mirror of https://github.com/phonopy/phonopy.git
50 lines
1.8 KiB
C++
50 lines
1.8 KiB
C++
.. _group_velocity:
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Group velocity
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---------------
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Method
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~~~~~~~
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Phonopy calculates group velocity of phonon as follows:
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.. math::
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\mathbf{v}_\mathrm{g}(\mathbf{q}\nu) = & \nabla_\mathbf{q} \omega(\mathbf{q}\nu) \\
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=&\frac{\partial\omega(\mathbf{q}\nu)}{\partial \mathbf{q}} \\
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=&\frac{1}{2\omega(\mathbf{q}\nu)}\frac{\partial[\omega(\mathbf{q}\nu)]^2}{\partial
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\mathbf{q}} \\
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=&\frac{1}{2\omega(\mathbf{q}\nu)}\left<\mathbf{e}(\mathbf{q}\nu)\biggl|
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\frac{\partial D(\mathbf{q})} {\partial
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\mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right>,
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where the meanings of the variables are found at :ref:`formulations`.
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Finite difference method
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~~~~~~~~~~~~~~~~~~~~~~~~~
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In the previous versions, group velocity was calculated using finite
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difference method:
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.. math::
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\mathbf{v}_\mathrm{g}(\mathbf{q}\nu) =
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\frac{1}{2\omega(\mathbf{q}\nu)}\left<\mathbf{e}(\mathbf{q}\nu)\biggl|
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\frac{\partial D(\mathbf{q})} {\partial
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\mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right>
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\simeq \frac{1}{2\omega(\mathbf{q}\nu)}
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\left<\mathbf{e}(\mathbf{q}\nu)\biggl|
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\frac{\Delta D(\mathbf{q})}
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{\Delta \mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right>.
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Group velocity calculation with the finite difference method is still
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able to be activated using ``GV_DELTA_Q`` tag or ``-gv_delta_q``
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option. :math:`\Delta\mathbf{q} = (\Delta q_x, \Delta q_y, \Delta
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q_z)` is described in Cartesian coordinated in reciprocal space. In
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the implementation, central difference is employed, and :math:`+\Delta
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q_\alpha` and :math:`-\Delta q_\alpha` are taken to calculate group
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velocity, where :math:`\alpha` is the Cartesian index in reciprocal
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space. :math:`\Delta q_\alpha` is specified in the unit of reciprocal
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space distance (:math:`\mathrm{\AA}^{-1}` for the default case) by
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``--gv_delta_q`` option or ``GV_DELTA_Q`` tag.
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