Schottky anomaly

The Schottky anomaly is an observed effect in solid state physics where the specific heat capacity of a solid at low temperature has a peak. It is called anomalous because the heat capacity usually increases with temperature, or stays constant. It occurs in systems with a limited number of energy levels so that E(T) is bounded. Cv =(dE/dT), hence bump.

This effect can be explained by looking at the change in entropy of the system. At zero temperature only the lowest energy level is occupied, entropy is zero, and there is very little probability of a transition to a higher energy level. With increasing temperature the probability of a transition, and thus an increase in entropy, goes up. As the temperature approaches the difference between the energy levels there is a broad peak in the specific heat corresponding to a large change in entropy for a small change in temperature. At high temperatures all of the levels are populated evenly, so there is again little change in entropy for small changes in temperature and thus a lower specific heat capacity.

$S=\int _{0}^{T}\!\left({\frac {C_{v}}{T}}\right)dT\,$

For a two level system the specific heat coming from the Schottky anomaly has the form:

$C_{\rm {Schottky}}=R\left({\frac {\Delta }{T}}\right)^{2}{\frac {e^{\Delta /T}}{[1+e^{\Delta /T}]^{2}}}\,$

Where k_BΔ is the energy between the two levels.

This anomaly is usually seen in paramagnetic salts at low temperature.

It was named after Walter H. Schottky.