# Magnetic Impurity Corrections to Specific Heat of Alumina (and Alumina Substrates)

## Specific Heat Anomalies in RuOx on Alumina

Scottky-type anomalies have been observed in a variety of RuO_2 thick film resistors on alumina substrates (Volokitin 1994) and alumina ceramics with magnetic impurities (Tarasenko 2010).

## Nuclear Schottky effect

In the presence of a magnetic field, the energy levels of a nucleus with spin $$I$$ split into $$2I+1$$ levels. This effect — known as the nuclear Zeeman effect — leads to hyperfine splitting of energy levels when the nuclei are in a local magnetic field and a field and temperature dependent contribution to the specific heat.

In closed form (Phillips 1971, Leyarovski 1988),

$$\label{eq:ClosedFormSolution}c_{N}=R\left(\frac{x}{2I}\right)^{2}\left[{\textrm{Csch}}^{2}\left(\frac{x}{2I}\right)-(2I+1)^{2}{\textrm{Csch}}^{2}\left[(2I+1)\left(\frac{x}{2I}\right)\right]\right]\\$$

where

$$\label{eq:ClosedFormVariable}x=\frac{g_{N}{\mu}_{N}IH}{k_{B}T}\\$$

The Zeeman splitting of the nuclear energy levels leads to a broad peak in the specific heat when $$k_{b}T$$ is on the order of the energy splitting $$\mu H$$, but because the energy splittings are small11the nuclear magneton is a factor of 1836.1 smaller than the Bohr magneton $$\mu_{B}$$, only the high temperature $$(\mu H/k_{B}T)^{2}$$ tail of specific heat peak is usually seen. In that limit, and assuming that the zero field field splitting (ZFS) is negligible in comparison to that from the applied field $$H$$, then (Lounasmaa 1974, Leyarovski 1988),

$$\label{eq:SchottkyTail}c_{N}(T,H)=\frac{{\lambda}_{N}}{\mu_{0}}\left(\frac{H}{T}\right)^{2}\textrm{ [J/ mol K]}\\$$

where $${\mu}_{0}=4\pi\cdot 10^{-7}\textrm{ H/m}$$ and the nuclear Curie constant $$\lambda_{N}$$ is given by

$$\label{eq:CurieConstant}{\frac{\lambda_{N}}{\mu_{0}}=N_{A}I(I+1)\frac{\left({\mu}_{N}g_{N}\right)^{2}}{3k_{B}}\left[\frac{\textrm{ J K}}{\textrm{mol T}^{2}}\right]}.\\$$

where here $$g_{N}=\mu/I$$ and $$\mu_{N}$$ is the nuclear magneton $$\mu_{N}=\frac{he}{4\pi M_{p}c}=5.05110^{-27}\textrm{ J/T}$$.

It is often convenient to express this Zeeman splitting of the nuclear energy levels in temperature units. In this case

$$\label{eq:NuclearEnergySplittingInKelvin}\Delta=g\mu_{n}H/k_{B}\textrm{ [K]}.\\$$

## Nuclear Specific Heat of $$\kappa$$-(BEDT-TTF)$${}_{2}$$Cu(NCS)$${}_{2}$$

$$\kappa$$-(BEDT-TTF)$${}_{2}$$Cu(NCS)$${}_{2}$$ — with chemical formula C$${}_{22}$$H$${}_{16}$$N$${}_{2}$$S$${}_{18}$$Cu$${}_{1}$$ (Urayama 1988) — has a molar mass \(