Phonon dispersion relation

Phonon dispersion relation

Yanming Wang, Saad Bhamla and Wei Cai

May, 2012

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$V({r_{i}})=V({R_{i}})+{\frac {1}{2}}\sum _{i,j}U_{i}^{T}\cdot \left[{\frac {\partial ^{2}V}{\partial r_{i}\partial r_{j}}}\right]\cdot U_{j}$

$\left[{\begin{array}{ccc}{\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial y_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial z_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial z_{j}}}\\\end{array}}\right]$