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\begin{eqnarray*}

  r_{c}^{fcc} & = & \frac{1}{2} \left(\frac{\sqrt{2}}{2} + 1\right) \mathrm{a} \simeq 0.8536 \:\mathrm{a} \\

  r_{c}^{bcc} & = & \frac{1}{2}(\sqrt{2} + 1) \mathrm{a} \simeq 1.207 \:\mathrm{a} \\

  r_{c}^{hcp} & = & \frac{1}{2}\left(1+\sqrt{\frac{4+2x^{2}}{3}}\right) \mathrm{a}

\end{eqnarray*}

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\begin{eqnarray*}

   U^{cond} = \displaystyle\sum_{i=1}^{N} u_{i}^{cond} \\ 

   U^{mech} = \displaystyle\sum_{i=1}^{N} u_{i}^{mech} \\

   U^{chem} = \displaystyle\sum_{i=1}^{N} u_{i}^{chem} \\

   U = \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech} + u_{i}^{chem}) \\

   \theta_{avg} = (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\

\end{eqnarray*}                           

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