# Formula Thermal Capacity (Einstein Approximation) Einstein temperature Temperature Number of particles

## Thermal capacity

`$$ C_{\text V} $$`

Thermal capacity indicates how well a material can store thermal energy at a constant volume \( V \). It is the derivative of the internal energy with respect to temperature.

The thermal capacity in the Einstein approximation is well suited when *optical* phonons dominate, because in contrast to *acoustic* phonons, they have a relatively flat dispersion relation. The unit of thermal capacity is \( \mathrm{J}/\mathrm{K} \).

## Einstein temperature

`$$ T_{\text E} $$`Unit

`$$ \mathrm{K} $$`

Einstein temperature of the considered crystal for which this temperature is characteristic. It is defined as \( T_{\text E} = \frac{\hbar \, \omega_{\text E}}{k_{\text B}} \), where \( \omega_{\text E} \) is the Einstein frequency. It is constant and is chosen to fit the material. In the Einstein approximation it is assumed that all \(3N\) vibrational states of the crystal have the same frequency, namely the Einstein frequency \( \omega_{\text E} \).

## Temperature

`$$ T $$`Unit

`$$ \mathrm{K} $$`

Absolute temperature of the crystal under consideration.

## Number of particles

`$$ N $$`Unit

`$$ - $$`

Number of particles in the crystal.

## Boltzmann Constant

`$$ k_{\text B} $$`Unit

`$$ \frac{\mathrm J}{\mathrm K} = \frac{\mathrm{kg} \,\mathrm{m}^2}{\mathrm{s}^2 \, \mathrm{K}} $$`

Boltzmann constant is a natural constant, which appears frequently in statistical physics and thermodynamics. It has the value: \( k_{\text B} ~\approx~ 1.380 \,\cdot\, 10^{-23} \, \frac{\text J}{\text K} \).