2013-06-20 14:44:20 +08:00
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.. _formulations:
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2017-08-26 14:18:01 +08:00
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Formulations
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=============
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2012-12-14 17:06:27 +08:00
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2016-10-12 13:58:02 +08:00
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.. contents::
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:depth: 2
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:local:
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2012-12-14 17:06:27 +08:00
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Second-order force constants
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2017-08-26 14:18:01 +08:00
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-----------------------------
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2012-12-14 17:06:27 +08:00
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Potential energy of phonon system is represented as functions of atomic
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positions:
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.. math::
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V[\mathbf{r}(j_1 l_1),\ldots,\mathbf{r}(j_n l_N)],
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where :math:`\mathbf{r}(jl)` is the point of the :math:`j`-th atom in
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the :math:`l`-th unit cell and :math:`n` and :math:`N` are the number
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of atoms in a unit cell and the number of unit cells, respectively. A
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force and a second-order force constant :math:`\Phi_{\alpha \beta}`
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are given by
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.. math::
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F_\alpha(jl) = -\frac{\partial V }{\partial r_\alpha(jl)}
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and
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.. math::
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\Phi_{\alpha\beta}(jl, j'l') = \frac{\partial^2
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V}{\partial r_\alpha(jl) \partial r_\beta(j'l')} =
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-\frac{\partial F_\beta(j'l')}{\partial r_\alpha(jl)},
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respectively, where :math:`\alpha`, :math:`\beta`, ..., are the
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Cartesian indices, :math:`j`, :math:`j'`, ..., are the indices of
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atoms in a unit cell, and :math:`l`, :math:`l'`, ..., are
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the indices of unit cells. In the finite displacement method, the
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equation for the force constants is approximated as
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.. math::
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\Phi_{\alpha\beta}(jl, j'l') \simeq -\frac{
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F_\beta(j'l';\Delta r_\alpha{(jl)}) - F_\beta(j'l')} {\Delta
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r_\alpha(jl)},
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where :math:`F_\beta(j'l'; \Delta r_\alpha{(jl)})` are the forces on
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atoms with a finite displacement :math:`\Delta r_\alpha{(jl)}` and
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usually :math:`F_\beta(j'l') \equiv 0`.
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2013-08-04 14:21:46 +08:00
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.. _force_constants_solver_theory:
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2018-05-30 07:02:58 +08:00
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2013-11-08 13:02:37 +08:00
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Modified Parlinski-Li-Kawazoe method
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2017-08-26 14:18:01 +08:00
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-------------------------------------
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2012-12-14 17:06:27 +08:00
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2013-08-04 14:21:46 +08:00
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The following is a modified and simplified version of the
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2016-10-12 13:58:02 +08:00
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Parlinski-Li-Kawazoe method, which is just a numerical fitting
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approach to obtain force constants from forces and displacements.
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2012-12-14 17:06:27 +08:00
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The last equation above is represented by matrices as
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.. math::
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\mathbf{F} = - \mathbf{U} \mathbf{P},
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where :math:`\mathbf{F}`, :math:`\mathbf{P}`, and :math:`\mathbf{U}`
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2013-08-04 14:21:46 +08:00
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for a pair of atoms, e.g. :math:`\{jl, j'l'\}`, are given by
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2012-12-14 17:06:27 +08:00
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.. math::
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\mathbf{F} =
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2012-12-14 17:06:27 +08:00
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\begin{pmatrix}
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F_{x} & F_{y} & F_{z}
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2012-12-14 17:06:27 +08:00
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\end{pmatrix},
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.. math::
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\mathbf{P} =
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2012-12-14 17:06:27 +08:00
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\begin{pmatrix}
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\Phi_{xx} & \Phi_{xy} & \Phi_{xz} \\
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\Phi_{yx} & \Phi_{yy} & \Phi_{yz} \\
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\Phi_{zx} & \Phi_{zy} & \Phi_{zz}
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2012-12-14 17:06:27 +08:00
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\end{pmatrix},
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.. math::
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2013-08-04 14:21:46 +08:00
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\mathbf{U} =
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2012-12-14 17:06:27 +08:00
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\begin{pmatrix}
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\Delta r_{x} & \Delta r_{y} & \Delta r_{z} \\
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2012-12-14 17:06:27 +08:00
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\end{pmatrix}.
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The matrix equation is expanded for number of
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forces and displacements as follows:
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.. math::
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\begin{pmatrix}
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\mathbf{F}_1 \\
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\mathbf{F}_2 \\
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\vdots
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\end{pmatrix}
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= -
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\begin{pmatrix}
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\mathbf{U}_1 \\
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\mathbf{U}_2 \\
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\vdots
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\end{pmatrix}
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\mathbf{P}.
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With sufficient number of atomic displacements, this
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may be solved by pseudo inverse such as
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.. math::
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\mathbf{P} = -
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\begin{pmatrix}
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\mathbf{U}_1 \\
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\mathbf{U}_2 \\
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\vdots
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\end{pmatrix}^{+}
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\begin{pmatrix}
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\mathbf{F}_1 \\
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\mathbf{F}_2 \\
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\vdots
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\end{pmatrix}.
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Required number of atomic displacements to solve the simultaneous
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equations may be reduced using site-point symmetries. The matrix
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2013-08-04 14:21:46 +08:00
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equation can be written using a symmetry operation as
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2012-12-14 17:06:27 +08:00
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.. math::
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2013-08-04 14:21:46 +08:00
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\hat{R}(\mathbf{F}) = -\hat{R}(\mathbf{U})\mathbf{P},
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2012-12-14 17:06:27 +08:00
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2013-08-04 14:21:46 +08:00
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where :math:`\hat{R}` is the site symmetry
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operation centring at
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:math:`\mathbf{r}(jl)`. :math:`\hat{R}(\mathbf{F})` and :math:`\hat{R}(\mathbf{U})` are defined as
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:math:`\mathbf{RF}(\hat{R^{-1}}(j'l'))` and :math:`\mathbf{RU}`,
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respectively, where :math:`\mathbf{R}` is the matrix
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representation of the rotation operation. The combined
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simultaneous equations are built such as
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2012-12-14 17:06:27 +08:00
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.. math::
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\begin{pmatrix}
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\mathbf{F}^{(1)}_1 \\
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\mathbf{F}^{(2)}_1 \\
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\vdots \\
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\mathbf{F}^{(1)}_2 \\
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\mathbf{F}^{(2)}_2 \\
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\vdots \end{pmatrix} = -
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\begin{pmatrix}
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2018-05-30 07:02:58 +08:00
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\mathbf{U}^{(1)}_1 \\
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2012-12-14 17:06:27 +08:00
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\vdots \\
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\mathbf{U}^{(2)}_1 \\
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\mathbf{U}^{(1)}_2 \\
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\mathbf{U}^{(2)}_2 \\
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2012-12-14 17:06:27 +08:00
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\vdots
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\end{pmatrix}
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\mathbf{P}.
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where the superscript with parenthesis gives the index of
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site-symmetry operations. This is solved by pseudo inverse.
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.. _dynacmial_matrix_theory:
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Dynamical matrix
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2017-08-26 14:18:01 +08:00
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-----------------
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2012-12-14 17:06:27 +08:00
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2013-08-04 14:21:46 +08:00
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In phonopy, a phase convention of dynamical matrix is used as follows:
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2012-12-14 17:06:27 +08:00
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.. math::
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:label: eq_dynmat
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2012-12-14 17:06:27 +08:00
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D_{\alpha\beta}(jj',\mathbf{q}) = \frac{1}{\sqrt{m_j m_{j'}}}
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\sum_{l'}
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\Phi_{\alpha\beta}(j0, j'l')
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\exp(i\mathbf{q}\cdot[\mathbf{r}(j'l')-\mathbf{r}(j0)]),
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where :math:`m` is the atomic mass and :math:`\mathbf{q}` is the wave
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2013-11-08 13:02:37 +08:00
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vector. An equation of motion is written as
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2012-12-14 17:06:27 +08:00
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.. math::
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\sum_{j'\beta} D_{\alpha\beta}(jj',\mathbf{q}) e_\beta(j', \mathbf{q}\nu) =
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m_j [ \omega(\mathbf{q}\nu) ]^2 e_\alpha(j, \mathbf{q}\nu).
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where the eigenvector of the band index :math:`\nu` at
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:math:`\mathbf{q}` is obtained by the diagonalization of
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:math:`\mathbf{D}(\mathbf{q})`:
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.. math::
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\sum_{j \alpha j' \beta}e_\alpha(j',\mathbf{q}\nu)^* D_{\alpha\beta}(jj',\mathbf{q})
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e_\beta(j',\mathbf{q}\nu') = [\omega(\mathbf{q}\nu)]^2 \delta_{\nu\nu'}.
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2018-05-30 07:02:58 +08:00
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2012-12-14 17:06:27 +08:00
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The atomic displacements :math:`\mathbf{u}` are given as
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.. math::
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u_\alpha(jl,t) = \left(\frac{\hbar}{2Nm_j}\right)^{\frac{1}{2}}
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\sum_{\mathbf{q},\nu}\left[\omega(\mathbf{q}\nu)\right]^{-\frac{1}{2}}
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\left[\hat{a}(\mathbf{q}\nu)\exp(-i\omega(\mathbf{q}\nu)t)+
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\hat{a}^\dagger(\mathbf{-q}\nu)\exp({i\omega(\mathbf{q}\nu)}t)\right]
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\exp({i\mathbf{q}\cdot\mathbf{r}(jl)})
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e_\alpha(j,\mathbf{q}\nu),
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where :math:`\hat{a}^\dagger` and :math:`\hat{a}` are the creation and
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annihilation operators of phonon, :math:`\hbar` is the reduced Planck
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constant, and :math:`t` is the time.
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2013-07-14 15:31:47 +08:00
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.. _non_analytical_term_correction_theory:
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Non-analytical term correction
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2018-05-30 07:02:58 +08:00
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-------------------------------------
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2013-07-14 15:31:47 +08:00
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2018-05-30 07:02:58 +08:00
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To treat long range interaction of macroscopic electric field
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2013-07-14 15:31:47 +08:00
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induced by polarization of collective ionic motions near the
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:math:`\Gamma`-point, non-analytical term is added to dynamical matrix
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(:ref:`reference_NAC`). At
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:math:`\mathbf{q}\to\mathbf{0}`, the dynamical matrix with
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non-analytical term is given by,
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.. math::
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D_{\alpha\beta}(jj',\mathbf{q}\to \mathbf{0}) =
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2015-09-25 16:22:43 +08:00
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D_{\alpha\beta}(jj',\mathbf{q}=\mathbf{0})
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2017-12-04 14:00:14 +08:00
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+ \frac{1}{\sqrt{m_j m_{j'}}} \frac{4\pi}{\Omega_0}
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2017-10-19 17:06:15 +08:00
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\frac{\left[\sum_{\gamma}q_{\gamma}Z^{*}_{j,\gamma\alpha}\right]
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\left[\sum_{\gamma'}q_{\gamma'}Z^{*}_{j',\gamma'\beta}\right]}
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{\sum_{\alpha\beta}q_{\alpha}\epsilon_{\alpha\beta}^{\infty}
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q_{\beta}}.
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2013-07-14 15:31:47 +08:00
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2018-05-30 07:02:58 +08:00
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Phonon frequencies at general **q**-points with long-range
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dipole-dipole interaction are calculated by the
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method of Gonze *et al.* (:ref:`reference_dp_dp_NAC`).
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2013-01-23 10:07:06 +08:00
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.. _thermal_properties_expressions:
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Thermodynamic properties
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-------------------------
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Phonon number
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~~~~~~~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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2013-03-25 18:52:41 +08:00
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n = \frac{1}{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1}
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2013-01-23 10:07:06 +08:00
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Harmonic phonon energy
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~~~~~~~~~~~~~~~~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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E = \sum_{\mathbf{q}\nu}\hbar\omega(\mathbf{q}\nu)\left[\frac{1}{2} +
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2013-03-25 18:52:41 +08:00
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\frac{1}{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1}\right]
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2013-01-23 10:07:06 +08:00
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Constant volume heat capacity
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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C_V &= \left(\frac{\partial E}{\partial T} \right)_V \\
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2013-03-25 18:52:41 +08:00
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&= \sum_{\mathbf{q}\nu} k_\mathrm{B}
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\left(\frac{\hbar\omega(\mathbf{q}\nu)}{k_\mathrm{B} T} \right)^2
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\frac{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B}
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2018-05-30 07:02:58 +08:00
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T)}{[\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1]^2}
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2013-01-23 10:07:06 +08:00
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Partition function
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~~~~~~~~~~~~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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2013-03-25 18:52:41 +08:00
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Z = \exp(-\varphi/k_\mathrm{B} T) \prod_{\mathbf{q}\nu}
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\frac{\exp(-\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B}
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2018-05-30 07:02:58 +08:00
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T)}{1-\exp(-\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)}
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2013-01-23 10:07:06 +08:00
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Helmholtz free energy
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~~~~~~~~~~~~~~~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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2013-03-25 18:52:41 +08:00
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F &= -k_\mathrm{B} T \ln Z \\
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&= \varphi + \frac{1}{2} \sum_{\mathbf{q}\nu}
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2013-03-25 18:52:41 +08:00
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\hbar\omega(\mathbf{q}\nu) + k_\mathrm{B} T \sum_{\mathbf{q}\nu} \ln
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2018-05-30 07:02:58 +08:00
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\bigl[1 -\exp(-\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T) \bigr]
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Entropy
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2017-08-26 14:18:01 +08:00
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~~~~~~~~
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2013-01-23 10:07:06 +08:00
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.. math::
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2017-10-19 17:06:15 +08:00
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S &= -\frac{\partial F}{\partial T} \\ &= \frac{1}{2T}
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\sum_{\mathbf{q}\nu} \hbar\omega(\mathbf{q}\nu)
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\coth(\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B}T)-k_\mathrm{B}
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\sum_{\mathbf{q}\nu}
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\ln\left[2\sinh(\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B}T)\right]
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2017-10-19 17:06:15 +08:00
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.. include:: thermal-displacement.inc
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2013-12-04 12:47:15 +08:00
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2017-10-19 17:06:15 +08:00
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.. include:: group-velocity.inc
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.. _physical_unit_conversion:
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Physical unit conversion
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-------------------------
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Phonopy calculates phonon frequencies based on input values from
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users. In the default case, the physical units of distance, atomic
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mass, force, and force constants are supposed to be
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:math:`\text{Angstrom}`, :math:`\text{AMU}`, :math:`\text{eV/Angstrom}`, and
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:math:`\text{eV/Angstrom}^2`, respectively, and the physical unit of the
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phonon frequency is converted to THz. This conversion is made as
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follows:
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Internally phonon frequency has the physical unit of
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:math:`\sqrt{\text{eV/}(\text{Angstrom}^2\cdot \text{AMU})}` in angular
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frequency. To convert this unit to THz (not angular frequency), the
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calculation of ``sqrt(EV/AMU)/Angstrom/(2*pi)/1e12`` is made. ``EV``,
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``AMU``, ``Angstrom`` are the values to convert them to those in the
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SI base unit, i.e., to Joule, kg, and metre, respectively. These values
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implemented in phonopy are found at `a phonopy github page
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<https://github.com/phonopy/phonopy/blob/master/phonopy/units.py>`_. This
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unit conversion factor can be manually specified. See
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:ref:`frequency_conversion_factor_tag`.
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The unit conversion factor in the ``BORN`` file is multiplied with the second
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term of the right hand side of the equation in
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:ref:`non_analytical_term_correction_theory` where this equation is written
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with atomic units (:ref:`Gonze and Lee, 1997 <reference_NAC>`).
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The physical unit of the part of the equation corresponding to force
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constants:
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.. math::
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\frac{4\pi}{\Omega_0}
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\frac{[\sum_{\gamma}q_{\gamma}Z^{*}_{j,\gamma\alpha}]
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[\sum_{\gamma'}q_{\gamma'}Z^{*}_{j',\gamma'\beta}]}
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{\sum_{\alpha\beta}q_{\alpha}\epsilon_{\alpha\beta}^{\infty} q_{\beta}}.
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is :math:`[\text{hartree}/\text{bohr}^2]`. In the default case for the
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VASP interface, internally :math:`\Omega_0` is given in
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:math:`\text{Angstrom}^3`. In total, the necessary unit conversion is
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:math:`(\text{hartree} \rightarrow \text{eV}) \times (\text{bohr}
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\rightarrow \text{Angstrom})=14.4`. In the default case of the Wien2k
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interface, the conversion factor is :math:`(\text{hartree}
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\rightarrow \text{mRy})=2000`. For the other interfaces, the
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conversion factors are similarly calculated following the unit
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systems employed in phonopy (:ref:`calculator_interfaces`).
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.. _definition_of_commensurate_points:
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Crystal structure
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------------------
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Coordinates in direct and reciprocal spaces
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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As usual, in phonopy, the Born-von Karman boundary condition is
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assumed. Basis vectors of a primitive lattice are defined in three
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column vectors :math:`( \mathbf{a} \; \mathbf{b} \; \mathbf{c}
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)`. Coordinates of a point in the direct space :math:`\mathbf{r}` is
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represented with respect to these basis vectors. The direct lattice
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points are given by :math:`i
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\mathbf{a} + j \mathbf{b} + k \mathbf{a}, \{i,
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j, k \in \mathbb{Z}\}`, and the points for atoms in a unit cell :math:`x
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\mathbf{a} + y \mathbf{b} + z \mathbf{a}, \{0 \le x, y, z
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< 1\}`. Basis vectors of the reciprocal
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lattice may be given by three row vectors, :math:`( \mathbf{a}^{*T}
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/\; \mathbf{b}^{*T} /\; \mathbf{c}^{*T} )`, but here they are defined
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as three column vectors as :math:`( \mathbf{a}^{*} \; \mathbf{b}^{*}
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\; \mathbf{c}^{*} )` with
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.. math::
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:label: eq_rec_basis_vectors
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\mathbf{a}^{*} &= \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot
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(\mathbf{b} \times \mathbf{c})}, \\
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\mathbf{b}^{*} &= \frac{\mathbf{c} \times \mathbf{a}}{\mathbf{b} \cdot
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(\mathbf{c} \times \mathbf{a})}, \\
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\mathbf{c}^{*} &= \frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \cdot
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(\mathbf{a} \times \mathbf{b})}.
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Coordinates of a point in the reciprocal space :math:`\mathbf{q}` is
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represented with respect to these basis vectors, therefore :math:`q_x
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\mathbf{a}^{*} + q_y \mathbf{b}^{*} + q_z \mathbf{c}^{*}`. The
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reciprocal lattice points are given by :math:`G_x\mathbf{a}^{*} + G_y
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\mathbf{b}^{*} + G_z \mathbf{c}^{*}, \{G_x,
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G_y, G_z \in \mathbb{Z}\}`. Following these definition, phase
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factor should be represented as :math:`\exp(2\pi
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i\mathbf{q}\cdot\mathbf{r})`, however in phonopy documentation,
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:math:`2\pi` is implicitly included and not shown, i.e., it is
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represented like :math:`\exp(i\mathbf{q}\cdot\mathbf{r})` (e.g., see
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Eq. :eq:`eq_dynmat`). In the output of the reciprocal basis vectors,
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:math:`2\pi` is not included, e.g., in ``band.yaml``.
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In phonopy, unless :ref:`primitive_axis_tag` (or ``--pa`` option) is
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specified, basis vectors in direct space :math:`( \mathbf{a} \;
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\mathbf{b} \; \mathbf{c})` are set from the input unit celll structure
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even if it is a supercell or a conventional unit cell having centring,
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therefore the basis vectors in the reciprocal space are given by
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Eq. :eq:`eq_rec_basis_vectors`. When using :ref:`primitive_axis_tag`,
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:math:`( \mathbf{a} \; \mathbf{b} \; \mathbf{c})` are set from those
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transformed by the transformation matrix :math:`M_\text{p}` as written
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at :ref:`primitive_axis_tag`, therefore :math:`( \mathbf{a}^{*} \;
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\mathbf{b}^{*} \; \mathbf{c}^{*} )` are given by those calculated
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following Eq. :eq:`eq_rec_basis_vectors` with this :math:`( \mathbf{a}
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\; \mathbf{b} \; \mathbf{c})`.
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Commensurate points
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~~~~~~~~~~~~~~~~~~~~
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In phonopy, so-called commensurate points mean the q-points whose waves are
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confined in the supercell used in the phonon calculation.
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To explain about the commensurate points, let basis vectors of a
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primitive cell in direct space cell be the column vectors
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:math:`(\mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \;
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\mathbf{c}_\mathrm{p})` and those of the supercell be
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:math:`(\mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \;
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\mathbf{c}_\mathrm{s})`. The transformation of the basis vectors from
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the primitive cell to the supercell is written as
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.. math::
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( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )
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= ( \mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \;
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\mathbf{c}_\mathrm{p} ) \boldsymbol{P}.
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:math:`\boldsymbol{P}` is given as a :math:`3\times 3` matrix and its
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elements are all integers, which is a constraint we have. The
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resolution for q-points being the commensurate points is determined by
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:math:`\boldsymbol{P}` since one period of a wave has to be bound by
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any of lattice points inside the supercell. Therefore the number of
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commensurate points becomes the same as the number of the primitive
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cell that can be contained in the supercell, i.e.,
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:math:`\det(\boldsymbol{P})`.
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Then let the basis vectors in reciprocal space be the column vectors
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:math:`(\mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \;
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\mathbf{c}^*_\mathrm{p})`. Note that often reciprocal vectors are
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deifned by row vectors, but column vectors are chosen here to
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formulate. Formally we see the set of besis vectors are :math:`3\times
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3` matrices, we have the following relation:
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.. math::
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( \mathbf{a}^*_\mathrm{p} \;
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\mathbf{b}^*_\mathrm{p} \; \mathbf{c}^*_\mathrm{p} ) = (
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\mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \;
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\mathbf{c}_\mathrm{p} )^{-\mathbf{T}}.
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Similarly for the supercell, we define a relation
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.. math::
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( \mathbf{a}^*_\mathrm{s} \;
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\mathbf{b}^*_\mathrm{s} \; \mathbf{c}^*_\mathrm{s} ) = (
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\mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \;
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\mathbf{c}_\mathrm{s} )^{-\mathbf{T}}.
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Then
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.. math::
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( \mathbf{a}^*_\mathrm{s} \; \mathbf{b}^*_\mathrm{s} \;
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\mathbf{c}^*_\mathrm{s} ) \boldsymbol{P}^{\mathrm{T}} = (
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\mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \;
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\mathbf{c}^*_\mathrm{p} ).
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To multiply an arbitrary q-point :math:`\mathbf{q}` on both sides
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.. math::
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( \mathbf{a}^*_\mathrm{s} \; \mathbf{b}^*_\mathrm{s} \;
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\mathbf{c}^*_\mathrm{s} ) \boldsymbol{P}^{\mathrm{T}} \mathbf{q} = (
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\mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \;
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\mathbf{c}^*_\mathrm{p} ) \mathbf{q},
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we find the constraint of a q-point being one of the commensurate points is
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the elements of :math:`\boldsymbol{P}^{\mathrm{T}} \mathbf{q}` to be integers.
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