From ec93d1038313bad94fda413d0c74611eb59213c0 Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Mon, 23 Jul 2012 17:09:02 -0400 Subject: [PATCH] Started discussion of S(a,b) tables in documentation. --- docs/source/methods/physics.rst | 75 +++++++++++++++++++++++++++++++-- 1 file changed, 72 insertions(+), 3 deletions(-) diff --git a/docs/source/methods/physics.rst b/docs/source/methods/physics.rst index dc40924ec5..d556018d4f 100644 --- a/docs/source/methods/physics.rst +++ b/docs/source/methods/physics.rst @@ -137,15 +137,76 @@ and accepting it with probability It is normally assumed that the velocity distribution of the target nucleus assumes a Maxwellian distribution in velocity. ---------------------------------------- -S(:math:`\alpha`, :math:`\beta`) Tables ---------------------------------------- +------------ +|sab| Tables +------------ + +For neutrons with thermal energies, generally less than 4 eV, the kinematics of +scattering can be affected by chemical binding and crystalline effects of the +target molecule. If these effects are not accounted for in a simulation, the +reported results may be highly inaccurate. There is no general analytic +treatment for the scattering kinematics at low energies, and thus when nuclear +data is processed for use in a Monte Carlo code, special tables are created that +give altered cross-sections and secondary angle/energy distributions for thermal +scattering. These tables are mainly used for moderating materials such as light +or heavy water, graphite, hydrogen in ZrH, beryllium, etc. + +The theory behind |sab| is rooted in quantum mechanics and is quite +complex. Those interested in first principles derivations for formulae relating +to |sab| tables should be referred to the excellent books by [Williams]_ and +[Squires]_. For our purposes here, we will focus only on the use of already +processed data as it appears in the ACE format. + +Each |sab| table can contain the following: + +- Thermal inelastic scattering cross section +- Thermal elastic scattering cross section +- Correlated energy-angle distributions for thermal inelastic and elastic + scattering + +Note that when we refer to "inelastic" and "elastic" scattering now, we are +actually using these terms with respect to the *scattering system*. Thermal +inelastic scattering means that the scattering system is left in an excited +state, not any particular nucleus as is the case in inelastic level +scattering. In a crystalline material, the excitation could be the production of +phonons. In a molecule, it could be the excitation of rotational or vibrational +modes. + +Both thermal elastic and thermal inelastic scattering are generally divided into +incoherent and coherent parts. Coherent elastic scattering refers to scattering +in crystalline solids like graphite or beryllium. These cross-sections are +characterized by the presence of "Bragg edges" that relate to the crystal +structure of the scattering material. Incoherent elastic scattering refers to +scattering in hydrogenous solids such as polyethylene. As it occurs in ACE data, +thermal inelastic scattering includes both coherent and incoherent effects and +is dominant for most other materials including hydrogen in water. + +The first aspect of using |sab| tables is calculating cross-sections to replace +the data that would normally appear on the incident neutron data, which do not +account for thermal binding effects. For incoherent elastic and inelastic +scattering, the cross-sections are stored as linearly interpolable functions on +a specified energy grid. For coherent elastic data, the cross section can be +expressed as + +.. math:: + :label: bragg + + \sigma(E) = \frac{\sigma_c}{E} \sum_{E_i < E} f_i e^{-4WE_i}. + +where :math:`\sigma_c` is the effective bound coherent scattering cross section, +:math:`W` is the effective Debye-Waller coefficient, :math:`E_i` are the +energies of the Bragg edges, and :math:`f_i` are related to crystallographic +structure factors. Since the functional form of the cross-section is just 1/E +and the proportionality constant changes only at Bragg edges, the +proportionality constants are stored and then the cross-section can be +calculated analytically based on equation :eq:`bragg`. ---------------------------------------------- Unresolved Resonance Region Probability Tables ---------------------------------------------- .. _NJOY: http://t2.lanl.gov/codes.shtml + .. _PREPRO: http://www-nds.iaea.org/ndspub/endf/prepro/ .. [SIGMA1] Dermett E. Cullen and Charles R. Weisbin, "Exact Doppler Broadening @@ -153,3 +214,11 @@ Unresolved Resonance Region Probability Tables .. [Gelbard] Ely M. Gelbard, "Epithermal Scattering in VIM," FRA-TM-123, Argonne National Laboratory (1979). + +.. [Williams] M. M. R. Williams, *The Slowing Down and Thermalization of + Neutrons*, North-Holland Publishing Co., Amsterdam (1966). + +.. [Squires] G. L. Squires, *Introduction to the Theory of Thermal Neutron + Scattering*, Cambridge University Press (1978). + +.. |sab| replace:: S(:math:`\alpha,\beta`)