The markov chain method for solving dead time problems in the space dependent model of reactor noise

Ann. Nucl. Energy, Vol. 24, No. 16, pp. 1301-1319, 1997

Pergamon

PII: S0306-4549(96)00107-7

~ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0306-4549/97 $17.00 + 0.00

THE MARKOV CHAIN METHOD FOR SOLVING DEAD TIME PROBLEMS IN THE SPACE DEPENDENT MODEL OF REACTOR NOISE S. B. D E G W E K E R Theoretical Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India (Received 18 November 1996)

Abstract--The discrete time Markov chain approach for deriving the statistics of time-correlated pulses, in the presence of a non-extending dead time, is extended to include the effect of space energy distribution of the neutron field. Equations for the singlet and doublet densities of follower neutrons are derived by neglecting correlations beyond the second order. These equations are solved by the modal method. It is shown that in the unimodal approximation, the equations reduce to the point model equations with suitably defined parameters. © 1997 Elsevier Science Ltd. 1. INTRODUCTION In two earlier papers Degweker (1997a, 1997b) here after referred to as I and II, respectively, we had developed the discrete time Markov chain method for solving problems related to the effect of a non-extending dead time on the statistics of time-correlated pulses. Applications of this technique for passive neutron assay of Pu by the variable dead time (VDC) method (Birkhoffet al., 1972; Ensslin et al., 1978; Lees and Houten, 1978, 1980) and for estimation of or, the prompt decay constant of a reactor by the dead time ot method (Srinivasan, 1967) were also discussed. For the latter problem it was assumed that the incoming pulses are adequately described by the point model. In practical applications, for measurement of a or other kinetics parameters by this method, space-energy effects may be significant and hence it is pertinent to consider these. Such an analysis has been the subject of investigation for a number of experimental methods for determination of ot and other kinetics parameters (Sheff and Albrecht, 1966; Williams, 1967; Natelson et al., 1966; Otsuka and Iijima, 1965; Borgwaldt, 1966; Saito, 1979). The general conclusion is that for reactors that are not too large and not too far from critical, the fundamental mode is dominant and with a careful definition of various parameters, the point model is valid. More recently, Munoz Cobo et al. (1987, 1988) have considered multi-mode expansions for calculating the variance, the cross power spectral density (CPSD) between two detectors and for the analysis of the 252Cf method for subcriticality measurement. All these studies are based on a study of the moments of the probability distribution. 1301

1302

S.B. Degweker

For the methods involving the use of the probability distributions themselves there have been fewer studies with regard to space energy effects. Bell (1965) has derived an expression for the probability generating function (PGF) of the number of neutrons in a region of a reactor while Saito (1969, 1979) has given an expression for the P G F of detected neutrons in the fundamental mode approximation with one energy group. Williams (1974) has discussed the problem of extinction probability. In these problems, the equations are non-linear and analytical solutions are not easily obtainable. Even in the point model, analytical solutions are obtained under the assumption that the P G F of the neutron multiplicity distribution in a fission is quadratic. In the space-dependent theory there is an additional problem--even with the above assumption, modal equations can be solved analytically in the single mode approximation, as we shall see later. This means that the theory would not be applicable to large or to highly sub-critical systems. In the dead time ot method, which is the subject of this paper, non-linearity appears not only in the determination of the various probabilities from stochastic transport theory but also in relating the measured count rates to these probability functions. We treat this problem by an extension of the techniques developed in I and II. In Section 2, we use the forward formulation, discussed in Degweker (1994), to derive the Markov Chain equation for the probability generating functional (PGFI) of followers. This equation is then transformed into equations for the singlet and doublet densities of the followers by representing the PGF1 of followers as an expansion in correlation functions truncated after the second order. These equations involve the ordinary neutron densities (in the presence of a detector) as inputs. The latter are derived, using the backward stochastic transport equation (Bell, 1965), in Section 3. Solution by expanding in modes and the reduction to a point model is discussed in Section 4. Finally, Section 5 presents some numerical results based on this theory.

2. T H E S P A C E - D E P E N D E N T MARKOV CHAIN E Q U A T I O N The space energy-dependent Markov chain will be formulated along the lines of I. The number of follower neutrons at the end of a dead time gate interval is the Markov variable used to describe the system. A slightly different formulation based on the total number of neutrons as the Markov variable, which may have some advantages in some problems, is sketched in the Appendix. Using the symbol x to stand for the space and velocity coordinates, we write Q~i+l)(xt,...xk) for the probability density that there are k followers at the end of the i-t- lth gate interval and located around the points x~ .... xk. Likewise we write Jkl(xl ..... Xl, lyl ..... Yt) to represent the transition probability matrix which gives the conditional probability that given I followers around the points yl ..... yt, at the end of any gate interval, there will be k followers around Xl ..... xk at the end of the next interval. For a discussion of such probability functions, the range of the variables xi and the normalisation of the probability function, the reader is referred to Degweker (1994). The equation of the Markov chain describing the detection in the presence of a nonextending dead time now takes the form:

Q¢'+*•,-, ~ l ..... x,) =

~l I ~

dyt...dy, Jkl (xl ..... xklyl ..... Yl)QIi)(Yl .... Yt).

(i)

The Markov chain method for solving dead time problems

1303

Since we are interested in the stationary solutions of this equation we shall henceforth drop the index i corresponding to the interval number. We also introduce the generating functional denoted by using the corresponding probability variable as a subscript. The PGF1 of Q and J are thus written as:

fQ[u] =

d x l . . . d x k Qk(xl ..... Xk)U(Xl)...u(x~)

Fj([u]Iyl ..... Yl) = Z

dxl ...dxk Jkt(xl ..... xk lYl ..... yt)U(Xl)...U(Xk).

(2)

(3)

k=O

On dividing equation (1) by k!, multiplying by u(xl)...u(xk), integrating over xl .... xk and summing over k we get:

fo[u] = ~= ~.. dyl ...dyt Fj([u]lyl .... yl)Qli)o'j .... Yl).

(4)

To derive an expression for F j we introduce the following probabilities in analogy with the corresponding functions in the point model. The probability PN~, defined as the probability of there being at any time N~ undetected chains of length v. In any chain of length v, the probability density of neutrons being located at xl ..... x~ is described by the function p~(xl .... x~) normalised so that

~.

dxt...dx~ p , ( x l ..... xv) = 1.

(5)

0) ~OM(XI XN; rlYl ..... YM; 0) is the probability that M neutrons at )'l .... YM at time 0 result in N neutrons at xl ..... XN at time r, without any detection, in a source-free medium. TNM(Xl ..... XN; rlyl ..... YM; 0) is the probability that M neutrons at yl .... YM at time 0 result in N neutrons at xl ..... XN at time r, with or without detection, in a source-free medium. The corresponding PGFIs are defined in a manner similar to that of J and are denoted by F~0~([u]; rJyj ..... YM) and Fr([u]); rJyj ..... YM), respectively. From the independence of neutron behaviour in a reactor it is clear that these PGFls can be written as a product of the corresponding 1 neutron PGFI. In other words, denoting the I neutron PGFls by G, we have: .....

Fv0~C[u]; rlyl ..... YM) = G~0~C[u]; rlyl)....Gvo,([u]; rlyM)

(6a)

S. B. Degweker

1304 and

FT([U]; fly1 ..... yM) = Gr([U]; riyl)....GT([U]; r[yM).

(6b)

With these definitions, we can derive expressions for A, the average rate of detection of new chains, as was done in I. A = Z

PN..Nv.~.

dXl...dxv

~d(Xl)pv(Xl ..... xv)

Nv, ~a

1

= )--~ R~ (v - 1)! v

I dxl...dx~ ~.d(xl)p~(xl ..... Xo)

(7)

= J p(l)(x) ~a(x)dx where p(t)(x) stands for the singlet density of all the neutrons belonging to undetected chains at any time and ~ ( x ) stands for the detection probability per unit time of a neutron and is the same as rEd(X). Since these chains may originate at any time in the past and from a source event that may be spatially distributed, we have oo

p(')(x) = I dr J G(1) o(X, r,y)S(y)dy

(8)

0

where S(y) is the source distribution and we use the symbols G (m) and G~") to represent the ruth order densities starting with a single neutron in a source-free medium. The subscript 0 indicates no detections in the relevant time interval. The transition matrix Jkl - or rather its PGF1 Fj - - is obtained by using essentially the same arguments that were employed in I. Fj consists of two factors: one as a result of the detection of the first neutron after the end of any gate interval and which also triggers the next gate; the other arises due to detections during this next gate which result in additional followers being produced. By following the arguments of I, keeping proper track of the space-velocity variables xi, we obtain the following equation for Fj: oc

r,'t lly,

.....

A,z,

.....

0

x [Jdzv2+l ~d(Zv2+I)T(O)+,.,(ZI ..... Zv2+l; r'yl ..... yl;0) (9) +

rn(Zv2-n+2 . . . . . 2v2) v2-n+l,l( 1..... Zv2-n+l; g / n =l

x e x p [ J~-'~ d yoild' 'r' d{ ~y n - ' ~ n (. y ' n = 2

..... Yn-l)(Fr([u],r[yl ..... yn-,) - 1)}]

The Markov chain method for solving dead time problems

1305

where ~ is defined by

~v(Xl

.....

Xv-I) = Z N~

PN~.N~,

J dx~,~.d(X~)p~(xl..... x~).

(9a)

The proposed solution method for equation (4) is similar to that employed in II. For this purpose we need to introduce the equivalents of the factorial moments and cumulants for the point model. These are clearly the generalised correlation functions (Van Kampen, 1983) defined by: /~I(Xl) : MI(XI)

(lOa)

~2(Xl, X2) : M2(Xl, x2) - M1 (Xl)M1 (x2)

(lOb)

and

where M1 and M2 are the singlet and doublet densities of the stationary distribution of followers. Then by analogy with the solution method employed in II for the point model, we represent the PGF1 of Q as follows:

fQ[u] = exp[l #l(xl)(u(xl ) -- l )dxl + ~ l #2(xl, x2)(u(xl ) - l ) (u(x2) - l )dxldx2] (11)

The equations for the singlet and doublet densities M1 (xl) and M2(xl, x2) are obtained by taking functional derivatives of equation (4) and setting u(x) -- 1. We thus obtain

MI(X)= ~._I1Jdyl...dyI

..... Yl), ~Fj([U]]TI ~u(x) Yl) .....

u(x)=l

, ,82Fj([u]IYl..... Yt) M2(xl, x2) = xl~=2l,!fj dyl...dYl QI(yl .... yl, ~ll ~-u(-~2) u(x,=l"

(12a)

(12b)

On carrying out the functional differentiation of Fs and making use of the independence of neutrons embodied in equation (6), we can write the following equations for MI and M2: A

Ml(x)---Id~lG(I)(x, rlXl)B(xl)dxl+ll(x) o where,

(13a)

1306

S.B. Degweker oo

0

+ J dyj dy2dx~dx2 ~.d(yDG°)(x, A lY,)G~O(Y~,rlXl)G~~)(y2,rlxDf'~(x,, x2, r) + A J dyldx, G(l)(x,Aly,)G~l)(y,,rlx,)f'o(x,, r) + l dy,dxlB(yl)GO)(x, Aly,)fo(r) ] M2(x, y) =

dr GO)(x,r]xl)[~(xl)dxl

i dt J G(1)(y, r[xl)~'(Xl)dXl

-l-[idTIG(l'(x,r,Xl)B(xl)dXl]ll(y) dr G(2)(x,y,rlxl

+

xOdxl + IidrlG(I)(x, rlXl)G(')(Y,rlx2)C(xl x2)dxldx21

oc

+ e

dr fo(x,,x2, x3, r) [a(ys)am(x, Alyl)G0)(y, 2xly2)

0

x a~'~(y,, rlx,)a~l)(y2, rlx2)a~')(y3, rlx3)dx, dx2dx3dy,dy2dy3

+Jf~(x,,x2, r) 2d(y3)G(')(x,Alyl)G(t}(y,Aly2) x {G~l)(y3,r[xl)G~2)(y,,Y2, fix2)+ 2G~I)(yl, rlxl)G~2)(y2,Y3, rlx2)}dxldx2dyldy2dy3

+ ]f~(xl , x2, r) ~.d(y2)G(Z)(x,y,A[yl)G~l)(y1, r [x I)UO r(l)-(Y2,rlx2)dxldxzdyldy2 + J f'a(Xl, r) Xd(Y3)G (I)(x, Alyl)G(1)(y, A[y2)G~3)(yl,Y2,Y3,r[xl)dxldy2dyldy3

+If'o(x,,r) 2d(y2)G(2)(x,y,Alyl)G~2)(yl,Y2,r[xi)dxtdyldy2 + A[If~(x,,x2, t)G(')(x, Aly,)G(')(y, Aly2)G~')(y,,rlx,)G~')(y2,rlxDdx,dx2dy,dy2

+Jf'o(Xi, r)a(1)(x, A]yl)G(I)(y, AlY2)G~2)(yl,Y2, rixl)dxldyldy2

l

+ fQ(xi, r)G(2)(x,y, Alyl)G(ol)(yl,rlxl)dxldyl

]

+ 2 If'p(Xl, r)B(yz)G(1)(x,A lYi)G(I)(Y,A ly2)G~l)(yi,rlxi)dxl dyl dy2 + JfQ(r)B(yl)GI2)(x,y, A lyl)dyl + JfQ(r)dO',, y2)a(')(x, Aly,)Gl'iO,, Aly2)dy,dy2]. (13b)

The Markov chain method for solving dead time problems

1307

The functions /~(y) and ~'(x,y) are related to the doublet and triplet densities of undetected chain at any time as follows:

/~(Y) = f ~-d(Yl)p(2)(Y, yl)dyl

(14a)

C'(X, y) ~---I ~-d(Yl)P(3)(X, Y, yl )dyl

(14b)

J

and the doublet and triplet densities can be calculated like the singlet density as follows: OO

pl2'(x, y ) =

J dr J G~2)(x, y, r]y,)S(y,)dy,

(15a)

o 04)

(15b) o

fQ(t x , "r), etc.,

stand for the functional derivatives offQ[u] evaluated at -C0/(x, r) is the probability that a neutron initially at the point x u(x) = G~°)(x, r) where c/0 results in no detections by the time r. Expressions forfQ, f'o, etc., are then easily written down as follows: The functions

II I~l(xl)h(xl,

fQ(r) = exp -

r)dxj

'I

+~ #2(xl, x2)h(xl,

r)h(x2, r)dxldx2

f'Q(X, r) = fQ(r)[Ul(X)- I #z(x, x2)h(x2, r)dx2 ]

]

(16a)

(16b)

(16c)

(16d)

where

h(x, r) = 1 - G~°)(x, r).

1308

S.B. Degweker

3. EQUATIONS FOR THE DENSITIES G (m) AND G~m) Before we can consider solution of the moments equations de:rived in the previous section we must obtain expressions for the densities G (m) and~ mG~m)G appearing in these equations. We note that it is necessary to obtain the densities alone since the others can be deduced from these by setting ~.d(x) equal to zero. The backward formalism of Bell (1965) is particularly convenient for this purpose. Let us consider a neutron introduced at time t at the point x and we ask for the number of neutrons nl, n2 and n3 in three small regions Rl, R2 and R3, having volumes V1, V2 and V3, around the points xl, x2 and x3, respectively, together with the number of counts no in a detector. The situation is described by the generating function H(uo, ul, u2, u3; x, t) where ui is the P G F variable corresponding to the number ni. H obeys the backward stochastic transport equation

1 OH

V Ot

f2.VH + ~,tH = ~,c + UO~d (17)

+ l ~s(V-+ v,)H(vl)dv, + ~ffJmlI ~ )

H(vl)dV, }

whereJ~m is the P G F of the multiplicity distribution of the number of neutrons in a fission. The final and boundary conditions are as follows: Final condition: H(uo, Ul, u2, u3; x, tf) = ui if x e Ri = 1 otherwise Boundary condition: H(uo, ul, u2, u3; Xb, t) = 1 for an outgoing neutron. The functions of interest to us are the averages < nl > /VI < nln2 > IV1V2, etc., conditional to there being no detections i.e. no = 0. These can be obtained by differentiating H w.r.t, ul, u2, etc., and setting uo = 0 and ul, u2 and u3 = 1. The equations for the required averages are immediately obtained by carrying out these operations on equation (17) for the PGF.

v Ot + L*h =

El(X)

h(vr)dgYdv' +Ed(X)

(18)

where h = 1 - G ~ °) and L* is the adjoint transport operator. The final and boundary conditions are, Final condition: h(x, t]) = 0 Boundary condition: h(xb, t) = 1 for an outgoing neutron.

1 oG" + L'G?

v Ot

(19)

=-Ef(x)v(v-~)[I~)G~l)(xl;x')d~2'dv'][[X(V')h(x')df2'dv']'LJ 4Jr (l) Final condition: Go (xl; x, tr) = 8(x - xt) Boundary condition: G~l)(xi; xb, t) = 1 for an outgoing neutron.

The Markov chain method for solving dead time problems

1309

1 8G~2)

v W + L*a

=

x(v')

(2~

,,lifo0

1

v'

(20)

G~')(x2; x')df2'dv ' .

tx,; x')d~2'dv'

Final condition: G~2) (xl, x2; x, tf) = 0 Boundary condition: G~2)(xl, x2; xb, t) = 1 for an outgoing neutron.

v Ot = - ~2f(x)v(v - 1)[ [ X(v') G~3)(x,, x2, x3; x')dfl'dv'

h(x')d~2'dv'

LJ 4~ V/

t

. . [ fX( )G(2),x x2;x,)df2ldv,J[X(v),.(O, x()

yI

(2)

,

,

,

yt

x(v)

!

(~)

x .ff)d~tdvl]

(21)

.

!

+Ef(x)v(v - 1) [ I X4-~ G~2)(x2, x3; x ' )am ' dv ,][x(vjL~__)~(,),_uo t~l,'x')d~dv ' ]

Final condition:

G~3)(xl, x2, X3; X, tf)

~- 0

Boundary condition: G~3) (Xl, X2, X3; Xb, t) = 1 for an outgoing neutron. The final and boundary conditions are easily written down from simple physical considerations and do not require further explanation. We have written down an equation for h = 1 - G~°) since it is required for solving the above system for the higher densities. This equation is obtained by putting Ul, u2 and u3 equal to 1 and u0 to 0 in equation (17), and truncating the expansion ofj)m(x) around 1 at the quadratic term. This is similar to the approximation made in II and is essential if any analytical results are to be obtained. Its effect on the accuracy of the results has been discussed in reactor noise literature and we have also shown that it is not very important.

4. S O L U T I O N BY M O D A L EXPANSION AND R E D U C T I O N TO THE P O I N T M O D E L We begin by solving the equation for h. The method is similar to that described in Bell (1965). We expand h in a set of complete eigenfunctions of the linearised adjoint transport operator.

h(x, t) -- E aj(t)4~7(x) J

(22)

S. B. Degweker

1310 where,

L ck~ =

(23)

Likewise we also define the direct eigenfunctions, 4~j obeying 1

(24)

= v j j. Following Bell (1965), we assume that the functions are normalised so that,

!

(25)

= a0

and Ot i ~--- - - O t i*.

(26)

Substituting the expansion (22) in equation (18), taking the scaler product with ~i and using the orthonormality relation (25), we obtain:

daJdt otjaj* = I v(v~-~-ej(x) qbj(x)

aidp~(x)

dfadv

ed(X)~b,(x)dx.

dx-

(27)

The only case in which a solution can be found analytically is when the sum on the right is truncated at the first term. The equation then becomes dao d---7- otao = 2a Ylao2

~-d

(28)

where we have written a for the lowest eigenvalue ao* and

~-d = f ~.d(X)C~O(X)dX. Y1 =

2ol Ef(x)d~o(x) qba(x)

The final condition on a0 is simply obtainable:

ao(tf)=

(29)

d~dv

dx

(30)

0. The following solution is then readily

(2~.jot)(l - e - u z l r ) a0 = (1 + e l ) - (1 - zl)e -~:,~

(31)

The Markov chain method for solving dead time problems

1311

where r = t f - t the time of interest in equations (13a) and (13b) and,

zl=

~/

4 Yl ,kd l+-Of

(32)

We next consider the equation for G~l). Assuming that this also attains the fundamental mode distribution in a short time, we can write

G~l)(xl ; x, t) = bo(t) ¢o(xl) ~ ( x )

(33)

where in view o f t h e final condition we have bo(tf) = 1. Substituting this f o r m i n equation (19) we get, db0 dt (l+2Ylao)abo=O. (34) Substituting a0 from equation (31), the above equation for bo is easily integrated to give,

4z~e-~Z, r 6o = ((1 + Zl) - (1 - zl)e-a-"r) 2"

(35)

In a similar manner assuming the following expressions for G~2) and G~3): G~2)(xl, x2; x, t) = co(t) ~bo(xl)4~o(x2).~b~(x)

(36)

1~1•2

G~)(xl, x2, x3; x, t)=do(t)~°(xl)~°(x2)¢°(x3).~(x)

(37)

VI V2V3

we obtain the following equations for co and do: _--d c__._z~_ (1 + 2 Ylao)otco = 2 Yiotb2o dt

da0

- - - (1 + 2Ylao)otdo = 6Yluboco dt

(38)

(39)

with final conditions, co(tf) = do((/) = 0. On substituting a0 and b0 obtained earlier, we obtain the following solutions for co and do:

Co =

16Ylz~e-~Z,~(1 _e-~Z, ~)

((I + z ~ ) - ( l

-~,)e-~:,~) 3

(40)

1312

S. B. Degweker

do

96 Y2z~e-~'z'~(1 - e -~z' r)2 ((1 4- Zl) - (1 -

(41)

zl)e-az'r) 4

We have already noted that the corresponding non-zero subscripted densities are obtainable from the ones already derived by setting ,ka = 0 which means Zl = 1. We thus obtain:

GO)(xl;x,

r) = ¢0(xO ¢~(x)e_~,

(42)

Vl

and 2Yl

G(2)(Xl, x2; x, r ) =

¢o(xl)¢o(X2)

~b~(x)e-~(l - e - ~ )

(43)

VI V2

Now we are in a position to simplify equations (13a) and (13b). It is clear from these equations that the x dependence of the functions M1 (xl) and M2(xl, x2) will be the same as that of GIt) and G (2), respectively. We therefore write

¢0(xO

MI(Xl) = m ! -

(44a)

Pl

M 2 ( X l , x2) = m2

¢0(xl)¢0(x2)

(44b)

I~1122

Substituting equations (44a,b) in equations (13a,b), taking the scalar product of equation (13a) by q~)(xl) and of (13b) by ¢~ (xl)¢~(x2) and making use of equations (22), (33), (36), (37), (42) and (43) and the orthonormality relation of equation (25) we get finally B ml = - ( 1 - e -~A) + e-~611

(45a)

o/

m2 = { B ( 1 - e-'~a) } 2+2~B~( l - e-'~a)e-'~AIl + B Y--!(1- e-'~'~)2

C(I - e -2~zx) 2or

+ e-2azxI2 (45b)

where,

oo

I, =

I -At e

tt

2

[OCp(a0)b0 + f'p(ao)co)~.d

+ Af'~(ao)bo + Bfe(ao)]dt

(46a)

0 oo

12 _j -- e

e(ao)bo3 + f ,,v(ao)(3boco + 2Yl(e '~zx-

1)b2) + f'e(ao)(do

+ 2Yl(e '*A

1)Co)}~.d

o

+ A {f'~(ao)b~ +f'p(ao)(co + B{ 2f'e(ao)bo

+ 2 Y, (e'~A - 1)bo) }

+ 2 Y1(e'*zx - 1)f(a0) } +

Cfp(ao)] dt. (46b)

The Markov chain method for solving dead time problems

1313

For singlet neutron sources, A, B, C are given by: A --

2XdS or(1 + zl)

(47a)

4XdSYI B -- or(1 + Zl) 2

C-

(47b)

8XdSY~ cffl + zl) 3

(47c)

where, S is the effective source strength given by

S = I S(x)¢~(x)dx. The functionsfp,fp,, etc are the same as the point PGF function of the followers and are given by fP(ao) = e x p [ - m t a o q- ~l (m2 -- m~)a2]

(48a)

f~(ao) = [ml - (m2 - m~)aollp(ao)

(48b)

f ~(ao) = (m2 - m~)fp(ao) + [m, - (m2 - m~)ao]f'e(ao)

(48c)

f'~'(ao) . 2 ( m. 2 m~)f'e(ao) . . + [ml

(m2

2 ,, ml)ao]fp(ao).

(48d)

Lex can be related to the solution ml and m2 of equation (45) by utilizing the following expression for P~,: OG

egf= Z k!(v2 - 1)! e-ACdr dzl...dz~2dxl...dXkQk(xl .... Xk) v2,k

'

J

1

0

X ~.d(Zw)TOv2,k(Z1. . . . . . . .

" r l x l . . . . . Xk;

(49)

O)

On simplification we obtain finally, O(3

Lex = T l e-ArfP(a°)dr- 1 0

(50)

1314

S.B. Degweker

The set of equations (45-50) show that, with a suitable definition of parameters, the fundamental mode approximation to the space-energy dependent Markov chain equation for the dead time problem is identical to the point model discussed earlier in I and II.

5. C O M P A R I S O N W I T H M O N T E CARLO S I M U L A T I O N S We have seen in the previous section that the space dependent Markov chain description reduces to the point model, with a suitably defined set of parameters, under the unimodal approximation. This assumption is expected to hold in small systems not too far from critical. In this section we show comparison between Monte Carlo simulations and the theory described in the previous section. The three systems for which we present such a comparison are bare homogeneous reactors described by one group diffusion theory. This is chosen merely as a matter of convenience, since the various integrals involved in the definition of the parameters can be evaluated easily, and does not constitute any limitations on the validity of the results (since we are mainly interested in the effect of the presence of higher modes). While the first of the three systems is a slab, the other two are cubes. The contribution of the higher modes is expected to be most significant in the third system. The prompt Keff of the three systems is in the range 0.9854).988. The migration area is chosen to be around 40 cm 2 and the neutron lifetime is about 60 #s, typical of a light water reactor. The mean free path is chosen to be sufficiently small so that transport corrections are negligible. Figures 1-3 show the variation of Lex for the three systems under study. The continuous line is the theoretical (fitted) result and the points correspond to Monte Carlo simulations. The fit is seen to be quite good. Table 1 shows a comparison of the ot estimated from the fitting and the direct Monte Carlo results. It is seen that while the first two systems show pretty good agreement, the third shows considerable difference between the results. This

1.0

0.8

O~•OO•00000 0.6

0.4

_

0/

o/

•fl,,.'" • • • •

0.2 - o / 0

0

I 2

I

4 Dead time (ms)

MonteCarlo Theory p

6

Fig. 1. Variation of the excess count loss per gate interval (Lex) with the applied dead time (A) for the case of a 100 cm slab reactor.

The Markov chain method for solving dead time problems

1315

gives an indication of the range of validity of the approximations made in deriving the point model. The calculations of Lex were done using a slightly modified version of the procedure described in II. The main point to be noted in this regard is that the probability generating function of the number of followers is a product of two factors: one arising due to the effect of the triggering pulse and the other due to leaders detected in the gate. An exact expression for the latter can be worked out by performing the summations and integrations in the argument of the exponential: 1.5 --

r1.0 /•

°/° /

f

/

/ ./

0.5

• O~ O~•--O--O-O--O

• • • •

Monte Carlo Theory

0 0

I

I

I

!

5

10

15

20

Dead time (ms) Fig. 2. Variation of the excess count loss per gate interval (Lex) with the applied dead time (A) for the case of a cubical reactor of side 50 cm. 1,0

•~

0.8

o''O'O'O "O

0,6

./ 0.4 o

/ • • • •

0,2

0 0

I 2

Monte Carlo Theory

I

I

I

4

6

8

Dead time (ms) Fig. 3. Variation of the excess count loss per gate interval (Lex) with the applied dead time (A) for the case of a cubical reactor of side 100 cm.

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S.B. Degweker

Table 1. Comparison of ct estimated by fitting Lex data obtained from a Monte Carlo simulation) to the present theory with the value obtained directly from the Monte Carlo simulation System

ct(sec- l)

Slab reactor (100 cm) Cubical reactor (50 cm side) Cubical reactor (100 cm side)

This gives

Direct Monte Carlo estimation

Fitting Lex data to the theoretical model

198 + 3

204 101 230

105+5 265 + 3

_ -S/YI or. (1 + Zl){1 + Y1(I - x)} + YI(1 - x)(1 - zl)e ,~6]

J Instead of using a factorial cumulant representation for the entirefQ(x), we use it only for the first factor and use the above expression for the second. An iterative scheme similar to that described in II for computing the factorial moments can then be worked out. This scheme is somewhat more accurate than the one described in II, particularly at large values of ~A.

6. CONCLUSION A space energy-dependent formulation of the dead time ot method using a non-extending dead time has been described. The method is a generalisation of the discrete time Markov chain approach. It has been shown that the unimodal approximation reduces to the point model with suitably defined parameters. Comparison of results with Monte Carlo simulations shows that the point model with parameters defined by our theory works well for small systems that are not very far from critical.

REFERENCES

Bell, G. I. (1965) Nuclear Science and Engineering 21,390. Borgwaldt, H. (1966) Report INR-4/66-5. Birkhoff, G., Bondar, L. and Coppo, N. (1972) Report EUR-4801e. Degweker, S. B. (1994) Annals of Nuclear Energy 21,531. Degweker, S. B. (1997a) Annals of Nuclear Energy 24, 1. Degweker, S. B. (1997b) Annals of Nuclear Energy 24, 375. Ensslin, N., Evans, M. L., Menlove, H. O. and Swansen, J. E. (1978) Nucl. Mat. Mngmnt 7, 43. Lees, E. W. and Houten, B. W. (1978) Report AERE-R 916B. Lees, E. W. and Houten, B. W. (1980) Report AERE-R 9701. Munoz-Cobo, J. L., Perez, R. B. and Verdu, G. (1987) Nuclear Science and Engineering 95, 83.

The Markov chain method for solving dead time problems

1317

Munoz-Cobo, J. L. (1988) NATO Advanced Research Workshop on Noise and Nonlinear Phenomena in Nuclear Systems, May 27-31, Valencia. Natelson, N., Osborn, R. K. and Shure, F. (1966) Journal of Nuclear Energy A/B 20, 557. Norelli, F., Jorio, V. M. and Pacilio, N. (1975) Annals of Nuclear Energy 2, 67. Otsuka, M. and Iijima, T. (1965) Nukleonik 7, 488. Saito, K. W. (1969) Journal of Nuclear Science and Technology 6, 604. Saito, K. W. (1979) Progress in Nuclear Energy 3, 157. Sheff, J. R. and Albrecht, R. W. (1966) Nuclear Science and Engineering 24, 246. Srinivasan, M. (1967) Nukleonik 10, 224. Van Kampen, N. G. (1983) Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam Williams, M. M. R. (1967) Journal of Nuclear Energy A/B 21,321. Williams, M. M. R. (1974) Random Processes in Nuclear Reactors. Pergamon, Oxford.

APPENDIX

Markov Chain Formulation with the Total Neutron Number at the End of a Gate Interval as the Markov Variable. In the main text and in I and II we had used the number of followers as the Markov variable. This gives, in a simple way, the required Markov chain equation in the passive neutron assay problem. However, for the corresponding reactor noise problem, the derivation of the transition matrix is quite involved. In this appendix we show that a formulation based on the total neutron number at the end of a gate interval the transition matrix is easily derived. We consider the point model of reactor noise without delayed neutrons. We show first how the total neutron number can be related to Lex. Suppose Q(m) is the stationary distribution of the total neutron number at the end of a gate. Then the average spacing between two counts 7 is given by:

? = A + ~ Q(m) I tP(#, O, tim, O, O, S)/zZddt m'la

(A1)

d 0

where POt, O, tim, O, O, S) stands for the probability of having # neutrons at t starting with m neutrons at 0 without any detections and in the presence of a source S. Using the fact that N6 is the inverse of ?, we can write L~x = T ( ? - A ) - 1.

(A2)

On partially integrating equation (AI), the expression for ~ can be rewritten as follows: Do

o<~

, = A + ~ Q(m) J P(#, O, tim, O, O, S)dt = A + J go(l, t)fQ[Go(1, t)]dt rn,la

0

(A3)

0

where the generating function g0(x, t) stands for the PGF of the number of neutrons at the time t under the condition that a source was introduced in the system at time 0; there

1318

S.B. Degweker

was no neutron at time 0; and there are no detections in the interval 0 to t. Go (x, t) is the PGF defined in I for the situation that there is one neutron at time 0 but there is no source and there are no detections in the interval 0 to t. The functions gl (X, t) and G1 (x, t) are the corresponding PGFs with detections permitted in the interval 0 to t. FinallyfQ is the PGF of the distribution Q. The transition matrix J,m can be written straightaway as follows:

Jnm = Z J P(Iz + 1, O, tim, 0, 0, S)(/x + 1)*dP(n, rid, AI#, 0, 0, S)dt

(A4)

nd'~ 0

We can convert the above expression in PGF form which involves only the functions G0(x, t), Gi (x, t), g0(x, t) and gl (x, t) to give: oo

Fj(x[m) = 3.dgl(z, A) J 0 [g0[G I(x, A), tllGotGl(x, A), t]}m]dt.

(A5)

0

The various PGFs describing the multiplying systems, which can be written down from the solutions obtained by Norrelli et aL (1975), are given below e-~t(1 - x)

Gl(X, t) = 1 -

1 + ~-~.f(l

Go(x, t) = 1 4 { Y I ( I

- x)

(A6)

- e-~t)(1 - x)

=b ½(I - Z l ) } { l ( l -- Zl) -- ½(I + z,)e -'~:'t } + ½(I - z,)z] YI(1 - e-'~="){ Y,(1 - x) + ½(1 - z,)} + Y,z,

gl(X, t ) = {I + Y I ( I - x)(l-e-(~t}

(

-s/r'(~

go(x,t)= 1+½(1 --71)+Zl Yl(l --x)(1 _e_~,:,t)J-s/r'~ex p ~_y~_l(1 - z l )

(A7)

(A8)

]

(A9)

where,

v(v2¢t-- 1) )~f

(AI 0)

zt = V/I -t 4Y~.da

(All)

ot = ~.c + ~-d -- ~-f(v -- 1)

(A12)

YJ

The method described for solution of the Markov Chain in II can also be applied to the above formulation. While no simplification in the calculations is expected, the most

The Markov chain method for solving dead time problems

1319

important advantage of this method is as follows. The use of "leaders" and "followers" in the earlier formulation implied the existence of chains independent of one another. Since no such ideas are used here, the present method is no longer restricted to branching processes. Thus any Markov process can be treated by the method. To summarise: the detector does not alter the Markov process being studied (except for providing additional absorption). However, the fact that a detection has taken place at any time changes our knowledge of the system at that time and therefore the a posteriori probability distribution at that time. Thus the neutron distribution immediately after a detection may be somewhat different from what it is at a randomly chosen time point. Such distributions can be found by explicitly modeling the detection process in the forward or backward Kolmogorov equations used to describe the process. The difficulty in modeling a non-extendible dead time arises due the fact that every detected neutron does not trigger a gate. The triggering pulses are a sample of all the pulses. However this is not a random sample but is biased in a complicated way. Hence it becomes difficult to determine the distribution of neutrons at the times that the triggering pulses (which are also the counts) occur. The Markov chain formulation essentially attempts to do this (at a time A later than the occurrence of the triggering pulse (count)).