= Schottky anomaly
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A <Schottky anomaly> is a peak in <heat capacity> caused by thermal occupation of a small set of separated <energy levels>. For $N$ independent two-level units with equal degeneracies and gap $\Delta>0$, write $x=\Delta/(k_BT)$; their <heat capacity> is $C=Nk_Bx^2e^{-x}/(1+e^{-x})^2$, with <Boltzmann constant> $k_B$. It vanishes at both low and high <temperature> and can spoil a lattice $T^3$ 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>.
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