Computer Programs

NAME OR DESIGNATION OF PROGRAM, COMPUTER, DESCRIPTION OF PROBLEM OR FUNCTION, METHOD OF SOLUTION, RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM, TYPICAL RUNNING TIME, UNUSUAL FEATURES OF THE PROGRAM, RELATED AND AUXILIARY PROGRAMS, STATUS, REFERENCES, MACHINE REQUIREMENTS, LANGUAGE, OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED, OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS, NAME AND ESTABLISHMENT OF AUTHOR, MATERIAL, CATEGORIES

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Program name | Package id | Status | Status date |
---|---|---|---|

FPSPH/DFPSPH | NEA-0593/01 | Tested | 28-JUN-1982 |

Machines used:

Package ID | Orig. computer | Test computer |
---|---|---|

NEA-0593/01 | IBM 370 series | IBM 370 series |

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3. DESCRIPTION OF PROBLEM OR FUNCTION

In the computation of Doppler- broadened resonance cross sections, use is made of the symmetric and anti-symmetric line shape functions. These functions usually denoted as Psi and Phi (Psi and Chi in Anglo-Saxon formalism) are defined in terms of the real and imaginary parts of the error function for complex arguments. They are the product of the convolution of a Gaussian function with the symmetric and anti-symmetric Breit-Wigner functions, respectively. FPSPH and DFPSPH compute these functions.

In the computation of Doppler- broadened resonance cross sections, use is made of the symmetric and anti-symmetric line shape functions. These functions usually denoted as Psi and Phi (Psi and Chi in Anglo-Saxon formalism) are defined in terms of the real and imaginary parts of the error function for complex arguments. They are the product of the convolution of a Gaussian function with the symmetric and anti-symmetric Breit-Wigner functions, respectively. FPSPH and DFPSPH compute these functions.

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4. METHOD OF SOLUTION

For (1+x**2) > 20 Beta**2, the calculation is based upon the asymptotic expansion:

Psi+(i*Phi) = 1/(1-ix)*(1-t+3t**2-3.5t**3+3.5+7t**4---)

with: t = 1/(2z**2) z = (1-ix)/Beta

The half-plane (Beta,x) is split in several parts, and use is made of PADE approximants.

For 1 + x**2 < 20 Beta**2, the calculation is based upon the relation with the erf function:

Psi + i*Phi = SQRT(Pi)/Beta*(e**(z**2))*(1-erf(z))

(z = (1-ix)/Beta, and erf(z) being calculated from its analytic expansion:

erf(z) = 2/SQRT(Pi)*z*e**(-z**2)*(1+z**2/3+z**4/(3*5)

+z**6/(3*5*7)+---)

PADE approximants are used to compute the expansion and e**z**2.

For (1+x**2) > 20 Beta**2, the calculation is based upon the asymptotic expansion:

Psi+(i*Phi) = 1/(1-ix)*(1-t+3t**2-3.5t**3+3.5+7t**4---)

with: t = 1/(2z**2) z = (1-ix)/Beta

The half-plane (Beta,x) is split in several parts, and use is made of PADE approximants.

For 1 + x**2 < 20 Beta**2, the calculation is based upon the relation with the erf function:

Psi + i*Phi = SQRT(Pi)/Beta*(e**(z**2))*(1-erf(z))

(z = (1-ix)/Beta, and erf(z) being calculated from its analytic expansion:

erf(z) = 2/SQRT(Pi)*z*e**(-z**2)*(1+z**2/3+z**4/(3*5)

+z**6/(3*5*7)+---)

PADE approximants are used to compute the expansion and e**z**2.

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10. REFERENCES

- For FPSPH, DFPSPH,

Rapport interne SERMA-T-382-CEN-Saclay (revised November 1979).

- D.R. Mathews:

"FSDP3. A Computer Program for Computing Cross Sections for

Resonance Parameters"

Gulf-GA-A12052 (1972).

- H. Henryson II et al.:

"MC2-2. A Code to Calculate Fast Neutron Spectra and Multigroup

Cross Sections"

ANL-8144 (ENDF 239) (1976).

- For FPSPH, DFPSPH,

Rapport interne SERMA-T-382-CEN-Saclay (revised November 1979).

- D.R. Mathews:

"FSDP3. A Computer Program for Computing Cross Sections for

Resonance Parameters"

Gulf-GA-A12052 (1972).

- H. Henryson II et al.:

"MC2-2. A Code to Calculate Fast Neutron Spectra and Multigroup

Cross Sections"

ANL-8144 (ENDF 239) (1976).

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NEA-0593/01

File name | File description | Records |
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NEA0593_01.001 | FPSPH SUBROUTINE | 220 |

NEA0593_01.002 | DFPSPH SUBROUTINE | 246 |

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- A. Cross Section and Resonance Integral Calculations
- P. General Mathematical and Computing System Routines

Keywords: Breit-Wigner formula, Doppler broadening, Gauss function, cross sections, error function, resonance absorption, resonance scattering.