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 + ) \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

$$U_{\rm vib}(T,V) = E_{\rm ZP}(V)+\frac{1}{N} \sum_{\mathbf{k},i} h \nu_{\mathbf{k},i}(V) \left[ \exp\left( -\frac{h \nu_{\mathbf{k},i}(V)}{ k_BT } \right) - 1 \right]^{-1}$$

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

$$F(T,V)=E_{\rm lat}(V)+E_{\rm ZP}(V)+k_B T \frac{1}{N} \sum_{\mathbf{k}, i} \ln \left[ 1 - \exp\left( -\frac{h \nu_{\mathbf{k},i}(V)}{ k_BT }\right) \right], $$ 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)