: : : . I OFI ORNL P 7:48 .. . > . . - ' EFE 1:33:33 ... f MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 war - : : This paper was submitted for publication in the open literature at least 6 months prior to the issuance date of this Micro- card. Since the U.S.A.E.C. has no evi- dence that it has been published, the pa- per is being distributed in Microcard form as a preprint. emo rie 9.. wees. La Na d . . + . . amer WSV NPC Whi WO "E ' . . *;-sern. . .- 1.. ... ver mer to com . . .. op 2 . . 1 LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representa- tion, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, appa- ratus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, "person acting on behalf of the Commission" includes any em- ployee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employ- ment or contract with the Commission, or his employment with such contractor. E . - .. - - A tua DRAL-P-448 MOTIF 1071F-8 DEC 30 11 ONE-VEIDCITY MONTE CARLO CALCULATIONS OF URANIUM METAI. CRITICAL GEOMDTRIES* ORNL P-740 * MASTER J. T. Mihalczo Oak Ridge National Laboratory Oak Ridge, Tennessee A wide variety of unmoderated and unreflected critical ex- This paper was submitted for publication in the open literature at least I forths prior to the issuance date of this Micro- card. Since the V.8.A.E.C. had no evi- donce that it has been publishod, the pa- per ir being distributed in Microcard form as a preprint. aloe s..* that tron dhe port may not bongo y en: fach- sponsored wort, Matches the Daited me contractor, to the action sployee of media contractor propera, bolo pogre or contract NW m Coc voch LEGAL NOTICE - y perman acting a bhall al do Cond A. Makes may narraty or reprowatation, arener or lapte, ma rupect to do CP. pero b The report we prepared um wood of Gover moy, completenes, or watalow of the taformation contatood bade mport, or des do w a of my taformation, appuntua, medhoda or proouw dieclosed in the my Vaduties no repect to the wool or for ong taormation, appartous, medhodha os procrus deolowed ba de report bomo, "perna mot mod ceploys or contractor of the Commission, or Hates, nor the cooleston, nor ploge or contractor of phe Coaunintonor destabios, or proridos noc.. to, ar teormation med ; or wa the cow.usdoo, ur Me employat va mucha contractor. a pointedy ond nowd te B. A w 11 geriments with geometrically complicated configurations of 93.2% 2354-enriched uranium metal (p = 18,75 g/cc) have been completed.2-3 Some of these experiments have been analyzed by Monte Cario methods with as detailed a treatment of the energy as the cross-section in- formation will allow. For calculations of multiplication constants only, however, a simpler one-velocity treatment is adequate, pro- viding that the geometry is handled exactly. The one-velocity calculations assume isotropic scattering and no nonproductive absorption. The input data required are the collision A Niz cross section and the production per collision, the latter compensating for the leakage. These are determined from the critical radius for a sphere, in mean free paths as a function of production rate per col- lision, obtained by an exact solution of the Boltzmann transport equation in one dimension, together with the critical radius de- APPROVED FOR PUBLIC RELEASE termined from a critical experiment in spherical geametry (Godiva). Since the enrichment (93.2 wt 2350) and the uranium density in the critical assemblies (o = 18.70 g/cc) for which these calculations are being made are slightly different than those in Godiva (93.8 wt% -330; p = 18.75 g/cc), small corrections were necessary. *Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. -2- The calculations were performed with the 05R Monte Carlo code which has a geometry routine that divides space into parallelepipeds and describes the material boundaries within the parallelepipeds by quadratic functions. The functions used for each parallelepiped can be independent. The geometries that were calculated were cylinders, parallelepipeds, cylindrical onnuli, combinations of cylindrical annuíi and cylinders or parallelepipeds, arrays of up to 64 cylinders arranged in cubic lattices, and one cabination of cylinders, parallelepipeds, and a hemisphere, ali of which have been made critical experimentally. The multiplication constant was computed by two methods: (1) by cal- culating the ratio of the number of neutrons produced from the source neutrons, and (2) by dividing the Iuel material into a number of regions and computing the probability that a neutron born in each region will produce a fission in all other regions. The largest eigenvalue of the matrix formed by these probabilities is the multi- plication constant. The results of the first method of calculation are given in Table I for all assemblies, along with some results of the second method. Considering the simplicity of the calculational method, the agreement between the calculated and experimental results is good and indicates that the one-velocity method may be used to predict the multiplication constants of critical experiments with complicated geometries from the results of a spherical experiment. - - . . . . ... ... .. .- .. Table I. One-Velocity Multiplication Constants for Uranium Metal Critical Geometries Geometry Multiplication Constanta Method 1° Method 2 Cylinder, 38.10-cm-diam x 7.65-cm-high 1.025 Two Coaxdal Cylinders . Bach 38.10-cm-diam x 6.04-cm-high, flat faces 12.27 cm apart Each 17.78-cm-diam x 7.31-cm-high, flat faces 0.86 cm apart Cylindrical annulus; 38.10-cm-OD, 22.86-cm-ID, 14.98-cm-high 1.013 0.986 1.022 -----------... . 1.021 1.014 0.992 .. Parallelepiped, 12.70 x 12.70 x 23.19 cm Two Parallelepipeds Each 20.32 x 25.40 x 7.94 cm, large flat faces 12.45 cm apart Each 20.32 x 25.40 x 5.08 cm, large flat faces 0.97 cm apart Cylinder and Cylindrical Annulus, Each 10.1l-cm-high 1.014 0.994 1.008 0.994 .-..---------- 0.995 Annulus: 38.1-cm-OD; 27.94-cm-ID Cylinder: 17.78-cm-diam Table I (cont.) Table I (cont.) . Geometry Multiplication Constanta Method 2 Method 1b Cylindrical Annulus and Parallelepiped, Each 12.98 cm high 1.002 Annulus: 38.10-cm-OD; 27.94-cm-ID Paralielepiped: 12.70-cm side length Cylindrical Annulus and Two Parallelepipeds 1.004 Annulus (two sections combined): 38.10-cm-OD, 27.94-cua-ID, 12.98 cm high Upper Section Parallelepipeds: 12.70-cm side length; No. 1, 7.62 cm high; No. 2, 11.18 cm high Lower Section .... . ... Table I. (cont.) Gecmetry Multiplication Constant Method 10 Method 2 (cm) Arrays of Cylindrical Units, Cubic Lattices Unit Unit Surface Array Diameter Height Separation (cm) (cm) 2 x 2 x 2 11.49 8.08 0.90 0.971 11.51 10.77 2.25 0.997 : 0.993 3 x 3 x 3 1.51 5.38 2.01 1.003 11.48 10.77 6.36 0.998 0.993 4 x 4 x 4 11.51 5.36 3.95 0.972 4 x 4 x 1 11.49 10.77 1.52 0.988 2 x 4 x 2 21.49 10.72 3.89 0.993 2 x 2 x 2 Arrays of Three Adjacent Coaxial Cylinders, Cubic Lattices , An 11.45-cm-diam x 5.38-cm-high cylinder between two 9.16-cm-diam x 4.32-cm-high cylinders 1.0115 A 9.12-am-diam x 4.32-cm-high cylinder between two 11.49-cm-diam * 2.69-cm-high cylinders 0.977 0.972 Eight mits of various shapes arranged in a circle and an irregularly 1.017 1.017 shaped centerpiece 0.997 a. 12000 neutron histories in 30 batches. b. Standard deviations vary from + 0.009 to + 0.014; the average of the multiplication constants for the different geametries is 0.9992. C. Description of the units, clockrise around the circle: (1) a 9.12-cm-diam x 12.98-cm-high cylinder; (2) an 11.48-cm-diam x 13.46-cm-hich cylinder; (3) & 9,12-cm-dian x 12.98-cm-high cylinder; (+) a 7.62 x 12.7 cm paralleiepiped 8.91 cm high with a 9.12-cm-diam x 4.32-cm-high* cylinder or top; (5) a 9.12-cm-diam x 2.98-cm-high cylinder; (6) an 11.48-cm-diam x 13.46-cm-high cylinder; (7) a 9.12-cm-diam x 16.98-cm-high cylinder; and (8) a 12.7 x 12.7 cm parallelpiped whose height varied in three steps (13.05, 13.38, and 12.15 cm). d. The centerpiece was an 1.113-chi-diam x 2.82-cm-high cylinder topped by a parallelepiped with & 12.70 x 12.70 cm base and a 5.72-сm height and by a hemisphere with a 6.07-cm radius. RATORETICES G. E. Hansen, Physics of Fast and Intermediate Reactors, Proc. OP. Seminar, Vol. , p. 453 IAEA, Austria (1962). John T. Mihalczo, Nucl. Sci. and ing. 20, 60-65 (1964), J. T. Thomas, Critical Three-Dimensional Arrays of Neutron- Interacting Units, Part II - U(93.2) Metal, ORNL-IM-868, July 1964. J. T. Mihalczo and D. C. Irving, Trans. And. Nucl. Soc. November 1964. B. G. Carlson and G. I. Bell, Proc. Intem. Conf. Peaceful Uses of Atomic Energy, and Geneva, 1958, 16, 535-49 (1959). R. R. Coveyou et al., 05R, A General Purpose Monte Carlo Neutron Transport Code, ORNL-3622 (in pre88). 6. November 13, 1964 : - . г. . . и . | . 1 . ки. - . i х .:А - + - •9. - ** Е. . : . 1 . " . 5., • te * * — ., , . . " : 3 .. + 2. т “ 1 . . 1. 2. - дог. . • . . ::":". са с . Ако ? .. .09 ар . . н• г :, • • 4 . ••. • се е ок. . .:: 1 1 4 . . 1 к . , : . . • Photo 66028 : : А . . ке 1 ін не " s 5, : . . .. . . . . , и . ••• ... : : 2. см.; ,, не . си :: Діагние: . * . :: . арга . 1 . Epir. д . 5 л . н . , ко, ки. не . 1 : ) А А, . А : . •6 АА Eril htter: 1. • . : с.::: ди. R 1 E : 4 : :: x Жуму 70AAA995.yo . . 6 " ::: . . 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