PN solutions of the time-dependent neutron transport equation with anisotropic scattering in a

homogeneous sphere

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J. Phys. D: Appl. Phys.33 (2000) 704–710. Printed in the UK PII: S0022-3727(00)08279-6

PN solutions of the time-dependentneutron transport equation withanisotropic scattering in ahomogeneous sphere

Cemal Yıldız

Department of Physics, Istanbul Technical University, 80626, Maslak, Istanbul, Turkey

Received 30 September 1999, in final form 24 December 1999

Abstract. In an earlier paper, the spherical harmonics method for the solution of thetime-dependent transport equation with the Marshak boundary conditions was presented inorder to investigate the effect of a strongly anisotropic scattering law on the slab thickness.Here, the previous work is extended to the study of the time-dependent problems in ahomogeneous sphere with the same scattering function as used in the previous work. Thetime-dependent neutron transport equation is solved in the manner used for a critical sphereand all radii for a given time-dependent system are determined by finding the critical radii forthe corresponding critical system.

ThePN calculations of the critical radii were carried out for various combinations of theanisotropy parameters and the fundamental time eigenvalues. Some indications of theaccuracy of the method were given for the problem of interest and the variation of the radiuswith anisotropic scattering was studied. We also obtained numerical values of the criticalradii in the range of(1−3) 6 α, β 6 1. Finally, some results were discussed and comparedwith those already obtained by various methods.

1. Introduction

In a previous work [1], we have shown how to use the classicalspherical harmonics method to solve the time-dependentneutron transport problem with anisotropic scattering. Thiswork will be extended here to solve time-dependent neutrontransport problems in a homogeneous sphere. The variousformulations of the spherical harmonics method (also calledthePN method) based on expanding the particles’ distributionfunction in Legendre polynomials are widely reported in[1–23]. The method was extensively developed and extendedto various one-dimensional geometries with various typesof boundary conditions [2, 8–13, 15–16, 22]. One of thedifficulties in the method is the formulation of the boundaryconditions, since the boundary conditions are imposed overonly half of the angular range. This introduces a degreeof arbitrariness into our choice of boundary conditions forthe PN equations. One possibility would be to choose anappropriate boundary condition for the considered problem.As mentioned in [1], the Marshak boundary conditions areagain imposed over half of the angular range for the problemcurrently under investigation. These types of approximateboundary conditions have already been proposed severaltimes [1, 2, 5, 15–18], especially for the numerical treatmentof anisotropic scattering. The most usual boundary conditionis that of a vacuum around the system, i.e. no incoming

particles. ThePN method with the Marshak boundaryconditions gives reasonable accuracy for sufficiently largeregions of the problem [2, 8, 13, 15–17] and converges rapidlywith very short computing times. It is generally believedthat the Marshak conditions will lead to better accuracyfor solutions whenN is low (N = 5 or 7, dependingon c = 1/(1 − 3) [3, 4, 15, 16, 17, 24]. We also believethat this procedure is adequate for the anisotropic scattering(a combination of linearly anisotropic and strongly forward-backward scattering) mentioned above, and was previouslyused in [1].

In this paper, we wish also to investigate the influenceof anisotropic scattering on the radius for a homogeneoussphere together with the accuracy of the present method.We compute the critical radii for various values of scatteringparameters. The connection between the eigenvalues arisingfrom the time-dependent and the stationary transport equationis used to transfer results from the critical to the time-dependent systems [7]. (Criticality-type eigenvalues areneeded for a variety of applications in reactor physics,for example in some pulsed neutrons experiments for thedescription of the phenomena. After a short neutron pulse,the neutron population decays with time. It is of interest toconsider the behaviour time of the angular flux in determiningthe desired population, criticality conditions and behaviourover later times.)

0022-3727/00/060704+07$30.00 © 2000 IOP Publishing Ltd

Time-dependent neutron transport equation

Many researchers [19–59] have studied the transportproblems for time-dependent systems using a variety ofmethods. In most cases the criticality and time eigenvaluesproblems have been considered with isotropic scattering.However sometimes linearly anisotropic scattering has alsobeen used [7, 17, 29–36]. Sahniet al [40] studied criticalityand the time eigenvalue spectrum in a homogenous slabwith strongly forward and backward scattering using theSNmethod. Similar work that is more closely related to thepresent work was performed by Sahni and Sjostrand [32–34].They computed the fundamental time constantλ (or rather3)in spheres with vacuum boundary conditions as a functionsof its size by the same method used in earlier calculations[40, 48].

2. The PN solutions

As described in [1, 2, 7, 28–48], there is formal equivalencebetween the transport equation for a critical system and fora system decaying over time. Since the basic aspects of thetransformations are so well known, our presentation here isbrief. The approach is the same as that used in [1], thenthe spherical harmonic expansion of the flux9(r, µ) can bewritten as [2, 31–34, 37–40]

9(r, µ) =N∑n=0

(2n + 1

2

)8n(r)Pn(µ). (1)

The angular flux9(r, µ) is positive for 06 r 6 R0 and−1 6 µ 6 1, µ is the cosine of the angle between theradial direction and the direction of neutron motion. TheradiusR0 is measured in mean free paths (mfp). Followingour previous work, with the spherical harmonics method,we express our approximatePN solution to the transportequation for one-speed neutrons in a homogenous, criticalsphere [1, 2, 5, 16, 21, 22] as

9(r, µ) =N∑n=0

2n + 1

2Pn(µ)

(N+1)/2∑j=1

AjGn(νj )

×[Kn+1/2(−r/νj ) + (−1)nKn+1/2(r/νj )]/√r (2)

where the following boundary and symmetry conditions atthe surface can be imposed:

9(R0, µ) = 0 9(r, µ) = −9(−r,−µ)−16 µ < 0. (3)

The arbitrary coefficientsAj are to be determined from theboundary conditions (3). In (2)Kn+1/2(r/ν) is the modifiedBessel function of the second kind andGn(ν) is the standardpolynomials satisfying the recursion relation

(n + 1)Gn+1(ν) + nGn−1(ν)

−(2n + 1){1− c(1− α − β)(δn0 + b1δn1)

−c[α + (−1)nβ]}Gn(ν) = 0 (4)

whereG−1(ν) = 0 andG0(ν) = 1 with Gn(−ν) =(−)nGn(ν). In thePN approximation, equation (4) is closedby the condition

GN+1(ν) = 0. (5)

Equation (5) leads the same eigenvaluesνj and eigenfunc-tionsGN+1(νj ) as in the plane case. Clearly equations (4)

and (5) give the well known eigenvalue problem for one-speed neutron transport. It can be solved in a number ofways [2, 8, 14, 15, 60]. The procedure, in this work, forsolving equation (5) is similar to that used in plane geom-etry. We have used a FORTRAN program (double pre-cision) in the IBM program package (subroutine POLRT,using the Newton–Raphson iterative technique, limited to36th order polynomials or less and able to determine rootwith 500 iterations on five starting values) [60]. For thispackage, a recursion formula involving only even poly-nomials was derived from equation (4) and used to in-put coefficients into package program. The recursion for-mula constitutes a fast and particularly accurate procedurefor computing the polynomialsGN+1(ν) and ν ∈ (νj ).The computation time is extremely short on Pentium ma-chine running at 166 MHz (for the results in table 2,the time required to compute the eigenvaluesνj is about3 s).

Finally, as the representation of9(r, µ) is inconsis-tent with the boundary conditions (3), we use the Mar-shak approximations [1, 2] to define the boundary condi-tions.

For the Marshak approximations, we multiply equa-tion (2) byP2k−1(µ) and integrate overµ from zero to oneto obtain the system of linear algebraic equations

(N+1)/2∑j=1

Aj

{ (N−1)/2∑n=0

Q(n, k)G2n(νj )Fj,2n(R0)

+G2k−1(νj )Fj,2k−1(T0)

}= 0 (6)

for k = 1, 2, . . . (N + 1)/2 and the(N + 1)/2 unknowncoefficientsAj . Note that hereN is an odd number andis some particular value ofk. The constants are:

Fj,2n =(νj

r

)2n{ n−1∑m=0

[(r

νj

)2n−2m

B2n,2m sinh

(r

νj

)

−(r

νj

)2n−2m−1

B2n,2m+1 cosh

(r

νj

)]+B2n,2m sinh

(r

νj

)}(7a)

Fj,2k−1 =(νj

r

)2k−1[ k−1∑m=0

(r

νj

)2k−1−2m

B2k−1,2m

× cosh

(r

νj

)−(r

νj

)2k−2m−2

B2k−1,2m+1 sinh

(r

νj

)](7b)

wheren = 0, 1, 2, . . . (N − 1)/2; k, j = 1, 2, . . . (N + 1)/2with

Q(n, k) = (4n + 1)(−1)n+k(2k − 1)!(2n)!

22k+2n−1(2n− 2k + 1)(k + n)[n!(k − 1)!] 2(8)

and

Bi,j = (i + j)!

j !(i − j)!(

1

2

)jB0,1 = 0. (9)

Clearly, the problem under consideration is now reducedto a linear system of algebraic equations (6) that is easily

705

C Yıldız

Table 1. The radiusR06∗s obtained inP3 andP7 approximations with different values of the decay constant3 and two values of the

anisotropy coefficientb1. It is assumed thatα = β = 0. Comparison with results of Carlvik [28], Dahlet al [37], Dahl and Sjostrand [38]and Sharma and Sahni [35].

SharmaDahl and and

This work This work Carlvik Dahlet al Sjostrand Sahnib1 3 P3 P7 [28] [37] [38] [35]

0.0 0.222 88 3.218 03 3.217 45 3.217 020.370 01 2.381 69 2.381 23 2.380 970.529 40 1.912 96 1.912 62 1.912 440.645 07 1.690 89 1.690 62 1.690 470.858 18 1.410 21 1.409 94 1.409 930.984 50 1.290 97 1.290 42 1.290 17

0.0 0.018 9229 12.501 33 12.500 065 — 12.5 —0.028 8448 10.001 29 10.000 64 — 10.0 —0.049 2855 7.501 26 7.500 62 — 7.5 —0.102 978 5.001 20 5.000 56 5.03.341 216 2.500 78 2.500 30 2.5

0.2 0.0330 10.340 01 10.339 16 — — 10.3410.1020 5.568 26 5.567 39 — — 5.56750.3124 2.910 46 2.909 65 — — 2.9090.62 97 1.891 36 1.890 87 1.89050.8700 1.538 57 1.538 05 1.538

0.0 0.1024 5.016 97 5.016 33 5.0140.4952 1.993 60 1.993 24 1.9880.6880 1.623 56 1.623 31 1.618

Table 2. The radiusR06∗s obtained in variousPN approximations as a function of the backward scatteringβ for three different degrees of

the linearly anisotropic scatteringb1, assumingα = 0 and the decay constant3 = 0.5.

β b1 P1 P3 P5 P7 P9 P11

0.00 −0.3 1.944 51 1.768 18 1.768 28 1.768 06 1.768 00 1.730 210.0 1.813 80 1.982 22 1.981 77 1.981 42 1.981 30 2.006 990.3 2.576 73 2.319 97 2.316 96 2.316 11 2.315 84 —

0.25 −0.3 1.835 68 1.651 02 1.650 21 1.650 12 1.650 11 —0.0 1.979 58 1.771 08 1.770 55 1.77 0 42 1.770 39 1.755 190.3 1.770 09 1.927 15 1.926 95 1.926 76 1.926 71 1.924 65

0.45 −0.3 1.761 25 1.565 96 1.561 91 1.561 77 1.561 43 1.561 370.0 1.850 05 1.636 97 1.632 81 1.632 64 1.630 04 —0.3 1.954 82 1.720 25 1.715 99 1.715 79 1.709 83 1.700 27

0.50 −0.3 1.744 08 1.553 55 1.540 34 1.539 74 1.500 03 1.500 220.0 1.821 65 1.607 86 1.601 25 1.600 60 1.605 56 1.600 730.3 1.911 34 1.681 21 1.671 20 1.670 48 1.667 34 1.644 48

0.75 −0.3 1.665 56 1.445 46 1.423 62 1.421 30 1.421 15 1.400 260.0 1.697 82 1.468 78 1.445 32 1.442 63 1.442 43 1.440 440.3 1.732 14 1.493 38 1.468 08 1.464 92 1.464 67 1.462 87

0.91 −0.3 1.620 80 1.381 85 1.335 42 1.322 64 1.318 81 1.300 220.0 1.631 24 1.388 79 1.341 00 1.327 52 1.323 34 1.302 910.3 1.641 90 1.395 84 1.346 64 1.332 39 1.327 84 1.305 15

0.95 −0.3 1.610 19 1.365 82 1.310 77 1.292 36 1.285 06 1.267 060.0 1.615 84 1.369 49 1.313 55 1.294 61 1.287 00 1.300 490.3 1.621 56 1.373 18 1.316 35 1.296 86 1.288 73 —

0.99 −0.3 1.599 79 1.349 71 1.285 08 1.259 66 1.247 30 1.248 450.0 1.600 90 1.350 40 1.285 58 1.260 02 1.247 58 —0.3 1.602 00 1.351 10 1.286 07 1.260 39 1.247 86 —

1.00 −0.3 1.597 23 1.345 67 1.278 51 1.251 16 1.237 34 1.224 230.0 1.597 23 1.345 67 1.278 51 1.251 16 1.237 34 1.224 230.3 1.597 23 1.345 67 1.278 51 1.251 16 1.237 34 1.224 23

solved by the standard method. Certainly, for the non-trivialsolution, the determinant of equation (6) must vanish, whichis the criticality condition for critical radiusR06t , and relatesthe critical dimensions to the parameterc (or3), vice versa.We note that, in writing equation (2), we have consideredthe same neutronic notation, consistent with earlier papers[1, 32–36, 40]. Then, for a certain critical sphere with

radiusR06t we can obtain the corresponding3 value andradius R06

∗s for the time-dependent case (as defined in

previous work, obviouslyc is the number of secondaryneutrons per collisions and the generalized decay constant3 = λ/(υ6∗s )−6∗a/6∗s = 1− 1/c with 6t = 6∗s (1−3),6t is the total macroscopic cross section and6∗s denotes thescattering cross section in a time-dependent system).

706

Time-dependent neutron transport equation

Table 3. The radiusR06∗s as a function of the anisotropy parameterα andβ for three different degrees of the linearly anisotropic scattering

b1, as calculated with theP7 approximation and assuming the decay constant3 = 0.1. Eitherα or β is fixed and the other allowed to vary.

α or β b1 α = 0.10 α = 0.30 α = 65 β = 0.00 β = 0.15 β = 0.50 β = 0.65

0.00 −0.3 4.720 74 5.270 80 7.068 49 4.505 19 4.333 84 3.982 08 3.837 360.0 5.324 46 5.939 30 7.942 37 5.083 26 4.752 06 4.160 24 3.946 760.3 6.263 62 6.978 99 9.304 59 5.982 74 5.328 75 4.366 10 4.066 40

0.05 −0.3 4.652 29 5.172 38 6.802 14 4.609 04 4.424 90 4.046 74 3.888 470.0 5.188 63 5.746 16 7.441 26 5.199 49 4.844 88 4.215 07 3.985 970.3 5.983 49 6.585 51 8.329 07 6.118 11 5.421 24 4.407 60 4.091 66

0.10 −0.3 4.586 22 5.078 08 6.552 82 4.720 74 4.522 09 4.113 64 3.938 310.0 5.062 16 5.679 47 7.022 30 5.324 46 4.943 44 4.270 84 4.022 830.3 5.737 84 6.251 40 7.597 11 6.263 62 5.518 68 4.448 71 4.112 97

0.20 −0.3 4.460 24 4.899 58 6.074 34 4.972 13 4.737 54 4.252 66 4.023 050.0 4.832 72 5.255 24 6.281 10 5.605 57 5.160 30 4.382 93 4.076 390.3 5.324 22 5.707 95 6.513 02 6.590 81 5.729 74 4.526 80 4.132 06

0.30 −0.3 4.340 93 4.730 53 5.504 45 5.270 80 4.987 87 4.391 09 4.015 830.0 4.628 39 4.979 74 5.549 02 5.939 30 5.407 68 4.486 41 4.031 620.3 4.985 56 5.276 03 5.594 55 6.978 99 5.964 93 4.588 70 4.047 55

0.35 −0.3 4.283 16 4.647 74 4.991 11 5.442 97 5.128 82 4.452 00 3.841 410.0 4.533 67 4.852 13 4.991 11 6.131 56 5.545 21 4.526 01 3.841 410.3 4.836 86 5.088 17 4.991 11 7.202 50 6.092 61 4.604 11 3.841 41

0.45 −0.3 4.169 91 4.480 03 5.848 24 5.449 67 4.482 540.0 4.355 61 4.608 02 6.583 76 5.851 90 4.505 130.3 4.570 13 4.748 50 7.727 95 6.367 43 4.527 99

0.50 −0.3 4.113 64 4.391 09 6.090 33 5.634 49 4.298 310.0 4.270 84 4.486 41 6.853 66 6.026 21 4.298 310.3 4.448 71 4.588 70 8.041 38 6.520 93 4.298 31

0.60 −0.3 3.998 87 — 6.688 77 6.064 220.0 4.105 58 — 7.520 07 6.415 820.3 4.221 89 — 8.814 80 6.843 86

0.70 −0.3 3.873 19 3.720 08 7.527 31 6.585 840.0 3.936 78 3.720 08 8.452 06 6.858 080.3 4.003 82 3.720 08 9.895 42 7.171 67

0.85 −0.3 3.57933 9.865 58 4.601 360.0 3.590 22 11.039 45 4.601 360.3 3.601 19 12.890 59 4.601 36

1.00 −0.3 51.319 490.0 51.319 490.3 51.319 49

3. Numerical results and discussion

In this section, in order to examine the validity and accuracyof the present method a computer program in FORTRANwas written to calculate the critical radii. For a fixedvalues of 3, a series of calculations were performedfor various combinations of the scattering parametersα(forward), β(backward) andb1 (linear anisotropy factors)on a Pentium machine running at 166 MHz. The resultsof our calculations are presented in tables 1–4 as a functionscattering parametersα, β andb1. They are also shown infigures 1–3.

In table 1 we present a comparison with other works[28, 35, 37, 38] obtained by different methods. For thiscomparison, we therefore take the values given in thesepapers as reference values. The agreement is generallywithin three or four significant decimal places. Table 1also gives an estimate of the values ofR06

∗s for the PN

method. In table 2 we report the values of the radii asa functions ofβ for three different degrees ofb1 and forvarious orders of thePN approximations. One observesa significant improvement between theP1 and P3 and avery good approximation is already reached for theP7

approximation. Furthermore, theP7 andP9 results agreegenerally within two or three significant figures. In the

P7 approximation, the radius is converges to two or threesignificant figures for small or intermediate values ofβ(or for large radii, sayR06

∗s > 1.4 mfp). Note that in

approximations up toP9 some results become uncertain asthe radius decreases.

In table 3 and 4 we list our converged radii alongwith some poor results calculated for the extreme case,3 = 1− α ± β for various combinations ofα, β, 3 andb1. Unfortunately, no direct comparison has been possiblefor these results. However, it can be seen that our resultsfollow the same general trend as those of Sahni and Sjostrand[34, 48]. As we indicated by the values compared in table 1and from the results in table 2, we believe that ourP7 resultsin tables 3 and 4 are correct at least in the first three significantfigures for3 6 0.50 and for all values ofα, β andb1. Forintermediate or larger values of3 (say, 0.50 6 3 6 0.90)the results are still acceptable forN 6 11 for all cases.However, it is always possible to obtain accurate results eitherby changing3 or by changing the values ofα, β andb1. Thedashes (—) in tables 3 and 4 indicate divergent results for3 = 1− α ± β. One might think that, for certain valuesof the radii, there are the singular points. It was pointedout by Sahni and Sjostrand [33, 34] that these are singularpoints of the Inonu transformations [52]. For these casesa clear convergence cannot be seen. Nevertheless, we have

707

C Yıldız

Table 4. The radiusR06∗s as a function ofα andβ for various values of the decay constant3 and for three different degrees of the linearly

anisotropic scatteringb1, as calculated with theP7 approximation and assuming eitherβ = 0 orα = 0. For eachb1 value the upper row isfor pure forward scattering (β = 0) and the lower row is for pure backward scattering (α = 0).

α or β b1 3 = 0.01 3 = 0.05 3 = 0.10 3 = 0.25 3 = 0.50 3 = 0.75 3 = 0.85

0.15 −0.3 16.618 02 7.090 77 4.841 34 2.861 24 1.882 00 1.401 24 1.313 3414.791 46 6.334 22 4.334 36 2.570 57 1.695 81 1.316 31 —

0.0 18.861 59 8.014 92 5.459 35 3.214 56 2.107 34 — 1.389 2916.298 76 6.959 01 4.753 22 2.810 24 1.844 80 1.432 46 —

0.3 22.385 59 9.455 51 6.420 64 3.765 44 2.461 82 — 1.476 0218.396 41 7.823 88 5.331 64 3.144 40 2.059 24 1.591 38 —

0.25 −0.3 17.647 89 7.507 43 5.114 65 3.010 92 1.933 12 — 1.558 2114.441 15 6.182 48 4.228 48 2.504 75 1.650 12 1.207 70 1.147 74

0.0 20.025 57 8.482 46 5.764 85 3.380 82 2.207 89 — 1.572 0015.647 96 6.682 77 4.563 52 2.695 57 1.770 42 1.283 49 1.197 60

0.3 23.758 01 10.001 59 6.776 11 3.958 17 2.578 12 1.979 60 1.760 5917.224 31 7.333 50 4.998 62 2.943 29 1.926 76 1.394 40 1.370 44

0.50 −0.3 21.418 87 9.010 37 6.090 33 3.536 13 2.288 28 1.754 74 1.614 0213.657 15 5.835 15 3.982 08 2.347 87 1.539 74 1.190 82 1.098 27

0.0 24.282 77 10.166 53 6.853 66 3.962 83 2.554 42 1.954 00 1.795 7914.304 17 6.102 72 4.160 29 2.447 60 1.600 60 1.235 22 1.138 36

0.3 28.770 24 11.965 47 8.041 38 4.632 21 2.978 61 2.275 49 2.090 4415.054 13 6.412 19 4.366 10 2.562 18 1.670 48 1.285 89 1.183 98

0.75 −0.3 29.675 15 12.180 66 8.098 60 4.576 17 2.892 60 2.187 08 2.002 8712.961 74 5.501 48 3.734 10 — 1.421 30 1.095 95 1.010 10

0.0 33.581 94 13.707 31 9.085 80 5.750 16 3.214 74 2.423 44 2.217 1933.228 75 5.610 35 3.805 17 — 1.442 63 1.110 72 1.023 23

0.3 39.682 82 16.082 77 10.629 64 5.957 24 3.740 29 2.816 00 2.575 2913.513 19 5.726 21 3.880 72 — 1.464 92 1.125 99 1.031 67

0.90 −0.3 45.051 86 17.680 63 11.440 44 6.206 58 3.799 46 2.820 42 2.568 5712.050 60 5.242 20 — 2.044 07 1.329 82 1.027 66 0.948 11

0.0 50.832 64 19.814 15 12.445 82 6.886 72 4.193 08 3.102 08 2.822 0612.615 91 5.279 20 — 2.054 76 1.335 45 1.031 53 0.951 57

0.3 59.827 35 23.161 06 14.281 73 8.006 92 4.863 94 3.593 03 3.267 0812.713 48 5.317 02 — 2.065 55 1.341 10 1.035 40 0.955 02

obtained some results (converged to, only one or two decimalplaces) for the case of3 = 1− α ± β, especially for theforward scattering. However, it is not always possible toobtain reasonable results for these extreme cases. Thesedepend on the values of3, α andβ. We do not discuss thesesingular points further here, but we note that the method doesnot give good results when3 = 1− α± β; the convergencealso becomes worse. When3 = 1 (i.e. corresponding to aninfinitec value for a critical system), the present method doesnot work with the Marshak approximations for the problemwe considered.

Although this paper concerns itself mainly with the rangeof the applicability of the present method, we also wish toexamine the effect of anisotropic scattering on the radius. Asit was pointed out in [1, 40, 54, 55, 59] the numerical resultsin table 2 indicate that for backward scattering (α = 0) thevalue ofR06

∗s decreases with increasingβ. Whenβ = 1,

all radii converge towards the same value (limiting value) forall values ofb1. The results listed in tables 3 and 4 show theeffects of anisotropic scattering on the radii. In contrast tothe slab case, the variation of the radius is monotonic withincreasing or decreasingα(β = 0).

A non-monotonic variation of the radius may beobserved only when bothα andβ are not equal to zero andα + β → 1: eitherα = β or the value ofβ is fixed and thevalues ofα are allowed to vary. This depends on the values of3 andb1. The corresponding curves are shown in figures 1and 2 to clarify these results.

Figures 1 and 2 also show the effect of the linearanisotropy on the radius. The addition of linear anisotropy

increases or decreases the radius with respect to the isotropiccase according to the positive or negative values ofb1,respectively.

Another important result is that the monotonic variationcontinues in the range of(1− 3) 6 α, β 6 1. This effectcan be observed from figures 1–3.

Figure 3 shows that the fundamental decay constantdecreases uniformly with the increasing size of the assemblyin all cases. This is becausecmonotonically decreases whilec6t increases [7]. Furthermore, the decay constant3 isalmost independent ofb1 in the case of backward scattering.For forward scattering, there is a significant contributionsof b1 to the variations of the radius. Clearly, for backwardscattering the radius is smaller than forward scattering.Similar conclusions can be drawn from figures 1 and 2.

4. Conclusions

We first conclude that the present method is suited and is,also, very convenient for the direct numerical technique inthe case of linear- or higher-order anisotropic scattering.This method gives convergent results over a wider rangeof parameters for anisotropic scattering and for sufficientlysmall radii (the corresponding eigenvalue3 6 0.5 orR06

∗s > 1.4 mfp withα, β 6 0.9). ThePN results with the

Marshak boundary conditions improve asN increases, butall calculations ultimately breakdown at sufficiently largeNas the radius decreases [15, 16]. Also, it was pointed out byAronson [15, 16] that, during the course of computations, the

708

Time-dependent neutron transport equation

Figure 1. The radiusR06∗s as a function of the anisotropy

parameterα andβ for three different degrees of the linearlyanisotropic scatteringb1 as calculated with theP7 approximationand assuming the decay constant3 = 0.50. Three cases areshown for eachb1 value: eitherα or β is fixed and the other isallowed to vary andα + β assuming thatα = β.

Figure 2. The radiusR06∗s as a function of the anisotropy

parameterα + β for three different degrees of the linearlyanisotropic scatteringb1 and for three different values of the decayconstant3, as calculated with theP7 approximation and assumingα = β.

breakdowns may occur for any radius for sufficiently largeN .Therefore the method appears to be inadequate for the cases:(i) for any radius, for sufficiently largeN (for the problemunder considerationN is greater than 11 or 13); (ii) over therange in radius from 0 to≈1.4 mfp for allN and for all cases;and (iii) for certain combinations of the values of3, α andβ, when3 = 1− α ± β.

Figure 3. The radiusR06∗s as a function of the decay constant3

for three different degrees of the linearly anisotropic scatteringb1

as calculated with theP7 approximation. The labels F and B referto forward and backward scattering, respectively.

Finally, we have also noted that the anisotropic scatteringhas an influence upon the radius. For fixed time-decayconstant3, it is shown that the variation of the radius ismonotonic in the case of forward and backward scattering.However, for a certain combination ofα, β andb1, it maybe possible to observe non-monotonic variation of the radius.That is, under some conditions the radius for a certain valueof b1 is a function of two parameters, viz the decay constant3 and the strength,α andβ, of the forward and backwardscattering components [1, 7, 34, 40, 59]. In other words, thevalue of the radius required to maintain criticality varies asa function of the scattering parameters. Such monotonic ornon-monotonic behaviours of the size is a function of thegeometry chosen. This may be understood by consideringthe change in the angular distribution [28, 40, 54–57].

References

[1] Yıldız C 1999J. Phys. D: Appl. Phys.32317[2] Marshak R E 1947Phys. Rev.71433[3] Davison B 1958Neutron Transport Theory(London:

Clarendon)[4] Garcia R D M, Siewert C E and Thomas J R Jr1994Trans.

Am. Nucl. Soc.71220[5] Lathrop K D 1965Nucl. Sci. Eng.21498[6] Bell G I and Glasstone S 1972Nuclear Reactor Theory(New

York: Van Nostrand-Reinhold)[7] Sahni D C and Sjostrand N G 1990Prog. Nucl. Energy23

241[8] Ganapol B D, Kelley C T and Pomraning G C 1993Nucl.

Sci. Eng.11412[9] Benassi M, Garcia R D M,Karp A H and Siewert C E 1984

Astrophys. J.280853[10] Garcia R D M and Siewert C E 1990J. Quant. Radiat.

Spectrosc. Transfer43201[11] McCormick N J and Siewert C E 1991J. Quant. Radiat.

Spectrosc. Transfer46519[12] Rulko R P, Larsen E W and Pomraning G C 1991Nucl. Sci.

Eng.10976[13] Larsen E W and Pomraning G C 1991Nucl. Sci. Eng.10949

709

C Yıldız

[14] Case K M and Zweifel P F 1967Linear Transport Theory(New-York: Addison-Wesley)

[15] Aronson R 1984Nucl. Sci. Eng.86136[16] Aronson R 1984Nucl. Sci. Eng.86150[17] Lee C E and Dias M P 1984Ann. Nucl. Energy17515[18] Garcia R D M and Siewert C E 1996Ann. Nucl. Energy23

321[19] Khouaja H, Edwards D R and Tsoulfaridis N 1997Ann.

Nucl. Energy24518[20] Neshet K, Siewert C E and Ishiguro Y 1977Nucl. Sci. Eng.

62330[21] Siewert C E 1993J. Quant. Radiat. Spectrosc. Transfer49

514[22] Ganapol B D and Kornreich D E 1996Ann. Nucl. Energy23

301[23] Garcia R D M and Siewert C E 1996Ann. Nucl. Energy23

321[24] Sanchez R and McCormick N J 1982Nucl. Sci. Eng.80481[25] Sahni D C, Dahl B and Sjostrand N G 1997Ann. Nucl.

Energy24135[26] Boffi V C, Molinari V G and Spiga G 1977Nucl. Sci. Eng.

64823[27] Sahni D C, Dahl E B and Sjostrand N G 1995Transp. Theory

Statist. Phys.241295[28] Carlvik I 1968Nucl. Sci. Eng.31295[29] Kschwendt H 1971Nucl. Sci. Eng.44423[30] Hembd H and Kschwendt H 1972J. Comp. Phys.10134[31] Kerner I O, Kiesewetter H and von Weber S 1967

Kernenergie6029[32] Sahni D C and Sjostrand N G 1998Transp. Theory Statist.

Phys.27499[33] Sahni D C and Sjostrand N G 1998Transp. Theory Statist.

Phys.27137[34] Sahni D C and Sjostrand N G 1999Ann. Nucl. Energy26

347[35] Sharma A and Sahni D C 1993Ann. Nucl. Energy20449

[36] Sjostrand N G 1992Exact Solution of Some One-SpeedNeutron Transport ProblemsCTH-RF-94 (Sweden:Chalmers University of Technology)

[37] Dahl E B, Protopopescu V and Sjostrand N G 1983Nucl.Sci. Eng.83374

[38] Dahl E B and Sjostrand N G 1989Ann. Nucl. Energy1652[39] Dahl E B and Sjostrand N G 1979Nucl. Sci. Eng.69114[40] Sahni D C, Sjostrand N G and Garis N S 1992J. Phys. D:

Appl. Phys.251381[41] Hembd H 1970Nucl. Sci. Eng.40224[42] Sjostrand N G 1975J. Nucl. Sci Technol.12256[43] Protopopescu V and Sjostrand N G 1981Prog. Nucl. Energy

7 47[44] Dahl E B 1985Ann. Nucl. Energy12153[45] Garis N S and Sjostrand N G 1989Kerntechnik54183[46] Garis N S 1991Kerntechnik5633[47] Dahl E B and Sahni D C 1990Ann. Nucl. Energy1743[48] Sahni D C and Sjostrand N G 1997Ann. Nucl. Energy24

1157[49] Williams M M R 1985Ann. Nucl. Energy12167[50] Siewert C E and Williams M M R 1977J. Phys. D: Appl.

Phys.102031[51] Spiga G and Vestrucci P 1981Ann. Nucl. Energy8 165[52] Inonu E 1973Transp. Theory Statist. Phys.3 107[53] Williams M M R 1978J. Phys. D: Appl. Phys.112455[54] Inonu E and Tezcan C 1978Transp. Theory Statist. Phys.7

51[55] Pomraning G C 1980Transp. Theory Statist. Phys.9 1[56] Tezcan C 1979Transp. Theory Statist. Phys.8 187[57] Tezcan C and Yıldız C 1986Ann. Nucl. Energy13345[58] Wosnicka U and Sjostrand N G 1997Ann. Nucl. Energy15

1277[59] Sahni D C and Sjostrand N G 1991Transp. Theory Statist.

Phys.20339[60] IBM Scientific Subroutine Package 1969 Tech. Pub. Dept.,

112E Post Road, White Plains, NY 10 601, USA

710