deletions | additions       diff --git a/When_an_external_field_B__.tex b/When_an_external_field_B__.tex  index 4970fc5..76d67ca 100644  --- a/When_an_external_field_B__.tex  +++ b/When_an_external_field_B__.tex 
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\subsection{Varying the Polarization time}  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 $M_{\infty}$. Eq. \ref{eq:growthrate} is used to describe the relationship between the voltage verses a changing polarization time. \\  The Curie law can derived using $M=n\mu$ where $$ represents the average value of $cos(\theta)$, measuring the alignment between the magnetic moment and the external field B, for all magnetic moments in the sample. The calculation of $$ using classical thermodynamics results in the Curie law. The Curie value can be calculated using the following equation:   \begin{equation} 
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M_o=\frac{n\mu^2B}{3kT}  \end{equation}  \subsection{Varying the current}  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)}\\  \\   \subsection{Studying Larmor precession}  Larmor precession is when a proton in a magnetic field experiences a magnetic torque that aligns the proton magnetic moment with the field. Due to the angular momentum and spin, the proton's motion is a precession about the magnetic field. Taking $B_{e}= \text{Earth's magnetic field} \approx 50 \mu T$ and $\gamma= 2.675 \cdot 10^8 \frac{1}{s\cdot T}$, mathematically the precession frequency would be around,  \begin{equation}  \label{eq:precession}