deletions | additions       diff --git a/The_relationship_between_the_polarization__.tex b/The_relationship_between_the_polarization__.tex  index 8d7ee1d..f81f58d 100644  --- a/The_relationship_between_the_polarization__.tex  +++ b/The_relationship_between_the_polarization__.tex 
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When an external field B is applied to a sample, the magnetization M exponentially approaches the equilibrium magnetization.  The relationship between the polarization time and the voltage is described by the growth rate. The growth rate of M(t) towards $M_{\infty}$ is described by the following equation, \begin{equation}  \label{eq:growthrate}  M(t)=M_{\infty}(1-e^\frac{t}{T_1})  \end{equation}  In the equation above, $M_{\infty}$ represents the equilibrium Curie value (Eq. \ref{eq:curielaw}). The time constant $T_1$ is known as the spin-lattice relaxation time. In other words, $T_1$ is the time it takes for the magnetization to exponentially approach $M_{\infty}$.   The Curie value can be calculated using the following equation:   \begin{equation}  \label{eq:curielaw}  M_o=\frac{n\mu^2B}{3kT}  \end{equation}  This equation can derived using $M=n\mu$ where $$ represents the average value of $cos(theta)$, measuring the alignment between the spin angular momentum and magnetic moment, for all magnetic moments in the sample. The calculation of $$ using classical thermodynamics results in the Curie law equation.  However, the same relationship described in Eq. \ref{eq:growthrate} can not be applied to the degree of polarization (or the magnetization). Theory predicts that magnetization follows the relationship described in Equation \textbf{(insert eqn)} because magnetization is effected by quantum factors within the atom as well as the applied magnetic field, so it cannot be as simply modeled by an exponential. \textbf{(cite textbook)}\\  \\