Einstein solid

In solid-state physics, the Einstein model ( Albert Einstein) describes a method to calculate the contribution of the lattice vibrations ( phonons) to the heat capacity of a crystalline solid. It is not as successful as the Debye model, since the Einstein model can be applied exclusively to optical phonons, while the latter also describes acoustic phonons.

Fundamentals of the Model

The lattice vibrations of the crystal are quantized, that is, the solid body can absorb vibration energy only in discrete quanta. This quantum is also called phonons. It then describes the solid state as a quantum harmonic oscillators consisting of N that can oscillate independently of each in three directions. The occupation probability of such a mode of vibration ( a phonon ) depends on the temperature T and obtains (since phonons are bosons ) of the Bose - Einstein distribution:

Thus, the internal energy U in the solid state yields to ( It the quantization of the harmonic oscillator was used):

Where N = number of particles.

The contribution is at the zero point energy. The contribution of phonons to the heat capacity is then:

With the Einstein temperature results in a simpler notation:

Failure at low temperatures

It arises in the limit of large and small temperatures:

As the Debye model gives the Einstein model, the correct high-temperature limit according to the Dulong - Petit law. The course of CV (T) described above for small temperatures, however, differs significantly from measurements. This is due to the assumption that all harmonic oscillators in the solid state would oscillate at a single frequency. The ratios in real solids are much more complicated.