Ann. Nucl. Eneryy, Vol. 20, No. 7, pp. 449-453, 1993 0306-4549/93 $6.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1993 Pergamon Press Ltd

E V A L U A T I O N OF F U N D A M E N T A L DECAY CONSTANTS BY THE S. M E T H O D

ANURADHA SHARMA and D. C. SAHNI

Theoretical Physics Division, Fifth Floor, Central Complex, Bhabha Atomic Research Centre, Bombay 400085, India

(Received 11 Auyust 1992; in revised form 1 December 1992)

Abstract--Fundamental time eigenvalues of the one-speed isotropic scattering neutron transport equation are computed for homogeneous spheres of different radii. Two methods, the integral transform method and the conventional S, method, are employed. It is found that the S, method gives very good agreement even above the Corngold limit, despite the negative effective total cross-section.

I. INTRODUCTION

In many applications, e.g. in the analysis of pulsed neutron experiments, one has to evaluate the fun- damental time eigenvalue of the neutron transport equation. This eigenvalue, or the decay constant, is defined as the smallest number 20 with a time behav- iour of the form exp ( - 2 0 0 for the neutron distri- bution. The corresponding angular flux is positive everywhere and for all velocities. Because of the geo- metrical complexity of the experimental setup and the energy dependence of the relevant cross-sections, one has to use well-known numerical methods, like the S, method, for solving the transport equation. Evalu- ation of the eigenvalue 20 involves a series of ko~ calculations, adding an estimated time absorption cross-section (-2/Vg) to both the absorption and the total cross-sections of the group g. The desired eigen- value 2o is then obtained by the requirement kerr = 1. If the decay is too fast (large value of 2), this may lead to a negative value of the total cross-section in some groups and in some regions. In the development of Sn codes, one usually demands positivity of all the fluxes. Moreover, one sweeps through the space angle mesh following the neutrons, calculations proceeding from left to right if the neutron direction cosine is positive and in the reverse direction if otherwise. This is done so that roundoffs etc. are also attenuated just like the neutrons. It is not clear if the negativity of the total cross-section causes any problems or interferes with the "negative flux fix up recipes" used in these general- purpose S, codes.

Carlvik (1968) reported very accurate values of the fundamental time eigenvalues for homogeneous slabs and spheres. Subsequently, the spectrum of a one- speed transport operator in homogeneous systems

with vacuum boundary conditions has been studied by many authors (e.g. Dahl et al., 1983). Sahni and Sjostrand (1990) have reviewed these computed results quite extensively. Most of these accurate results have been obtained using Carlvik's method or its close analog, the integral transform method (Hembd, 1970). However, we have not found any comparison of these benchmark results with conventional methods, like the S, method, of solving the transport equation. In this paper we present such a comparison for a homo- geneous, isotropically scattering sphere, varying the size of the sphere over a wide range. Our aim is to see if the eigenvalues, as well as the eigenvectors, are predicted accurately by the S, method.

2. THEORY

2.1. The S, method

We consider the source-free, time-dependent, one- speed, neutron transport equation with isotropic scattering in a homogeneous sphere of radius "a". Thus, we have (with the usual notation) :

1 ~u? v ~ - + (~" V)~(r, ~ , t)

+ a(r)~F(r, [~, t) = 4~ 1 ~P(r, fl ' , t) dfl ' ;

(1)

along with the vacuum boundary condition at the surface of the sphere,

~F(a, ~ , t) = 0, ~ - a < 0. (2)

For convenience, we use spherical polar coordinates and take the origin at the centre of the sphere. Thus,

449

450 ANURADHA SHARMA and D. C. SAI4NI

the position vector a is on the surface. Since we are interested in the fundamental mode of this equation, we have

qJ(r, ~ , t) ---- exp (--).ot)~(r, ~" r/r)

= exp(--).ot)~k(r, tt), (3)

where ~(r, tt) is positive for 0~<r~<a and - 1 ~< # ~< 1 and satisfies the equation

c3~b 1 _#z (3~b tt~r + - - + ( a - ).o/V)t,k(r, it) r Oft

~O(r,/~') d/t' (4)

and the boundary condition

~k(a, t t ) = 0 , --1 ~<#<0 . (5)

In order to solve this time-independent equation by the Sn method, we consider the following ke~ (fun- damental) eigenvalue problem :

c94b (1 _#2) ~3~b /ZOrr + - - r 0/~ +ZqS(r,/1)

I' = 1 (~ +vYdko.) J- , q~(r, ~,') d . ' ; (6)

with the same boundary and positivity conditions. The total cross-sections 2 is varied while the scattering and fission cross sections, E~ and vEf, are defined so that

E = cr-).iv, Z~ + vEf = G, (7)

i.e. we split the scattering cross-section G into two parts Z~ and vZf. The time absorption ().0/v) is obtained from E and a when k~ = 1. Note that we have to explore negative values of E if ). > va. In order to see how it affects the S. solution, we briefly recapitulate the theory of the S. method as given by Lathrop (1965). In the S. method (code DTF IV), equation (6) is replaced by the neutron balance equa- tion over a discrete space angle (i, m) mesh, where re(ri, r~+ l) and tt belongs to an interval of weight co,.. This balance equation reads

{corn/t,.(Ai+ ,Ni+ I,m-- AiNi,,.) + [(G.+ I/z,/+ il2Ni+ ,lZ~+ 112

-- ~,.- i/z/+ ll2Ni+ 112,,.- I/2)] + ~-~COm Vi+ ll2Ni+ l12.m}

= comVi+l/2(]~s+V~f/keff)~Ne+u2,m,. (8)

Here Ai+ 1 and Ai are the areas of the surface of radii r~+ ~ and r~, V~+ l/2 is the volume of this spherical shell and/z~ is a representative direction of motion in the interval of weight COm. The ~ coefficients account for the angular derivative terms in equation (6) and hence 0~1/2 and ~u+ ~/2 both vanish. Equation (8) is

solved recUrsively in conjunction with the diamond difference relations

Ni+ l,m + Ni,,. = Ni+ l/2,m+ l/2

+Ni+~/2,,,-~/2 = 2N~+~12,,,. (9)

Since the two fluxes say, Ni, m and Ni+ ,/2,m-- ~/2, are known from either the boundary conditions or the previous calculations in a neighbouring cell, we have

N,+ ,/Z,m = [ I~.,I(A,+, +A,)N~,,. + (~,.+ 1/2,~+ l/Z

+~,. ,12.,-I/z)(ltcom)Ni+ ,12.,.-llZ

-F Vi+ l/2(Es + vErlkefr)

× ~ N,+,n,m,]t[l~ml(A,+, +At) m ' = 1

-1- (0~m+ I/Z,I+ 1/2 + am- 112,1 1/2)

+ZV,+ ,/2]- (lO)

IfZ is negative there is a possibility that the denomi- nator becomes negative. This will lead to negative values for N~+ 1/2,,. if the numerator is positive. The extrapolated fluxes Ni+ l/z,m+ 112 and N~+ 1,,, [use of difference relation (9)] will also be negative and call for negative flux fixup recipes. We are interested whether it really happens when Z is large and negative, i.e. while computing 2 eigenvalues for small assembhes.

We have used the standard program DTF IV for this purpose (Lathrop, 1965) and the well-known quad- rature sets such as P,, DP, and S, up to the 16nth order. The cross-sections cr and G were both taken to be unity, while the radius "a" was varied over a wide range from 0.007 to 5 m.f.p. The specific values were taken from the paper by Sahni et al. (1991) based on the integral transform method. We used 10 spatial intervals, equally spaced, for the discretization of the r variable.

2.2. The integral transform (IT) method

We wish to compare the results of the S, method with those given by a very accurate method--the IT method. As mentioned in the Introduction, Carlvik's (1968) method yields very accurate criticality eigen- values. This method also solves the criticality (time- independent) problem. One expands the total flux in Legendre polynomials of spatial coordinate r/a. The method is applicable for both positive and negative (even complex) values of E. Its close analog, the IT method as originally developed by Hembd (1970) and others, is limited to the case when Re E > 0. It has been extended recently to the region Re Z < 0 by Sahni et al. (1991). Essentially one starts from the

Fundamental time eigenvalues by the S, method 451

integral transport equation, which is equivalent to equation (6) and its boundary condition, namely

cE ('a ('tr +r'l ds rp(r) = 2-J0/ r'p(r')dr' dlr-,'l | exp (--sZ) s ' (11)

The lowest criticality eigenvalue "c" determines the scattering cross-section as (equal to cZ) for which equation (11) admits a positive solution. The total flux p(r) is related to the angular flux ~b(r, #), defined in the last section, by the relation

f_ p(r) = ~b(r,/t) d#. (12) I

Using the representation of Gradshteyn and Ryzhik (1965),

I ' '+ ' ' as 4fo~ as exp ( - sE) = ~ exp ( - sE) - - dlr-r'l S S

x fo ° dk sin sk sin kr sin kr" ~7 , (13)

where "b" is any number greater than "2a", we derive the integral equation :

F(k) = - (E~ + vY~f sin kr sin k'r drF(k')

dk'f k" ,Jo exp ( - sz) dS sin × (14)

The function F(k) is the Fourier transform of the total flux :

f: F(k) = rp(r) sin kr dr. (15)

One expands the transform F(k) in the series

F(k) = ~22 ,=o ~ ~ U g J(z3+ 3~c2)(ak) ' (16)

which corresponds to the series expansion

1 ~ rp(r) = ~aag~=o(--)gx/4y+ 3UaP2o+,(r/a ) (17)

for the total flux. Substituting equation (16) in equa- tion (14), one obtains a homogeneous matrix equation for the coefficients Ug. The matrix elements are evalu- ated analytically. The largest eigenvalue of this matrix equation yields the criticality eigenvalue Es + vZr for given values of Y. and "a" and the corresponding eigenvector Ug. The method is very fast and accurate, as very few terms are needed in expansion (16) which converges rapidly. Equation (7) then determines the

desired 2o. Using equation (17) the averages of the form indicated by equation (18) can be computed easily. We normalize the flux to a unit fission source, as is done in DTF lV. Thus, the coefficient Uo is chosen as (x/3/47t)(1/(a3/2vZf). Further, the DTF IV output gives the eigenvalue 20 that corresponds to k~fr = 1 and the cell average flux ~b in the ith mesh (ri, ri+ 0. We interpret this quantity as

fi'+'r2dp(r,l~)dr

= e ( 1 8 ) ir i+ I r 2 dr dr

to compare the results with those given by the IT method. These averages can be computed easily.

3. RESULTS, DISCUSSION AND CONCLUSIONS

The present problem involves the evaluation of the fundamental time eigenvalue in a sphere, a bounded geometry. Thus, as shown by Jorgens (1958), the eigenvalue 2o and a corresponding eigenvector which is positive for all space points and all directions do exist. Thus, the Sn method should work, at least when the space angle mesh is sufficiently refined (high-order S,). However, the intermediate eigenvectors, where we add different values of the time absorption 2Iv and find kcfr and a corresponding (positive or at least non- negative) eigenvector, may not exist for a sufficiently broad range of 2Iv. Indeed, we find this situation to be true. For very small assemblies we find that such attempts usually fail unless we are able to guess the desired value of 2o in a very narrow interval. Within this narrow range of 2, ke, is a very rapidly varying function of 2. Moreover, in low-order S, calculations, the computed value of 20 is significantly different from the true decay constant as found by the IT method. Thus, the range of 2 values is not known, even if the exact value of 20 is known.

The results of our calculations are presented in Tables 1-5. Table 1 gives the values of the fundamental decay constant 2o in units of scattering mean-free- path 1/E~ for different sphere radii. We take the neu- tron speed v = 1 and the absorption cross-section to be zero. For computations with the S, method we have used $4, S16 , P3, P~5, DP~ and DP 7 quadrature sets and compared their results with those given by IT 7 and IT l0 approximations. These correspond to retaining 7 and 10 terms in expansions (16) or (17). It can be seen that the results of IT 7 and IT~o are identical to the given number of digits. Moreover, we have checked the accuracy of the ITs0 method with Carlvik's (1968) results to the given number of digits.

452 ANURADHA SHARMA and D. C. SAHNI

Tablel . Fundamenta ld~ayconstant20~ranas~mblyofradius(aZ0

aE $4 $16 P3 Pi5 DPI DP7 IT7 ITlo

0.007 614.41 697.29 622.04 701.91 647.05 702.27 716.0 716.0 0.0302 118.23 130.81 119.29 131.59 122.99 131.69 133.3 133.3 0.1097 25.76 27.90 25.92 28.05 26.52 28.07 28.35 28.35 0.1919 12.89 13.83 12.96 13.90 13.22 13.91 14.03 14.03 0.4853 3.835 4.048 3.850 4.064 3.904 4.068 4.091 4.091 0.9703 1.444 1.504 1.448 1.508 1.462 1.510 1.515 1.515 1.618 0.6673 0.6880 0.6691 0.6893 0.6726 0.6898 0.6911 0.6911 1.988 0.4823 0.4952 0.4835 0.4959 0.4853 0.4962 0.4971 0.4971 4.100 0.1443 0.1463 0.1446 0.1462 0.1443 0.1463 0.1464 0.1464 5.014 0.1014 0.1024 0.1016 0.1024 0.1012 0.1024 0.1025 0.1025

Table 2. Flux profile ( x 104) for an assembly of radius 0.0302 m.f.p.

S. No. x~ = ri/a x,+~ = r~+ i/a $4 Sj6 DP l DP7 ITjo

1 0.00 0.10 1.24109 1.70342 1.42895 1.72579 1.84508 2 0.10 0.20 1.36005 1.85092 1.57044 1.86268 1.90245 3 0.20 0.30 1.50784 1.98528 1.69898 1.98852 2.01796 4 0.30 0.40 1.73956 2.18016 1.88115 2.17897 2.20111 5 0.40 0.50 2.05440 2.44848 2.14658 2.44637 2.46524 6 0.50 0.60 2.46493 2.81745 2.53757 2.81493 2.82962 7 0.60 0.70 3.01430 3.31609 3.09211 3.31225 3.32165 8 0.70 0.80 3.78614 3.98514 3.86175 3.97978 3.97904 9 0.80 0.90 4.91453 4.87698 4.92925 4.87410 4.85330

10 0.90 1.00 6.60956 6.05900 6.43015 6.06732 6.01268

Table 3. Flux profile ( x 10 z) for an assembly of radius 0.1919 m.f.p.

S. No. xi = r~/a x~+ i = ri+ l/a $4 S~6 DP~ D P 7 ITs0

1 0.00 0.10 1.16563 1.34718 1.23940 1.35269 1.42855 2 0.10 0.20 1.22111 1.40712 1.30271 1.40614 1.43692 3 0.20 0.30 1.26241 1.43614 1.33587 1.43077 1.45308 4 0.30 0.40 1.32101 1.46914 1.37115 1.46247 1.47769 5 0.40 0.50 1.39337 1.50551 1.41350 1.50050 1.51136 6 0.50 0.60 1.47560 1.55037 1.47372 1.54756 1.55457 7 0.60 0.70 1.56497 1.60539 1.55569 1.60424 1.60814 8 0.70 0.80 1.66368 1.67192 1.66005 1.67154 1.67260 9 0.80 0.90 1.77926 1.75119 1.78136 1.75187 1.74795

10 0.90 1.00 1.92340 1.83356 1.91005 1.83716 1.83016

Table 4. Flux profile for an assembly of radius 0.9703 m.f.p.

S. No. x~ = rda Xi+l = r,+ ~/a S 4 816 D P t OPT ITs0

1 0.00 0.10 1.77400 1.86669 1.80916 1.86058 1.93180 2 0.10 0.20 1.79332 1.88653 1.83142 1.87597 1.90902 3 0.20 0.30 1.76783 1.85326 1.80177 1.84129 1.86426 4 0.30 0.40 1.72463 1.79414 1.74663 1.78362 1.79797 5 0.40 0.50 1.66251 1.70956 1.66784 1.70284 1.71133 6 0.50 0.60 1.58161 1.60424 1.57251 1.60121 1.60508 7 0.60 0.70 1.47942 1.48086 1.46268 1.48066 1.48062 8 0.70 0.80 1.35249 1.33900 1.34084 1.34050 1.33851 9 0.80 0.90 1.19691 1.18291 1.20235 1.18479 1.17814

10 0.90 1.00 1.00890 1.99141 1.01500 1.99417 0.99046

We, therefore, take them as reference values. We see from Table 1 that the S~6 approximation is quite accurate with error of the order of 1/2%, except for very small radii when it is around 2%. The $4 approxi- mation is relatively inaccurate, with error ranging from 1 to 10%. We also see that the DP, quadrature

coefficients yield somewhat better results, while the S, and P, sets are quite similar.

In Tables 2, 3, 4 and 5 we compare the flux profiles as given by different approximations for assemblies of radii 0.0302, 0.1919, 0.9703 and 1.988 m.f.p., respec- tively. We see from these tables that the St6 approxi-

Fundamental time eigenvalues by the S, method

Table 5. Flux profile ( x 10-1) for an assembly of radius 1.988 m.f.p.

453

S. No. xj = ri/a xi+ 1 = r~+ i/a S, S~6 DPj DP 7 IT10

1 0.00 0.10 2.67234 2.74896 2.69759 2.73476 2.28203 2 0.10 0.20 2.66525 2.74171 2.69149 2.72451 2.76816 3 0.20 0.30 2.58367 2.65279 2.60616 2.63648 2.66676 4 0.30 0.40 2.45764 2.51223 2.47109 2.49941 2.51851 5 0.40 0.50 2.28904 2.32470 2.29060 2.31704 2.32815 6 0.50 0.60 2.08352 2.09870 2.07487 2.09554 2.10040 7 0.60 0.70 1.84482 1.84260 1.82937 1.84272 1.84180 8 0.70 0.80 1.57532 1.55920 1.56201 1.56151 1.55803 9 0.80 0.90 1.27342 1.26146 1.27949 1.26383 1.25350

10 0.90 1.00 0.92923 0.92223 0.93829 0.92501 0.92005

mat ion predicts flux profiles with an accuracy ranging from 2 to 10%, the larger error being for extremely small radii. The accuracy of the S , approximat ion ranges from 5 to 25%. One should not be surprised by the large errors for very small systems. It should be remembered that for such small systems the angu- lar flux varies very rapidly with spatial and direction coordinates. In fact, these cases appear to be outside the range of applicability of the S, method ; al though, the results show that even in these cases the method works reasonably well. Indeed, the broad features of the flux shapes, small at the centre and large at the edges, are predicted even by low-order S, approximation.

Acknowledgements--We wish to thank Drs V. Kumar and

S. D. Paranjape for help with the IT method calculations and many useful discussions.

REFERENCES

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Nuel. Sci. Engn 9 83, 374. Gradshteyn I. S. and Ryzhik I. M. (1965) Tables oflntegrals,

Series and Products, Sect. 3.763(2), p. 422. Academic Press, London.

Hembd H. (1970) Nucl. Sei. Engng 40, 224. Jorgens K. (1958) Communs Pure Appl. Math. II, 219. Lathrop K. D. (1965) Nucl. Sci. Engn 9 21,498. Sahni D. C. and Sjostrand N. G. (1990) Prog. Nucl. Energy

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