Phonon dispersion relation: Difference between revisions
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For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1 |
For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1 |
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<math>V(\{ \vec{r}_{i} \}) =V(\{ \vec{ r}_{i} \}) + \frac{1}{2}\sum_{ij} \vec{u}_{i}^{T}\cdot [\frac{d^{2}V}{d \vec{r}_i \vec{r}_j}]\cdot \vec{u}_{j} </math> |
<math>V(\{ \vec{r}_{i} \}) =V(\{ \vec{ r}_{i} \}) + \frac{1}{2}\sum_{ij} \vec{u}_{i}^{T}\cdot [\frac{d^{2}V}{d \vec{r}_i \vec{r}_j}]\cdot \vec{u}_{j} </math> (1) |
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where <math>V</math> is the potential function, <math>\vec{ r}_{i}</math> is the coordinate of the <math>ith</math> atom, <math>\vec{ R}_{i}</math> is the position of the $ith$ atom at emuilibrium and <math>\vec{u}</math> is a small displacement from the equilibrium position: <math>\vec{u}_{i} = \vec{r}_{i} - \vec{R}_{i} </math>. |
where <math>V</math> is the potential function, <math>\vec{ r}_{i}</math> is the coordinate of the <math>ith</math> atom, <math>\vec{ R}_{i}</math> is the position of the $ith$ atom at emuilibrium and <math>\vec{u}</math> is a small displacement from the equilibrium position: <math>\vec{u}_{i} = \vec{r}_{i} - \vec{R}_{i} </math>. |
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In Equation 1, the linear terms in the Taylor expansion are at 0K with the minimum energy state and the higher order terms are neglected using the Harmonic approximation. The second derivative of the potential energy evaluated at the equilibrium position and is called the '''force constant matrix''' or the '''Hessian matrix''' and is obtained from MD++ using the '''calHessian''' function. |
In Equation 1, the linear terms in the Taylor expansion are at 0K with the minimum energy state and the higher order terms are neglected using the Harmonic approximation. The second derivative of the potential energy evaluated at the equilibrium position and is called the '''force constant matrix''' or the '''Hessian matrix''' and is obtained from MD++ using the '''calHessian''' function. |
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\frac{\partial^2V}{\partial y_i\partial x_j} & \frac{\partial^2V}{\partial y_i\partial y_j} & \frac{\partial^2V}{\partial y_i\partial z_j} \\ |
\frac{\partial^2V}{\partial y_i\partial x_j} & \frac{\partial^2V}{\partial y_i\partial y_j} & \frac{\partial^2V}{\partial y_i\partial z_j} \\ |
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\frac{\partial^2V}{\partial z_i\partial x_j} & \frac{\partial^2V}{\partial z_i\partial y_j} & \frac{\partial^2V}{\partial z_i\partial z_j} \\ |
\frac{\partial^2V}{\partial z_i\partial x_j} & \frac{\partial^2V}{\partial z_i\partial y_j} & \frac{\partial^2V}{\partial z_i\partial z_j} \\ |
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\end{array}} \right] </math> |
\end{array}} \right] </math> (2) |
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Revision as of 05:29, 15 June 2012
Phonon Dispersion Relation
Yanming Wang, Saad Bhamla and Wei Cai
May, 2012
METHODS
For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1
(1)
![{\displaystyle V(\{{\vec {r}}_{i}\})=V(\{{\vec {r}}_{i}\})+{\frac {1}{2}}\sum _{ij}{\vec {u}}_{i}^{T}\cdot [{\frac {d^{2}V}{d{\vec {r}}_{i}{\vec {r}}_{j}}}]\cdot {\vec {u}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcb6a0039cf26ae5fb951642edb5fa6a7a927fc)
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is the potential function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ r}_{i}} is the coordinate of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ith} atom, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ R}_{i}} is the position of the $ith$ atom at emuilibrium and is a small displacement from the equilibrium position: .
In Equation 1, the linear terms in the Taylor expansion are at 0K with the minimum energy state and the higher order terms are neglected using the Harmonic approximation. The second derivative of the potential energy evaluated at the equilibrium position and is called the force constant matrix or the Hessian matrix and is obtained from MD++ using the calHessian function.
(2)
![{\displaystyle K_{ij}=[{\frac {d^{2}V}{d{\vec {r}}_{i}{\vec {r}}_{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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdfcf64909139b50c1a1ccabd7e03f14090b726a)