# Formula Debye Approximation for Thermal Capacity Debye temperature Number of particles Energy ratio

## 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 internal energy with respect to temperature.

The thermal capacity in the Debye approximation is well suited when *acoustic* phonons dominate because, unlike *optical* phonons, the acoustic dispersion relation is reasonably linear. The unit of thermal capacity is \(
\mathrm{J}/\mathrm{K} \).

## Debye temperature

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

`$$ \mathrm{K} $$`

Debye temperature of the considered crystal for which this temperature is characteristic. It is defined as \( T_{\text D} = \frac{\hbar \, \omega_{\text D}}{k_{\text B}} \), where \( \omega_{\text D} = v_{\text s} \, k_{\text D} \) is the Debye frequency and \( v_{\text s} \) is the speed of sound in the crystal. Thus, in the Debye approximation, it is assumed that all dispersion branches \( \omega(k) = v_{\text s} \, k \) of the crystal are linear. With \( k \) as the angular wave vector.

## Temperature

`$$ T $$`Unit

`$$ \mathrm{K} $$`

Absolute temperature of the crystal under consideration.

## Number of particles

`$$ N $$`Unit

`$$ - $$`

Particle number of the crystal under consideration.

## Energy ratio

`$$ u $$`Unit

`$$ - $$`

It is defined as:

`\[ u := \frac{\hbar \, \omega_{\text D}}{k_{\text B} \, T} \]`It was simply defined to make the Debye formula more compact.## 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 physical constant and is often used in statistical physics and thermodynamics. It has the value: \( k_{\text B} ~\approx~ 1.380 \,\cdot\, 10^{-23} \, \frac{\text J}{\text K} \).