爱因斯坦模型

• 晶格中的每一个原子都是三维量子谐振子
• 原子不互相作用；
• 所有的原子都以相同的频率振动（与德拜模型不同）。

热容（微正则系综）

$C_V = \left({\partial U\over\partial T}\right)_V.$

$T$是系统的温度，可以从求出：

${1\over T} = {\partial S\over\partial U}.$

$N^{\prime} = 3N$

SHO的可能的能量为：

$E_n = \hbar\omega\left(n+{1\over2}\right)$

$\varepsilon = \hbar\omega$

$\Omega = {\left(q+N^{\prime} - 1\right)!\over q! (N^{\prime} - 1)!}$

$S/k = \ln\Omega = \ln{\left(q+N^{\prime} - 1\right)!\over q! (N^{\prime} - 1)!}.$

$N^{\prime}$是一个很大的数，把它减去一总体上没有任何影响：

$S/k \approx \ln{\left(q+N^{\prime}\right)!\over q! N^{\prime}!}$

$S/k \approx \left(q+N^{\prime}\right)\ln\left(q+N^{\prime}\right) - N^{\prime}\ln N^{\prime} - q\ln q.$

$U = {N^{\prime}\varepsilon\over2} + q\varepsilon.$

${1\over T} = {\partial S\over\partial U} = {\partial S\over\partial q}{dq\over dU} = {1\over\varepsilon}{\partial S\over\partial q} = {k\over\varepsilon} \ln\left(1+N^{\prime}/q\right)$

$U = {N^{\prime}\varepsilon\over2} + {N^{\prime}\varepsilon\over e^{\varepsilon/kT} - 1}.$

$C_V = {\partial U\over\partial T} = {N^{\prime}\varepsilon^2\over k T^2}{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT} - 1\right)^2}$

$C_V = 3Nk\left({\varepsilon\over k T}\right)^2{e^{\varepsilon/kT}\over \left(e^{\varepsilon/kT} - 1\right)^2}$

热容（正则系综）

$Z = \sum_{n=0}^{\infty} e^{ - E_n/kT}$

$E_n = \varepsilon\left(n+{1\over2}\right)$

\begin{align} Z & {} = \sum_{n=0}^{\infty} e^{ - \varepsilon\left(n+1/2\right)/kT} = e^{ - \varepsilon/2kT} \sum_{n=0}^{\infty} e^{ - n\varepsilon/kT}=e^{ - \varepsilon/2kT} \sum_{n=0}^{\infty} \left(e^{ - \varepsilon/kT}\right)^n \\ & {} = {e^{-\varepsilon/2kT}\over 1 - e^{ - \varepsilon/kT}} = {1\over e^{\varepsilon/2kT} - e^{ - \varepsilon/2kT}} = {1\over 2 \sinh\left({\varepsilon\over 2kT}\right)}. \end{align}

$\langle E\rangle = u = - {1\over Z}\partial_{\beta}Z$

$\beta = {1\over kT}.$

$u = -2 \sinh\left({\varepsilon\over 2kT}\right){ - \cosh\left({\varepsilon\over 2kT}\right)\over 2 \sinh^2\left({\varepsilon\over 2kT}\right)}{\varepsilon\over2} = {\varepsilon\over2}\coth\left({\varepsilon\over 2kT}\right).$

$C_V = {\partial U\over\partial T} = -{\varepsilon\over2} {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}\left( - {\varepsilon\over 2kT^2}\right) = k \left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}.$

$C_V = 3Nk\left({\varepsilon\over 2 k T}\right)^2 {1\over \sinh^2\left({\varepsilon\over 2kT}\right)}.$

参考文献

• "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme", A. Einstein, Annalen der Physik, volume 22, pp. 180-190, 1907.