Quantum isochoric molar heat capacity of solids

To derive the heat capacity of a solid, one should account for quantum effects. Albert Einstein used the same model as in the classical case, namely the atoms are harmonic oscillators with three degrees of freedom, located in the nodes of the crystal lattice, performing thermal oscillations around the equlibrium positions with the same frequency. But he used a more correct expression for the energy of the oscillators, and although the result still only qualitatively describes the heat capacity of solids, it is a big achievement and the result has correct asymptotic behaviour for \(T \to 0\).

Notation:

1. \(R\) (R) is molar_gas_constant.

Links:

1. Wikipedia, equivalent formula in the box.

isochoric_molar_heat_capacity

molar_heat_capacity at constant volume.

Symbol:

c_Vm

Latex:

\(c_{V, m}\)

Dimension:

energy/(amount_of_substance*temperature)

reduced_photon_energy

Reduced photon energy, defined as the ratio of photon energy \(\hbar \omega\) or \(h \nu\) to thermal energy \(k_\text{B} T\).

Symbol:

x

Latex:

\(x\)

Dimension:

dimensionless

law

c_Vm = 3 * R * x^2 * exp(x) / (exp(x) - 1)^2

Latex:

\[c_{V, m} = 3 R \frac{x^{2} \exp{\left(x \right)}}{\left(\exp{\left(x \right)} - 1\right)^{2}}\]