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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.

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  • $\begingroup$ The nonzero chemical potential in this model is that of a solid, not of a phonon gas. $\endgroup$
    – Gec
    Nov 25, 2018 at 20:44
  • $\begingroup$ @Gec Only the phonon gas has thermal contribution to the heat capacity, and nothing else. $\endgroup$ Nov 25, 2018 at 22:28
  • $\begingroup$ Initial question was about chemical potential. $\endgroup$
    – Gec
    Nov 26, 2018 at 5:46

1 Answer 1

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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 $\mu = F/(3N)$.

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  • $\begingroup$ Thank you for pointing out mistakes. I revised accordingly. $\endgroup$ Nov 26, 2018 at 9:41
  • $\begingroup$ On "So the number of phonons in the Einstein solid is not equal to N nor to 3N". The mean number of phonons in the Einstein's solid is $ 3N/(\exp (\beta \hbar \omega)-1) $. $\endgroup$ Nov 26, 2018 at 14:01
  • $\begingroup$ @Quanhui Liu , that's right, yes. I said that just for example. $\endgroup$
    – Gec
    Nov 26, 2018 at 14:06

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