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# 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),

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

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 $$m_{mole}$$ 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

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

where

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

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, $$\alpha_{Cu}=6.18\cdot 10^{-9}\left[\frac{\textrm{J K}}{{\textrm{T}}^{2}}\right]$$.

### References

1. Norman E. Phillips. Low-temperature heat capacity of metals. C R C Critical Reviews in Solid State Sciences 2, 467–553 (1971). Link

2. E.I Leyarovski, L.N Leyarovska, Chr Popov, O Popov. High field magnetocalorimetry below 1 K: specific heat of copper at 14 T. Cryogenics 28, 321–335 (1988). Link

3. O. V. Lounasmaa. Experimental principles and methods below 1 K / O. V. Lounasmaa.. (1974).

4. Hatsumi Urayama, Hideki Yamochi, Gunzi Saito, Kiyokazu Nozawa, Tadashi Sugano, Minoru Kinoshita, Shoichi Sato, Kokichi Oshima, Atsushi Kawamoto, Jiro Tanaka. A new ambient pressure organic superconductor based on BEDT-TTF with Tc higher than 10K (Tc=10.4K).. Chem. Lett. 55–58 (1988). Link

5. Gladys H. Fuller. Nuclear Spins and Moments. Journal of Physical and Chemical Reference Data 5, 835 (1976). Link