Nuclear Schottky Corrections to Specific Heat Measurements

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),

\begin{equation} \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]\\ \end{equation}


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

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),

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

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

\begin{equation} \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]}.\\ \end{equation}

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

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

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 \(m_{mole}\) of \(832.98\textrm{ g/mole}\). Both the hydrogen and copper nuclei will contribute significantly to the nuclear specific heat.

The predominant stable isotope of hydrogen is \({}^{\ 1}_{\ 1}\textrm{H}\), which has a nuclear spin \(I=1/2\) (in units of \(\hbar\)) and a nuclear magnetic moment \(\mu=2.79278\) (in units of nuclear magnetons), resulting in a nuclear g-factor \(g=\mu/I=5.586\) (Fuller 1976). In deuterated samples — where \({}^{\ 1}_{\ 1}\textrm{H}\) is replaced by \({}^{\ 2}_{\ 1}\textrm{H}\) – the larger nuclear spin \(I=1\) and the smaller nuclear magnetic moment \(\mu=0.85742\) result in a much smaller nuclear g-factor \(g=I/\mu=0.857\).The two most common stable copper isotopes are \({}^{\ 63}_{\ 29}\textrm{Cu}\) and \({}^{\ 65}_{\ 29}\textrm{Cu}\). \({}^{\ 63}_{\ 29}\textrm{Cu}\) has a nuclear spin \(I=3/2\) and a nuclear magnetic moment \(\mu=2.2228\) while \({}^{\ 65}_{\ 29}\textrm{Cu}\) has a nuclear spin \(I=3/2\) and a nuclear magnetic moment \(\mu=2.3812\) (Fuller 1976).

Substituting \(g_{N}=\mathrm{1.5}\) — the weighted average nuclear g-factor for these two most common Cu isotopes (Leyarovski 1988)— into Eq. \ref{eq:CurieConstant} gives a theoretical value \(\lambda_{\mathrm{Cu}}=3.93\cdot 10^{-12}\textrm{ K}{\textrm{ m}}^{3}{\textrm{ mol}}^{-1}\). Similarly, the corresponding Curie Constant for H is \(\lambda_{\mathrm{Cu}}=10.91\cdot 10^{-12}\textrm{ K}{\textrm{ m}}^{3}{\textrm{ mol}}^{-1}\).

Experimentally, (Leyarovski 1988) find a value of \(\lambda_{\mathrm{Cu}}=4.03\cdot 10^{-12}\textrm{ K}{\textrm{ m}}^{3}{\textrm{ mol}}^{-1}\) upon fitting Eq. \ref{eq:SchottkyTail} to their measurements of the specific heat of Cu taken between 0.3 K and 1 K in an applied field of 14 T.

In our Jan 2015 heat capacity run, we used a 0.40 mg sample of \(\kappa\)-(BEDT-TTF)\({}_{2}\)Cu(NCS)\({}_{2}\). In our July 2015 run, we used a 0.50 mg sample. The nuclear contribution of the Cu and H nuclei to the measured heat capacity of \(\kappa\)-(BEDT-TTF)\({}_{2}\)Cu(NCS)\({}_{2}\) is predicted to be

\begin{equation} \label{eq:NuclearCuContribution}C={\alpha}\frac{H^{2}}{T^{2}}\mathrm{\ [J/K]}\\ \end{equation}


\begin{equation} \label{eq:NuclearHeatCapacity}\alpha=m_{mole}\left(\frac{{\lambda}_{Cu}+{\lambda}_{H}}{{{\mu}_{0}}}\right).\\ \end{equation}

For the Jan 2015 run, \(\alpha_{Cu}=4.94\cdot 10^{-9}\left[\frac{\textrm{J K}}{{\textrm{T}}^{2}}\right]\). For the July 2015 run, \(\alpha_{Cu}=6.18\cdot 10^{-9}\left[\frac{\textrm{J K}}{{\textrm{T}}^{2}}\right]\).