mirror of https://github.com/phonopy/phonopy.git
100 lines
3.4 KiB
ReStructuredText
100 lines
3.4 KiB
ReStructuredText
.. _thermal_displacement:
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Mean square displacement
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--------------------------
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From Eq. (10.71) in the book "Thermodynamics of Crystal", atomic
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displacement, **u**, is written by
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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_\nu(\mathbf{q})\right]^{-\frac{1}{2}}
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\left[\hat{a}_\nu(\mathbf{q})\exp(-i\omega_\nu(\mathbf{q})t)+
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\hat{a}^\dagger_\nu(\mathbf{-q})\exp({i\omega_\nu(\mathbf{q})}t)\right]
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\exp({i\mathbf{q}\cdot\mathbf{r}(jl)})
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e^\alpha_\nu(j,\mathbf{q})
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where *j* and *l* are the labels for the *j*-th atomic position in the
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*l*-th unit cell, *t* is the time, :math:`\alpha` is an axis (a
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Cartesian axis in the default behavior of phonopy), *m* is the atomic
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mass, *N* is the number of the unit cells, :math:`\mathbf{q}` is the
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wave vector, :math:`\nu` is the index of phonon mode. *e* is the
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polarization vector of the atom *jl* and the band :math:`\nu` at
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:math:`\mathbf{q}`. :math:`\mathbf{r}(jl)` is the atomic position and
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:math:`\omega` is the phonon frequency. :math:`\hat{a}^\dagger` and
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:math:`\hat{a}` are the creation and annihilation operators of
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phonon. The expectation value of the squared atomic displacement is
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calculated as,
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.. math::
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\left\langle |u^\alpha(jl, t)|^2 \right\rangle = \frac{\hbar}{2Nm_j}
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\sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1}
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(1+2n_\nu(\mathbf{q}))|e^\alpha_\nu(j,\mathbf{q})|^2,
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where *n* is the phonon population, which is give by,
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.. math::
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n_\nu(\mathbf{q}) =
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\frac{1}{\exp(\hbar\omega_\nu(\mathbf{q})/\mathrm{k_B}T)-1},
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where *T* is the temperature, and :math:`\mathrm{k_B}` is the
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Boltzmann constant. The equation is calculated using the commutation
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relation of the creation and annihilation operators and the
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expectation values of the combination of the operations, e.g.,
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.. math::
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[ \hat{a}_\nu(\mathbf{q}), \hat{a}^\dagger_{\nu'}(\mathbf{q'}) ]
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= \delta(\mathbf{q}-\mathbf{q}')\delta_{\nu\nu'},
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[ \hat{a}_\nu(\mathbf{q}), \hat{a}_{\nu'}(\mathbf{q'}) ] = 0,
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[ \hat{a}^\dagger_\nu(\mathbf{q}), \hat{a}^\dagger_{\nu'}(\mathbf{q'}) ] = 0,
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\langle|\hat{a}_\nu(\mathbf{q})\hat{a}_{\nu'}(\mathbf{q'})|\rangle
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= 0,
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\langle|\hat{a}^\dagger_\nu(\mathbf{q})\hat{a}^\dagger_{\nu'}(\mathbf{q'})|\rangle
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= 0.
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Mean square displacement matrix
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--------------------------------
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Mean square displacement matrix is defined as follows:
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.. math::
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\mathrm{B}(j, t) = \frac{\hbar}{2Nm_j}
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\sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1}
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(1+2n_\nu(\mathbf{q}))
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\mathbf{e}_\nu(j,\mathbf{q}) \otimes \mathbf{e}^*_\nu(j,\mathbf{q}).
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This is a symmetry matrix and diagonal elements are same as mean
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square displacement calculated along Cartesian x, y, z directions.
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Projection to an arbitrary axis from the Cartesian axes
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--------------------------------------------------------
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In phonopy, eigenvectors are calculated in the Cartesian axes that are
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defined in the input structure file. Mean square displacement along an
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arbitrary axis is obtained projecting eigenvectors in the Cartesian
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axes as follows:
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.. math::
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\left\langle |u(jl, t)|^2 \right\rangle = \frac{\hbar}{2Nm_j}
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\sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1}
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(1+2n_\nu(\mathbf{q}))|
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\hat{\mathbf{n}}\cdot\mathbf{e}_\nu(j,\mathbf{q})|^2
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where :math:`\hat{\mathbf{n}}` is an arbitrary unit direction.
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.. |sflogo| image:: http://sflogo.sourceforge.net/sflogo.php?group_id=161614&type=1
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:target: http://sourceforge.net
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|sflogo|
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