Thermodynamic-properties

One of the important motivations for lattice dynamical calculations of crystalline solids is that the harmonic approximation gives access to thermodynamic properties including the zero-point energy and the free energy as a function of temperature. CASTEP lattice dynamics calculations can be followed by a thermodynamics calculation to calculate the zero-point energy and temperature dependent free energy, entropy, and specific heat

$$\begin{aligned} F &= E_{\text{elec}} + k_{B} T \sum_{{\mathbf{q}},i} ln \left [ 2 \sinh \left (\frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T} \right ) \right ],\\ S &= \frac{1}{2T} \sum_{{\mathbf{q}},i} \hbar \omega_{{\mathbf{q}},i} \coth \left (\frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T} \right ) - k_{B} \sum_{{\mathbf{q}},i} ln \left [ 2 \sinh \left (\frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T} \right ) \right ],\\ C_V &= \sum_{{\mathbf{q}},i} k_{B} \left ( \frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T} \right )^2 \exp{\left (\frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T}\right )} /% \left [\exp{\left (\frac{\hbar \omega_{{\mathbf{q}},i}}{ k_{B} T}\right )} -1 \right ]^2 . \end{aligned}$$

The thermodynamics calculation follows a previous phonon calculation. It is selected by setting the parameter

task : THERMODYNAMICS

This may be used in a continuation run from a previous phonon calculation, where the value of the continuation parameter is the name of the previous .check file. Alternatively it may be configured as a new run from scratch by setting the remainder of the parameters exactly as if this were a phonon task, and the phonon calculation will be performed first. Only the phonons defined on the “fine” set of phonon kpoints will be used to compute the free energy as it is normally expected that a thermodynamics calculation will follow an interpolation or supercell calculation. However provided that the standard and fine sets of phonon k-points are identical, it may also be used following a standard phonon calculation.

This task will compute and print the free energy, entropy and specific heat plus the vibrational atomic displacement parameters (“ADP”s) in the range of temperatures specified by the parameters thermo_t_start and thermo_t_stop. The number of temperatures is set by one or other of the parameters thermo_t_spacing or thermo_t_npoints. All temperatures are absolute and the default unit is K. The results of the calculation are written to the .castep file.

The harmonic approximation free energy is only defined if all frequencies are greater than or equal to zero. Any zero or imaginary frequencies are automatically omitted from the calculation and a warning message is printed. It is the responsibility of the end user to check that the computed free energy is not rendered meaningless by the presence of an imaginary mode.