Talk:Quasi-harmonic approximation

Missing internal energy contribution

Dear all,

If I'm not wrong there is a missing T in U(T,V). The internal lattice energy is also T-dependent if one includes the internal energy of the harmonic oscillator (the internal energy of the harmonic oscillator is not only given by the zero point vibrations...U = (1/2 + <n>) \hbar \omega).

I would suggest to split the U(V) introduced in the article into a "static" lattice contribution $E_{\rm {lat}}(V)$ (i.e., the usual T=0 equation of state) PLUS the internal energy due to lattice vibrations: $U(T,V)=E_{\rm {lat}}(V)+U_{\rm {vib}}(T,V)$

The second term should read as

$U_{\rm {vib}}(T,V)=E_{\rm {ZP}}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}h\nu _{\mathbf {k} ,i}(V)\left[\exp \left(-{\frac {h\nu _{\mathbf {k} ,i}(V)}{k_{B}T}}\right)-1\right]^{-1}$

If one combines internal energy and entropy (F=U-TS), one arrives at the well known expression for the free energy

$F(T,V)=E_{\rm {lat}}(V)+E_{\rm {ZP}}(V)+k_{B}T{\frac {1}{N}}\sum _{\mathbf {k} ,i}\ln \left[1-\exp \left(-{\frac {h\nu _{\mathbf {k} ,i}(V)}{k_{B}T}}\right)\right],$ where the last term is sometimes mistakenly identified as the entropy due to lattice vibrations.

Unfortunately I do not have the book by Martin Dove (given as reference) at hand, so I can't say how its explicitly written there.

Best greets, Fritz Körmann — Preceding unsigned comment added by 193.175.131.12 (talk) 09:46, 17 April 2014 (UTC)

You are correct, the original way the article was written was wrong. I have edited it accordingly. JanJaeken (talk) 17:27, 21 November 2019 (UTC)