[cig-commits] [commit] doc_updates: Clarified use of Einstein model in HP2011 (deb742c)

[cig_noreply at geodynamics.org]

Repository : https://github.com/geodynamics/burnman

On branch  : doc_updates
Link       : https://github.com/geodynamics/burnman/compare/3899cc014a0ad3bc3a78bb1af28e94a6fb931390...deb742ca521994741721a482198472fd89cf31da

>---------------------------------------------------------------

commit deb742ca521994741721a482198472fd89cf31da
Author: Bob Myhill <myhill.bob at gmail.com>
Date:   Tue Dec 30 00:14:41 2014 +0000

    Clarified use of Einstein model in HP2011


>---------------------------------------------------------------

deb742ca521994741721a482198472fd89cf31da
 sphinx/background_thermoelastics.txt | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/sphinx/background_thermoelastics.txt b/sphinx/background_thermoelastics.txt
index 84ccc50..3bf0077 100644
--- a/sphinx/background_thermoelastics.txt
+++ b/sphinx/background_thermoelastics.txt
@@ -81,11 +81,12 @@ Mie-Grüneisen-Debye (thermal)
 HP2011 (thermal correction to Modified Tait)
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
-Thermal pressure can be incorporated into the Modified Tait equation of state, replacing :math:`P` with :math:`P-P_{\textrm{thermal}}` in Equation :eq:`mtait` :cite:`HP2011` 
+The thermal pressure can be incorporated into the Modified Tait equation of state, replacing :math:`P` with :math:`P-P_{\textrm{thermal}}` in Equation :eq:`mtait` :cite:`HP2011`. Thermal pressure here is calculated using a Mie-Grüneisen equation of state and an Einstein model for heat capacity, even though the Einstein model is not actually used for the heat capacity when calculating the enthalpy and entropy (see following section).
 
 .. math::
-    P_{\textrm{thermal}} &= d \int_{T0}^T \frac{(\frac{\Theta}{T})^2\exp(\frac{\Theta}{T})}{(\exp(\frac{\Theta}{T})-1)^2} dT, \\
-    d &= \alpha_0K_0 \frac{(\exp(\frac{\Theta}{T})-1)^2}{(\frac{\Theta}{T})^2\exp(\frac{\Theta}{T})}
+    P_{\textrm{thermal}} &= \frac{\alpha_0 K_0 E_{\textrm{th}}}{C_{V0}}, \\
+    C_{V0} &= 3 n R \frac{(\frac{\Theta}{T})^2\exp(\frac{\Theta}{T})}{(\exp(\frac{\Theta}{T})-1)^2}, \\
+    E_{\textrm{th}} &= \int_{T0}^T \frac{1}{C_{V0}} dT
 
 :math:`\Theta` is the Einstein temperature of the crystal in Kelvin, approximated for a substance :math:`i` with :math:`n_i` atoms in the unit formula and a molar entropy :math:`S_i` using the empirical formula