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Journal of Science and Technology
1 (2), 2007, 293-301
©BEYKENT UNIVERSITY
UN APPROXIMATION TO NEUTRON
TRANSPORT EQUATION IN SLAB GEOMETRY;
COMPUTATION OF GENERAL EIGENVALUE
SPECTRUM
Ahmet BÜLBÜL
Fikret ANLI*
ahmetbulbul@gmail.com
fikretanli@gmail.com
Kahramanmaraş Sutcu Imam University, Faculty of Sciences and Letters,
Department of Physics, 46100 Kahramanmaraş, Turkey
ABSTACT
General theoretical scheme for the UN method and evolution of general
eigenvalue spectrum are described. Firstly, applicability of UN method to one
dimensional slab geometry neutron transport problem is discussed and then
eigenvalue spectrum is calculated for isotropic scattering with different values
of parameter c (where c is the mean number of secondary neutrons per
collision and known as the fundamental eigenvalue). Results of the UN method
showed that both discrete and continuum eigenvalues are present as in the case
of PN method. All the results computed by both methods i.e. UN and PN
methods are presented side by side in the Tables for the comparisons.
Keywords : Chebyshev Polynomials, neutron transport in slab geometry,
eigenvalue spectrum.
1. INTRODUCTION
In applied science, some special classical orthogonal polynomials such as
Legendre, Chebyshev [1, 2], Hermite etc. have a great importance. Especially,
Legendre polynomials have an extensive usage area in the solution of physics
and engineering problems. In neutron transport theory, Legendre polynomials
have an extraordinary importance. Angle and position dependent angular flux
of neutrons are expanded [3, 4] in terms of Legendre polynomials and then the
method is called PN approximation. As well known, the variables or the roots
of both polynomials Legendre and Chebysev are in the interval of [-1, 1]. If the
PN method gives the appropriate solutions, then the Chebyshev polynomials
* Corresponding Author: Phone:+90 344 2191305
293
Un Approximation To Neutron Transport Equation In Slab Geometry; Computation Of
General Eigenvalue Spectrum
should also give the appropriate solutions for the neutron transport equation in
plane geometry.
The Chebyshev polynomial method was first proposed by Aspelund [5] and
Conkie [6]. By TN method (first kind Chebyshev polynomial approximation),
the necessary condition for reactor criticality and the values of extrapolated
end point were discussed by Yabushita [7]. In his work, it was shown that TN
method gave the results closer to exact value than PN method for the strong
absorber but for weak absorber PN method gave the results closer to exact
value than TN method. In recent published papers, Anli et al. showed that the
eigenvalue spectrum and critical half thickness of neutron transport equation in
slab geometry can be calculated by modified TN method [8, 10].
One of the important problem in the solution of neutron transport equation is
the calculation of eigenvalues. As well known, these eigenvalues depend on
the c values. In nuclear reactor physics, the important coefficients such as
diffusion length, diffusion coefficient and buckling also depend on the
parameter c which is known as the fundamental eigenvalue in the neutron
transport theory.
The spherical harmonics approximation, PN method, [3, 4, 11] has been
explored in the early years of the nuclear age for the solution of neutron
transport equation. In the present work, we describe a new theoretical scheme
for the solution of neutron transport equation in plane geometry. We expanded
the neutron angular flux in terms of Chebyshev polynomials of the second kind
instead of Legendre polynomials and then we called this application as UN
method.
2. THEORY
2.1. P N APPROXIMATION
We consider the one-dimensional Boltzman integro-differential equation. This
equation, for isotropic scattering and with no sources present, is
dw( x,u)
¡ 1 ^ ^ — +
dx
Jt^(X,/U)
=
J
s
2
1
-1
,
,
/ ) d/u ,
- a < x < a,
- 1 < / < 1.
(1)
where y/(x, ¡¡¡) is the angular flux or flux density of neutrons at position x
traveling in direction p, a T and a S are total and scattering differential cross
section, respectively.
In order to solve Eq.(1), it is well known that in the PN approximation the
angular flux is expanded in terms of Legendre polynomial as
294
A. Bülbül and F. Anlı
M) = £
(x)Pn (M),
2
n=0
-
a < x < a, -1 <M< 1.
(2)
The orthogonality and recurrence relations of Legendre polynomials are given
by, respectively
1
|2/(2n +1) ,
=\0K
'
[0
,
n^ m
( 2 n + 1 ) MPn ( M ) = ( n + 1 ) Pn+i ( M ) +
(M)
J Pn M) Pm (PW
-
n =m
(3)
(4)
and then, inserting Eq.(2) into Eq.(1) and using the recurrence and
orthogonality relations of the Legendre polynomials defined in Eq.(4) and
Eq.(3) respectively, we obtain the PN moments of the angular flux as
d<(- —
x) +
—
< < .( .x .) = < < (. x. ).
(5a)
dx
dx
+ 2<
2
and in general, assuming <
d
(x)
(n +1) < + '
dx
0
+ 3 C T t < ( x) = 0
(5b)
dx
d
(x)7 + 3 < ( x 7)
+3
+ 5o T $ 2 (x) = 0
dx
dx
d
+n <
n- (x)
'
dx
(5c)
(x) = 0
+ a T (2n + 1)<(x) = <s<a(x)8na
,
(5d)
For the solution of Eqs.(5) we assume [11,12]
<n ( x ) = Gn
( V ) ex
p(&T
x / V)
(6)
and putting this in Eqs(5) we obtain analytical expressions for all Gn(V) as,
G0V) = 1, G J ( V ) = - V ( 1 - c)
(7a)
and in generally,
(n + 1)Gn+1(v) + nGn-1(v) + ( 2 n + 1 ) G ( V ) - v c G 0 { v ) 8 n 0 = 0,
n = 1,2,...N
(7b)
where c = aS / <JT, a n d Sn0 is the kronecker delta and V is a constant
appearing as an eigenvalue associated with the eigenfunction GN (V) . The
eigenvalues Vk, k
= 1,2,.. N +1 can be calculated by setting GN+j (V) = 0.
295
Un Approximation To Neutron Transport Equation In Slab Geometry; Computation Of
General Eigenvalue Spectrum
2.2. CHEBYSHEV POLYNOMIAL APPROXIMATION (U N
APPROXIMATION)
We expand the neutron angular flux in terms of Chebyshev polynomial as,
N
2 /
W ( x , n ) = - V -l - V 2
nn ( x ) U n (M), _ a < x < a, _1 <M< 1
(8)
n
n=0
The orthogonality and recurrence relations of Chebyshev polynomials of the
second kind are given by, respectively
I
f
f Un MU
-
m
(^)V1
7
- V
dV
in/2,
I tt / 2,
'
[0,
nn == mm
= \
2 v U n (M) = Un+i (M) + Un-1 (V),
(9)
n^ m
- 1<V <1
(10)
Inserting Eq.(8) into Eq.(1) and using the recurrence and orthogonality
relations of the Chebyshev polynomials of the second kind defined in Eq.(10)
and Eq.(9) respectively, we obtain the UN moments of the angular flux as,
^
+ 2aT O 0 (x) = 2aS O 0 (x)
dx
dO 2 ( x) d O 0 ( x) „
^
+
— + 2ot Oj(x) = 0
dx
dx
(11a)
(11b)
and in general, assuming O _ ( x ) = 0
dx
+
dx
+
(x) = n
t
f
n +1
^
«
,
(.10
Here we clarify an important point; As well known, in the diffusion
approximation it is assumed that all the flux moments O N + 1 ( x ) equal to zero
for all N > 1. Applying this rule to Eq.(11b) for n = 1, which is the
U1 approximation, we obtain
O1
where O 0 ( x )
.
1 d O 0 (x)
(x) = _ 2aT
dx
(12)
is the scalar flux and O j ( x ) i s the neutron current at the
position x . The general relation between the scalar flux and current is given
by the Fick's law [3, 4, 11] which is defined for a general geometry as
296
A. Bülbül and F. Anlı
J ( r ) = -D V O ( r )
(13)
where D is the diffusion coefficient and physically Eq.(13) is equivalent to
Eq.(12) for one dimension. If Eq.(12) is used in Eq.(11a) it is obtained that
d
°02(x)
dx
-4a T 2 (1 - c ) O 0 ( x ) = 0, c = cts/ Gt
(14)
which is the neutron diffusion equation with no source. As a result, in the
U1 approximation diffusion coefficient and diffusion length are found to be
D = 1 /(2o> ) and
L = 1 / ( 2 o y VT - c )
respectively,
but
in
the
P approximation (Legendre polynomial approximation) these are given by
D = 1/(3<rT ) and L = 1/(ct tA /3(1 - c)) .
For the solutions of Eqs.(11), again we assume [11,12]
o n (x) = An (v)exp(^Tx/v)
(15)
and putting this in Eqs.(11) we obtain analytical expressions for all An (v) as
Ac(v) = 1
A1 (v) = -2v(1 - c)A0 (v),
An+1 ( v ) + 2vAn ( v ) + An-! (v) =
(16a)
c = aS / aT
1 + (-1) n
n +1
(16b)
(v), n > 1
( 1 6 c)
As A n (v) depends on v, we obtain the permissible values of V from
Eq.(16c), that is since O N + 1 (x) is neglected in the UN approximation, we
should thus have AN+1 (v) = 0 . Hence the permissible values of V will be
obtained from the solution of
AN+1(v) = 0
(17)
The other way to compute V eigenvalues is to use the determinant of the
coefficients matrix of An (v) , that is
[M(v)]A(v) = 0,
where
A(v)
is
the
column
(18)
vector
given
by
[A0(v), Aj(v),...., AN+1(V)]T and M(v) is a (N +1) x (N +1) square
matrix which is given explicitly as;
297
Un Approximation To Neutron Transport Equation In Slab Geometry; Computation Of
General Eigenvalue Spectrum
2v(1 - c)
1
M (v) =
2
—vc
3
0
2
—vc
5
1
0
0
2v
1 0
1
2v
1
0
1
2v
0
0
1
1 + (-1) n
2n +1
0
0
0
0
0
0
1 0
2v
(19)
1
1
2v
If one solves the equation det M ( v ) = 0 for any order N+l, it is clear that
same results are obtained with the results of Eq.(17).
The discrete
[vk < 0, vk > 0] and continuum
[_ 1 <vk
< 1] eigenvalues for different
values of c are given in the Tables for any given value of the approximation
order N.
3. RESULTS AND DISCUSSION
In this work, we investigated the applicability of second kind of Chebyshev
polynomial approximation method for the solution of neutron transport
equation in slab geometry and then we attempted to compute the general
eigenvalue spectrum of neutron transport equation. In all Tables, generated
numerical results were calculated by using Maple Software. The numerical
results for c>1 are presented in Table 1 and c<1 are presented in Table 2.
Results obtained by both methods are given side by side for the comparison. It
is seen that the results obtained by UN method are in good agreement with the
results obtained by PN method.
As seen Eqs.(16) the analytical expressions for all An ( v ) are present and by
using these analytical expressions the discrete and continuum v eigenvalues
can be calculated by setting AN+1 ( v ) = 0 for various c values. This comes
from essential idea of the UN method for which O N + 1 ( x ) = 0 as in the case
of PN method. As seen in Table 1, when c>1 then one pair of the roots is
purely imaginary and the others are the pairs in the interval [-1,1]. These are
298
A. Bülbül and F. Anlı
the expected results, because it is possible to see in literature [4,11] that, when
c>1 asymptotic roots or discrete eigenvalues are purely imaginary. When c=1
then one pair of the roots is + ro i and the remaining roots are the pairs in the
interval [-1,1]. As seen in the Table 2, all the roots are real and absolute value
of only one pair of them is greater than 1 and these are known as the discrete
eigenvalue of transport equation. Some values of V lie on the real line
between - 1 and +1 and these are called continuum eigenvalues. Thus, for c<1
it is clear that the discrete eigenvalues may change between V=1 and V = ro.
When c = 0 , then all the roots of AN+1 (V) = 0 are identical to the roots of
UN+I (V) = 0 , i.e. to the roots of second kind Chebyshev polynomials. For N
even, AN (V) = 0 gives symmetric eigenvalues as ±Vk, therefore the system
has N eigenvalues. For odd N, AN (V) = 0 gives the eigenvalues such that one
of them is zero and remaining N -1
pairs are symmetric according to
V = 0 . Only positive values of Vk are presented in the Tables. As seen in the
Tables, UN
approximation also gave the eigenvalues; two of them discrete
and the others are the continuum as in the case of PN approximation. It should
be noted that when - 1 < V < 1, the continuous solutions of the neutron scalar
flux change very faster with x than the asymptotic solutions.
As a result, neutron transport equation can also be solved by the Chebyshev
polynomials (with both kinds TN and UN) not only by Legendre polynomials.
For the solution of neutron transport equation, we suggested the appropriate
expansion as seen in Eq.(8) but we could have suggested the expansion as
2
N
Such
an
expansion
causes
some
complications to compute the eigenvalues. If the above expansion was used,
we could not obtained the exact analytical expressions of An (V) in terms of
the variable V.
299
Un Approximation To Neutron Transport Equation In Slab Geometry; Computation Of
General Eigenvalue Spectrum
Table 1 Eigenvalue spectrum for c>l
c
N=2
N=5
N=S
N=ll
N=14
U„
5.0000001
PK
U„
5.7735031 5.7504991
0.633694
0.00
PK
5.7505401
0.716347
0.00
U„
5.7505401
0.865812
0.598709
0.209668
PK
5.750 5401
0.906866
0.629102
0 22479:
U„
PK
5.750 5401 5.750 5401
0.931581
0.954936
0.792056
0.813683
0.56349 1
0.591"":
0.255496
0.311306
0.00
O.OO
U„
57505401
0.959067
0.872811
0.721127
0.5:41:1
0.343589
0.116907
PK
5.7505401
0.973685
O.E89087
0.752162
0.571145
0 356664
0.121260
1.5811391
1.8257421
I.7554901
0.614266
0.00
1.7566251 1.7566771
0.728131 0.B65654
0.606066
0.00
0.213453
1.7566521
0.910018
0.636523
022315"
17566521
0.93305 2
0.795904
0.568315
0.303404
O.OO
1.7566521
0 956155
0.817375
0.597041
0 3 1 : : Ol
O.OO
I.7566521
0.53948
0.874 858
0.732476
0.558149
0.346705
0.118140
1.7566 521
0.9742 E2
0.B51038
0.755538
0575208
0 360038
0.122579
0.7071071
0.8164971 0.6789531
0.678953
0.00
0 6 854921 0.6856501
0.76224« 0.880252
0.00
0.630480
0.230921
0 6 892061
0.91 El 35
0.660647
0249488
0.6852131 0 6 8 9 I 2 9 i
0.937600
0.959270
0.80682 1
0.827801
0.585237
0 614884
0.320050
0 332281
0.30
O.OO
0.601241
0.962251
0.880554
0.742760
0.572099
0.5:5:: 5
0.123854
0.6851311
0.97:81:
0.856308
0.765635
0 5851::
0373807
0.128685
0.5590171
0.6454971 0.4826441
0.694943
0.30
0.4910741 0.5010701
0.77605" 0.884472
0.00
0.642030
0.244121
0.5033951
0.921234
0.671839
0264647
0.5031071
0.939373
0.81133 7
0.593900
0.331223
O.OO
0.5027761
0.960465
0.832105
0.623635
0 343529
O.OO
0.5027801
0.5631:6
0.8K2S23
0.747368
0.579224
0.367706
0.128356
0.50281 H
0.976405
0.B5842:
0.770040
0.596315
0 382454
0.133495
0.5000001
0 5 7 7 3 5 0 1 0.4004461
0.702631
0.00
0.4094941 0.4309501
0 7 8 2 3 8 7 0.886432
0.00
0.647670
0.252612
0.4303421
0.922652
0.677270
0 274250
0.4154481 0.428B631
0.94015"
0.961016
0.813456
0.834132
0.598314
0.628003
0.337785
0 350129
O.OO
O.OO
0.4289301
0.563;""
0.883881
0.749599
0.5K2855
0.372418
0.131432
0.4289871
0.976684
0.E89410
0.772150
0.599912
0 387381
0.1367B2
1.01
1.1
1.5
13
2.00
Table 2 Eigenvalue spectrum for c<l
N=2
PK
5.000000
:.773503
N=5
UH
PK
5.796687
5.796725
0.631204
0.713470
O.OO
0.00
N=8
UH
PH
5.796729
5.796729
0.864832
0.906054
0.596976
0.627344
0.208837
0 223844
N=11
UH
PK
5.796729
5.796729
0.530942
0.954620
0.791100
0.812764
0.562368
0 590:::
0.29S62 5 0 310415
0.00
0.00
1.000000
1.154701
1.281508
0.55811:
O.OO
1 2 8 8294
0.671S71
0.00
1.288811
0.848576
0.573762
0.199204
1289450
0.8914:1
0.603419
0 2125:"
1.289424
0.923109
0.776072
0.547630
0.288218
0.00
1 289463
0.948640
0.798197
0.573975
0 299747
0.00
1.289459
0.954668
0.863:19
0.716470
0.540912
0.3345:9
0.113462
1 289464
0 970573
0.880137
0.739002
0.;;7582
0 346632
0.117579
0.707107
0.816497
1-009084
0.561322
O.OO
1.028875
0.621177
0.00
1.036339
0.820973
0.54706 1
0.185902
1.042231
0.861401
0 575296
0 201936
1.042263
0.906126
0.7530:1
0.530834
0.27-721
0.00
1.044032
0.93:":;
0.775433
0.554621
0288965
0.00
1.043706
0.944086
0-846572
0.700382
0.526674
0.325480
0.110319
1.044323
0.9616"!
0.863919
0.721557
0.542854
0 336963
0.114224
0.577350
0.666667
0.912414
0.527837
O.OO
0.944149
0.575157
0.00
0.965375
0.785956
0 521696
0.181389
0.980110
0.824300
0.548405
0.192206
0.982708
0.88280 5
0.727743
0.51447 8
0.267869
0.00
0.990749
0.906021
0.750336
0 535683
0 278838
0.00
0.990103
0.926057
0.825458
0.683457
0.512593
0.5169::
0.107342
0.99:116
0.942094
0.845118
0.702901
0 528253
0327695
0.111050
0.500000
0 577350
0.86602 5
0.500000
O.OO
0.906180
0.538469
0.00
0.539653
0.766044
0.500000
0.173648
0.960290
0.796666
0.525532
0.183435
0.965926
0.86602 5
0.707107
0.500000
0 . 2 5 ! 819
0.00
0.978229
0.887063
0.7301;:
0.519096
0269543
0.00
0.978148
5 . 9 0 545
0.809017
0.6691:1
0.500000
0.309017
0.10452 8
0.986284
0.92843:
0.827201
0 687293
0 515249
0.51911:
0.10805:
0-99
0.75
0.50
o.::
0.00
300
N=14
Pk
5.796729
0 9755:1
0.888593
0.751337
0.570200
0 355905
0.120970
5.796729
0.95884 1
0.872283
0.728316
0.553220
0.342889
0.116636
A. Bülbül and F. Anlı
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301