A modelled theory of pulsed-neutron experiments in fast subcritical assemblies

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T. A. E. C.

ÇEKMECE NUCLEAR RESEARCH AND TRAINING CENTER

I

s t a n b u l

-

t u r k e y

ÇN A EM -R-132

A MODELED THEORY OF PULSED - NEUTRON EXPERIMENTS

IN FAST SUBCRITICAL ASSEMBLIES

By

U. Adalıoğlu and J. J. Duderstadt

P. K. 1, Hava Alanı,

Istanbul,

Turkey

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 3(2&3), 147-166 (1973)

A MODELED THEORY OF PULSED-NEUTRON EXPERIMENTS IN FAST SUBCRITICAL ASSEMBLIES

Ulvi Adalioglu

Çekmece Nuclear Research Center Istanbul, Turkey

and

James J. Duderstadt

Department of Nuclear Engineering The University of Michigan

Ann Arbor, Michigan

ABSTRACT

The time behavior of a neutron pulse in a fast subcritical assembly is modeled by approximating the spatial dependence by a single diffusion mode and using a simple model of inelastic scattering. This model allows an explicit calculation of the time response of a detector in such a system. Of particular interest is a study of the detector response in those situations in which the flux does not decay in an exponential fashion.

I . INTRODUCTION

In an earlier study [1] the time-decay constants characterizing the time behavior of a neutron pulse in a subcritical fast assembly were investigated by analyzing the time eigenvalue spectrum of the transport

operator for fast neutrons. Although this analysis was simplified some

what by assumptions of isotropic scattering and a simple model of inelastic scattering, it was still of a rather formal nature and only revealed more

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ADALIOGLU AND DUDERSTADT

general qualitative features of the eigenvalue spectrum. Perhaps of most interest was the conclusion that, for sufficiently small concentrations of fissile material and/or sufficiently small assemblies, there were no point eigenvalues in the spectrum of the. transport operator and hence no truly asymptotic exponential decay of the neutron pulse.

To continue our investigation of the time behavior of pulsed fast systems, we shall now fall back on much simpler approximate models of the

neutron-transport process. Such modeled studies are quite common in the

study of relaxation problems in transport theory and allow one to use the standard tools of complex-variable theory to study the time eigenvalue spectrum, in contrast with the rather elaborate machinery of functional analysis required in the more general analysis of the transport operator.

Similar models ware first applied to the study of pulsed fast systems by Storrer [2], Stievenart [3], and Cadilhac et al. [4], who studied a single-mode diffusion model within both multigroup and continuous energy- dependence treatments. More recently D o m i n g , Nic.olaenko, and Thurber

[5-7] have developed and studied in considerable detail a variety of

diffusion models with various treatments of the energy dependence, including

elastic slowing down aa described by the Grueling-Goertzel kernel. By using

analytic continuation techniques, these authors were able to treat the situation in which no discrete decay constants exist for intermediate times. These authors have also studied models in which both exact elastic scattering

and inelastic scattering by discrete nuclear levels were included [7]. How

ever, the generality of this model restricted their conclusions to be of a very qualitative nature.

Our efforts here will be specifically concerned with models that treat inelastic slowing down since this process is thought to be of major

importance in determining the time behavior of fast assemblies. We shall

employ the simple evaporation model of inelastic neutron scattering and treat spatial neutron transport with a single diffusion mode. It is of particular interest to evaluate the predictions of such models in light of the more general theory developed in Ref. 1.

II. A MODELED STUDY INCLUDING INELASTIC SCATTERING

We begin by assuming that the spatial dependence of the neutron density can be adequately described by a single diffusion mode such that the transport

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PrtSf.ti-NTnKON EXPERIMENTS equation reduces to | | + v[Et (v) + D E 2 ]N(v,t) | dv' v'Z (v',v)N(v',t) + vx(v) I s j v dv' v'T (v’)N(v',t).

(

1

)

We shall model the inelastic scattering kernel after Okrent [8] as

in Y L s (v’v) v'Z (v ')g(v)/h(v') for v' > v 0 for v' < v where g ( v ) = A v 2e x p , h (v ) = | dv' g ( v ’), 0 (

2

) (3)

while (v) 3s assumed to vanish below seme inelastic scattering threshold

v. .

i n

Using Laplace transforms, one can immediately solve E q . (1) for the transformed detector response following a neutron pulse at time t = 0 with a fission spectrum speed dependence x(v):

where R(s) dv vI^(v)N(v,s) 0 F(B.s) D(B,s)’ (4) F ( B , s ) fCO 0 dv

vZd(v)

s + v i t (v) + vDB2 X ( v ) «O

0

■ v l ' d v ' x ( v * ) $ ( v f , s ) e x p ■ I d v "

J

g f

/7

' * ( V " * S ) V

.

v ' “ (5)'

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while the so-called dispersion law D(B,s) for this model is given as

D(B,s) -- 1 - p (B, s) , p(B,s) vEf (v) dv s + v l t (v) + v DB2’

X(v)

dv' x(v ')$(v '»s) exp dv"(l d ^ " * (v" >s) (

6

) Here v Z i / v ^ ( v ) /h(v) * ( V ’S)

5

7

+ v i t (v) + vDB^

Of course the singularities of [D(B,s)] 1 correspond to the eigenvalue

spectrum of the modeled transport operator in Eq, (1). Hence we need to

study D(B,s) as a function of the complex variable s. One can show from Eq. (6) that

(i) p(B,s), and hence D(B,s), is defined in the complex s-plane cut

is

along the negative real axis from -X^ to where

min v £[0,»)

[vZ (v) + vDB2] . _{( 8)}

(ii) p(B,s) is a monotonically decreasing function of s = B = real for 3 e[ - X *,»)•

(iii) p(B,3) < “ • In fact, if we model the speed dependence

of the cross sections as

Bf (v) = X_/v, £ (v) = Z + £°/v, D(v) = D,

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PULSED-NEUTRON EXPERIMENTS

£in<V > "

Lin for V ^ V in*

l 0 for v < v1qS

£d (v) =

f

2d for v “ vd>

0 for v < v^, (9)

then we can explicitly calculate this limit as

\>A, P (B !- Ajj) j. + x>B r " , , v a.e. dv i M + ---i in o v [ i p r 5 i ^ T 2 ,v dv g(v) in v[h(v) ]n ' v dv*

v ( v ' )

th(v')]l-n n = in (E + DB^) •

(

10

)

We can now apply these properties of p(B,s) in our search for the zeros of D(B,s), that is, the discrete eigenvalues of this model. First note that one can demonstrate that the only possible zeros must be real. Hence we can confine our search to s = B = real for B e T -A*,»). Since

L B

p(B,s) decreases monotonically from a finite value at B = -A*, there can be at most one discrete eigenvalue, call it A^ (see Fig. 1). (Note that p(B,g) plays the same role as the associated eigenvalues Kj(g) in our more

general spectral theory [l].)

*

Now it is evident that p(B,Ag ) decreases in all cases with either increasing B 2 or decreasing E^. Hence for sufficiently small systems (large B2) or small concentrations of fissionable material, the dispersion law D(B,s) will have no zeros ■— that is, no discrete eigenvalues. Further more, if we increase E^ sufficiently, we can drive A n into the right half

plane— corresponding to a critical or supercritical system. Hence this

simple model faithfully mirrors the qualitative behavior of the more general transport description [1],

The disappearance of discrete decay constants is controlled by the

^ A

(7)

Fig. 1. The eigenvalue spectrum of the model when a point eigenvalue

e x ists.

corresponds to v -*■ 0 or, mere effectively, to the energy dependence of the cross sections in the thermal energy range (E < 1 e V ) , and since in fast systems essentially all neutrons have energies in excess of several

thousand electron volts, w e might suspect that the bound on the disappearance of discrete eigenvalues will not be relevant to fast systems (or present in such a "weak" sense as to be undetectable). It would be desirable to fix up our model to avoid these difficulties arising from low-energy behavior.

Of course one scheme would be to simply truncate the speed range to v e [vc ,=°), where v c is some cutoff speed (say, 10 keV) below which the neutron density can be assumed to be zero. Such a cutoff model runs into

mathematical difficulties, however. In particular, one finds that in such

a cutoff model X(v)___ v 2 c (v) - Ac as v -*■ v , c

(ID

where X = min [vE (v) + v D B 2 ] C ve[vc>°°)

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P UL S E D -NEÜTRON EXPERIMENTS

and hence p(0,“) diverges as |1 -* -A . From Fig. 2 we see that this implies that a discrete decay constant will exist for any size assembly with any finite amount of fissile material— in sharp contrast to our earlier analysis. This feature is also present in more elaborate studies of the spectrum of the transport operator (such as those in Ref. 1) when a speed cutoff is

used. This feature of the cutoff-speed model is certainly as unappealing

as the influence of low-energy neutron-cross-section behavior on the point eigenvalue spectrum of the full-speed-range model we have considered.

It is possible to construct a cutoff model that avoids these

mathematical idiosyncrasies, however. Such a model is motivated by

recognizing that both the fission spectrum x(v) and the inelastic scattering

spectrum g(v) also vanish for neutron energies in the keV range. Indeed,

it is this fact, plus the absence of appreciable neutron moderation, that gives rise to the fast-neutron spectrum in the systems of interest and makas any model that depends on the cross-sec.tlon behavior for thermal

neutrons highly susnect. Hence, if we choose the cutoff sufficiently

low for the fission and inelastic scattering 3pectrums to vanish for v < vc> the divergence Eq. (11) in p(B,g) as v v c will disappear, and

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ABALIOGLU AND DUDERSTADT

Fig. 3. Modeled cross-section behavior with the corresponding s-plane

structure.

the cutoff model will yield a mathematical behavior very similar to that of the model that allows all v e [0,“).

Hence we shall utilize a cutoff fission and inelastic scattering spectrum in conjunction with our cutoff-speed model:

Xc (v) = A f (v - v c>2 exp(-v2/v|) (12a)

and

g c (v) = A i (v - v c) 3 exp(-v2/v2), (12b)

where

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(Of course, if such cutoff models are to be useful, our results should be insensitive to the specific values chosen for v .) When it becomes

c

necessary to use explicit forms for the cross sections, we shall use the simple forms introduced in E q . (S).

ill. ANALYTICAL STUDIES OF THE DETECTOR RESPONSE

We can easily adapt Eos. (4) through (7) for the time response of a detector following a neutron pulse at t = 0 with a fission speed depencence to account for the cutoff-speed model to find

F (B - f ■“> v£,(v) dv d s4vZ (vJ+vDB^ J v t Xc lv> + dv' xc (v ')*(v',s) x exp f d v " - V i _ * * 8 C dv7 7 - 4 (v",s) (13) f=° vE,(v)v D(B,s) = 1 - i dv j_^ uv s+vE"t (v)+vDB2 X_(v) +

r

dv' x„(v')4(v’ ,s) x exp 1 dgc 4

(V" , S)

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Once again we find chat R(s) is analytic in the s-plane cut along the negative real axis. However, since. E is discontinuous at v = v^n , we now find that the cut is fragmented into two segments,

Tj s {s e [-X1,-Xc ]}

and

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ADALIOGLU AND DUDERSTADT where A, = V. (e + E, + 1 in'- a f DB 21 = v. E , in 1 (15a) and X2 H V in (Ea + E f + r in + ^ S v in£2 (15b>

Once again we can show that p(B,s) is a raonotonically decreasing function of Re{s) = A and that p(B,-Ac ) < Hence this model exhibits the desired behavior that when B 2 > B * 2 or E^ < £*, p(B,~Ac ) < 1, and hence there will be no discrete root of D(B,s), that is, no discrete time eigenvalue.

To explicitly perform the inversion of R(s), we deform the Bromwich contour around the pole s = - A n and the branch cut r = r, + r„ to write

r 0 1 2 0+1“ R( t ) = 2 ^ r f ds e S t R(s) a-i” = F(B 3D/3s -If)) s=-A 0 ds e ot:R(s) . r (16)

To evaluate the contribution from the cut, we need to calculate the limiting values of R(s) as s -*• - A± ie. Using

-A+ie + vE -A + vE ± iir5(-A + vE) , (17)

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F(B,-X±ie) = FR (X)±iFI (X) (18a)

and D(B,-X±1e) = DR (X)±iDI (X), (18b) where Fr(X) e EjP 1 vx (v) dv _{v Z - X}— r + ^_d vd 1 dv' v'xc (v')/h(v’) v'E, - X V i 2 V I dv[ 6(v - v ^ M v ) + g (v) ] x exp dv" 4(v",s) (19) Fj (X) = rrZd dv vxc (v)6 (vEj - X ) ,

(

20

)

Dr(X) = l-vEf 1 vx (v) dv —t t--- - - vE, vEj - X f dv' v ,xc (v')/h(v') v l 2 - X V | dv[5(v-vi)h(v) + g(v)] x exp dv" *(v",s)

(

21

)

Dj (X) = TTVEf rv i dv vxc (v)5(vE - X) ( 22)

for s e [-Xj.-X ] = P j , with similar expressions holding for the limiting values on V2■ Using Eqs. (16) and (18), we can write the detector response for the case in which there is no discrete exponential decay

R(t) = | dX e XtK(X), B 2 > B*2 or Ef < E*, r

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ABALIOCLU AND DUDERSTABT

where

K(x)

1 Dr(\)Fi(X) + Di(X)Fr(X)

* Dr(X) + Dİ£(X) (24)

Since we are interested primarily in the long-time behavior of this detector response, we can confine our attention to the contribution from

the neighborhood of X - ~^c ' this case, K(X) reduces to

F (X)D (X)

K(X) - D^(X) "+ D](X)" • (25)

If we define X -• X + xi . then K (k) becomes

c t

K(<) v Xc (vc + k)

V K)

b|(<) + I)2(<) • (26)

An examination of this expression reveals the following:

1. The amplitude function K(A) has a modified fission spectrum shape.

2. The amplitudes exhibit a maximum close to Xc

-3. The maximum approaches Xc as the concentration of fissile material

is decreased until Z^ » Z|.

4. When Z^ = Z£, the function K(X) -* «• as X -*■ X . 5. When Z, # Z * , the function K ( X ) -*■ 0 as X -*■ X .

r r c

6. For Zj < Z|, the maximum In K(X) moves away from the point X = Xc .

In Fig. 4 the function K(X) has been shown for values above and

below Z£. The variation of the point X at which K(X) is a maximum may be

P

considered as kind of a "continuation" of the discrete decay constant X Q for If < Z*.

Using these results, we can estimate the asymptotic form of the

response R(t) for large times. For a fuel concentration corresponding

to Z,, let k be the solution of

(14)

Fig. 4. The continuum amplitude function K(X) versus X.

w

0,

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Then ‘the Taylor series expansion of about k

gives k (i.e., about X - X ) p p 3DR ” »<'> ' <* - V J T K“ K P (28)

and the amplitude function assumes the following approximate Lorentzian

form: K (k) a.

W

w

r(Kp)

(.< -

<

)2

+ rz(< )

p p (29) where r <*p>

W

1

(30)

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can be identified as the "half-width" of the Lorentzian. Using this form, one can find that the asymptotic form for the detector response is

where X, = A + * , £ , .

1 c 1 1

This asymptotic behavior of R(t) clearly indicates that the long-time behavior of the flux in a subcritical fast assembly with £j < £* will indeed be nonexponential in nature.

In the above analysis the idea of a "continuation" of the discrete eigenvalue into the continuous spectrum for < 1* or B 2 > B*2 has been developed. D o m i n g , Nicolaenko, and Thurber 5 used a more elegant adaptation of the idea by analytically continuing the dispersion law D(B,s) to calculate the position of the amplitude peak in K(A). We shall

apply a very similar technique to our inelastic scattering model. First

note that the analytic continuation of the dispersion law D(B,s) across the cut Tj can be easily constructed by first rewriting D(B,s) in terms of Cauchy integrals as

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X c

(32)

Then note that the analytic continuation across the cut is just

D~(B,s) = 1 - i(i(s) -- (33)

c

Here

X = vE (34a)

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PULSED-NEITCRON EXPERIMENTS 'P(s) 1 v'xc (v')/hc (v') s + v 'Z2 V* y v"g (v")/h (v") X dv [ 5 (v - Vj^hçCv) + g„(v)[ exp l. in dV" s + v"E v i V 2 +

In fact, one can expand D~(B,s) about the branch point s = -X by using

a Puiseaux expansion 5 D ± (B,s) 1 - ! t ( s ) - P , ( B , - X c ) + ( s + Xc ) p J ( B , - X c ) + 6*(B) (s + Xc )2 in(s + Xc) ± 2Trie*(B) (s + Xc)2 . (35) Here 0 *(B ) vZf A f Y " 7 XC exp (36) + +

Now for < E*, the roots s of D (B,s) = 0 are very close to -Xc * This

allows one to use an iteration scheme to solve Eq. (35) for Sq. Hence the asymptotic estimate of the detector response becomes

R(t)

' f

X, -Xt

1 „ e

dA ~/ r . 4 - \ '2_{T x + s p s + s'};=F7 + 0(e X2t) ,

which for large times yields

(37)

R(t) ^ t 1 exp(-Xc t) [1 - exp(-XJt + Xc t)] (38)

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TABLE 1

Number Densities of 2 35U

Parameter s • 0 s = c

«;

0.16163 x 1023 N* 0.13514 x io2: X 0.73215 x IQ7 0.67196 x io7 C

IV. SOME NUMERICAL STUDIES

It is useful to adopt a somewhat more quantitative approach by

actually calculating the time decay constants and time response predicted

by the modeled theory for a typical subcritical system (although it would

certainly be unreasonable to expect our analytical analysis to yield any

more than a qualitative agreement with the observed time response).

We idealize the system as a 20-in.-diameter sphere of pure iron

moderator with a small amount of 235U as the fissionable material. A

truncated speed range will correspond to 17 keV < E < 10 MeV, and the

inelastic scattering threshold energy for iron is E^ = 0.86 MeV. The

cross sections are to be taken as constants (actually the average of the

Yiftah-Okrent-Moldauer set [9] over a fission spectrum).

The number densities of 235U, N^ and N*, for which the discrete

roots of D(B,s) are 6 « X^ = 0 and s « X^ = - *c » respectively, are given

in table 1. The decay constant -Xc < Xg < 0 for N* < N^ < N® is determined,

and Fig. 5 shows a plot of X^ versus N^..

In the case of N^ < N*, one should search for the roots of Eq.- (32).

Since <Ks) is analytic along s t [-X,-Xc ], it is reasonable to assume that

i|i(s) is a smoothly varying function of s for Re{s} e [ — X ,—X ]. Hence as a

first approximation we replace i|i(s) by *(sR ), and to first order in

|sI/(sR -Xc)jEq. (32) is approximated by two real equations.

- »(Sr)

1 9(B,X)-0(B,s_)

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PULSED-MEUI&ON EXPERIMENTS Fig. 5. s I 2tt9(B,sr ) 6 (B,X)-9(B,s )-1

» - •»>*

(40) st 0 = - ( S R + i s p . (41)

Table 2 gives the solutions of Eci. (36) for several < N*, and Fig. 6

shows the bifurcation of the discrete root.

The time response of the assembly using these analytically continued

roots and the asymptotic estimate

R(t) ^ R 0 dx c X e~Xt (X + s R )2 + Sj.2 (42) is shown in F i g . 7.

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ADALIOGLU AND DUDERSTADT ' l m ( s ^ x105 1.0

\

0.9

V

0.8

'

r

x107 R e ( s ) *

Fig. o. Bifurcation of the discrete eigenvalue when < N^.

ta ble 2 N f 0.1343 x 1023 0.1345 x 102 3 0.1347 x 1023 0.1350 x 1023 A c 0.67008 x 107 0.67053 x 107 0.67098 x 107 0.67166 x io7 X1 0.47701 x i09 0.47733 x 10® 0.47765 x 108 0.47814 x io8 SR 0.81065 x 107 0.80994 x 107 0.80705 x 107 0.80577 x 107 Si 0.98545 x 10s 0.96861 x 1 0 s 0.92280 x 1 0 5 0.89557 x 105

These calculations reveal a pseudo-exponential decay for those situa tions in which E^ < E*. Indeed, at first glance the time response appears

to be exponential. However, closer examination indicates a nonexponential

curvature to the decay curves. Because of the fact that Eqs. (39) tnrough (40) are only valid for E^ £ E^, only a limited range of fissile-material

concentrations could be investigated. These results agree qualitatively

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V. CONCLUDING REMARKS

In this study w e have attempted to formulate for neutron-pulse decays

in fast subcritical assemblies a simple model that explicitly takes into

account inelastic scattering. Using a single-diffusion-mode treatment of

the spatial dependence and modeling the inelastic scattering by a simple

separable kernel, w e have been able to verify the major conclusions of the

m o r e general spectral analysis of the transport operator for fast neutrons.

It w as felt advisable to modify the model somewhat in order to eliminate

any dependence on low-energy neutrons. This w e accomplished by Using a

truncated speed range, along w i t h truncated fission and inelastic scattering

spectra. The predictions of such a model seem to b e in reasonable agree

ment w i t h b oth the m ore formal theory and experimental measurements.

REFERENCES

1. U. Adalioglu and J. Duderstadt, T ransport Theory and Statistical

P h y s i c s . 2, 275 (1S72).

2. F. Storrer, in Pulsed Neutron R e s e a r c h , Vol. II (International Atomic

Energy Agency, Vienna, 1965).

3. M. Stievenart, in Pulsed Neutron R e s e a r c h , Vol. II (International

Atomic Energy Agency, Vienna, 1965).

4. M. Cadilhac, P. Govaerts, P. Hammer, M. Moore, and B. Nicolaneko, in

Pulsed Neutron R e s e a r c h , Vol. II (International Atomic Energy

Agency, Vienna, 1965).

5. J. J. D o m i n g , B. Nicolaenko, and J. K. Thurber, Trans. Am. Nucl.

S o c . . 12, 251 (1969).

6. J. J. D o m i n g , B. Nicolaneko, and J. K. Thurber, Trans. Am. Nucl.

S o c . , 12, 656 (1969).

7. J. J. D o m i n g , B. Nicolaneko, and J. K. Thurber, Trans. Am. N ucl.

S o c . . 12, 657 (1969).

8. M.M.R. Williams, The Slowing Down and Thermallzation of Neutrons

(North-Holland, Amsterdam, 1966).

9. Yiftah, D. Okrent, and P. Moldauer, Fast Reactor Cross Sections