状態方程式

 

  V_0 = V_{(T = T_0, P=0)}, \\ V = V_{(T = T_0, P)} \\  K_0 = K_{(T=T_0, P=0)}, K'_0 = (\partial K_0 / \partial P)_T,K''_0 = (\partial^2 K_0 / \partial P^2)_T \\  a=\left(\cfrac{V_0}{V}\right)^{1/3} \\
とすると、


2nd-order Birch-Murnaghan
  P_{(T=T_0, V)} = \cfrac{3K_0}{2}  \left( \cfrac{V_0}{V} \right)^{5/3}  \left[ \left( \cfrac{V_0}{V} \right)^{2/3} - 1 \right]  = \cfrac{3K_0 }{2} \left\{a^5 ( a^2 - 1)\right\}


3rd-order Birch-Murnaghan
  P_{(T=T_0, V)} = \cfrac{3K_0}{2}  \left( \cfrac{V_0}{V} \right)^{5/3}  \left[ \left( \cfrac{V_0}{V} \right)^{2/3} - 1 \right]  \left[ 1 + \cfrac{3}{4} (K'_0-4) \left\{ \left(\cfrac{V_0}{V} \right)^{2/3} -1 \right\} \right]\\ \  = \cfrac{3K_0}{2} \left\{a^5 ( a^2 - 1)\right\} \left\{ 1 + \cfrac{3}{4} (K'_0-4) (a^2 -1) \right\}


4th-order Birch-Murnaghan
  P_{(T=T_0, V)} = \cfrac{3K_0}{2}  \left( \cfrac{V_0}{V} \right)^{5/3}  \left[ \left( \cfrac{V_0}{V} \right)^{2/3} - 1 \right]  \left[  1 +  \cfrac{3}{4} (K'_0-4) \left\{ \left(\cfrac{V_0}{V} \right)^{2/3} -1 \right\} +  \cfrac{9K_0 K''_0 + 9{K'_0}^2 - 63K'_0 + 143}{24} \left\{ \left(\cfrac{V_0}{V} \right)^{2/3} -1 \right\}^2  \right]\\ \  = \cfrac{3K_0}{2} \left\{a^5 ( a^2 - 1)\right\} \left\{ 1 + \cfrac{3}{4} (K'_0-4) (a^2 -1)  + \cfrac{9K_0 K''_0 + 9{K'_0}^2 - 63K'_0 + 143}{24} (a^2 -1)^2  \right\}


T-dependence-Birch Murnaghan
  V_{(T,P=0)} = V_{(T_0, P=0)} \exp{\displaystyle\int_{T_0}^T a + b T^2+c/T^2 dT}\\  K_{(T,P=0)} =K_{(T_0, P=0)} + (\partial K_{(T,P=0)}/ \partial T) (T-T_0)


Vinet
  P_{(T=T_0, V)} = 3K_0  \left(\cfrac{V}{V_0}\right)^{-2/3}  \left\{ 1- \left(\cfrac{V}{V_0}\right)^{1/3} \right\}  \exp \left[ \cfrac{3}{2} (K'_0-1) \left\{ 1- \left(\cfrac{V}{V_0}\right)^{1/3} \right\} \right] \\ \  = 3K_0 a^{2} \left\{ 1- 1/a \right\}  \exp \left[ \cfrac{3}{2} (K'_0-1) \left( 1- 1/a \right) \right]


Mie-Grüneisen
  \gamma = \gamma_0 \left( \cfrac{V}{V_0}\right)^q, \ \ \theta = \theta_0 \exp \left( \cfrac{\gamma_0-\gamma}{q} \right)\\  P_{th} =P_{(T, V)} - P_{(T=T_0, V)}= \cfrac{9nz \gamma R} {N_A V \theta^3}  \left\{  T^4 \displaystyle\int_0^{\theta/T} \cfrac{z^3}{\exp{(z)}-1} dz  - T_0^4 \displaystyle\int_0^{\theta/T_0} \cfrac{z^3}{\exp{(z)}-1} dz  \right\} \ \ \ \ {\rm [Pa]}  \\ \\  R:{\rm\ Gas\ constant, 8.314 4621(75) [N\ m\ K^{-1}\ mol^{-1}] }\\  N_A:{\rm\ Avogadro\ constant, 6.02214129(27) \times 10^{23} [mol^{-1}]}\\  z:{\rm\ Number\ of\ formula\ in\ unit\ cell}\\  n:{\rm\ Atoms\ per\ formula}\\  T_0:{\rm\ Standard\ temperature [K]}\\  T:{\rm\ Target\ temperature\ [K]}\\  V_0:{\rm\ Standard\ unit\ cell\ volume\ [m^3]}\\  V:{\rm\ Target\ unit\ cell\ volume\ [m^3]}\\  \theta_{0}:{\rm\ Debye\ Temperature\ at\ standard\ volume\ [K]}\\  \gamma_{0}:{\rm\ Gruneisen\ parameter\ at\ standard\ volume}\\  q:{\rm\ Volume\ dependence\ of\ Gruneisen\ parameter}\\


Pt
  \begin{array}{ccccccccccc}  {\rm\ Author(s) }               & K_0      &   K'_0  & z & n &T_0\ [K]& V_0\ [A^3]& \theta_0\ [K]&\gamma_0   & q     & comment\\  \hline\hline  {\rm\  Matsui\ et\ al.\ (2009)} &273       &5.20     & 4 & 1 & 300    & 60.38      & 230         &   2.70    &  1.10 &V+M\\  {\rm\  Fei\ et\ al.\ (2007)}    &277       &5.08(2)  & 4 & 1 & 300    & 60.38      & 230         &   2.72(3) &  0.5 &V+M\\  {\rm\  Zha\ et\ al.\ (2008)}    &273.5(10) &4.70(6)  & 4 & 1 & 300    & 60.38      & 230         &   2.75(3) &  0.25(V/V_0) &V+M\\  \hline  \end{array}


NaCl B2
  \begin{array}{ccccccccccc}  {\rm\ Author(s) }               & K_0      &   K'_0  &z  &n  &T_0\ [K]& V_0\ [A^3]& \theta_0\ [K]&\gamma_0 & q    & comment \\  \hline\hline  {\rm\ Sakai\ et\ al.\ (2009)}   &47.00(46) &4.10(2)  & - & - & -      & 37.73(4.05)& -           & -       &  -   & 3BM \\  {\rm\ Sakai\ et\ al.\ (2009)}   &40.40(54) &5.04(4)  & - & - & -      & 37.73(4.05)& -           & -       &  -   & V \\  {\rm\ Ueda\ et\ al.\ (2008)^1}  &28.45(31) &5.16(4)  & - & - & 300    & 41.115    & -            & -       &  -   & V \\  \hline  \end{array}\\  1: Vinet + (\partial P/ \partial T)(T-300), \partial P/ \partial T = 0.00468(4)


NaCl B1
  \begin{array}{cccccccccccc}  {\rm\ Author(s) }               & K_0      &   K'_0  &  K''_0    & z & n  &T_0\ [K]& V_0\ [A^3]& \theta_0\ [K]&\gamma_0   & q     & comment \\  \hline\hline  {\rm\  Matsui\ et\ al.\ (2012)} &23.7      &5.14(5)  &-0.392(21) & 4 & 2  & 300    & -         & 279          & 1.56      & 0.96  & 4BM+V \\  {\rm\  Decker\ (1971)^1}        &23.70(1)  &4.91(1)  &-0.267(2)  & 4 & 2  & 300    & -         & 279          & 1.59      & 0.93  & 4BM+V \\  \hline  \end{array}\\  {\rm\ 1:\ Recalculated\ by\ Matsui\ et\ al.\ (2012)}


Al2O3
  \begin{array}{cccccccccccc}  {\rm\ Author(s) }                    & K_0      &   K'_0  &  \partial K_{(T,P=0)}/ \partial T & a                & b                    & c     &T_0\ [K]& V_0\ [cm^3/mol]&  comment \\  \hline\hline  {\rm\  Dubrovinsky\ et\ al.\ (1998)} &258(2)    &4.88(4)  &-0.020                             & 2.6\times10^{-5} & 1.81(9)\times10^{-9}  & -0.67 &300     & 25.59(2)       & T-dependence-BM \\  \hline  \end{array}\\


Diamond
  \begin{array}{cccccccccccc}  {\rm\ Author(s) }                    & K_0      &   K'_0  & V_0\ [cm^3/mol]&  comment \\  \hline\hline  {\rm\  Occelli\ et\ al.\ (2003)} &446(1)    &3.0(1)   & 3.4170(8)        & 3BM \\  \hline  \end{array}\\

 Posted by at 11:33 AM