update screening doc

Docs/source/screening.rst

@@ -4,8 +4,45 @@ Screening of Reaction Rates
 
 .. index:: SCREEN_METHOD
 
-Screening of reaction rates can be computed using several different methods,
-controlled by the make parameter ``SCREEN_METHOD``.  For example,
+Introduction
+------------
+
+Plasma screening is the enhancement of nuclear reaction rates, :math:`R`
+due to the Coulomb coupling of the surrounding plasma and electrons
+and ions. The enhancement of the reaction rates is done via the form:
+
+.. math::
+   R_{\mathrm{scr}} = R \exp{(h)}
+
+where :math:`R_{\mathrm{scr}}` is the screened reaction rate and :math:`h`
+characterizes the magnitude of the screening.
+Plasma screening can be broken up to different regimes depending
+on the Coulomb coupling parameter, :math:`\Gamma`,
+of the reaction rate reactants. Generally, :math:`\Gamma \ll 1` and
+:math:`\Gamma \gtrsim 1` correspond to weak and strong screening regimes,
+respectively. :math:`\Gamma` is defined as:
+
+.. math::
+   \Gamma = \alpha(Z_1, Z_2) \Gamma_e
+
+where :math:`\alpha(Z_1, Z_2)` characterizes the Coulomb strength of
+the reactants, its definition can vary slightly depending on the
+screening routine. :math:`\Gamma_e` is the Coulomb coupling parameter
+depending only on the thermodynamic conditions.
+
+.. math::
+   \Gamma_e= e^2 \frac{\sqrt[3]{4 \pi n_e / 3}}{k_B T}
+
+where :math:`e` is the electron charge and :math:`n_e` is the
+electron number density.
+
+
+Screening Options
+-----------------
+
+The screening enhancement factor can be can be computed using
+several different methods, controlled by the make parameter ``SCREEN_METHOD``.
+For example,
 
 .. prompt:: bash
 
@@ -19,19 +56,24 @@ The options are:
 
   This is the screening routine from the Kepler stellar evolution code
   and is the default used with the distributed versions of the "aprox"
-  family of reaction networks.  It uses the screening described in
-  :cite:`graboske:1973` for the weak limit and :cite:`jancovici:1977`,
-  :cite:`alastuey:1978`, :cite:`itoh:1979` for the strong limit. The
-  overall procedure is described in :cite:`Wallace:1982`.
+  family of reaction networks. In the weak screenng regime,
+  :math:`\Gamma < 0.3`, it uses screening described in
+  :cite:`graboske:1973`. In the strong screening regime,
+  :math:`\Gamma > 0.8`, it uses screening described in
+  :cite:`jancovici:1977`, :cite:`alastuey:1978`, :cite:`itoh:1979`.
+  For the intermediate screening regime, :math:`0.3 < \Gamma < 0.8`,
+  a weighted blending between the weak and strong screening are used.
+  The overall procedure is described in :cite:`Wallace:1982`.
 
   This is the default screening method.
 
 * ``chugunov2007`` :
 
   This implements the screening of :cite:`chugunov:2007`, following
-  :cite:`yakovlev:2006` to extend to binary mixtures.
+  :cite:`yakovlev:2006` to extend to binary mixtures. It is suitable
+  for :math:`\Gamma \lesssim 600`.
 
-* ``chugunov2009``
+* ``chugunov2009`` :
 
   This implements the screening of :cite:`chugunov:2009`.  The main
   difference is that the 2007 one calculates an effective coupling
@@ -42,17 +84,23 @@ The options are:
   This includes the portion in the appendix that blends in the weak
   screening limit.
 
-* ``chabrier1998``:
+* ``chabrier1998`` :
 
-  This implements the screening of :cite:`Chabrier_1998` as well as the quantum corrections for strong screening according to screen5. This is suggested in the appendix of :cite:`Calder_2007`. This screening is compatible with NSE calculations unlike ``screen5``, ``chugunov2007``, and ``chugunov2009``. This screening should be valid in the weak screening regime, :math:`\Gamma < 0.1`, and strong screening regime, :math:`1 \lesssim \Gamma \lesssim 160`.
+  This implements the screening of :cite:`Chabrier_1998` as well as
+  the quantum corrections for strong screening according to screen5,
+  which is suggested in the appendix of :cite:`Calder_2007`.
+  This screening is compatible with NSE calculations unlike ``screen5``,
+  ``chugunov2007``, and ``chugunov2009``. This screening is valid in the
+  weak screening regime, :math:`\Gamma < 0.1`, and strong screening regime,
+  :math:`1 \lesssim \Gamma \lesssim 160`.
 
 * ``null`` :
 
   This disables screening by always returning 1 for the screening
   enhancement factor.
 
 Runtime Options
-----------------
+---------------
 
 .. index:: screening.enable_chabrier1998_quantum_corr