A program for generating pointwise weighting functions

Ann. Nucl. Energy, Vol. 20, No. 9, pp. 605 609, 1993 Printed in Great Britain. All rights reserved

0306-4549/93 $6.00+0.00 Copyright © 1993 Pergamon Press Ltd

A P R O G R A M F O R G E N E R A T I N G POINTWISE WEIGHTING FUNCTIONS A. D. CALDEIRA and E. S. CHALHOUB Instituto de Estudos Avanqados (IEAv), Centro T6cnico Aerospacial (CTA), 12231-97(b--S~oJos6 dos Campos, SP, Brazil

(Received3 February 1993) Abstract--The ACES program, which generates pointwise functions for weightinggroup constants through

a combination of tabulated values and/or mathematical expressions,is presented. It is a powerful tool, due to its flexibility,especiallywhen tabulated values are required to describe the weighting function.

1. I N T R O D U C T I O N

Although the actual weighting function to be used for producing group constants from evaluated nuclear data files should be application dependent, i.e. for each case it should be obtained from a transport calculation, it is often possible to obtain accurate group constants for a particular application if the shape of the weighting function is reasonably well known. Along these lines, are the classical Maxwellian + l/E+fission spectrum which is used for weighting cross sections for thermal and epithermal systems and the Watt spectrum for fast systems. The ACES program (Caldeira and Chalhoub, 1993) was developed with the objective of generating a function, by a combination of tabulated values and/or mathematical expressions, to be used as a pointwise weighting function. This program, due to its flexibility, can be an important tool for researchers involved in the generation of group constants.

2. T H E ACES P R O G R A M

The program starts by calculating the continuity constants for joining the specified weighting functions. For the function specified in the first interval, the continuity constant is set equal to one and the remaining constants are obtained by imposing continuity of the weighting functions at the interface points between adjacent intervals in a sequential order. After this step, the area under each weighting function is obtained through trapezoidal integration, for pointwise options, or analytical integration for analytical options. The inverse of the total area is used as a normalization constant and the products of the continuity constants by the normalization constant are stored in the A array. Once the normalization step is completed, the program calculates the weighting function at the number of points per interval specified by the user. If, with this number of points, the accuracy criteria is not attained, the number of points per interval is increased by 40% for a new calculation, except for intervals where tabulated, 1/(tro+at(E)), and constant functions are specified. The execution terminates when the deviation from unity of the normalized area under the weighting function is within 0.1%. This strategy permits the user to choose the importance of an interval, i.e. if an interval is more important than another and consequently should be represented in more detail then its initial number of points should be greater than the number of points specified for the other intervals.

The ACES program calculates energy-dependent weighting functions utilizing tabulated values and/or mathematical expressions. The desired energy range can be divided into up to a maximum of 8 intervals (I), and a selected function can be assigned to each one. The functions to be used by the program are defined by input parameters and are stored in the 2.1. Built-in functions I F U N array. In each interval, the energy points are distributed The ACES program is prepared to accept 8 options. according to lethargy constant sub-intervals, which Depending on the needs of the user, other options can are obtained by dividing the interval by the number be introduced with little effort. The options are as of desired sub-intervals ( P O I N T S - 1). follows : 605

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2.1.1. Tabulated ( I F U N ( I ) = 1). For this option, the program reads the number of points given and the energy and corresponding weighting function tabulated values. The number of points for this option is not changed for a new calculation. 2.1.2. 1/(tro+tr,(E)) ( I F U N ( I ) = 2). For this option, the program reads the number of points given, the a 0 value, the energy and the corresponding total cross-section values. The number of points for this option is not changed for a new calculation. 2.1.3. Constant ( I F U N ( I ) = 3). For this option no other input data are required. The program attributes, internally, the value 2 to the number of points, which is not changed for a new calculation. 2.1.4. 1/E ( I F U N ( I ) = 4). For this option, the program only reads the number of desired points. 2.1.5. Maxwellian spectrum ( I F U N ( I ) = 5). For this option, described by f ( E ) = A(1)x//Eexp ( - E / C M X 1 ) ,

(1)

the program reads the number of desired points and the Maxwellian spectrum (Magurno, 1980) constant, CMX1 in units of eV. 2.1.6. Evaporation spectrum ( I F U N ( I ) = 6). For this option, described by

f ( E ) = A ( I ) E e x p (--E/CEV1),

(2)

the program reads the number of desired points and the evaporation spectrum constant, CEV1 in units of eV. 2.1.7. Watt spectrum ( I F U N ( I ) = 7). For this option, described by

f ( E ) = A(I) exp (--E/CWT1) sinh ~

,

(3) the program reads the number of desired points and the Watt spectrum (Magurno, 1980) constants, CWT1 and C W T 2 in units of eV and 1/eV, respectively. 2.1.8. Madland-Nix spectrum ( I F U N ( I ) = 8). For this option, described by

f ( E ) = 0.5A (I)[N(E, C M N I ) + N(E, CMN2)],

(4)

where

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X, = ( x / ~ - ~ N ) 2 / C M N 3

(6)

X2 = ( ~ F E + ~ / r ~ A I ) 2 / C M N 3 ,

(7)

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the program reads the number of desired points and the Madland-Nix spectrum (Madland and Nix, 1981) constants, CMN1, CMN2 and CMN3, in units of MeV. In equation (5), E1 represents the exponential integral and GI the incomplete gamma function.

3. RESULTS

To illustrate the versatility of the program ACES, Fig. 1 shows a graphical representation of 8 functions generated by ACES, Fig. 2 shows a graphical representation of the peaks of 4 fission spectrum weighting functions and Fig. 3 shows a Maxwellian + 1/E+ fission spectrum.

4. FINAL COMMENT The weighting functions generated by the program ACES can be utilized as input weighting functions for programs that perform group constants generation. As an example, in the case of the NJOY system (MacFarlane et al., 1982), the weighting function can be specified as input data through the variable IWT. If the value for this variable is set equal to one, the weighting function is read in as a smooth weighting function.

Acknowledgements--The authors wish to thank Roberto D. M. Garcia, head of the Nuclear Data Center, for his interest in this work and many helpful suggestions. Thanks are also due to Edwany Abranches Cavalcante Seito for her expert typing.

REFERENCES

Caldeira A. D. and Chalhoub E. S. (1993) ACES : a pointwise weighting function generator. Technical Note, IEAv/CTA, Brazil. MacFarlane R. E., Muir D. W. and Boicourt R. M. (1982) The NJOY nuclear data processing system, Vol. I : user's manual. Report LA-9303-M ENDF-324. Madland D. G. and Nix J. R. (1981) New calculation of prompt fission neutron spectra and average prompt neutral multiplicities. Report LA-UR-81-2968. Magurno B. A. (1980) Status of data testing of the ENDF/ B-V reactor dosimetry file. Report BNL-NCS 26760.