Started discussion of S(a,b) tables in documentation.

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`)