragraphs, cases were analysed
77??77
O
+1+1+1+1+1+1
O
77??77 m m m m m m
IIIIII 30
.8 .,-
+1÷1+1+1+1+1
~
M
~
.
ED
?7?777 s I0
e~
8
08
SKEWNESS -02
i I
2
3
I
I
I
I
4 5 T6H 7 8 I (ORDER OF E MATRIX)
9
Fig. 4. Central m o m e n t s for ensemble 2 as function of the order of the matrix. The size of the symbols correspond to the 68.3~o confidence limits, the continuous line is from the probability distribution, equation (52).
532
F. C. D1FILIPPO Table 6. Results corresponding to changes in fuel concentration Matrix element (0, 0)
Mean Standard deviation Skewness Kurtosis
Reactivity
Theory
Simulation
Theory
2 x 2 Matrix simulation
0.00 2.96E - 3 0.00 3.00
(3.22_+ 9.13)E- 5 (2.89 + 0.60)E - 3 (-4.00 + 11.0)E- 2 2.88 __+0.26
1.59E - 4 2.95E - 3 3.10E-4 3.00
(2.08 + 0.91)E - 4 (2.88 + 0.60)E - 3 ( - 1.80+ l l . 0 ) E - 2 2.86 + 0.25
The errors correspond to 68.3~ confidence limits.
relative to cross-sectional variation without reference to any particular variation of composition. To illustrate a case with explicit reference to random dispersal of fuel, a Monte Carlo simulation of variations in fuel concentration was made. The reference reactor is described in Table 2; in this case a correlation between variation of absorption and fission macroscopic cross sections appears according to equation (26). The rest of the parameters were: ct=0.175, N = 100 and a Gaussian distribution for the relative change in fuel concentration (standard deviation 5.77~). The results are similar to those quoted above, there is an asymptotic behavior similar to the cases shown in Fig. 2. The consistency between the theory and the simulations is shown in Table 6, the variances of the matrix elements were calculated with the help of equation (23)-(26).
averages reproduce the numerical results in cases with small perturbation. The probability distribution of reactivities was related in an explicit way with a stochastic parameter (ao0) which is determined by the random distribution of materials in a reactor. In this way it is possible to relate the probability to find the system in a given reactivity range with the degree of randomness, this relationship might be important in the safety analysis of nuclearreacting systems with inherent randomness. An important limitation of the results presented in this report is the use of one-energy-group approximation, further investigations are needed in order to include fluctuations effects in the moderation process in the present model. Acknowledgement--This work is dedicated to the memory of Kurt Wilckens.
CONCLUSION REFERENCES
A space-domain reactor-noise problem was related to the theory of random matrices. Simulations based on random matrices have been shown to give good results in comparison to simulations based on numerical solution of the diffusion equation, therefore it is possible to apply mathematical techniques developed for nuclear-physics problems to the calculation of the probability distribution of reactivities. This distribution is not Gaussian and illustrates the non-linear relationship between variations of reactor-material composition and reactivity. The shape of the theoretical distribution obtained compares quite well with a Monte Carlo simulation, but the theoretical
Lynn J. E. (1968) Theory of Neutron Resonance Reaction, Chapt. V. Clarendon Press, Oxford. Porter C. E. (1965) Statistical Theories of Spectra: Fluctuations. Academic Press, New York. Porter C. E. and Rosenzweig N. (1960) Suomol. Tiedeakat. Toim., A VI, No. 44. Van Dam H. (1977) Prog. nucl. Energy l, 273. Wilks S. S. (1943) Mathematical Statistics. Princeton University Press, New Jersey. Williams M. M. R. (1974) Random Process in Nuclear Reactors, Chapt. 8. Pergamon Press, Oxford. Yamada S., Nishimura M. and Sumita K. (1980a) Ann. nucl. Energy 7, 561. Yamada S., Yamagol C. and Sumita K. (1980b) Ann. nucl. Energy 7, 655.