demo.tex

\documentclass{equation-sheet}
\input{macros}

\begin{document}

\subsection{Exam 1 Material}

Coulombic attraction:

\begin{equation*}
	\begin{aligned}[l]
		F_c &= \frac{-k}{a^2} \\
	\end{aligned} 
	\qquad \qquad
	\begin{aligned}[r]
	    a &= \text{Separation distance} \\
		k &= k_0 q^2 \cdot z_1 \cdot z_2 \\
		k_0 &= \scinote{9}{9} \, V m C^{-1} \\
		q &= \scinote{1.6}{-19} C \\
		z &= \text{particle charge} \\
		& \text{(e.g. +1, -1)}
	\end{aligned}
\end{equation*}

\begin{equation*}
    \begin{aligned}
        F &= \deriv{E}{a}
    \end{aligned}
    \qquad \qquad
    \begin{aligned}
        F &= \text{Bonding force} \\
        E &= \text{Bonding energy}
    \end{aligned}
\end{equation*}

Thus, zero force occurs at minimum bonding energy.

\sepspace

\secbreak

\subsubsection{Crystal Structures}

\begin{itemize}
    \item Lattice direction ex: $[3 \bar 1 5]$
    \item Family of lattice directions: structurally identical, only differ by basis vectors
    \item Cubic system ex: Cubic$<100>=[100],\, [0 \bar 1 0],$ etc...
    \item Lattice planes: measured by normal vectors, a.k.a. intercepts$^{-1}$
    \item Lattice plane family: All have same planar density
    \item Linear/planar density: \# of atoms centered on a line/ plane per unit length/area
\end{itemize}

\begin{equation*}
    \text{Theoretical density: } \rho = \frac{nA}{V_c N_A} 
\end{equation*}
\cutspace
\begin{align*}
    n&= \text{Atoms/unit cell} \\
    A &= \text{Atomic weight }(amu) \\
    V_c &= \text{Volume of unit cell }(m^3) \\
    N_A &= \text{Avogadro's number }(atoms/mol)
\end{align*}

\begin{tabular}{|c|c|c|c|}
\hline
Structure & $a$ vs. $r$ & APF & Atoms/Cell \\
\hline
bcc & $a = \frac{4r}{\sqrt{3}}$ & 0.68 & 2 \\
\hline
fcc & $a = \frac{4r}{\sqrt{2}}$ & 0.74 & 4 \\
\hline
hcp & $a = \sqrt{2r}$ & 0.74 & 6 \\
\hline
\end{tabular}

\end{document}