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PARAMETRIC REPRESENTATION of the SHIELDING FACTOR CURVES Is Rat 3 a Tomic E Nergy Go

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PARAMETRIC REPRESENTATION of the SHIELDING FACTOR CURVES Is Rat 3 a Tomic E Nergy Go IA-1295 pSKI S ©^cu^SKiii I __.. PARAMETRIC REPRESENTATION OF THE SHIELDING FACTOR CURVES Y. GUR and S. YIFTAH is rat 3 Atomic Energy Go mm isnon PARAHE'IRIC REPRESENTATION OF THE SHIEIDING FAC '.'OR CISVhb ^i- SAL and S- YiiLah ed in pare by GlK, K«rnl jeschungsi'-entram, K.irlsruhe, Gurm;my Israel Atomic Energy Commission January 1974 LCS9TENTS Page INTRODUCTION 1 THE CONCEPT OF A SELF-SHIELDED MULT1GROUP CONSTANTS SET 2 TEMPERATURE INTERPOLATION 4 BACKGROUND CROSS SECTION INTERPOLATION 6 PARAMETRIC REPRESENTATION OF TEMPERATURE DEPENDENT SELF- SHIELDING FACTORS 11 Hyperbolic Tangent Representation 12 Tangent Representation 13 CALCULATIONS 14 Accuracy 15 REFERENCES 17 APPENDIX 18 PARAMETRIC REPRESENTATION OF THE SHIELDING FACTOR CURVES Y. Gur and S. Yiftah ABSTRACT Two new methods for a parametric represen­ tation of the temperature dependent self-shielding factor curve are given. The concept of the self- shielding factor is described in detail and current conventional methods of interpolation between two tabulated values of the shielding factor curve are reviewed. Two methods for the parametric represen­ tation of the curve are suggested. Of the two, one is found to fit the data very well. A complete list of parameters for the temperature dependent self-shielding factor curves, using the better method, is given for U-235, U-238, Pu-239, Pu-240, Pu-241, and Pu-242. INTRODUCTION With the publication of the Bondarenko cross sections1" in 1964, the shielding factor method for the generation of multigroup constants (?) received wide recognition. Since then, the methods have been extended and use of the shielding factor approach has steadily increased primarily due to its speed and ease of application. According to the concept of the self-shielded raultigroup cross section set (described in tiie next section) a typical set can be divided into two parts: infinitely diluted cross sections and scattering matrix elements, which are constants, and self-shielding factors (ssf) which are curves given in tabular form- Since the ssf are tabulated curves of two variables, temperature and background cross section, the user must interpolate i between given values. There is as yet no single or accepted solution to the problem of interpolation, which is reviewed in succeeding sections. Parametric representation is a natural means of denoting a curve. Two methods for parametric representation of the ssf curvee are given, both of which use three temperature dependent parameters. Two sets of these parameters (a total of six parameters) completely define the temperature dependent self-shielding factor curve per energy group. Details of computation and use of these parameters are given for both methods- Of the two methods, one is much more effective than the other in fitting the analytic curve to the data. Calculations performed in connection with the better representation and its accuracy are discusse. in the last section. In the Appendix, a complete list of parameters of the temperature dependent, ssf curve, represented by the better method, is given for U-235, U-23S, Pu-239, Pu-240, Pu-241 and Pu~2->2, A table, showing the accuracy of the fit obtained using the parameters given in the Appendix, is available, upon request, from the authors, THE CONCEPT OF A SELF-SHIELDED MDLTIGROUP CONSTANTS SET Only a brief outline of the shielding factor method will be given here as a complete and detailed exposition can be found in Rafs. (1) and (4). First, shielding factors and infitnitely diluted cross sections are generated for all isotopes of interest. The shielding factor is defined by the equation f - ?\ (1) where: o effective resonance self-shielded cross section of x,g type x in group g <o > infinitely diluted cross section of the same reaction, namely: r hg j av(E)4>(E)dE — = -J& (2) x,g I "B lg - 3 - E. , E, == lower and higher energy boundaries of group g 4>(E) «• weighting flux, defined by <P(E) = l/J:tlE) (3) for self-shielding factors calculation In zhe resonance region this (4) definition of the weighting flux is more realistic than the one fl 31 commonly used ' J #(E) = 1/E £ (E) where E is energy and 2(E) is the macroscopic total cross section of the mixture. I (E) is now approximated as follows; E <E> = N(0 (E) + a ) (A) where: N = atom density (atoms/cc) of the isotope considered a (E) = total microscopic cross section of the isotope considere c = constant background cross section that represents the mixture by a=4 I N- o~ - (5) other isotopes where: N. = atom density of the i-th isotope o . = effective total cross section of the i-th isotope, in group g. For each isotope, shielding factors and infinitely diluted ::OFS sections are computed for each energy group and reaction type. The sei shielding factors are computed for selected temperatures and background cross sections. Usually, three temperatures and up to eight background cross sections are considered. The generated tables are used for the calculation of mixture (4) dependent effective cross sections as follows : - It _ Obtain, for heavy elements 0a . -• U(X - N.... o. ,)/N. (6) or, for light and structured elements: o • * U££) - N. 5. CJ .J/N. c <7) o,g,i t,g, i si t,g,i li a . = background cross section for the i-th isotope in energy group g I = total macroscopic cross section of the mixture in group g- C. = average lethargy gain per collision for the i-th isotope. C^t,g " I Hi Sl "t.g.l For the first iteration the effective total microscopic cross section is replaced by the infinitely diluted cotal microscopic cross section. In most cases * , Eq (6) is used for all elements. After o . has been determined, for group g and isotope i , the shielding factors are obtained from the tables. To date, as shielding factors are given in tables only for certain temperatures and background cross sections, interpolation muse be used in intermediate cases. TEtlPERATURE INTERPOLATION Below are two simple schemes which may serve for interpolation of shielding factors , i> in T , as well as for extrapolation, provided the temper <•• cure is not toe far from the range of tabulation. a) fCc0,T) * a(o ) In T + b(o ) b) ln(f (c ,T>) = a(o ) In T + b (o ) Given f(o ,T.) and f (o ,T_) , f(a ,T) is expressed an followb 1V1 0 1 o 2 o scheme "a": and for scheme "b": L(T) f(VT) = f£ao,TI)-[f(ao,T2Vf(00.T1)) LCT) = ln(T/T VlnOyT^) It was established that both methods can be used for the temperature interpolation as follows, f (a .,T.) was calculated for all types of x,gx 0,1' 2 reactions and all fissile and fertile isotopes. All groups in the 25 Bondarenko energy group structure, and seven background cross sections (0 . = HPi-2\ where i=l,...,7) for three temperatures (300, 900 and o ,1 15001:) were considered. The input data was taken from Ref. (4).f(c .,300) and f(a .,1500) were considered as f(a .,Tn) and f(o . ,T0) from which A 0,1 o,i 1 o,i 2 f (a ,1=900) was calculatedfby Eq. (8) and Eq. (9). A The relative deviation of f (a .,T-900) from f(a .,900) was o,x o,i ^ calculated. It was found that in general the deviation of f and the maximum deviation calculated by the first method are somewhat larger than those calculated by the second method. In Table 1 the maximum deviation for each f of each isotope is x given. The basic data of Pu-240 in ENDF/B-III shows that Pu-240 has a wide resonance whose center is in group 23 and whose wing contri­ butes most of the cross section in group 24. This explains the anomalously large deviations in the behavior of the shielding factors found in these two groups. fce designated f to be the value obtained by interpolation to distinguish it from f obtained directly from tables in Ref. 4. - 6 - TABLE 1 Maximum relative deviation of Interpolated shielding factors from directly calculated shielding factors at 900°K Maximum relative deviation (percent) of Isotope total elastic f !f fission . capture method method method method a b a b a b a b U-235 2 2 0-2 0.2 2.5 2 3.2 2.6 U-238 2.2 1.5 1.55 1.2 -- 3.6 3.6 Pu-239 1.6 1.4 0.4 0.4 3 2.6 3.3 2.6 Pu-240* 2.25 2 1.5 1.5 4.7 3.2 4.4 3.9 Pu-241 1.2 1 1 1 1.45 1.35 2.7 2.4 Pu-242 2.A 2.1 1.1 1 1 - - 3.8 3.2 * In groups 23 and 24 of Pu-240 Che maximum relative deviation is in the order of 10-16%. The second method is setter in these cases also. BACKGROUND CROSS SECTION INTERPOLATION f 3—8) Various authors suggest different o interpolation schemes, (7) ° Segev gives a dense grid of o , which differs from element to element (e.g. a = 10, 50, 100, 300, 600, 1000 for Pu-239; a = 100, 300, 600, o o 1000, 3000, 6000 for Pu-240) so that any interpolation would be satis­ factory. (4) Gur, Yiftah and Segev give tables of shielding factors, f(o ), with a recommended interpolation scheme of log (f(a )), linear in log (oo). Hardie and Little recommend, and use in 1DX, an interpolation scheme of f(a ) linear in log (a ), and suggest f(10 )=1 to be used for cross section in the shielding factor cables.
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