added calc of macro xs section

docs/source/methods/cmfd.rst

@@ -4,7 +4,9 @@
 Nonlinear Diffusion Acceleration - Coarse Mesh Finite Difference
 ================================================================
 
-This page section discusses how nonlinear diffusion acceleration (NDA) using coarse mesh finite difference (CMFD) is implemented into OpenMC. Before we get into the theory, general notation for this section is discussed.
+This page section discusses how nonlinear diffusion acceleration (NDA) using
+coarse mesh finite difference (CMFD) is implemented into OpenMC. Before we get
+into the theory, general notation for this section is discussed.
 
 --------
 Notation
@@ -111,3 +113,92 @@ in detail in the following sections.
 .. tikz:: Flow chart of NDA process 
    :libs: shapes, snakes, shadows, arrows, calc, decorations.markings, patterns, fit, matrix, spy
    :include: cmfd_tikz/cmfd_flow.tikz
+
+Calculation of Macroscopic Cross Sections
+-----------------------------------------
+
+A diffusion model needs macroscopic cross sections and diffusion coefficients to
+solve for multigroup fluxes. Cross sections are derived by conserving reaction
+rates predicted by MC tallies. From Eq. :eq:`eq_neut_bal`, total, scattering
+production and fission production macroscopic cross sections are needed. They are
+defined from MC tallies as follows:
+
+.. math::
+   :label: xs1
+
+   \overline{\overline\Sigma}_{t_{l,m,n}}^g \equiv
+   \frac{\left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+   \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+   {\left\langle\overline{\overline\phi}_{l,m,n}^g
+   \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle},
+
+.. math::
+   :label: xs2
+
+   \overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow g} \equiv
+   \frac{\left\langle\overline{\overline{\nu_s\Sigma}}_{s_{l,m,n}}^{h\rightarrow
+   g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+   {\left\langle\overline{\overline\phi}_{l,m,n}^h
+   \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+
+and 
+
+.. math::
+   :label: xs3
+
+   \overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow g} \equiv
+   \frac{\left\langle\overline{\overline{\nu_f\Sigma}}_{f_{l,m,n}}^{h\rightarrow
+   g}\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}
+   {\left\langle\overline{\overline\phi}_{l,m,n}^h\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}.
+
+In order to fully conserve neutron balance, leakage rates also need to be
+preserved. In standard diffusion theory, leakage rates are represented by
+diffusion coefficients. Unfortunately, it is not easy in MC to calculate a
+single diffusion coefficient for a cell that describes leakage out of each
+surface. Luckily, it does not matter what definition of diffusion coefficient is
+used because nonlinear equivalence parameters will correct for this
+inconsistency. However, depending on the diffusion coefficient definition
+chosen, different convergence properties of NDA equations are observed.
+Here, we introduce a diffusion coefficient that is derived for a coarse energy
+transport reaction rate. This definition can easily be constructed from
+MC tallies provided that angular moments of scattering reaction rates can
+be obtained. The diffusion coefficient is defined as follows:
+
+.. math::
+   :label: eq_transD
+
+    \overline{\overline D}_{l,m,n}^g =
+    \frac{\left\langle\overline{\overline\phi}_{l,m,n}^g
+    \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle}{3
+    \left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g
+    \overline{\overline\phi}_{l,m,n}^g
+    \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle},
+
+where
+
+.. math::
+   :label: xs4
+
+   \left\langle\overline{\overline\Sigma}_{tr_{l,m,n}}^g
+   \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle
+   =
+   \left\langle\overline{\overline\Sigma}_{t_{l,m,n}}^g
+   \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle
+   \\ -
+   \left\langle\overline{\overline{\nu_s\Sigma}}_{s1_{l,m,n}}^g
+   \overline{\overline\phi}_{l,m,n}^g\Delta_l^u\Delta_m^v\Delta_n^w\right\rangle.
+
+Note that the transport reaction rate is calculated from the total reaction rate
+reduced by the $P_1$ scattering production reaction rate. Equation :eq:`eq_transD`
+does not represent the best definition of diffusion coefficients from MC;
+however, it is very simple and usually fits into MC tally frameworks
+easily. Different methods to calculate more accurate diffusion coefficients can
+found in [Herman]_.
+
+----------
+References
+----------
+
+.. [Herman] Bryan R. Herman, Benoit Forget, Kord Smith, and Brian N. Aviles. Improved
+            diffusion coefficients generated from Monte Carlo codes. In *Proceedings of M&C
+            2013*, Sun Valley, ID, USA, May 5 - 9 2013.