A NEW APPROACH FOR CALCULATING AVERAGE CROSS SECTIONS IN THE UNRESOLVED ENERGY REGION

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1 Nuclear Matheatical Coputational Sciences: A Century in Review, A Century Anew Gatlinburg, Tennessee, April 6-, 2003, on CD-ROM, Aerican Nuclear Society, LaGrange Park, IL (2003) A NEW APPROACH FOR CALCULATING AVERAGE CROSS SECTIONS IN THE UNRESOLVED ENERGY REGION L. C. Leal M. E. Dunn Oak Ridge National Laboratory Oak Ridge, TN 3783 US LealLC@ornl.gov; DunnME@ornl.gov ABSTRACT A novel approach for calculating average cross sections in the unresolved resonance region is proposed. The ethodology is based on the well-known probability table technique. Two new features are added to the probability table ethod in the way the energy level spacing is sapled: firstly, a constraint is iposed in the sapling of the energy level spacing, that is, in addition to sapling fro a Wigner distribution the 3 statistic test of Mehta-Dyson is also added; secondly, instead of generating a sequence of level spacing in a energy range (ladders) in the present ethod the level of spacing is sapled around a reference energy. The ethodology was ipleented tested in a coputer code naed URR (Unresolved Resonance Region). It has also been incorporated in the codes RACER AMPX. Key Words: cross section, unresolved, probability table. INTRODUCTION The foralis developed based on the R-atrix theory for treating the resonance in the resolved resonance region cannot be utilized when the average resonance width is of the order or larger than the average level spacing. One cannot deterine suitably the resonance paraeter for an individual resonance but rather resonance paraeter, which would represent a cluster of resonance. For instance, for the fissile fertile nuclides the unresolved energy region starts in the kev region extends up to few hundreds kev. An alternative for representing the cross section in the unresolved region would be to use a pointwise tabulation. However this approach would iply an overwheling aount of data storage. The foralis ostly used for calculating cross sections in the unresolved energy region is based on calculations done using average resonance paraeters. The calculation is expected to include the Doppler the self-shielding effects, which are iportant effects in the deterination of reactor ultiplication factors. The calculation of average resonance paraeters is based on the statistical distributions of the resonance paraeters. A coputer code naed URR (Unresolved Resonance Region) [] was developed to study the effect of various assuptions on the statistical distributions of unresolved resonance paraeters on the calculation of self-shielding factors self-indication ratios. The code uses the Monte Carlo ethod to generate probability tables for calculating of average cross sections. The self-shielding factors are calculated using the Bondarenko foralis. All cross section calculations are done based on the single-level Breit-Wigner approxiation. The calculations can be done at a specified energy, teperature cross section dilution. The ain characteristic of the URR code is that, contrary

2 L.C Leal M. E. Dunn to existing probability table codes, the resonance energies are not sapled fro consecutive points on a statistical ladder, but rather pairs of resonance are generated around the energy of interest. This ethod has been ipleented in the coputer codes RACER [2,3] AMPX [4]. In addition to the odification in the sapling of the energy level spacing the resonance energies generated are also copared with the corresponding energy levels on the 3 statistic test of Mehta-Dyson. 2. BRIEF OVERVIEW OF THE THEORETICAL DISTRIBUTIONS OF THE RESONANCE PARAMETERS 2. LEVEL SPACING DISTRIBUTION LAW The spacing between two consecutive resonance energies for the sae total angular oentu parity exhibits ro behavior. For a set of n resonance energy levels, E, E 2,..., E n, where the level spacing between two consecutive energies, E k E k-, is D k, the average level spacing is <D>, the probability distribution function predicted by the Wigner law[5] is p(x) dx ' π x 2 2 π x exp (& ) dx, () 4 where x ' D k /<D>, <D> is the average level spacing. The Wigner probability distribution function has the following property: 0 4 p(x) dx ' 0 4 xp(x)dx '. (2) The second oent of the Wigner distribution is given by x 2 ' 0 4 x 2 p(x)dx ' 4 π. (3) Equation () was the first atheatical prediction of the level spacing distribution to provide excellent agreeent with experiental results; it has triggered a series of investigations on the subject of the statistical distribution of resonance paraeters. Although other accurate level spacing distributions have been proposed, Wigner s law is the ost widely used is suitable for practical applications. 2.2 RESONANCE WIDTH DISTRIBUTION LAW Systeatic easureents of the resonance widths show strong fluctuations aong resonances of the sae angular oentu parity. The definition of resonance width involves two other quantities, naely the reduced widths, γ λc, the penetration factor, P c, which are related according to the equation Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

3 A New Approach for Calculating Average Cross Sections Γ λ ' jc (2P c ) γ 2 λc, (4) where λ refers to the energy levels in the copound nucleus c refers to the particle channel. One should expect that the fluctuations are connected to either the reduced widths, γ λc, or to the penetration factors, P c. However, it is iprobable that the fluctuations are due to the penetration factors since they are sooth functions of energy. Therefore, the observed fluctuations are caused by the reduced widths, γ λc ; these, in turn, are related to the projection of the eigenfunctions of the Hailtonian of the copound nucleus on the nuclear surface. This projection involves an integration of any uncorrelated contributions, positive negative, over the high-diensional phase space of the copound nucleus. It then follows fro the central liit theore that the distributions of γ 2 λchave a Gaussian distribution with zero-ean. Therefore, the distribution function of the reduced widths can be written as P(γ λc ) d γ λc ' γ 2 λc exp(& ) d γ 2 π <γ 2 λc > 2<γ 2 λc > λc, (5) where <γ 2 λc > is the average value of γ2 λ c. The probability distribution function of the resonance widths, Γ λ,can be derived fro Eq. (3) as follows: The statistical theore states that if y is a variable that is the su of squares of ν norally distributed zero-ean independent variables, then y is distributed according to a χ 2 distribution with ν degrees of freedo. Therefore, the distribution of Γ λ is p ν (x) dx ' ν 2 G(ν/2) (ν x/2) ν 2 & exp(&νx/2) dx, (6) where x ' Γ λ /<Γ >, G(ν/2) is the atheatical gaa function, <Γ > is the average value of the width taken over a given energy range. For ν ', Eq. (6) is the well known as the Porter- Thoas[6] distribution law of the neutron width. It is generally accepted that fission is a fewchannel process, that there are only a liited nuber of effectively open channels; 2 or 3 degrees of freedo (ν ' 2orν ' 3) are usually assued in the fission width distribution. In the neutron capture event, a large nuber of capture channels are opened; the gaa width distribution is represented by a χ 2 distribution with a large nuber of degrees of freedo ( ν64),which corresponds to a Dirac-delta function centered at Γ γ ' < Γ >. The χ 2 distribution function has the following property: Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

4 L.C Leal M. E. Dunn 0 4 pν (x) dx ' 0 4 xpν (x) dx '. (7) The second oent of a χ 2 distribution with νdegrees of freedo is given as x 2 ' 0 4 x 2 p ν (x) dx ' 2 ν %. (8) 2.3 DYSON AND MEHTA LONG-RANGE CORRELATION OF 3 STATISTICS TEST The 3 statistics test, introduced by Dyson Mehta, [7] provides a easure of the ean-square deviation between the nuber of observed energy levels in the energy interval to the best fit to the straight line, as a function of energy, given as ae% b. Strictly speaking, the definition is 3 ' Min (a, b) N(E) & ae&b 2 de, (9) 2 L where N(E) is the corresponding cuulative nuber of energy levels as a function of energy. The Dyson Mehta 3 test predicts that the theoretical average value < 3 > is given as < 3 > ' π 2 ln(n) & , (0) with variance V 3 '.69/π 4. Here n is the nuber of energy levels observed in the interval to. For practical applications, the coefficients a b in Eq. (9) are deterined according to the following conditions: M 3 M a ' 0, () Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

5 A New Approach for Calculating Average Cross Sections M 3 Mb ' 0. (2) These conditions lead to the following equations: a E 2 de % b EdE ' N(E) de, (3) a EdE % b de ' N(E) de. (4) The following identities will be used in evaluating a b: de ' E f &, (5) EdE ' (E 2 f & E 2 i )/2, (6) E 2 de ' ( E 3 f & E 3 i )/3. (7) If the energy levels in the range to are nubered fro l '&L to l ', then the following relations also hold: N(E) de ' j E l% E l lde' j l (E l% & E l ), (8) Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

6 L.C Leal M. E. Dunn N(E) EdE ' j E l% E l lede ' j l ( E 2 l% & E 2 l )/2, (9) N 2 (E) EdE ' j l 2 (E l% & E l ). (20) The syste of Eqs. (3) (4) can be written as α a % β b ' γ (2) α 2 a % β 2 b ' γ 2, (22) in which the Greek sybols are defined as α ' (E 3 f & E 3 i )/3, (23) α 2 ' β ' (E 2 f & E 2 i )/2, (24) β 2 ' &, (25) γ ' jl l (E 2 l% & E 2 l )/2, (26) γ 2 ' jl l (E l% & E l ). (27) Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

7 The solution for a b is then A New Approach for Calculating Average Cross Sections a ' γ & γ 2 β / β 2 α & α 2 β / β 2, (28) b ' γ 2 β 2 & α 2 β 2 γ & γ 2 β / β 2 α & α 2 β / β 2. (29) Substituting these definitions into Eq. (9) leads to the expression for the 3 test: 3 ' & N 2 (E) de & γ a & γ 2 b, (30) or 3 ' & j &L l 2 (E l% &E l ) & γ a & γ 2 b, (3) where a b are given by Eqs. (28) (29), γ γ 2 by Eqs. (26) (27). 3. SAMPLING TECHNIQUE TO GENERATE RESONANCE ENERGY AROUND THE REFERENCE ENERGY For a reference energy E, as tabulated in the Evaluated Nuclear Data Files (ENDF), series of resonance surrounding the reference energy is selected fro the probability distribution P(x)dx'xW(x)dx (32) where W( x) is the level spacing distribution law given in Eq. (Wigner distribution). The saple variable x is defined as the ratio between the level spacing D the average level spacing <D> ( x = D/ < D > ). The separation between the first two resonances around s sapled fro Eq. (32) pair of resonances are assigned above below E respectively as, E+ sd E+ ( s ) D. The value s is a ro nuber between 0 is deterined fro a unifor distribution as p(x)' dx s (33) Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

8 L.C Leal M. E. Dunn This procedure is repeated ore resonances are included around the reference energy. The total nuber of resonances (pairs of resonances) around the reference energy is arbitrary. While this ethod is very efficient for generation of the values needed for creating probability tables for cross section calculations it was originally developed for testing interference effects of the elastic channel in the cross sections. The long range interference effect in the elastic channel ay require large nubers of resonance pairs around the energy of interest for the deterination of the average cross sections. The resonance energy sequence saple as described is also tested with the Dyson Mehta 3 test as indicated in Eqs. (3) (4). The 3 test was successfully used in the evaluation of the 235 U cross sections.[8] It is also included in the coputer code SAMDIST[9] which is used for calculating statistical distribution of R-Matrix resonance paraeters. To test the unresolved resonance ethodology in the coputer code URR, coparisons of the experiental average neutron cross section for 235 U in the energy region fro 3 kev to 20 kev is shown is Figure. The upper curve represents the total cross section easured by Harvey et al.[0], whereas the iddle botto curves are respectively the fission capture cross sections of Weston et. al.[] Solid curves are the average calculations with URR the vertical lines are the experiental data. The average input data used in the calculations are of an ongoing 235 U unresolved evaluation done at ORNL. The URR result are excellent. Figure. Coparisons of average cross sections calculated with the URR coputer code with experiental data for 235 U. Upper curve is the total cross section, iddle botto curves are respectively fission capture cross sections. Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9

9 A New Approach for Calculating Average Cross Sections 3. CONCLUSIONS This work describe a novel approach for calculating average cross sections based on the probability table ethod. Two new features are established in the way the energy level spacing of the resonances are sapled. The ethod has been ipleented a coputer code naed URR has also been included in ajor coputer codes, such as RACER AMPX. Coparisons of the average cross sections calculated with URR for 235 U are copared with experiental results the results are excellent. ACKNOWLEDGMENTS This work was sponsored by the Office of Environental Manageent, U. S. Departent of Energy, under contract DE-AC05-00OR22725 with UT-Battelle, LLC. REFERENCES. L. C. Leal, G. desaussure, R. B. Perez, URR Coputer Code: A Code to Calculate Resonance Neutron Cross-Section Probability Tables, Bondarenko Self-Shielding Factors Self-Indication Ratios for Fissile Fertile Nuclides, ORNL/TM-297, Martin Marieta Energy Systes, Inc., Oak Ridge National Laboratory (August 989). 2. F. B. Brown T. M. Sutton, Monte Carlo Fundaentals, KAPL-4823, Departent of Energy - Office of Scientific Technical Inforation (996). 3. T. M. Sutton F. B. Brown, Ipleentation of the Probability Table Method in a Continuous-Energy Monte Carlo Code Syste, International Conference on the Physics of Nuclear Science Technology, p , Long Isl, New York, October 5-8, M. E. Dunn N. M. Greene, AMPX-2000: A Cross-Section Processing Syste for Generating Nuclear Data for Criticality Safety Applications, Trans. A. Nucl. Soc., 86, 8-9 (2002). 5. E. P. Wigner, Conf. On Neutron Physics by Tie-of-Flight, Gatlinburg, TN, 956, ORNL-2309, Union Carbide Corp., Nucl. Div., Oak Ridge National Laboratory, 957, pp C. E. Porter R. G. Thoas, Fluctuations of Nuclear Reaction Widths, Phys. Rev. 04, (956). 7. F. J. Dyson M. L. Mehta, Statistical Theory of the Energy Levels of Coplex Systes, J. Math. Phys. 4, 70 (963). 8. L. C. Leal, G. desaussure, R. B. Perez, An R-Matrix Analysis of the 235 U Neutron Induced Cross Sections up to 500 ev, Nucl. Sci. Eng. 09, -7 (99). 9. L. C. Leal N. M. Larson, SAMDIST: A Coputer Code for Calculating Statistical Distributions for R-Matrix Resonance Paraeters, ORNL/TM-3092, Lockheed Martin Energy Research Corporation, Oak Ridge National Laboratory (Septeber 995). 0. J. A. Harvey, N. W. Hill, F. G. Perey, G. L. Tweed, L. C. Leal, Proc. Int. Conf. On Nuclear Data for Science Technology, Mito, Japan (May 30-June 3, 988).. L. W. Weston, J. H. Todd, Nucl. Sci. Eng., 88, 567 (984). Aerican Nuclear Society Topical Meeting in Matheatics & Coputations, Gatlinburg, TN, /9