Generation of leading coefficients of orthogonal polynomials with an application to anisotropic scattering of neutrons

Ann. nucl. Energy,Vol. 11, No. 10, pp. 535-537, 1984

0306-4549/84 $3.00 + 0.00 Copyright © 1984 Pergamon Press Lld

Printed in Great Britain. All rights reserved

GENERATION OF LEADING COEFFICIENTS OF ORTHOGONAL POLYNOMIALS WITH A N A P P L I C A T I O N TO A N I S O T R O P I C S C A T T E R I N G O F N E U T R O N S R. OFEK Department of Nuclear Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 84120 Israel

(Received 14 May 1984) Abstract--A simple method to generate leading coefficients for high-order sets of orthogonal polynomials, by derivation of recurrence expressions for these coefficients, is developed. The method is applied to Legendre, Hermite, Chebyshev and Laguerre polynomials. The method may be used in calculations of high anisotropic neutron-scattering transfer cross-sections, where the angular distribution of the scattered neutrons is given in the ENDF/B files for most materials as coefficients of an expansion into Legendre polynomials. INTRODUCTION

PARITY OF ORTHOGONAL POLYNOMIALS

Due to the completeness property of orthogonal polynomials, any well-behaved (at least piecewise continuous) function F(x) can be approximated by a series

Most of the orthogonal polynomials have the property of parity, namely

p.(x) = ( - 1)" P . ( - x). Therefore

F(x) = ~ a.p.(x) n

. . ~ Y~ c.,,x' = ( - 1)" y~ c,,,(-x)' = (-1)"+'c..S

to any desired degree of accuracy, where p.(x) are polynomials of degree n that are given as

i-0

i-0

i-0

and

p.(x) = ~ C.jx i.

(1)

n

Z[1-(-

i=0

1 ,~n + i ~jt~.,~x i

=0

i=0

The leading coefficients C.,i up to a certain index are tabulated in many mathematical h a n d b o o k s - - u p to Pro in Abramowitz and Stegun (1970). However, if some calculations are connected with the expansion of a function into a highorder or orthogonal polynomials, or furthermore, if the highorder set of leading coefficients is to be prepared as input data for any computing programme, the determination of the leading coefficients and the formation of them as input data becomes a tedious task. In the present work, some recurrence expressions, for the leading coefficients of all the well-known sets of orthogonal polynomials, are derived that enable a simple evaluation of these coefficients. This method may be applied in calculations of group transfer cross-sections of elastic scattering and inelastic scattering of neutrons from a discrete level, where the angular distributions of the scattered neutrons for each prescattering energy are given in the ENDF/B and other neutrondata libraries for most materials as coefficients of truncated expansions into Legendre polynomials. The angular distribution of elastically-scattered neutrons from z4ZPu for example, with pre-scattering energies > 12 MeV, is expanded in ENDF/B-IV into Legendre polynomials in the centre-ofmass co-ordinates system up to P x9. Or, as in another example, the transformation matrices of Legendre coefficients from the centre-of-mass to the laboratory system for neutron elastic scattering from H or C are given in ENDF-B/V by order of 21 ×21. Furthermore, leading coefficients of Legendre polynomials may be needed for transport calculations, where the neutron source is very spatially anisotropic, and hence the transport equation has to be solved for high n in the P , approximation.

for each value of x. The above-mentioned expression implies that for orthogonal polynomials which possess parity, the leading coefficients C., i = 0 if n + i is odd. Legendre, Chebyshev and Hermite polynomials have the property of parity, and therefore the general form of these polynomials is ~0 + ct2xz + '" for even n and ~ x + ~3x3 + ... for odd n. On the other hand Laguerre polynomials do not possess parity, and therefore the general form of these polynomials is ~e + ~ ~x + ~2x2 + "' '. LEGENDRE POLYNOMIALS One of the recurrence formulae for Legendre polynomials is

(n+l)P.+l(x)=(2n+l)xP.(x)-nP. l(x).

(2)

Ifa Legendre polynomial is written in the form of equation (1), and substituted in equation (2), it follows that n+l

n

n

1

( n + l ) Z C.+1,,x ' = ( z n + l ) Z C.., x'+~-n Z C. . . . . ~'. i

O

i=O

~=O

(3t Equation (3) should hold for each value of x such that 1 ~< x ~< 1, then after equating coefficients of like powers it implies that -

(n+l)C.+l,i+ x =(2n+l)C..i-nCn_l,i+ 1.

(4)

Another recurrence formula of Legendre polynomials is

P'.+ l(x)-P, l(x) = (2n+ l)P.lx). 535

(5)

536

Technical Notes

If a Legendre polynomial is written in the form of equation (1) and substituted in equation (5) it follows that

and i+1 C..~ = - - - C . _ 1 , ~ +

n-2 i=0

n+i

~+

n

L (i + l)Cn+ Li+ l x l - ~ (i + l)C.-1,i+ l xi i=0

C._L~_ p

(13c)

n

Laguerre polynomials =(2n+l)

L C.,i x~, (6)

n2

C,,,i = m

i=0

n--I

which implies that

(i+ 1)(C.+L,+,-C._L,+ 0 = (2n+ 1)C..i.

(7)

C.+ta+t-

(14a)

Cn-

(14b)

and f Nin 2

From equations (4) and (7) it follows that n+i+l i+1

C._ t,i

Cn'i= --~T)

C.,~

(8)

1,i-1

Associated Laouerre polynomials C=.,. = C°.,.6o,m = ( - 1)"3o,,.

and n+i+l

C"+2'i =

(15a)

and

n - i + 2 C..i.

(9)

In addition, the leading coefficient Co,o is determined by the orthogonality of Legendre polynomials : Co.o = 1.

(10)

Thus, by using equations (8)-(10) and using the property of parity of Legendre polynomials, which implies that C..~ = 0 if n+i is odd, all the leading coefficients of Legendre polynomials up to any order can be calculated.

RECURRENCE EXPRESSIONS FOR THE LEADING COEFFICIENTS OF OTHER SETS O F ORTHOGONAL POLYNOMIALS

By applying the recurrence formulae of other sets of orthogonal polynomials to equation (1), some recurrence expressions for the leading coefficients of these sets are derived. By using these expressions and taking Co, o = 1, all the leading coefficients can be calculated.

Cnm+1 = (i +

1)C.~,,+I

(15b)

Recurrence expressions for the leading coefficients of associated Legendre and Gegenbauer polynomials may be derived in a similar way. ANALYTICAL EVALUATION

O F T H E ANGULAR INTEGRAL OF NEUTRON GROUP TRANSFER CROSS-SECTIONS

In this section an analytical evaluation of the angular integral of group elastic-scattering transfer cross-sections is introduced. The angular integral [whose integrand is composed of two Legendre polynomials: one w i t h / t L (the cosine of the angle of scattering in the laboratory system) as an argument, and the other with/*c (the cosine of the angle of scattering in the centre-of-mass system) as an argument] can be decomposed into double power series with #L and #c each multiplied by Legendre leading coefficients. The angular integral for the angular distribution given in the centre-of-mass system, is defined as (Hong and Shultis, 1982)

YI,m =

Hermite polynomials

Pt(ItL)Pm(#c) d#c,

(16)

,1 #~ln

Cn+l,i+ 1 =

2(n + 1) Cn, i i+1

(1 la)

where #~ax and #~,~nare the maximum and minimum values of #c permitted by the group structure. Then, after decomposing each Legendre polynomial in the form

and

n

2(n+ 1)(n+2) Cn+ 2, i =

n-i+2

P.(x) = ~ C.,ix ~(with C., i = 0 if n + i is odd), C.j.

(lib) it implies that

Chebyshev polynomials of the first kind Ca, t = n C n - l,o, Cn+

1,n+ 1 =

2C.,.

Y,.m= (12a) (12b)

and

n+i-2

i+1 Cn, i = - - - C n _ l , i +

n--1

1 + - -n _ ~ - C . - t . , - p

i=o

(12c)

c,,, ~ cm,j ~=o

~=o

#L ~ di,c.

(17)

J ~a""

But since for elastic scattering #L is related to Pc in the form (Williams, 1966) ~L --

1 + A#c ~/1

+ A2+

2A#c'

(18)

Yt,mcan be written as

Chebyshev polynomials of the second kind C.a = ( n + 1)C._Lo, C.+L.+ t = 2C.,.

(13a) (13b)

Yl,,. = L Cti ~, ~=o

' j=o

C,.j

~

k

A Q,,j+k,

(19)

where the integrals Q~,;+k can be evaluated as indefinite

Technical Notes integrals, and are given by Q,,, =

,~,, (1 + A 2 + 2Apc)i/2 "

537 REFERENCES

(20)

Thus, the method of generation of the leading coefficients of Legendre polynomials developed in this work can serve for calculation of group transfer cross-sections.

Abramowitz M. and Stegun I. A. (1970) Handbook qf Mathematical Functions, 9th edn. Applied Mathematics Series 55, NBS, Washington, D.C. Hong K. J. and Shultis J. K. (1982) NucL Sci. Engng 80, 570. Williams M. M. R. (1966) The Slowing Down and Thermalization of Neutrons. North-Holland, Amsterdam.