deletions | additions       diff --git a/section_Nuclear_Schottky_effect_In__.tex b/section_Nuclear_Schottky_effect_In__.tex  index ac03b7c..27e34b7 100644  --- a/section_Nuclear_Schottky_effect_In__.tex  +++ b/section_Nuclear_Schottky_effect_In__.tex 
...
\section{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. 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 small, only the high temperature $(\mu H/k_B T)^2$ tail of specific heat peak is seen. In that limit limit~\  \cite{Lounasmaa1974, Leyarovski_1988},neglecting the contribution of effective fields of the dipole and quadrupole moments in Cu in comparison to that from the applied field $H$,  \begin{equation}  \label{eq:SchottkyTail}  C_N(T,H) = \frac{{\lambda}_N}{\mu_0}\left(\frac{H}{T}\right)^2 \textrm{ [J/ mol K]} 
...
\label{eq:CurieConstant}  \lambda_N = \mu_0 N_A I (I+1)\frac{\left({\mu}_N g_N\right)^2}{3 k_B}.  \end{equation}  In this expression, we are assuming that the contribution ofthe effective fields of the dipole and quadrupole moments in Cu are negligible in comparison to that from the applied field $H$.  Substituting $g_N = \mathrm{1.5}$ --- the average nuclear g-factor for the two most common Cu isotopes ${}^{63}\mathrm{Cu}$ and ${}^{65}\mathrm{Cu}$ \cite{Leyarovski_1988}--- into Eq.~\ref{eq:CurieConstant} gives a theoretical value $\lambda_{\mathrm{Cu}} = 3.93 \cdot 10^{-12} \textrm{ K}{\textrm{ m}}^3 {\textrm{ mol}}^{-1}$. Experimentally, \cite{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.