A Schottky anomaly is a peak in heat capacity caused by thermal occupation of a small set of separated energy levels. For independent two-level units with equal degeneracies and gap , write ; their heat capacity is , with Boltzmann constant . It vanishes at both low and high temperature and can spoil a lattice fit in between. Localized electronic or nuclear levels can contribute, including in calorimeter addenda; their presence must be checked rather than attributed automatically to phonons.
For independent units with lower and upper energy levels separated by , let their degeneracies be and . Their shifted partition function is , with . Differentiation givesThe heat capacity vanishes at both low and high temperature. Degeneracy changes both the maximum and its temperature; the equal-degeneracy result is not universal. Independent lower-level degeneracy also leaves residual entropy in the ideal model.
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The Schottky anomaly refers to a specific behavior observed in the heat capacity of certain materials, particularly in ionic or non-metallic solids, at low temperatures. Named after physicist Walter H. Schottky, the phenomenon arises due to the presence of localized states or defects within the material's crystal structure. In these materials, as the temperature decreases, the heat capacity does not follow the expected behavior for standard Debye or Einstein models, which predict a decrease in heat capacity with decreasing temperature.