Einstein Solid

Definition

A solid with \(N\) atoms, each with 3 vibrate directions which are harmonic oscillations, and they do not interact with each other. They all oscillate at the same frequency.

Entropy

Think of the separation as tail and each energy unit as heads. The microstate of the solid at \(q\) number energy unit \(\epsilon\) is equivalent to the case of \(q\) heads in \(N + q - 1\) coins. Thus, $$ \bbox[5px,border:2px solid #666] { \Omega(q) = C^{N+q-1}_{q} = \frac{(N+q-1)!}{q!(N-1)!} } $$ For more than 1 Einstein solid, since they are independent, the total multiplicity will be the product of their own multiplicity. The entropy is $$S = k \ln \left(\frac{(N+q-1)!}{q!(N-1)!} \right)$$ When \(N >> 1\), by Stirling's approximation, the entropy is roughly equal to $$S/k \approx (q + N)\ln(q + N) - N\ln N - q\ln q$$

Internal Energy

When \(q \gg N\), \begin{align} \ln(q + N) &= \ln\left(a\left(1+\frac{N}{q}\right)\right) \\ &= \ln q + \ln\left(1 +\frac{N}{q}\right)\\ &\approx \ln q + \frac{N}{q} \end{align} Thus, the entropy is approximately equal to \begin{align} S/k &\approx (q + N)\left( \ln q + \frac{N}{q}\right) - N\ln N - q\ln q\\ &= N\ln\frac{q}{N} + N + \frac{N^2}{q} \\ &\approx N\ln\frac{q}{N} + N \end{align} As there are \(q\) unit of \(\epsilon\) energy, the internal energy of the Einstein solid is \(U = q\epsilon\) $$S/k \approx N\ln U - N\ln(\epsilon N) + N$$ Applying temperature definition, \begin{align} \frac{1}{T} &= \frac{\partial S}{\partial U} \\ &= Nk\left(\frac{1}{U}\right) \end{align} Thus, $$ \bbox[5px,border:2px solid #666] { U = NkT }$$

Chemical Potential

The chemical potential of Einstein solid is \begin{align} S &= k\left[(q + N)\ln(q + N) - q\ln q - N\ln N\right]\\ \mu &= -T\left(\frac{\partial S}{\partial N}\right)_q \\ &= -kT\ln\left(\frac{q + N}{N}\right) \end{align}