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

Formula: Debye Approximation for Thermal Capacity
Dispersion relation of lattice vibrations for monatomic crystal lattice

Thermal capacity

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

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.


Absolute temperature of the crystal under consideration.

Number of particles

Particle number of the crystal under consideration.

Energy ratio

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

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