The theory of space dependent reactor noise analysis using gamma radiation

Journal of Nuclear Energy, Vol. 25, pp. 637 f~ 655. Pergamon Press 1971. Printed in Northern Ireland

THE THEORY OF SPACE DEPENDENT REACTOR ANALYSIS USING GAMMA RADIATION

NOISE

LJ. KOSTIC*and W. SEIFRITZ Institut fuer Kerntechnik,

Technische Universitaet

Hannover,

Germany

(Received 13 May 1971) Abstract-The space dependent theory of the stochastic fluctuations of the prompt gamma flux in a homogeneous infinite multiplying medium at zero power is developed in the frequency domain. The Langevin technique has been used to obtain the cross power spectral density as well as the spatial coherence function of the prompt gamma flux fluctuations between two space points in the medium. For this purpose, a new quantity, the so-called Green’s function of the gamma flux, has been introduced as the response of the prompt gamma flux to the injection of a neutron into the system. The treatment is based on the one and two group energy diffusion model. Special emphasis is devoted to the spatial correlation range of the gammas compared with the neutrons by introducing micro- and macroscales as measures of spatial coherence. The theory is applied numerically to an infinite light water-uranium-235 system to illustrate the effects of gamma rays upon the spatial correlation range. INTRODUCTION THE field of the reactor noise analysis of zero power systems it is a well-known fact that the stochastic behaviour of the fission branching processes gives rise to fluctuations in the fission rate and reactor power, respectively. From the analysis of the behaviour of these fluctuations, either in the time or the frequency domain, information is obtained about the parameters of reactor kinetics, such as prompt neutron decay constant, reactivity, absolute reactor power and also coupling parameters in the case of coupled core systems. To date, studies of fluctuations mentioned above have been conducted mostly from the point of view that an experimental investigation will be based upon the direct detection of neutrons. However, the first basic theoretical paper which suggest that, in principle, it is possible to perform reactor noise experiments by detecting gamma radiation instead of neutrons was published by GELINAS and OSBORN (1966). They assumed a homogeneous and one speed point reactor model without delayed neutrons and delayed gammas, and derived both an expression for the variance-tomean ratio and the power spectral density resulting from prompt fission gamma detection events. From these relations it could be concluded that information relevant to kinetic reactor parameters may be obtained by photon observation as efficiently as in neutron detecting noise experiments, because the prompt time behaviour of the neutron branching processes is also contained in the prompt fission gamma distribution of the reactor. On the basis of this theory some experimental investigations have been performed. The first attempt to measure and interpret gamma ray noise was carried out by KENNEY (1967). The Cerenkov light produced in the pool water around a swimmingpool reactor was monitored by a simple light-sensitive photomultiplier detector. Its fluctuating output current was analysed for its frequency spectrum. Due to the large index of refraction of water it was possible to discriminate all gammas in the energy range below 0.5 MeV. From this and from the relatively high reactor power level at

IN

* On sabbatical leave from Boris KidriE Institute of Nuclear Sciences, Belgrade, Yugoslavia. 4

637

638

LJ. KOSTICand W. SELFRITZ

which measurements were performed the contribution of the delayed gammas from fission products was discriminated sufficiently well. A comparison with neutron detection measurements shows similar fluctuation spectra which confirm the theoretical predictions. It was also shown that the information in the gamma ray distribution was detectable up to a distance of 30 cm from the core. Another experiment was performed by Roux et al. (1968). A multisection ionization chamber-whose one part was sensitive to neutrons and the other to gammaswas positioned in the vicinity of a reactor operating at high power. Both neutron and gamma spectra were measured with the conclusion that the two spectra are almost identical, regardless of their interpretability in terms of simple models. Furthermore, LEHTO and CARPENTER(1968) and LEHTO (1967) used a gaseous Cerenkov detector to sense the high energy prompt fission gammas in the presence of a high fission product gamma field. These experiments were carried out in a pool-type reactor with a large fission product inventory. They extended the point reactor model for gamma cross spectral density measurements and in this way succeeded in determining the prompt neutron decay constant, /3/l, of the delayed critical reactor. Other experimental work in this field is that by Baloh, Kenney and Schultz. BALOHand KENNEY(1969) used two gamma ray collimated detectors to monitor local artificial disturbances inside an operating reactor by analysing gamma radiation emitted from the core. Interesting measurements using the same cross correlation technique were carried out also by KENNEYand SCHULTZ (1969). These authors measured the local power level at various spots inside a reactor core. The results indicated that a plot of power distribution through a reactor could be obtained to a core depth of approximately 10 cm by simple detectors, attached to the outside of the core. The plot obtained by gamma rays was similar to the other one obtained by a cobalt wire neutron activation. The most interesting aspects concerning gamma radiation in reactors were treated by OSBORN(1968). In his paper, Osborn presented, the theoretical model with the space correlation functions appropriate to high energy prompt gamma rays in thermal reactors. Fast fission, (n, 7) reactions, delayed neutrons and delayed gamma rays are neglected, and only the prompt gammas with energies higher than 4 MeV are studied. He suggested that the spatial range of the correlation in the gamma ray distributions may be much wider than in the neutron distribution. Therefore, it may be expected that gamma ray noise measurements will be somewhat less sensitive to space effects, and more realistically interpreted in terms of simple models, than neutron noise measurements. In conclusion of all these theoretical and experimental investigations it can be said that information about the kinetic reactor parameters can be obtained by photon observation just as efficiently as by neutrons. Besides, the advantage of gamma detection lies in the possibility of observing kinetic core parameters at points remote from the core. In this paper, the fluctuations of the gamma rays are studied under a new aspect. The space dependent theory of fluctuations of the gamma ray flux in an infinite multiplying medium at zero power is developed from the Langevin method. This method as applied here furnishes the cross spectral density function of the prompt gamma flux in terms of the system gamma Green’s function and an input neutron noise correlation function. The gamma Green’s function is a new quantity which has

The theory of space dependentreactor noise analysisusing gamma radiation

639

been introduced as the response of the prompt gamma flux to the injection of a neutron into the system. With the aid of this synthesized Green’s function we obtain the cross power spectral density function of the fluctuations of the gamma flux between two space points as well as the spatial coherence function by normalizing the cross power spectral density on the auto power spectral density. Macro and micro scales are then defined as measures of the spatial coherence of the gamma (and neutron) fluctuations. This theoretical concept has been numerically applied to an infinite homogeneous system to illustrate the effects of gamma rays upon the spatial correlation range between two space points in the medium. The system chosen is a mixture of 235Uand light water. The influences of delayed gammas as well as delayed neutrons were neglected and these prompt fission gammas were assumed to have a mean energy of 1 MeV. A table is included in which the scales of coherence of the gamma and neutron fluctuations are compared with each other. THEORETICAL

MODEL

1. The prompt gamma Green’sfunction in the time domain It is well known that in the theoretical treatment of space dependent stochastic processes the input-output relation for an infinite, homogeneous and stationary system-which is perturbed by a stochastically driving force-is given by the following convolution integral O(P, t) =

1

d3?

all apace F’ of BOUrCes

1

dt’G(I? - i’l, t - t’) . I(?‘, t’),

(1)

all time 1’

where O(F, t) = space dependent stochastic output function at the space point i at time t; I(?‘, t’) = space dependent stochastic driving force or input function at the space; point i’and time t’ < t; G(IJ - i’l, t - t’) = the weighting function or Green’s function of the system; i.e. the response of the system due to a Dirac d-function excitation at the space point F’at time t’. When this general formalism is applied to the neutron noise phenomenon in a multiplying medium, the above equation yields the stochastically varying neutron flux response, Q(P, t), due to a random input noise source, S,(i’, t’), which perturbs. the system. In the compressed notation of the convolution integral of equation (1) this relationship can be written as @(f, t) = G,(r, t)*S,(?, t), E

(2)

in which G,(IP - ?‘I, t - t’) means the Green’s function of the neutron flux, i.e. the response of the neutron flux due to the injection of one neutron at the space point i’ and at time t’ into the system. Starting from this basic principle, we now define the space and time dependent source strength ,S, of the prompt gamma rays with energy E which are produced afterthe injection of a neutron into a thermal reactor system. This intensity can obviously

LJ. KOSTIC and W. SEIFRITZ

640

be presented by the following relation S,(li -

i’l, t -

t’, E) =

Q(E)G,(li - ?‘I, t - t’),

cmp,as;;&,

(3)

in which Q(E) is the expression related to the number of prompt gammas per MeV energy interval formed by fission and (n, y)-capture events. In general, Q(E) can be written as Q(E) = &Y,(E)

+ iXf;y. i=l

Y:,;(E),

gammas ~

(4)

cm MeV

X, = macroscopic thermal fission cross-section, cm-l; Y,(E)

= number of prompt gammas from thermal fission of the fuel emitted per

MeV energy interval at energy E, F; z$, = macroscopic cross-section for capture reactions (including fuel) in the i-th material, cm-l;

Y$\ = number of prompt gammas from (n, y)-events in the i-th material emitted per MeV energy interval at energy E, ‘s. The total energy given off by the prompt fission gammas is 7.46 MeV per fission on the average (Nuclear Engineering Handbook, 1968; Reactor Handbook, 1962). Similarly, there are 7.51 photons per fission on the average and the mean energy per photon is about 0.9 MeV. One of the commonly used simple analytical formulae with an accuracy of about 120 per cent is (Nuclear Engineering Handbook, 1968; Reactor Handbook, 1962). Y,(E) = 7.5emE, O-2 MeV < E < 7 MeV.

(5) The distributions of the prompt gamma rays from (n, y)-reactions, Y$(E), in the reactor materials are more complicated. Graphs illustrating the appearance of the spectra up to 10 MeV for various reactor materials are given in the Reactor Handbook (1962). It can be seen that these spectra are either continuous or discrete. In this context, the most important of these spectra is the capture gamma line spectrum for light water which is given by Y:;‘(E)

= 6(E - 2.23 MeV)

(6)

resulting from neutron capture in hydrogen. Returning to equation (3) we obtain the Green’s function of the prompt gamma flux, G,, in the system by convoluting S, with an appropriate gamma attentuation kernel through G,(@’ -

7’1, t -

t’, E) =

s

d3X(p 18 - il)S,( I? - ?‘I, t - t’, E),

allspacei of

gammas

cm2sec MeV *

(7)

The theory of space dependent reactor noise analysis using gamma radiation

This synthesized gamma flux at i? of of a neutron at the For the gamma

641

gamma Green’s function can be interpreted as the response of the those gammas in the energy interval dE around E to the injection space point i’ and at the time t’ into the system. attentuation kernel K(,u Iw - r-1) we use the well-known relation K(p

IR - ?I) =

e-“I~-~l

1 4~ 18 - iI2 ’cm2 ’

(8)

which consists of the standard expression for the point source with absorption. ,u is the linear attentuation coefficient depending on the composition of the reactor medium and the gamma energy E. Empirical values for different reactor media are tabulated, e.g. in the Nuclear Engineering Handbook (1968) or the Reactor Handbook (1962). If equations (3), (7) and (8) are summed up, the gamma Green’s function becomes the convolution of the neutron Green’s function with the gamma attenuation kernel G,((R

-

t’, E) = Q(E)

?‘I, t -

s

all space i of gamma BO”lm!S

1H-i

e-”

d3i

1

4r 1X - r12

G,(li - ?‘I, t - t’).

(9)

For completeness it is noted that an alternate possibility of defining the Green’s function for prompt reactor gammas is an introduction of the dose rate Green’s function by using a gamma attenuation kernel like K(p

18 - ?I) = B(p IR - ii).

e-PII”-‘I

with a gamma ray build-up factor B which can be approximated exponential functions as follows B(p IR _ 71) = Ale-alPIR--il

(10)

4n IR - ii”’

by a sum of two

+ (1 _ ~l)e-aWIR--il~

(11)

(A,, al, a2

are dimensionless constants depending, like p, on the properties of the medium and on the gamma energy.) The corresponding relations of the dose rate Green’s function are then given either in terms of rads/hr or rontgen/hr-using equation (7) together with equation (IO)-by D,(IW - ?‘I, t - t’) = 5.767 x 1O-5

G,((R

-

P’I,

t - t’, E), F

(12)

all eamma

energies 23

or Dy( IR - J’I, t - t’) = 6.883 x 1O-5

J

dEE~)ai*

all gamma euergeis E

. G,(@ - ?‘I, t - t’, E), ;, where ,uJp is the energy absorption mass attenuation coefficient in cm2/g.

(13)

642

LJ. KOSTIC and W. SEIFRITZ

2. Frequency response function of the prompt gammajux Obtaining an expression for the cross power spectral density function of the gamma flux fluctuations between two space points R, and & requires a transformation of the expression for the gamma Green’s function from the time domain into the frequency domain. This transformation is carried out by means of the Laplace transform of equation (9) G,(]R - J’], iw) = LZ[G,(18 - P’j, t - t’].

(14)

For simplicity’s sake the variable E has been left out this time. As can be seen, the only term which must be transformed is the Green’s function of the neutron flux G,(]i - ?‘, t - t’). This stems from the reasonable assumption of an infinitely short time of flight of the gamma rays as against the time scale in neutron diffusion processes. In Table 1 below, neutron flux Green’s functions in the frequency domain for both the one group diffusion and two group diffusion models with their characteristic constants are given for angular frequencies high compared with the reciprocal time constants of the precursors (W > 3r,, prompt kinetics approximation). The derivation of these functions is given in the Appendix. Substitution of one of the two neutron frequency response functions of Table 1 in equation (9) will supply an explicit expression of the gamma Green’s function in the frequency domain or frequency response function of the gamma flux. If one begins with the simplest case, i.e. using the one group diffusion model, the gamma Green’s function in the frequency domain will be G:“(IR -

i’l,

co >

d3_ 1 e-K+-?j e-‘IR-il r---p 4nD jr - P’(47r IR - iI2 *

AJ = Q(E) s

w)

all spaoe i of gaInInS SOUrcB[I

[Superscript (1) means that G, is related to the neutron frequency response in the oneenergy group approximation.] This convolution integral has been evaluated by a novel integration technique by transforming equation (15) into prolate spheriodal coordinates which are known from the theory of spheroidal wave functions (Handbook of Mathematical Functions, 1965). This is based on the principle of identifying i? and i’ with the foci of a family of confocal ellipsoids. The situation is shown in Fig. 1 together with the relations between the distances ]J - i’] and ]iF - ?I and the spheroidal coordinates [, 7, 4. If the transformation is carried out, equation (15) results in G(l)(IR _ ?,I cc)s1,) V

3

1

e-blR-f'irt _ Q(E) * ild5drle-"in-p1i. (16) 8?TDss1 -1 t+q '

with P+K

b=p-K 2

a=2;

*

This equation can again be integrated with the aid of exponential integrals of the n-th kind which are defined through E,(z) =

m ‘5 s1

dt.

(17)

643

The theory of space dependent reactor noise analysis using gamma radiation TABLE I.-ONE

AND

TWO

GROUP NEUTRON

RESPONSE FUNCnONS

FREQUENCY

One group diffusion model Gfl,(,F n

_

w

p,,

gA,)

_

’

1 e;+i’l

neutrons

47rD Ir - i’j

’cm2sec sec

= frequency response of the thermal flux at P due to the injection thermal neutron of the space point P’.

of a

Two group diffusion model

onse of the thermal flux at P due to the injection of a fast neutron at the space point i’.

Symbols used:

L2, T = thermal and fast diffusion area, cm2 M2 = L2 + 7 = migration area p = resonance escape probability k = 1 - b(l - $) = infinite delayed multiplication const. % = reactivity, in dollars ($ 2 0) a. = cr,(l - $) = prompt neutron decay constant, set-’ c(, = ,6/I = prompt neutron decay constant at delayed critical, D = thermal diffusion constant, cm.

set-1

The remarkable result of this procedure is (p+d e-”

G;'(IK

-

?‘I, w > AJ =

p-if

1

Q(E)

47rD@-P’(‘/A+K

I-,’

s

e(2K'pfK)tEl(t) dt,

(18)

0

which means that it is possible to express the frequency response function of the prompt

LJ. KosnC and W. SEIFRITZ

644

d37 =((*-$1 d{ dq dq5.f’

Neutron hnput source at 7 Gamma source at 7 Gamma frequency response at E Distance of foci =2f

FIG. I.-Prolate

spheroidal coordinates.

gamma flux by a product of the frequency response function of the neutron flux Gk” for the space point i? and the function kL+K)IR--l”I

&dp+~,t~~(~) dt

w([R - ?‘I, K, ,!_A) = -

,

(19)

which can be considered to be a weighting for G,. For discussion of the characteristic differences in the neutron and gamma frequency responses it is only necessary to consider equation (19) which can be simplified in various limiting cases. For example, in the frequency range Ai < o < u where K is a real quantity we obtain: (I) ,U <

R~K

(small gamma attenuation)

lP’(IR - 7’1,ReK) -+ E

[ Y

Rf?KpLi,l

e-2tEl(t) dt

(2) p < ReK (strong gamma attenuation)

= y

[l - E,(,D IR - ?‘I)]

1 =y Q(E) for cK > R~TK IR - i’l P (3) P = ReK (gamma attenuation comparable with the neutron attenuation Retc) W”‘( IR _

?‘I,

ReK)

=

s2ReK’R-“‘e-t&(t)

Q(E) 2ReK

dt.

o

In Fig. 2, these three weighting functions are shown as a function of the dimensionless distances, ReK IR - P’I and ,U 18 - P’I. In limiting cases it is seen qualitatively that

The theory of space dependent reactor noise analysis using gamma radiation

645

/_L(R-7{

FIG. 2.-Weighting functions for the neutron frequency response function in the frequency range Ai
Frequency range G,

(,?e
1)

h-w-a

Distance from neutron s-source,

Re Kj/+7’(

FIG. 3.-Gamma frequency responses for two limiting cases in comparison with the neutron frequency response based on the one-energy group diffusion model.

the amplitude of the neutron frequency response function must be weighted with a small value at short distances and with a large value at long distances, respectively, to construct the corresponding gamma frequency response. This means that the curvature of the gamma frequency response is not as strong as in the neutron frequency response (Fig. 3). The conclusion of this fact and a specific example of the comparison of G, and G, will be given in the next section. In an analogous way we received also the gamma frequency response by using the frequency response function from the two group model. The result for the frequency

LJ. KOSTIC and W. SEIPRITZ

646

range izi < UJ < l/l in which IQ and ICYare real quantities is given by the following expression analogous to equation (19). Gp’(IR - ?‘I, iii < w < l/1) --lQIR-?l

1

-I(21R-r 1

=C&-.--w,(@ - %4 [ IR - J’(

with

‘G) - ++

r

w,(IR - i’l,,% ‘%) ,

(20)

(P+Kl) IR-i,I

w,(

IR

k&(IR

-

-

f’l,

,U, K1)

=

?'l,,U,K2) =

Q(E) P + KI

-

Q(E)

!-4++2

e(2K1h+~1)~~l(~)d~

s 0

(p+dIR-1"i

e(2KZlPfKZ)fEl(t) dt,

so

where C, K~ and Kg are explained in Table 1. Obviously, it is also possible to simplify the above expression when comparing p with pi and K~, respectively. On the other hand, equation (20) contains also the one-energy group expression when p = 1 and 7 --f 0. In this case, K1 -+ K, W, -+ W, and K~ --f co resulting in equation (18). 3. The crosspower spectral density function, CPSD, and the spatial coherence function, SCF, of the fluctuations of the prompt gamma flux The cross power spectral density function, CPSD, of the fluctuations of the prompt gamma flux between two space points R, and 3, is now expressed by the following equation of definition CPSD,(@,

- &I, w) =

d3?‘G,*(@, - 71, w)G,(@, - i’l, w)l s allspaceof

(21)

m?l.ltmn BO"lYXB

in

gammas 2 per Hz bandwidth ( cm2sec 1

and I is the input neutron cross spectral density function which, on the one hand, does not depend on the space variable J’ in an infinite medium and, on the other hand, is independent of the frequency in the range w > li. Its dependence on the mean neutron density A, the neutron lifetime I and, particularly, on the effective multiplication constant k,, has been carefully studied by SHEFF(1965) and is given by I = f [k&D

+ 2[1 -

k&l

-

b)]] m ; k,,fD,

in

(n~~3~~~)2

;

(24

ti is the mean number of neutrons emitted per fission and D is the Diven-factor N 0,8. Substituting equation (16) for the frequency range iii < w < a (plateau region), in which G, is a real function (K = ReK), into equation (21) and again applying the method of prolate spheroidal coordinates with respect to the distances I& - ?‘I and IR, - ?‘I we obtain the following expression for the cross power spectral density

641

The theory of space dependent reactor noise analysis using gamma radiation

function of the gamma flux between two space points & and & in the one-energy group diffusion model

x [eC(a1ea2)[~+~+

(t2--$)] -e&a1+a2’[~-~+ in

($---$)I)

gammas ~

[ cm’sec

1 2

per

Hz bandwidth,

where

a2= $1R2- RI . bGI - t2) + WV,- ~~11

(23)

[a, b are known from equation (16)]. A special case of the cross power spectral density in the previous equation is the auto spectral density of the gamma noise for I& - A,1 -+ 0 which gives

1

Corresponding expressions for the auto and cross spectral density functions APSDF) and CPSDP) for the two energy group diffusion model can easily be obtained in an analogous manner. The first step is the synthesization of the gamma frequency response, Gp) by replacing Gp) in equation (15) by Gr). The result has the same mathematical structure as equation (16) but consists of two additive terms. The cross power spectral density function CPSD,(2)is then obtained by evaluating equation (21) for G, = Gy). The final expressions are not given here explicitly because they are rather lengthy. In order to get an idea of the spatial range of the neutrons in comparison with the gammas in a multiplying medium we now introduce a new dimensionless quantity called spatial coherence function, SCR, which is defined in general by normalizing the cross spectral density function on the auto power spectral density (25) This definition is used for both the neutron and gamma fields. We introduce two scales of the spatial correlation range, the ‘integral scale’ or ‘macroscales’ and the ‘microscale’. The terminology for these quantities is borrowed from the theory of turbulence (HINZE, 1959), where these scales are normally used to interpret measured auto-correlation functions in the time domain of the fluctuations of velocity components or of the temperature in a turbulent flow. Here, we define the ‘integral scale’

648

LJ. KOSTIC and W. SEIFRITZ

as the most important scale for the spatial correlation range in a correlated neutron or gamma field in the following manner: R(w) =

s0

m[SCF (I& - &I) 1d I& - 8i1, cm,

(26)

which is a measure of the global or integral spatial coherence of the stochastic field. The ‘microscale’ is defined through the curvature of SCF at zero distance 1

.

1-

n”(o) =,I3~9&0

I~~~~l~,--~,l,~)l,~;~~~ I& - R,j2

(27)

This scale is a measure of the Constance or flatness of the SCF for small distances Ii?, - R,j. Whilst the two scales for the neutrons A,(l) , A(l) in the one and two A and RF), 3Li2) energy group models can be given analytically (see Appendix), the corresponding scales for the gammas A,(I) , l(l)A and A?), A?) must be evaluated numerically by means of equations (25)-(27). Examples of spatial correlation scales will be given in the next section when a specific multiplying system is considered. Numerical example. The previous theory is now applied numerically to a simple but clearly defined multiplying medium to illustrate the differences in the spectral correlation ranges of prompt thermal neutrons and gamma rays. Our infinite homogeneous medium consists of a mixture of light water and uranium-235. The nuclear data for the delayed critical system are summarized in the following Table 2. In this medium, the contributions to the yield of the total prompt gamma source TABLE 2.--PARAMETERSOF THE INFINITEHOMOGENEOUS MULTIPLYINGMEDIUM Neutronics Kinetic parameters

Static parameters

ken =

5. C,R

--Qr=l

1 = kB (I = 1.061 x lo-4 set u = 2.2 X lo5 cm/set p = 0.0064

f = 2.47 neutrons/fission NHO 2 = 1123.4 N26

CL,= p/l = 60.32 see-1

C,* = 0.04282 cm-’

a = a,(1 -

$)

C,25 = 0.017338 cm-* C (I26 = 0.02063 cm-’

K(& < w Q a) = fi

= 0.039 cm-l L K~ = 0.01369 cm-l

Ctz” = 0.02219 cm-l xl=+ ,n,y, = 0.00329 cm-l f = 0.482, LH~O= 2.85 cm L=L~~o.dl

K~ = 0.52067 cm-l

-f=2.5cm

7 =30cm8 MB=L2+r Gamma parameters

p = (ruhvater= 0.0706 cm-l (1 MeV-gammas)

($=O)

The theory of space dependent reactor noise analysis using gamma radiation

649

due to equation (4) originate from two sources, i.e. the fuel and the light water. The energy distribution of the gamma source is composed by the continuous spectrum of the prompt fission gamma rays Y,(E) and the line spectra of the capture gammas from light water and uranium. To simplify this situation, we find a mean energy E of the gamma energy source distribution by

s s

EQ(E>dE Q(E)dE

or

s

’

dE E[X,“YD(E) + xi:, . Y;:,(E)

+ c::Zy’. Y;;‘(E)]

1 MeV f

dE[E;5Y,(E) + xi:, . Y::,(E) + x:2,“.

(28)

Y:z,“(E>l

using Y,(E) and Yi$O(E) from equations (5) and (6). The macroscopic cross-sections for fission and capture reactions have been taken from Table 2. According to a recent publication on the level scheme of uranium-236 (MATUSSEK et al. 1970) it was possible to approximate the capture spectrum of uranium-235 by a simple line spectrum Y(25)(E) N d(E - 6.396 MeV) (29) KY because this 6.396 MeV transition from the 6.54 MeV capture state of uranium-236 is the only possible electric dipole transition to known low energy levels. Therefore, this gamma ray is only visible in the higher energy range. The mean prompt gamma energy obtained of about 1 MeV is a little bit higher than the mean energy of prompt fission gammas (0.9 MeV), since the capture gammas harden the total gamma energy distribution. However, the predominant gamma source strength in this system originates from prompt fission. The value of the gamma attenuation constant ,u for this mean gamma energy E is also given in Table 2. Clearly, ,Dis taken for the pure light water because the effect of the fuel on the gamma attenuation can be neglected due to the poor concentration (NHs0/Nz5 N 1123). After the assumption that all prompt gammas start with the same energy B, we calculated the gamma frequency response function for the one energy and two group energy diffusion model with equations (18)-(20). In Fig. 4, these functions are shown together with the corresponding neutron frequency responses. It is seen from these graphs that the amplitudes of the gamma frequency response functions are absolutely greater than those of the corresponding neutron response curves. A comparison of the curvatures of corresponding neutron and gamma responses clearly shows that the curvatures of the neutron curves are greater than those of the gamma curves, which indicates that the gamma wave penetrates deeper into the medium than the neutron wave, which finally results in greater spatial coherence of the gamma field. It is seen also that the two group diffusion model gives a more realistic behaviour of the neutron and gamma frequency response (deeper penetration into the medium) than the one group diffusion model, since T is greater than L2 (7 = 30 cm2, L2 = 4.2 cm2). Only in cases in which T < L2 do the expressions based on the one group model coincide with those on the two group model.

650

LJ. KosnC

and W. SEIPIUTZ

0.015 -

03

One group diffusion

model

G$‘( IR-/I,

Xi<
G;‘(/ R-r’].

Ai<< WCCal

Two group

diffusion

model

‘&D

001 K, =o-01369cmd ~~=0~52067cm-’

(I MeV)=O-0706cm-’

p

0005.

t

L 0

20

10

30

40

Distance

50

0

from neutron

IO

20

S-source,

30

40

I%?I,

50

60

cm

FIG. 4.-Neutron and gamma frequency response functions for the one- and two-group diffusion model in the infinite H,0-as5U delayed critical system.

The spatial coherence functions, SCF, based on the one and two group diffusion models SCF;? (I& - &I, Li < w < a) and SCFV) (I& - &,I, f, < cc)< l/1) for the prompt gamma rays within the critical water-uranium-235 medium have been calculated by the numerical evaluation of equation (25) together with equations (23) and (24). The integrals in these equations have been monte carloed on the IBM 1130. The uniformly distributed random floating point numbers between 0 and 1 are generated with the subroutine RANDU which is specific to the IBM 1130 and which is mathematically based on the power residue method (IBM). In Fig. (5), the spatial coherence functions for the gamma noise based on the one and two group diffusion models, SCF:) and SCFF), are shown together with the corresponding spatial coherence functions of the neutronic noise, SCFA’) and SCFk2), as a function of the absolute distance between two space points I& - &I in the delayed critical medium. In this I.0

R\\

Neutron

H,O - E35U system delayed crificol c#=o,

field

n’i’= 25.6 cm n’,‘i

.-g ii 5 ?j 0.5 5 k?

75.2 cm

A$)=29

cm

Gamma

field

,u (I MeV)

=O 0706

140 cm

160

cm-’

z 5 2 ::

0

20

FIG. %-The

40

60

80

100 Distance

120 Izz-Fl,

I80

200

spatial coherence functions, SCF, of the neutron and gamma fields in the infinite delayed H,0-Ja6U system.

i

The theory of space dependentreactor noise analysisusing gammaradiation

651

figure, the differences between the one and two group treatment are evident. In the same figure, a table contains all the parameters of interest, i.e. the macro and microscales of the spatial coherence in the frequency range izi < CI)< cz, (for the one group model) and Ai < cc)< l/Z (for the two group model). CONCLUSIONS From the theoretical investigations discussed above we conclude: (1) The spatial correlation range given in terms of the macroscale A of the stochastic field is in general greater for the gammas than for the neutrons. This also implies that a gamma sensitive detector ‘sees’a greater reactor volume than a neutron detector. However, the differences in the correlation ranges between the gamma and neutron fields are not so great when considering only a mean prompt gamma energy in the vicinity of about 1 MeV. For the larger part of the prompt gamma energy distribution (E > 5 MeV) and for a subcritical medium ($ < 0) these differences become greater because the gamma attenuation coelhcient ,u is smaller than for 1 MeV gammas and K becomes greater in the subcritical case. (2) In a qualitative transfer of the results on thermal finite reactor systems we found for all important reactor types (LWR’s, HWR’s, GCR’s) that K is much smaller than p in the delayed critical state. This fact is true also and in particular for fast systems with their metallic cores, although there are some difficulties of defining a corresponding quantity for the migration area. Thus, we conclude from Fig. 3 that the shape of the gamma frequency response in the frequency range li < o < tc is practically identical with the shape of the neutron frequency response resulting in an identical spatial coherence function with the same correlation parameters. (3) Considering the prompt neutronic coupling mechanism in a multiplying infinite system, the macroscale of the spatial coherence function of the neutron field A=

~ :/-

S)

can be related heuristically to the mean spatial extension of a neutron chain, because the branching processes are responsible for the spatial correlation between two space points. It is seen that A, on the one hand, depends on the diffusion parameters L2 and T and, on the other hand, on the dollar reactivity of the system, %,and in the ‘worth’ of the dollar p. From this, we can conclude that-for the analysis of reactor noise measurements in the prompt kinetics approximation at finite systems-the reactor can be considered as a ‘point reactor system’ if all geometric dimensions of this system are small compared with A. (4) Finally, it should be noted that it is also possible to extend this theoretical model to (1) make it include delayed neutrons (2) finite systems using corresponding neutron Green’s functions and space dependent cross power spectral density function Z(i’) of the input neutron source, present in the system (SHEFF, 1965) (3) neutron-gamma correlation analysis using the following definition of the neutron-gamma cross power spectral density function : CPSD,,, (RI, &, m) =

d%‘G,*(i?,, f, o)G,(&, s T’

i’, w)l(J’).

(30)

652

LJ. KOSTIC and W. SEIPR~TZ

REFERENCES BAL~H F. J. and KENNEYE. S. (1969) Nucl. Apphc. 6,232. GELINASR. J. and OSBORNR. K. (1966) Nucl. Sci. Engng 24,184. Handbook of Mathematical Functions (1965) Edited by M. ABRAMOW~TZ and A. STEGUN, p. 752. Dover, New York. HINZE I. 0. (1959) Turbulence. McGraw-Hill, New York. IBM-Application Program H 20-0225-2. 1130 Scientific Subroutine Package. KENNEYE. S. (1967) Neutron noise, waves and pulse propagation. USAEC, Conf. 660206, p. 399. KENNEYE. S. and SCHULTZM. A. (1969) Nuct. Applic. 6,238. LEHTOW. K. (1967) Ph.D. Thesis, University of Michigan. LEHTOW. K. and CARPENTERJ. M. (1968) Nucl. Sci. Engng 33,225. MATUSSEKP., MICHAELISW., WEITKAMPC. and WODA H. (1970) Safeguards techniques, p. 113. IAEA Proc. of a Symp. held in Karlsruhe, IAEA-VIENNA. MOORE M. N. (1960) Neutron wave optics and sonics in Japan-United States Seminar on Nuclear Reactor Noise Analysis, Tokyo and Kyoto. Nuclear Engineering Handbook (1968) Edited by H. ETHERINGTON,pp. 7-72. McGraw-Hill, New York. OSBORNR. K. (1968) Japan-United States Seminar on Nuclear Reactor Noise Analysis, p. 25, Tokyo and Kyoto. Reactor Handbook (1962) Edited by E. P. BLIZARDand L. S. ABBOTT,Vol. 3 Part B, p. 50. Interscience, New York. Reactor Handbook (1962) Edited by H. SOODAK,Vol. 3, Part A, p. 14. Interscience, New York. Roux D. P., FRY D. N. and ROBINSONJ. C. (1968) ORNL-TM-2144. SHEFFI. R. (1965) Ph.D. Thesis, University of Washington. TRUBEYD. K. (1970) Nucl. Applic. 9,439.

APPENDIX Neutron frequency response functions and spatial coherence functions for prompt neutrons A. The one-group dtjiision model The expression for the infinite medium Green’s function of the neutron flux on the basis of the one-energy group diffusion model is defined as the solution of the one-group diffusion equation with a a-function input source = --6(? - i’)s(t - t’).

DAG - ; G - ;;

64.1)

The result in the time domain, Gpl((i - i’l, t - t’) is known from the literature (SHEPP, 1965) and will not be repeated here. The Laplace transform of this function, i.e. the Green’s function of the neutron flux in the frequency domain-or the so-called frequency response function of the neutron flux-is obtained by both Laplace transforming equation (A.l) with respect to time (t +s), and Fourier transforming it with respect to space (i -+ B). The one transformation back into the space domain G:)(lP - P’I, s) = 2pz ,:_

i,, . 12

sin B I? - J’I Gz’(B*, s) dB

with

(A.3 results in G”‘(,P _ ,?,, o) = 1 eIK’P;PI’ n ~xD Ir -r’, with s = io.

(A.3)

K is the complex material buckling K = Reu + ilmrc

=;

+a”+ J-

7

+

o()V

+

i(l/a” + or2 - a)‘ta].

(A.4)

The theory of space dependent

653

reactor noise analysis using gamma radiation

The cross power spectral density function of the neutron flux is defined as CPSD;$?l,

&, w) =

* J

dSi’G;‘*(IR1 - 71, o)G;‘((&

- 71, 0).

I.

(A.9

all Spwx F of neutron sonNe8

The asterisk denotes the conjugate complex quantity and Z is the input neutron cross spectral density given in equation (22). The integration of equation (A.5) becomes very simple, if one profits from the elliptical symmetry of the problem, i.e. uses the same integration technique with prolate spheroidal coordinates which has been described already. One obtains the following real expression CpS,,‘“(IR n

2

_

jj

,

1,

(A.61

w

Normalization of this equation on its auto-power generally results in the spatial coherence function

spectral density, APSDy’(w)

= CPSDc’(0,

which, only ‘low frequency neutrons’ in the frequency range 1: < w 4 CLare considered, to SCFF’(I& - R,I, li <
w),

is simplified (A.@

The macroscale A of the neutronic correlation field defined in equation (26) is found to be Arc tgg _\;‘(ji, < W) =

Imlc

(A.% K

and

qvi
kK=

L dB(1-b)’

respectively.

A microscale 1:) cannot be defined for the one-energy group model because, in this case, the curvature of the SCF at zero distance is not negative. B. The two-group d@ision model The differential equation for the Green’s function of the thermal flux, GF’(li - i’l, I - t’), i.e the response of the thermal flux to the injection of a fast neutron in the case of a two-group treatment is given by 1 aG(lP - 71, f -

u

at

t’)

= DAG((P - 71, t - t’)

I kC, / P

dSPOG(IfO- 71, f - r’)P(\i - To\) I

all space P,

-E:,G(li [k = 1 - /?(l - 6) = infinite delayed multiplication

- 71, t - r’) + S(li - i’l)s(r - r’)

(B.1)

constant].

The second convolution term on the right hand side is the slowing down density of fast neutrons which are born through thermal fission at f0 with a source strength of (kC,/p)G(f,,, t). Fast fission is neglected. P(f - r0 ) denotes an appropriate displacement kernel, i.e. the probability that a fission neutron born at f,, will become a thermal neutron in the unit volume at i. The slowing down time is assumed to be zero because this time can be neglected with the diffusion time. In the two-group model, the displacement kernel is expressed by P(lp _ f0,) = c~.

e+-~o~~~~ IF - Fool

where 7 is the slowing down area andp is the reasonance escape probability. I,

(B.2)

654

LJ. KOSTIC and W. SEIFRITZ

The source term of equation (B.l) is the source distribution of thermal neutrons resulting from the injection of a localized fast neutron at i’ into the system. Its intensity is determined by S(li - i’l) =

dVJ’(I? - i&&IF0 - P’l)

s

(B.3)

all space P, p z-

e-Ii-i’I/d/;

4rri-

IP - P’I

(B.4)

.

Submitting equation (B.l) together with equations (B.2) and (B.4) to a two-fold transformation as equation (A.l), carrying out the Laplace transform with respect to time (I --f s) and the Fourier transform with respect to space (? -+ B), we obtain the characteristic equation P

GC’(B2, s) =

making use of the fact that the convolution

1

(B.5)

in equation (B.1) yields the product

=a

pG:'(BL,s)&. The backward Fourier transformation Gr)(li - ?‘I,s)=

T

from the B-domain into the space domain P by ’ jmB sin BIP - P'I Gp'(B',s)dB 2P2 If - i’J 0

now yields the final expression for the frequency response function of the thermal flux substituting s by io (B.7) with C=

4aDr

03.8)

[(

and the complex material bucklings, K~and K~ K12

=

[(

1 jwl 2 l +L‘J

1 1 +io[ KS2Z -2 [( 7+;

+;) - 4T-y+g] 1 + iol

(B.9)

1 1 + iwl -z;;a-;)Z+;]. ) + J(

(B.lO)

It should be noted that this procedure described above is a practical application of a general formalism given by MOORE (1968) to treat space-time-energy excitation experiments as well as dispersion laws in neutron wave optics: e.g. the characteristic equation for the poles of G(Ba, s) in equations (A.2) and (B.5) is obtained by zeroing the denominators of these equations, which results in the dispersion laws of the field, for it imposes a relation between the wave number B and the frequency s. The cross power spectral function CPSDF’ of the thermal flux for two space points R1 and & is obtained by introducing equations (B.7)-(B.lO) into the equation of definition [equation (A.S)]. The integrals can also be easily evaluated by using prolate spheroidal coordinates. The final result for 0 < Illis

e-&,-R,I - e-rll&-Rli 4 i- I& ___&I K12- Jr22 Normalization

on its auto-power

1. (B.11)

spectral density provides the real spatial coherence function

The theory of space dependent TABLE

3.--SCALES OF NEUTRONIC

COUPLING THERMAL

Parameter LZlT

(cm-l) 1~~(cm-l) R~K cm-’ Y 10-S K~

65.5

reactor noise analysis using gamma radiation

235U + HZ0 0.14 0.014 0.5207

IN SEVERAL INFINITE DELAYED

2W + c

‘WJ + D,O

6.7 0.0016 0.0559

39

CRITICAL

SYSTEMS

*W + Be

12 0.002 0,093

I.62

2.07

24 0.0032 0.12 5.26

One-group model 619

25.6

484

190

Two-group model Macroscale A& < w < l/l) (cm) Microscale n(ni << w < l/l) (cm)

with the approximations

75.2

636.5

531

343

29

272

180

125

for the bucklings

Forr+OitisseenthatK,-+coandM” ---f L2 and we obtain the SCF for the one-energy group model in equation (A.8) with ReK = Q. The macroscale or integral scale A is given by integrating equation (B.12) over all distances from zero to infinity [see equation 261, which yields (B.13) whereas the microscale 1 is obtained by evaluating the procedure of equation (27) like &,‘(%i < o < l/l) Y 6 J ,(lL”

s) .

(B.14)

In Table 3 above we computed and summarized the characteristic coupling parameters for a series of intinite thermal systems in the delayed critical state. It is seen that the graphite moderated system shows the best neutron coupling (with a macro scale greater than 6 m) whereas light water systems show relatively poor coupling because the migration area is so small.