* . ! TOF L. ORNL P 2084 . • * Los . - i pri 1: 2:1 11:25 11.4 1.6 no MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 ... . Cover P 2884 Corf-660303-30 comes to MAY 5.-1966 CFSTI PRICES H.C. $ 1,00 ; MN.50. IMPOSSIBLE LEGENDRE COEFFICIENTS* C. D. Irving, R. R. Coveyou, and R. D. MacPhersoni Dak Ridge National Laboratory Oak Ridge, Tennessee : . . in RELEASIZD VOR ANNOUNCEMENT IN NUCLEAR SCIENCE ABSTRACTS . LEGAL NOTICE This report mo prepared us an account of Government sponsored work. Neither the United Suatos, por the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accu- racy, completeness, or usefulness of the information containod in this report, or that the uso of any information, apparatus, method, or proceso disclosed in this report may not infringe privately owned rights; or B. Assumos vay Habilities with rospect to the use of, or for damages resulting from the use of any information, apparatus, method, or procers disclosed in this roport. As used in the abovo, "person acting on to all of the Commission" includes any en- ployeo or contractor of the Commission, or employee of such contractor, to the oxtent that such employee or contractor of the Commission, or employee of such contractor prepares, dissorainalns, or provides access to, any information pursuant to his employment or contract with the Commission, or bir employment with such contractor. : : 2 1 *Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. i : IMPOSSIBLE LEGENDRE COEFFICIENTS D. C. Irving, R. R. Coveyou, and R. D. Mac Pherson Oak Ridge National Laboratory Oak Ridge, Tennessee It has long been realized that Legendre polynomial expansions of differential cross sections can result in nega- . tive cross-section values due to the truncation of the infinite series. It is also possible that a given set of Legendre coeffi- cients may result in negative cross sections because they do not represent any physically possible, everywhere positive angular distribution. The nathematical criteria that must be satisfied in order that there exist an everywhere positive distribution having a given set of Legendre coefficients are discusseå. A check of Legendre coefficients from several published sources has revealed a surprisingly large number of instances where the co- efficients are impossible. A computer program 18 offered which will examine a set of Legendre coefficients to determine if they can be derived from a possible distribution. One important part of a neutronics calculation is the treatment of scat- . tering angular distribution. These are most often expressed through a Legendre expansion of the distribution in cosine of the scattering angle. We shall write - - ha en Preto (o) , .. (1) .. where u is the cosine of the scattering angle, and * w *--..:. dus = 1 . Then we expand f'(w) in the form . f(u) = (24 + 2) pe Pelu). (26+ (3) Since an infinite number of coefficients, f, would require too much memory . space, the expansion is truncated in same convenient and/or reasonable order and the approximate distribution **(w) - (244) Pe Pe (w) 24 + 1 (4) is used in the calculation. As f(u) is a probability density, f(u) 2 C for -l sa s t l . (5) . Unfortunately this restriction does not apply to f*(u). The truncated expan- sions can, and often do, assume negative values. For some computer programs . and problems this negativity can be tolerated; for others, it cannot. In general, a penalty such as decreased precision, must be paid for the presence of negative cross sections. That negative cross sections can arise from truncation of the Legenare expansion is well known. It has not been generally appreciated that there is a second source of negativity; the specified Legendre coefficient may be impossible, that is, an everywhere-positive angular distribution having the given Legendre coefficients in its expansion may not exist. This is not just a theoretical source of negative cross sections; a check of the purported Legendre coefficients of the scattering angular distribution for 15 elements in an evaluated library available at Oak Ridge revealed that six elements had impossible Legendre coefficients at one or more energies. A list of the re- sults appears in Table 1. The region of impossible coefficients in the ura- nium cross sections resulted in much wasted effort and computer time in a series of calculations until the true reason for the negative cross sections was found. Suppose then we are given fo, ..., fn, purporting to be some of the coef. ficients in the expansion of the scattering angular distribution in terms of Legendre polynomials. How do we decide whether or not there exists a scatter- ing angular distribution having these numbers as coefficients ? Since each Legendre polynomial Pe(u) satisfies -] s Pelu) s 1 for -] s H si and f, is the expected value of P, (u), we have . Itals i for all l. . This necessary condition is far fron sufficient; * P*(x) = { u is a ready counter example. We seek here necessary and sufficient conditions. The Legendre coefficients of a probability density confined to lui si are uniquely determined by the moments Many wech Plus) que : . and conversely; we shall hence work with these moments. 2 . 1 A T .. TABLE 1. RESULTS OF A CHECK FOR IMPOSSIBLE LEGENDRE COEFFICIENIS Number of Energy Values Number of Legendre Coefficients Number of Points With Impossible Coefficients Member of Points With Negative Cross Sections Element Source of Data 6 101 124 : Deuterium Carbon Nitrogen 0 6 101 0 E. 116 151 235 0 0 5002 Oxygen Sodium Magnesium Aluminum Silicon Sulfur Potassium 136 .. .. 0 0 . pū oo oooo oo w wo ñ ñ @ & co o HD nū to ū ū UNC 5038 GA 2156 UNC 5002 UNC 5038, Joanou and French NDA 2133–4 . UNC 5002 NDA 2011-3, NDA 2111-3 NDA 2133-4 UNC 5002 Caswell, J. of Research NDA 2111-3 GA 2156 J. T. Mihalczo J. T. Mihalczo 136 197 141 159 136 48 39 43 0 0 0 Calcium 0 0 Iron. . Zirconium Uranium-235 Uranium-238 1 1 We begin by calculating a sequence Q(u), Q. (u), ... of polynomials ortho- gonal with respect to f(u). If we define the notation scu) - {*W ) , (10) . then we wish to determine Qe Qie .. such that (11) with the normalization condition Ann = 1, and E[Qrz () Alc(u)] = ônk No . (12) We find that, since I(u) is non-negative, NA = ECQ (u)] > 0. (13) From the properties of orthogonal polynomials we know that an arbitrary polynomial of order n, An(u)s may be expanded in terms of the Q polynomials, R(u) = Ink Of (u) i (14) k=0 and that Qn (H) is orthogonal to all polynomials of lower order. It follows that E[Rm(u) Qn (w)] = 0 for m O we must have any = 0. Now we set (20) 0_1 = -dn, n-2 and we can write 2, 6-1) = (x - Why? 2-2 (w) - o -1 2-alue) NS This is a recurrence relation which determines our sequence of polynomials. Since E[ely (H) 22-2()] = 0 = E(v @u-z(v) 2-2(x)] - Why EC 2-1 (w) -2(x)} - O -2 =(4,-2(k) 2-2(x)] = EC4_(w) Cu u2() )3 - . = -20 -MA-2 . . VA N o n-1 n-2 We have 1 - 12/22 · Iet Let . +3806W .sy. I tu, hak : . kao 1 ..: Then Et qu) - 1)] = 0 = E[Q, (H) x4-2] = [y=2(a) w My – Hn -7(H) Why - -1 ER-(w) wun-; = In My N-1 - 0h-1 n-1 -1 In La-1 n-IM- IN-2 (25) poga nul Nnel Thus Q (umay be determined by quantities up to order Ny and Ige which are determined by moments up to Mr. and Mr.7, respectively. Thus ä (u), Q.(u), ...., Rn (u) are determined by the first 2n-1 moments of f(u). Qu(w) has n distinct, real roots in the interval (-i, +1). To prove this we assiime that an (u) has only s changes of sign in the interval («l, +1) at the points rzo rzy .., Ig: If we set 0() = (u - ry)(x - 52) (4 - 73) ... (u - Fg), (26) then (u) Qu (u) does not change sign in the interval (-1, +1) and hence ECO(u) Q (4)] +0. (27) (u) is a polynomial of order s sn. Since Qu (w) is orthogonal to all polynom- ials of order less than n, we must have s = n; thus proving the assertion. We have now established two necessary conditions on the first 2n-1 moments, Mo Mz, ..., Manage of a physical angular distribution: A) NZ, NZ., MA-1 > 0 and ... B) Q (H) has n roots in the interval (-1, +1). '. To see that conditions A and B are sufficient, we need only observe that the roots of Qu (u), 190 fpj ..., In when used with weights - 12 * 12 (28) provide us with the Gaussian quadrature having f(u) as a weighting function. That is, e tem w twa. - * ... ( k=1 . where Roma (w) is an arbitrary polynomial of order 2n-1 or lees. In particular . M = ud f(u) due = du = (Full Pc Por és 2n-1 or is 2n-1 . (30) k=] Wasanii Therefore the discrete distribution defined as having probabilities p, , P.,..:P at the points rl,r2, ..., In has the moments Mo, M7, ..., Maral and we can then exhibiü a non-negative distribution contained in the interval (-1, +2) which has the desired moments. The present 05R library of evaluated cross sections was examined to deter- mine if conditions A and B were satisfied by the Legendre coefficients. The results, summarized in Table 1, show a surprising number of failures in the data which was compiled from several sources. A common cause of impossible ........ coefficients seems to be extrapolation from experimentally measured coeffi- . cients. A second source of error was found in the uranium cross sections which were, in part, based on Legerdre coefficients published by A. B. Smith. He had usei a least-squares technique to fit a legendre expansion to his . measured data. This is equivalent to a high-order polynomial fit, & technique known to be unstalle. The expansions may have fitted his data from 200 to 145° quite well but took sharp negative excursions from 1450 to 180°. Since only the Legendre fits and not the actual measured data were published, it is difficult to correct this error. ..:•.!" .. We have prepared a subroutine, LEGCK, which will perform the calculations outlined above and determine if a given set of Legendre coefficients satisfies conditions A and B. In addition, LEGCK will search for negative excursions in the expansion (these may be due to truncation of the expansion as well as im- possibility of the coefficients) and calculate the total amount of negative probability. This program is available from the authors in versions for either the CDC 2604 or the IBM 7090. References: 1. A. B. Smith, "Elastic Scattering of Fast Neutrons from 11235," Nucl. Sci. Eng, 18, 128; also A. B. Smith, "The Scattering of Fast Neutrons from Natural Uranium," Nucl. Phys. 47, 633. A mere LIN .. END a .. . r DATE FILMED 6 / 16 /66 - - - 4.