Cullen, D.E.

Ozer, O.

Weisbin, C.R.

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Thie u a preprint of a paper intended for publication in a journal or proceeding a. Since changes may be made before publication, thia preprint i* made available with the understaoetinf that Ji will not be cited or reproduced without the permiaaion of the author. UCRL - 76629 PREPRINT Si.i:JM30J---/f MASTBI LAWRENCE UVERMORE LABORATORY Tabular Cross Section File Generation and Utilization Techniques D. E. Cullen, 0. Ozer, C. R. Weisbin March 1975 This paper was prepared for submittal to Proceedings of Conference on Nuclear Cross Sections and Technology March 3-7, 1975, Washington, D. C. TABULAR CROSS SECTIOH FItE GENERATION AND omiOTrOri TECIWIWES D.E. Cullen, Lawrence Livermor* laboratory, l ivemore, California 0. Ozer, Electric Power Research Institute, Palo Alto, California CO. Welsbin, Oak Ridge Hational Laboratory, Oak Ridge, Tennessee Criteria of Importance to the generation of linear cross section tabulations are presented. Algorithms for reducing or thinning such tables within a desired accuracy criterion and the irapltmentation of paging techniques for efficient utilization of large data tables are reviewed. (Cross sections, tables, pre-processing, relative error, l in rtzatlon. thinning, paging.) Introduction for «any applications i t is necessity to reduce the ENOF/B data f i les Into a tabular fora with a linear functional variation specified between the data paints. Such tabular data bases are presently In use at a number of national laboratories. Tabular representation of the files greatly simplifies the requirements for computer programs util izing such data. The straight forward task of displaying a cross section, which may require a rather expensive resonance calculation when the base ENOF/8 r i le is used »s the starting point, is reduced t i a simple table look-up and interpolation problem for tabular f i l es . Great reductions in computing time can be realized in various codes (such as. the point Monte Carlo Code REP1) which have been designed to do repeti­tive crass section calculations at a large number of energies. When generated at a temperature of O'K data tables can be easily Ooppler broadened to any desired tempera­ture using numerical kernel broadening techniques.*' Numerical broadening has the advantage of not being restricted to the "resonance range" and not ignoring the contribution of the "smooth" background correction terms which in fact often exhibit considerable struc­ture. These advantages become particularly significant 1n the case of high temperature or CTR applications and for light materials for which no explicit resonance parameters are given on the ENDF/B f i l e . The generation and use of such tabular fi les requires a number of special considerations: Tabular f i les are generated within a specified control­led accuracy criterion. {The accuracy is defined as the fractional deviation from the "true" value which would be obtained from the ENDF/B f i l e . ) Thus at any stage the upper bound on the processing error is known. The accuracy criterion must be selected in a man­ner that wi l l insure the adequacy of the tabular files In the widest range of applications yet result in the smallest number of points. Typically tables with 10000 to 40000 points per reaction type are generated when major isotopes are reconstructed at 0°K with an accuracy of . IX. As the experimental resolution is improved and the unresolved resonance range is further reduced3 f i les with 150000 to 200000 points should be expected. rttxMxl *ee«f i t j BuaCrr «r palm* U t** U**l»ttm &»5 stm | ft 0-5 W M zm 1 1.0 msi •M j U I S *-Q wit %TT 1 J I M S.O H I ! yu 1 3100 Such high accuracy files should be considered as a •aster data base and need to be generated only once. For aany applies ions data with considerably lower accuracy may be sufficient. Thinning techniques are used to reduce the size of the tables depending on the required accuracy. Doppler broadened tables should be thinned even I f the accuracy is the sane because the smoothing effect of the Doppler process results in fewer paints being -equired to represent a function within the same acc-racy. Finally special paging techniques for utilizing tabular data f i les a ection (or page) at a time should be incorporate into codes operating on this data base since even . ith a virtual core capability i t is often more efficient to have control over the amount of data brought in core at any one time. Points to be considered during the generation, thinning and paging of tabular data flies tre discussed in the following sections. Tspperatur* z» 1000 Kueoer of points necessary to represent the U-2M total cross section within U fHtof/t- l» Cite) 12)53 nil 5040 Taala I I . SsoothlAo effect of the Ooppler process. Tafcle I . acttrwtnt* «r taalr >|I< M accaraqr Generation The major consideration in generating tabular data files is the selection of a proper convergence or accuracy criterion. (/hen reducing ENDF/B files into a linearized tabular form two problems are encountered: a) Conversion of cross section data which is already in tabular form but may have a logarithmic or semi logarithmic functional dependence between the tabulated points into a linear tabulation ^Linearization) ») Processing of data given in parametric form (e.g. resonance range data) into tables with an energy grid selectee1 in a manner to insure the fewest number of points for a desired accuracy. Both of these problems can be considered as approxi­mating • function f(E) defined for a l l E, by another 7(E) such that: ?(E<) s f(Ef) at the tabulated points Ci , and the deviation of f(E) In areas between the tabulated points is less than a specified accuracy criterion. For-inherently positive functions such as a cross section the convergence between tabulated points should be determined on the basis of the fractional difference / I.e. |f(E) - TIEII « c f t £ ) > * s oppose"! to *"• absolute difference |f(E) - 7 { f ) | < c. Criteria based on the fractional difference ere consistent with uncertainties 1n the experimental data and result In a more uniform distribution of data points in lex as well as high enss section areas. A numerical "interval halving" algorithm incor­porated into the code RiSEND' and compatible with the fractional difference criterion has been found well suited for the generation of tabular files in the resonance range as well as for the linearization of already tabular EHOF/B data f i les. This algorithm consists of the following steps: a) A set of in i t ia l energy "node points" is selected. For tabular data, nodes are located at the edges of nonlinear Interpolation ranges. For resonance data nodes are set at the peaks of resonances and mid-points between the peaks. b) The exact cross section f(E) is calculated at the node points and TTO is set equal to f (E) . c) The value of the approximate function f(E> at the mid-point between two nodes is obtained by linear interpolation and compared with the exact value f(E) at the same point. I f the fractional difference at the mid-point is larger than the desired convergence criterion the mid-point is defined to be a new node and the process repeated in the halved interval. When a difference smaller than the convergence criterion is reached the interval is assumed to have converged and the process is repeated between the next set of nodes. Flsura 1 . C«Mr«clo« of EMrgr C*14 la Fratua USDV. When generating tahular cross section files from resonance parameter data, the convergence test should be applied to each of the individual reaction types separately and not be limited to the total cross section, since variations in reaction types with a low cross section (e.g. Interference dips in elastic sidttering) Unu to be washed out when contributions from other, larger reactions a n added on. For the process of linearization of data tables an algorithm based on the analytic determination of the location of the maximum in the absolute difference |f(E) - f (E) | has been developed ..nd implemented in the code MINX.5 However i^e interval halving technique is found to be preferable since i t is dependent on the fractional error and is somewhat simpler and faster. An extension of the MINX algorithm to fractional errors has not been found to be practical. Thinning The thinning procedure is logically equivalent to the process of linearization since one starts with a function f(E) which is given as a table of values and a linear interpolation law between the tabulated values, and one tries to determine a function f (E) . such that: a) f(E) Is also a table of values and linear Interpolation laws. b) The fractional difference between the "exact" f(E) and the approximation f(E) Is less than some desired allowable error c for al l energies. I.e. |f(E) - T ( E ) | < c f (E). However an additional criterion 1s added: c) The tabulated values of f(E) must constitute the smallest possible sub set of the tabulated vales in f(E). Thus a l l values in the tabulation of f(E) that an be approximated (to within c) by linear interpolation between adjacent points must be eliminated. The following multipoint algorithm has been developed for thinning data f i les. a) The f i rs t and third point of a table are selected and a test is made to determine whether point number two can be approximated by linear interpolation between points 1 and3. b) I f the test 1s successful the test interval 1s extended to include point 4 and again points 2 and 3 are tested to see whether they can be approximated by an Interpolation between points 1 and 4. c) Generalizing to N points, i f the points 2, 3, . . . (N-l) can be approximated by interpo­lating between points 1 and N the algorithm proceeds to N + 1 points. d) I f any one (or more) of the points 2, 3, . . . (N - l 1 fa i l the Interpolation test between points 1 and N the test interval has been extended too far. However since the previous test (with N-l points) must have been successful for the algorithm to reach this far al l the points between 1 and N-l can be removed leaving only the end points 1 and N-1. The algorithm is repeated with points N-l , N and n* 1 . The multipoint algorithm is considerably different from the "three-point" thinning technique used in the earlier versions of the ENOF/B system routine CROP.6 The three point algorithm operates by discarding point 2 i f i t can be interpolated between the adjacent points 1 am? 3 then proceeds to test point 3 by interpolating between points 1 and 4. I f point 3 is rejected there remains no guarantee that point 2 which was rejected earlier can be obtained by interpolating between the remaining points 1 and 4. This method produces particularly erroneous results when thinning points located very closely on a singly varying smooth curve. The multipoint thinning algorithm is not limited to linear tables but can be used with any interpolation law. A generalized,version has been implemented in a revised subroutine CROP-Two types of paging can be considered. 1 . Physical Paging I f one recognizes that only a section of a targe data table H i l l be used in a computation for a reasonable lengtH of time f. program may be intt ial ly designed to keep only a portion of the table (a page) core resident at any one tine. For applications requiring each portion of a table only once (e.g. group averaging, Doppler broadening) this physical paging •ay consist of simply allowing for the reading of a table a section at a time. Applications using dif­ferent portions of a table repeatedly may require the use of scratch flies or logic to randomly access different sections of a table. Z. logical Paoina Programs written without • paging capability can easily be converted to a logical form of paging with only minor coding modifications. Removing the name of an array to be paged from a l l COMHOK and DIMENSION statements Is equivalent to replacing that array by a function. On* only needs to add a function by the same name to determine which section of the data table the desired point is located I n , load that section i f not already resident in core and return the desired value from the table. Such logical paging has been Incorporated In code CHECKER and same versions af ET06-3' programs with a paging capability are not only more versatile but may actually be more economical to use. The size of a page should be selected depending on the type of computer environment the program is to be used in. a) In single user batch type of en environment 1/8 operation should be minimized by making the pages as large as possible within the available core. As the program is expanded, the page size can be reduced to allow I t to f i t In the same core. b) In a multi-user time slice environment page size 1s determined by the amount of data that can be used within a single time slice. Ideally the size of a page 1n a time sharing environment should be such that the program can be finished and ready to read the next page at the end of its time slice. 3. Page Overlap For applications requiring the use of two or more adjacent table entries simultaneously i t is necessary to structure the paging logic in such a manner that the last few points of a page are repeated at the beginning of the next page. The purpose of ' this overlap is to avoid the repeated reading of fages that would result when the program attempts to use points locatmi in •*•» adjacent pages. Conclusion The methods described in this paper have been Implemented into a number of computer programs. The programs given on Table I I I in combination supply al l the software necessary for processing the ENDF/8 data flies into a linear tabulation at any desired temperature, or to a multigroup cross section form. ft"!"-Miliar! INF/1 MATM-tiMrlMlilll data I«.i. tw-aei Rfcanitrvcti ftlntutlt tWF/a 4aU ttttta f>M AntnfMt •araawtir deia •aaalir hraadaw ltMtrlnf utaflar Doyi data Sjltrc fae tit* aiaaacHai, OsaiaetlflstltJ niiMiNt. Uftci r i l l * Uf iui r,|in| Hvalcil reflaf UawriilH, IMafttaf ftiyttul fa|lnt IftlMlf. PtotlCII P i l l * m l * I I I . t i l l df prdfraaa, Mttlf 'H taMar data files References Ml. Rothensteln, "Monte Carlo Code for the Calculation of Resonance Reaction Rates and Effective Resonance Integrals Based on ENOF/8 Data (REPCDC)" BM-13851 (1969). *0.£. Cull en. C.R. Weisbin, R. 0. Wright. J.E.White. "Comparison of Doppler Broadening Methods", Proceedings of this conference. *6. Oe Siassure, R.B. Perez, "Representation of Neutron Cross Sections in the Unresolved Resonance Region", Proceedings of this conference. *0. Ozer, "RESEND: A Program to Pre-process ENDF/B Materials with Resonance Files into a Pointwise Form" BNL-17134, Brookhaven National Laboratory (1972). s t .R. Weisbin, P.O. Soran, R.E. HcFarlane, O.R. Harris, R.J. LaBauve. J.5. Kenoricks, J.E. White. "MINX, a Hultigroup Interpretation of Nuclear Cross Sections from ENDF/B* To be published as Los Alamos National Lab. report. Also ASS Trans. 16, 127 (1973). «fl. Ozer, Ed. "Description of EHOF/B Processing Codes and Retrieval Subroutines," BNL-50300, (REACT-TECH HO 4500) EWF 110, Broofchaven National Laboratory (1971). 'M. Raymund, "ETOG-3 and ETOT-3, Advances in Processing EKOF/B-III Libraries" ANS Trans. J£, 323 (1973). •O.E. Cullen, "Program SCOPE" BNl-WOO (1972). 'D.E. Cullen "Program LINEAR" To be published as UC1D report. '"O.E. Collen "Program SIGHA1" UCID 16426 University of California Lawrence Livermore Laboratory (1974). *Work was perfonnsd under the auspices of the U.S. Energy Research and Developient Administration NOTICE "This report was prepared as an account of r*k sponsored by the Uniled Sraies Government, Neither the Limed Stale, nor Ihe Uniled Stale, Alomie Energy Commission, not any of idea employees, noi any at ihcir contractors, subcontractor*. o< their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for Ihcaccuracy.cempieteness or usefulness of any information, apparalus. product or process disclosed, or rcpiesenls that its use would not infringe privaiely-owned nejils" 

Tabular cross section file generation and utilization techniques

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Tabular cross section file generation and utilization techniques

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