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\subsection{Varying the Magnetic Polarization}   The relationship described in Eq. \ref{eq:growthrate} can not be applied to the degree of polarization or magnetization. In order to determine the relationship between magnetic field and degree of magnetization, we varied the applied magnetic field. When varying the magnetic field, the fractional populations are affected by the g-factor, or spectroscopic splitting factor. The fractional populations can be described by the lowerand upper states-- $N_1$ represents the lower  state $N_1$  and$N_2$ represents the  upper state. state $N_2$.  The difference between these two states multiplied by the magnetic moment describes the magnetization of the field. The following relationship, the Brillouin function, can be expressed in the equation below \cite{Kittel_1953}, \begin{equation}   \label{eq:tanh}  M=(N_1-N_2)\mu= N\mu\cdot\frac{e^x-e^{-x}}{e^x+e^{-x}}=N\mu \tanh{x}