Oiy TRE CALCULAl’ION OF NUCLEAR REACTOR DISTURRANCES RY TRE VONTE CARLO METHOD*

G. A. MIKHAILOV

Che lyabins k

(Received 23 Apri 1 1445)

THE universality of the Monte Carlo method enables us to use for nuclear

reactor disturbances the general formula for small disturbances first

obtained in [l]. The neutron fission importance function f*(x) required

for the calculations is defined in phase space X of coordinates r and

velocities v and satisfies the relation

kf” = iV*f*,

where k is the effective multiplication factor in the reactor, and N the

neutron multiplication operator per generation, i.e. the densities V,,,(X)

and v,,,+1(x) of the number of fission neutrons in generations with numbers

m and m + 1 respectively are connected by

v,,,+f = NV,.

This note is concerned with evaluating 6k by the Monte Carlo method in

the case of a small disturbance.

The importance function can be approximated as

f” z fn’ = N*“fo*, fo* > 0.

We consider the system of f,unctions {pi(x)?, i = 1, 2, .,, , m, ortho-

normalized with weight p(x). The best mean square approximation of fsn

by linear combinations of functions pi is given by the following coeffi-

cients:

ai = (Nf7’fO*, ppi) = (fo”, N*ppi),

where, in accordance with the meaning of the operator M’, the scalar

* %h. vychis 1. Vat. mat. Fiz. 6, 2, 380 - 3x4, lgfig.

Calculation of nuclear reactor disturbances 2 ‘G 9

product is to be understood as the integral of the product of the func-

tions.

Evaluation by the Monte

!vhere tr is the probability

Carlo method of functionals of the type

(f, N”v),

density in X, was discussed in [2]. III our case the function PO; is not a probability density, but at a suitable

stage the sampling can be carried out with respect to some density V(X),

by averaging the quantities obtained with the weight ppi/u. 4s a result,

we get the following for the cosfficients a;:

j,(E)

ai = ME 2

P(xO(E))Pi(xO(E)) fo*cx;))t~l) ,

js-1 v (x0 6% ) I

where $ is a random trajectory, x0(2) is the initial point of the tra-

jectory E, chosen with respect to the density U(T), and r,j, j = 1. 2,

. . 4 , j,(t), are the points of space X at which n-th generation neutrons

on c are generated.

Vote 1. Computational practice shows that, if we take

fo”(x) = gi(s) = 1,

the function f*5(~) as a rule provides a satisfactory approximation to

the importance function f’*(n). The rapid convergence of the iterations

here is in some degree explained by the relationship

fi*(s) = ,W”gil(s) = k(z),

where k(x) is the multiplication factor per generation for the neutron

source at the point X.

/Vote 2. The importance function has a “good” dependence (in the sense

of being able to be approximated with the aid of familiar orthogonal

function systems) on the spatial coordinates and neutron velocity direc-

tion, and conversely, a “poorS1 dependence on the square of the velocity

modulus, i.e. on the neutron energy E. It is therefore much simpler to

estimate the integral importance function p*(r) for the neutrons of one

fission, which can be done provided the number and “spectrum” of the

fission neutrons is not dependent on the velocity of the fissioning

neutron, as will be assumed in future.

Let I(X) be the complete macroscopic section, a,(x) the scattering

cross-sect ion, af(x) the fission cross-section, P,(x, z’) the distribution

270 G.A. Wikhailov

density with respect to r’ of the neutrons when scattered at the point

.x = (r, v) of the phase space, pf(x, x ‘) the distribution density over 1

% ’ of the number of neutrons when fissioning at

uJ&, 5’) = aJ(z)p;(l, z’),

u&G 2’) = wb)pt’(z, 2’1,

and n(x) is the neutron density in the reactor.

the point n,

Let the reactor system be subjected to a small disturbance. We intro-

duce the notation

Il(bz)= 1 n(s)6X(z)kf*(s)ds,

X

Z2(6w,)= 1 j n(x)dw,(x, x’)kf’(x’)dx dz’,

XX

13(&q) = 1 1 n(z)bIq(s, t’)f*(z’)dz dx’,

XX

D= ss

n(x)uq(x, x’)f’(x’)dx dx’.

XX

In view of our previous assumption regarding the independence of pf(x,x’)

on the fissioning neutron velocity and assuming also that pf(x, n’) is

unchanged by the disturbance, we have

Iz(dlur)= { n(x)8Bf(X)(p*(r)dx,

i

D =‘~‘n(z)c~~(x)(~‘(r)dx. i

The quantities II, 12, 1, and D have a simple physical meaning: II is

the change in the number of collisions with weight kf’, I2 is the change

in the number of scattered neutrons with weight kf*, I-J is the change in

the number of fissions with weight 9.. and fl is the total number of

fissions in the entire system with weight Q*.

In our notation, the formula for small disturbances

effective multiplication factor is

6k = $I-&(6X)+ Zz(Gw,)+ 13(hwf)l.

l-11 for the

Calculation of nuclear reactor disturbances 271

We consider a spatial disturbance of the system, depending on the

small parameter h. Let the length oE the segment along the neutron tra-

jectory in the disturbed region be given up to higher order infinite-

simals by

6b(z) = a(z)h,

when the neutron crosses the boundary of the disturbed region at the

point X. The probability of collision in the disturbed region is. up to

higher order quantities,

61(s) S(s) = cz(s)h Z(s).

To compute the number of scattered neutrons in the disturbed region,

a “forced” collision with weight ah1 is executed. To estimate the deriva-

tive dk/dh, we have to use the weight aI instead of ahE. The function f’ required here can be estimated by the probabilistic method, using the

law of conservation of importance in a critical system when the function

q* is computed. For this, the neutron trajectory is continued as far as

fission, absorption or emission, and q*(r) at the point of fission, or

0 in the case of absorption or emission, gives an estimate for kf’.

We can now state an algorithm for computation by the Monte Carlo

method from the formula of small disturbances, using the probabilistic

estimate for the function f*. .4 chain of neutron multiplications is

simulated. A satisfactory method for stochastic simulation of the multi-

plication chain is provided by the “lexicographic” scheme 121, which en-

ables us to construct a tree of trajectories, allowing for all the

randomly occurring “branches” as far as generation of a previously

assigned number. The computational algorithm described here is imple-

mented by starting from the generation of a large number NO of fissions,

when the neutron distribution becomes close to “stationary” with density

n(x). Experience shows that, if the initial fission neutron generation

is selected uniformly with respect to the mass of the active zone, it is

sufficient to take No = 5.

The sum of all fissions is computed with weight 9,. thus yielding an

estimate of D. On reaching the boundary of the disturbed region, the

following operations are performed before tracing the trajectory further.

1. We construct a trajectory in the undisturbed system with weight

-a6zh, where 61 = x - 1.‘. 1 is the new section, and 1’ the old section

in the disturbed region.

2. We construct a trajectory with aleight &h, starting from the

“forced” collision in the new medium.

272 G..-I. Hikhai lov

3. We construct a trajectory with weight

“forced” collision in the old medium.

-d’h, starting from the

A fission absorption or emission, provides the end of the supplement-

ary trajectory.

The sum of all the fissions with weight y* from the supplementary

neutrons gives an estimate of the square brackets in the formula for Sk.

If direct estimation is used for f*, a scheme is obtained with a much

smaller probability of error. In this case, however, there may be a

large constant displacement because of the poor approximation of f* with

respect to the variable E.

As an example, we computed the derivative with respect to the radius

of the effective neutron multiplication factor

dk k’ = -

dR R=Q,

for the one-group model of a spherical reactor with constants

ot = 0.25, of = 0.056, oe = 0; v = 2.5.

Using the improved diffusion method proposed in [3], we obtained for

‘kp = 8.977 the value of the derivative

kn’ = 009% . .

The functions q”(r) and f”(r, p), r= lrl, p = (r/r, I), were estimated

by the Monte Carlo mt;chod, where I is the unit vector in the direction

of neutron motion. The Legendre polynomials of degrees 0, 2, 4 in r

(even because of the system symmetry) and degrees 0, 1, 2, 3 in l.~ were

used as the orthogonal systems. All the possible products of these poly-

nomials were used for f*(r, p). The computations showed that the fourth

iteration is satisfactory, since

The computation using the function q*(r) consists here in executing a

V forced” collision with weight CJ~/COS 8 on reaching the neutron boundary of the reactor, where 8 is the angle of release, while the number of

fissions from all the supplementary neutrons is computed with weight

T*(r). We thus obtained

k,’ = O.GQ82.

The computations using direct estimation for f*(r, u) gave

Calc’ulat ion of nut lear reactor disturbances 273

kz’ = 0.0938.

The estimates for k,’ and k2’ were obtained with respect to the same

trajectories, Analysis of the calculstions shows that the error proba-

bility is much less in the estimation of h,’ than of tl’.

Trans lated by D.E. &-own

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