Quasi-harmonic approximation

The quasi-harmonic approximation expands upon the harmonic phonon model of lattice dynamics. The harmonic phonon model states that all interatomic forces are purely harmonic, but such a model is inadequate to explain thermal expansion, as the equilibrium distance between atoms in such a model is independent of temperature.

Thus in the quasi-harmonic model, from a phonon point of view, phonon frequencies become volume-dependent in the quasi-harmonic approximation, such that for each volume, the harmonic approximation holds.

For a lattice, the Helmholtz free energy F in the quasi-harmonic approximation is

${\displaystyle F(T,V)=E_{\rm {lat}}(V)+U_{\rm {vib}}(T,V)-TS(T,V)}$

where E_lat is the static internal lattice energy, U_vib is the internal vibrational energy of the lattice, or the energy of the phonon system, T is the absolute temperature, V is the volume and S is the entropy due to the vibrational degrees of freedom. The vibrational energy equals

${\displaystyle U_{\rm {vib}}(T,V)={\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {1}{2}}\hbar \omega _{\mathbf {k} ,i}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {\hbar \omega _{\mathbf {k} ,i}(V)}{\exp(\Theta _{\mathbf {k} ,i}(V)/T)-1}}={\frac {1}{N}}\sum _{\mathbf {k} ,i}[{\frac {1}{2}}+n_{\mathbf {k} ,i}(T,V)]\hbar \omega _{\mathbf {k} ,i}(V)}$

where N is the number of terms in the sum, ${\textstyle \Theta _{\mathbf {k} ,i}(V)=\hbar \omega _{\mathbf {k} ,i}(V)/k_{B}}$ is introduced as the characteristic temperature for a phonon with wave vector k in the i-th band at volume V and ${\textstyle n_{\mathbf {k} ,i}(T,V)}$ is shorthand for the number of (k,i)-phonons at temperature T and volume V. As is conventional, ${\textstyle \hbar }$ is the reduced Planck constant and k_B is the Boltzmann constant. The first term in U_vib is the zero-point energy of the phonon system and contributes to the thermal expansion as a zero-point thermal pressure.

The Helmholtz free energy F is given by

${\displaystyle F=E_{\rm {lat}}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}{\frac {1}{2}}\hbar \omega _{\mathbf {k} ,i}(V)+{\frac {1}{N}}\sum _{\mathbf {k} ,i}k_{B}T\ln \left[1-\exp(-\Theta _{\mathbf {k} ,i}(V)/T)\right]}$

and the entropy term equals

${\displaystyle S=-\left({\frac {\partial F}{\partial T}}\right)_{V}=-{\frac {1}{N}}\sum _{\mathbf {k} ,i}k_{B}\ln \left[1-\exp(-\Theta _{\mathbf {k} ,i}(V)/T)\right]+{\frac {1}{NT}}\sum _{\mathbf {k} ,i}{\frac {\hbar \omega _{\mathbf {k} ,i}(V)}{\exp(\Theta _{\mathbf {k} ,i}(V)/T)-1}}}$,

from which F = U - TS is easily verified.

The frequency ω as a function of k is the dispersion relation. Note that for a constant value of V, these equations corresponds to that of the harmonic approximation.

By applying a Legendre transform, it is possible to obtain the Gibbs free energy G of the system as a function of temperature and pressure.

${\displaystyle G(T,P)=\min _{V}\left[E_{\rm {lat}}(V)+U_{\rm {vib}}(V,T)-TS(T,V)+PV\right]}$

Where P is the pressure. The minimal value for G is found at the equilibrium volume for a given T and P.

Once the Gibbs free energy is known, many thermodynamic quantities can be determined as first- or second-order derivatives. Below are a few which cannot be determined through the harmonic approximation alone.

• Dove, Martin T. (1993). Introduction to lattice dynamics, Cambridge university press. ISBN 0521392934.