with changing volume (i.e. 's). For example, one can show that if one defines as the weighted average where 's are the partial vibrational mode contributions to the heat capacity, such that Proof To prove this relation, it is easiest to introduce the heat capacity per particle ; so one can write This way, it suffices to prove Left-hand side (def): Right-hand side (def): Furthermore (Maxwell relations): Thus This derivative is straightforward to determine in the quasi-harmonic approximation, as only the are V-dependent. This yields See also Debye model Negative thermal expansion Mie–Grüneisen equation of state External links Definition from Eric Weisstein's World of Physics References Condensed matter physics Dimensionless numbers of thermodynamics