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Added docs for conversion of PlankEinstein and PlanckEinstein2 forms to generalized form
Signed-off-by: Ian Bell <ian.h.bell@gmail.com>
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@@ -794,6 +794,47 @@ public:
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\f[
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\alpha^0 = \displaystyle\sum_i n_i\log[c_i+d_i\exp(\theta_i\tau)]
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\f]
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To convert conventional Plank-Einstein forms, given by
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\f$
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\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( b_k/T \right)^2\exp \left( b_k/T \right)}{\left(\exp \left(b_k/T\right) - 1 \right)^2}
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\f$
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and
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\f$
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\alpha^0 = a_k\ln \left[1 - \exp \left( \frac{-b_k\tau}{T_c} \right) \right]
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\f$
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use \f$c = 1\f$, \f$d = -1\f$, \f$n = a\f$, \f$\theta = -\displaystyle\frac{b_k}{T_c}\f$
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To convert the second form of Plank-Einstein terms, given by
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\f$
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\frac{c_p^0}{R} = a_k\displaystyle\frac{\left( -b_k/T \right)^2\exp \left( b_k/T \right)}{c\left(\exp \left(-b_k/T\right) + 1 \right)^2}
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\f$
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and
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\f$
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\alpha^0 = a_k\ln \left[c + \exp \left( \frac{b_k\tau}{T_c} \right) \right]
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\f$
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use \f$c = 1\f$, \f$d = 1\f$, \f$n = -a\f$, \f$\theta = \displaystyle\frac{b_k}{T_c}\f$
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Converting Aly-Lee tems is a bit more complex
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Aly-Lee starts as
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\f[\frac{c_p^0}{R_u} = A + B\left(\frac{C/T}{\sinh(C/T)}\right)^2 + D\left(\frac{E/T}{\cosh(E/T)}\right)^2\f]
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Constant is separated out, and handled separately. sinh part can be expanded as
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\f[B\left(\frac{C/T}{\sinh(C/T)}\right)^2 = \frac{B(-2C/T)^2\exp(-2C/T)}{(1-\exp(-2C/T))^2}\f]
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where
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\f[n_k = B\f]
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\f[\theta_k = -\frac{2C}{T_c}\f]
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\f[c_k = 1\f]
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\f[d_k = -1\f]
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cosh part can be expanded as
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\f[D\left(\frac{E/T}{\cosh(E/T)}\right)^2 = \frac{D(-2E/T)^2\exp(-2E/T)}{(1+\exp(-2E/T))^2}\f]
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where
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\f[n_k = -D\f]
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\f[\theta_k = -\frac{2E}{T_c}\f]
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\f[c_k = 1\f]
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\f[d_k = 1\f]
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*/
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class IdealHelmholtzPlanckEinsteinGeneralized : public BaseHelmholtzTerm{
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