NAME OR DESIGNATION OF PROGRAM, COMPUTER, NATURE OF PHYSICAL PROBLEM SOLVED, METHOD OF SOLUTION, RESTRICTIONS, TYPICAL RUNNING TIME, FEATURES, AUXILIARIES, STATUS, REFERENCES, REQUIREMENTS, LANGUAGE, OPERATING SYSTEM, OTHER RESTRICTIONS, NAME AND ESTABLISHMENT OF AUTHOR, MATERIAL, CATEGORIES

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Program name | Package id | Status | Status date |
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SUMMIT | NESC0056/01 | Tested | 01-JUL-1964 |

Machines used:

Package ID | Orig. computer | Test computer |
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NESC0056/01 | IBM 7090 | IBM 7090 |

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3. NATURE OF PHYSICAL PROBLEM SOLVED

The program evaluates the differential energy-transfer cross section for scattering by a crystalline moderator, utilizing the so-called phonon expansion. The scattering kernel for a 1-phonon change in energy is added to that for a 2-phonon energy exchange, and so on. This program has been used to determine scattering matrices for beryllium, graphite, and oxygen. Sigma(E(0) to E)/sigma(0)=(((M+1)/M)**2) * SQRT(E/E(0)) * 1/2 the integral from -1 to 1 of sigma(E(0) to E,cos(theta)) D(COS(theta)) where E(0) and E are the initial and final energies, theta is the angle of scattering, sigma(0) the free-atom cross section, and M the ratio of the mass of the scattering nucleus to that of the neutron.

The program evaluates the differential energy-transfer cross section for scattering by a crystalline moderator, utilizing the so-called phonon expansion. The scattering kernel for a 1-phonon change in energy is added to that for a 2-phonon energy exchange, and so on. This program has been used to determine scattering matrices for beryllium, graphite, and oxygen. Sigma(E(0) to E)/sigma(0)=(((M+1)/M)**2) * SQRT(E/E(0)) * 1/2 the integral from -1 to 1 of sigma(E(0) to E,cos(theta)) D(COS(theta)) where E(0) and E are the initial and final energies, theta is the angle of scattering, sigma(0) the free-atom cross section, and M the ratio of the mass of the scattering nucleus to that of the neutron.

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4. METHOD OF SOLUTION

The general method used to evaluate the scattering kernel involves two different expansions. The one that is to be used for given initial and final energies and for a given angle of scattering is determined by the magnitude of the free-atom recoil energy, (K*K)/2M. If (K*K)/2M is smaller than some quantity which is controlled by input, we use the phonon expansion. Each of the first N(PHO) terms in this expan- sion, where N(PHO) is an input number, is computed by numerically performing the convolution integral which determines the N-phonon cross section. The terms of order N greater than N(PHO) are approximated by means of the central-limit theorem. For larger values of (K*K)/2M, we use the short-collision-time approximation for the low-frequency modes, making the phonon expansion only for the high-frequency modes. In this case, the central-limit theorem is used to approximate the contributions from the high-frequency modes for all values of N.

The general method used to evaluate the scattering kernel involves two different expansions. The one that is to be used for given initial and final energies and for a given angle of scattering is determined by the magnitude of the free-atom recoil energy, (K*K)/2M. If (K*K)/2M is smaller than some quantity which is controlled by input, we use the phonon expansion. Each of the first N(PHO) terms in this expan- sion, where N(PHO) is an input number, is computed by numerically performing the convolution integral which determines the N-phonon cross section. The terms of order N greater than N(PHO) are approximated by means of the central-limit theorem. For larger values of (K*K)/2M, we use the short-collision-time approximation for the low-frequency modes, making the phonon expansion only for the high-frequency modes. In this case, the central-limit theorem is used to approximate the contributions from the high-frequency modes for all values of N.

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6. TYPICAL RUNNING TIME

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no. of energy temp. time* no. of

type points (deg. K) (centihr.) E(o) to E

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L-dependent 10 to 0.50 eV 300 4.6 55

isotropic

crystal 10 to 0.50 eV 300 1.9 55

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*total time to load = 36 seconds

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no. of energy temp. time* no. of

type points (deg. K) (centihr.) E(o) to E

-----------------------------------------------------------

L-dependent 10 to 0.50 eV 300 4.6 55

isotropic

crystal 10 to 0.50 eV 300 1.9 55

-----------------------------------------------------------

*total time to load = 36 seconds

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NESC0056/01

File name | File description | Records |
---|---|---|

NESC0056_01.001 | SOURCE & DATA | 3397 |

NESC0056_01.002 | OUTPUT | 5017 |

Keywords: beryllium, cross sections, crystals, graphite, moderators, oxygen, scattering.