Why the chemical potential of phonon gas in Einstein 's solid model is not zero

Question

In Einstein’s model of solid, each atom in the solid is considered to be an independent three-dimensional quantum harmonic oscillator with characteristic frequency $ω$ that is constant. Each degree of freedom as a separate one-dimensional harmonic oscillator and the energy levels are familiar: $ E_n=(n+1/2)ℏω $. Ignoring the zero-point energy as an irrelevant constant term, we take the energy levels as $E_n= n ℏω $. The partition function for a single 1d oscillator is: $Z_1=1/(1−\exp(−βℏω))$. The mean number of phonon is $<n>=1/(\exp(βℏω)−1)$. It is clearly the Bose-Einstein statistics with zero chemical potential.

However, the free energy is: $$F=−3Nβ^{−1} \log Z_1=3Nβ^{−1}\log(1−\exp(−βℏω)).$$ The chemical potential $$μ=F/N=3 β^{−1}\log(1−\exp(−βℏω)) $$ which is nonzero. This nonzero chemical potential is given in many textbooks (such as Pathria 3rd ed, Pauli, etc.) to discuss the solid-vapor phase transition.

1. Major question: Why the chemical potential of phonon gas in Einstein 's solid model is not zero?

2. Minor question: In Einstein’s model of solid, the pressure is zero. But during the solid-vapor phase transition, the vapor pressure is never zero! The condition for phase transition requires that the solid pressure is equal to vapor pressure.

Answer 1

The energy of a system of many interacting particles is often approximated by the energy of a system of noninteracting quasiparticles. Then the partition function of a system of physical particles is equal to the grand partition function of a system of quasiparticles with zero chemical potential. Particles and quasiparticles should be distinguished. The number of physical particles $N$ is fixed, but the mean number of quasiparticles depends on the temperature. So the number of phonons in the Einstein solid is not equal to $N$ nor to $3N$.

The chemical potential of a system of physical particles is $\partial F/\partial N \neq 0$.

I think there are mistakes in your formulas. First, the zero-point energy is finite in this model due to the finite number of oscillators. This energy is usually taken into account. Second, you need the chemical potential of a physical system of $N$ atoms, not the chemical potential of a system of $3N$ imaginary oscillators. Hence the right formula is $\mu = F/N$, not $\mu = F/(3N)$.