Einstein model

In solid-state physics , the Einstein model (according to Albert Einstein ) describes a method to calculate the contribution of the lattice vibrations ( phonons ) to the heat capacity of a crystalline solid . Since the Einstein model can only be applied to optical phonons , it is not as successful as the Debye model , which describes acoustic phonons .

Basics of the model

The lattice vibrations of the crystal are quantized , i.e. H. the solid can only absorb vibrational energy in discrete quanta . These quanta are also called phonons . The solid is then described as consisting of N quantum harmonic oscillators , each of which can oscillate independently in three directions. The occupation probability of such an oscillation mode (a phonon) depends on the temperature T and follows (since phonons are bosons ) the Bose-Einstein distribution : ${\displaystyle \hbar \cdot \omega _{\mathrm {E} }}$  ${\displaystyle \langle n\rangle }$ 

${\displaystyle \langle n\rangle ={\frac {1}{\exp \left({\frac {\hbar \cdot \omega _{\mathrm {E} }}{k_{\mathrm {B} }\cdot T}}\right)-1}}}$

With

• ${\displaystyle \hbar }$: reduced Planck's quantum of action
• ${\displaystyle \omega _{\mathrm {E} }}$: Einstein frequency ( angular frequency )
• ${\displaystyle k_{\mathrm {B} }}$: Boltzmann constant .

This results in the internal energy  U in the solid body (the quantization condition of the harmonic oscillator was used):

${\displaystyle {\begin{aligned}U&=3N\cdot \hbar \cdot \omega _{\mathrm {E} }\cdot \left(\langle n\rangle +{\frac {1}{2}}\right)\\&=3N\cdot \hbar \cdot \omega _{\mathrm {E} }\cdot \left[{\frac {1}{\exp \left({\frac {\hbar \cdot \omega _{\mathrm {E} }}{k_{\mathrm {B} }\cdot T}}\right)-1}}+{\frac {1}{2}}\right]\end{aligned}}}$

With

• ${\displaystyle N}$: Particle number .

The contribution indicates the zero point energy . ${\displaystyle {\frac {\hbar \cdot \omega _{\mathrm {E} }}{2}}}$

The contribution of the phonons to the heat capacity is then:

${\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V={\rm {const}}}=3N\cdot {\frac {(\hbar \cdot \omega _{\mathrm {E} })^{2}}{k_{\mathrm {B} }\cdot T^{2}}}\cdot {\frac {\exp \left({\frac {\hbar \cdot \omega _{\mathrm {E} }}{k_{\mathrm {B} }\cdot T}}\right)}{\left[\exp \left({\frac {\hbar \cdot \omega _{\mathrm {E} }}{k_{\mathrm {B} }\cdot T}}\right)-1\right]^{2}}}}$

With

• ${\displaystyle V}$: Volume .

The Einstein temperature results in a simpler notation: ${\displaystyle \Theta _{\mathrm {E} }={\frac {\hbar \cdot \omega _{\mathrm {E} }}{k_{\mathrm {B} }}}}$

${\displaystyle C_{V}\left(T\right)=3N\cdot k_{\mathrm {B} }\cdot \left({\frac {\Theta _{\mathrm {E} }}{T}}\right)^{2}\cdot {\frac {\exp \left({\frac {\Theta _{\mathrm {E} }}{T}}\right)}{\left[\exp \left({\frac {\Theta _{\mathrm {E} }}{T}}\right)-1\right]^{2}}}}$

Failure at low temperatures

Like the Debye model, the Einstein model provides the correct high temperature limit according to the Dulong-Petit law :

${\displaystyle T\rightarrow \infty :\ \ C_{V}\rightarrow 3\cdot N\cdot k_{\mathrm {B} }}$

In the Limes of lower temperatures, the following results:

${\displaystyle T\rightarrow 0:\ \ C_{V}\propto e^{-\Theta _{\mathrm {E} }/T}\rightarrow 0}$

This course of  C _V (T) for low temperatures deviates considerably from measurements. This is related to the assumption that all harmonic oscillators in the solid state would vibrate with a uniform frequency. However, the conditions in the real solid are much more complicated.

literature

• "The Planck theory of radiation and the theory of specific heat", A. Einstein, Annalen der Physik, volume 22, pp. 180–190, 1907. Online