223 lines
8.8 KiB
ReStructuredText
223 lines
8.8 KiB
ReStructuredText
.. index:: fix phonon
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fix phonon command
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==================
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Syntax
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""""""
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.. parsed-literal::
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fix ID group-ID phonon N Noutput Nwait map_file prefix keyword values ...
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* ID, group-ID are documented in :doc:`fix <fix>` command
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* phonon = style name of this fix command
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* N = measure the Green's function every this many timesteps
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* Noutput = output the dynamical matrix every this many measurements
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* Nwait = wait this many timesteps before measuring
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* map_file = *file* or *GAMMA*
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.. parsed-literal::
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*file* is the file that contains the mapping info between atom ID and the lattice indices.
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.. parsed-literal::
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*GAMMA* flags to treate the whole simulation box as a unit cell, so that the mapping
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info can be generated internally. In this case, dynamical matrix at only the gamma-point
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will/can be evaluated.
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* prefix = prefix for output files
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* one or none keyword/value pairs may be appended
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* keyword = *sysdim* or *nasr*
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.. parsed-literal::
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*sysdim* value = d
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d = dimension of the system, usually the same as the MD model dimension
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*nasr* value = n
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n = number of iterations to enforce the acoustic sum rule
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Examples
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""""""""
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.. code-block:: LAMMPS
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fix 1 all phonon 20 5000 200000 map.in LJ1D sysdim 1
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fix 1 all phonon 20 5000 200000 map.in EAM3D
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fix 1 all phonon 10 5000 500000 GAMMA EAM0D nasr 100
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Description
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"""""""""""
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Calculate the dynamical matrix from molecular dynamics simulations
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based on fluctuation-dissipation theory for a group of atoms.
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Consider a crystal with :math:`N` unit cells in three dimensions labeled
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:math:`l = (l_1, l_2, l_3)` where :math:`l_i` are integers. Each unit cell is
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defined by three linearly independent vectors :math:`\mathbf{a}_1`,
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:math:`\mathbf{a}_2`, :math:`\mathbf{a}_3` forming a parallelepiped,
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containing :math:`K` basis atoms labeled :math:`k`.
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Based on fluctuation-dissipation theory, the force constant
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coefficients of the system in reciprocal space are given by
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(:ref:`Campana <Campana>` , :ref:`Kong <Kong>`)
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.. math::
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\mathbf{\Phi}_{k\alpha,k^\prime \beta}(\mathbf{q}) = k_B T \mathbf{G}^{-1}_{k\alpha,k^\prime \beta}(\mathbf{q})
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where :math:`\mathbf{G}` is the Green's functions coefficients given by
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.. math::
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\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) = \left< \mathbf{u}_{k\alpha}(\mathbf{q}) \bullet \mathbf{u}_{k^\prime \beta}^*(\mathbf{q}) \right>
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where :math:`\left< \ldots \right>` denotes the ensemble average, and
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.. math::
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\mathbf{u}_{k\alpha}(\mathbf{q}) = \sum_l \mathbf{u}_{l k \alpha} \exp{(i\mathbf{qr}_l)}
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is the :math:`\alpha` component of the atomic displacement for the :math:`k`
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th atom in the unit cell in reciprocal space at :math:`\mathbf{q}`. In
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practice, the Green's functions coefficients can also be measured
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according to the following formula,
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.. math::
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\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) =
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\left< \mathbf{R}_{k \alpha}(\mathbf{q}) \bullet \mathbf{R}^*_{k^\prime \beta}(\mathbf{q}) \right>
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- \left<\mathbf{R}\right>_{k \alpha}(\mathbf{q}) \bullet \left<\mathbf{R}\right>^*_{k^\prime \beta}(\mathbf{q})
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where :math:`\mathbf{R}` is the instantaneous positions of atoms, and
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:math:`\left<\mathbf{R}\right>` is the averaged atomic positions. It
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gives essentially the same results as the displacement method and is
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easier to implement in an MD code.
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Once the force constant matrix is known, the dynamical matrix
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:math:`\mathbf{D}` can then be obtained by
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.. math::
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\mathbf{D}_{k\alpha, k^\prime\beta}(\mathbf{q}) =
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(m_k m_{k^\prime})^{-\frac{1}{2}} \mathbf{\Phi}_{k \alpha, k^\prime \beta}(\mathbf{q})
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whose eigenvalues are exactly the phonon frequencies at :math:`\mathbf{q}`.
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This fix uses positions of atoms in the specified group and calculates
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two-point correlations. To achieve this. the positions of the atoms
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are examined every *Nevery* steps and are Fourier-transformed into
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reciprocal space, where the averaging process and correlation
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computation is then done. After every *Noutput* measurements, the
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matrix :math:`\mathbf{G}(\mathbf{q})` is calculated and inverted to
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obtain the elastic stiffness coefficients. The dynamical matrices are
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then constructed and written to *prefix*\ .bin.timestep files in binary
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format and to the file *prefix*\ .log for each wave-vector
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:math:`\mathbf{q}`.
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A detailed description of this method can be found in
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(:ref:`Kong2011 <Kong2011>`).
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The *sysdim* keyword is optional. If specified with a value smaller
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than the dimensionality of the LAMMPS simulation, its value is used
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for the dynamical matrix calculation. For example, using LAMMPS to
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model a 2D or 3D system, the phonon dispersion of a 1D atomic chain
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can be computed using *sysdim* = 1.
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The *nasr* keyword is optional. An iterative procedure is employed to
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enforce the acoustic sum rule on :math:`\Phi` at :math:`\Gamma`, and the number
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provided by keyword *nasr* gives the total number of iterations. For a
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system whose unit cell has only one atom, *nasr* = 1 is sufficient;
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for other systems, *nasr* = 10 is typically sufficient.
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The *map_file* contains the mapping information between the lattice
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indices and the atom IDs, which tells the code which atom sits at
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which lattice point; the lattice indices start from 0. An auxiliary
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code, `latgen <https://code.google.com/p/latgen>`_, can be employed to
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generate the compatible map file for various crystals.
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In case one simulates a non-periodic system, where the whole simulation
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box is treated as a unit cell, one can set *map_file* as *GAMMA*, so
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that the mapping info will be generated internally and a file is not
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needed. In this case, the dynamical matrix at only the gamma-point
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will/can be evaluated. Please keep in mind that fix-phonon is designed
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for cyrstals, it will be inefficient and even degrade the performance
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of LAMMPS in case the unit cell is too large.
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The calculated dynamical matrix elements are written out in
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:doc:`energy/distance\^2/mass <units>` units. The coordinates for *q*
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points in the log file is in the units of the basis vectors of the
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corresponding reciprocal lattice.
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Restart, fix_modify, output, run start/stop, minimize info
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"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
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No information about this fix is written to :doc:`binary restart files <restart>`.
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The :doc:`fix_modify <fix_modify>` *temp* option is supported by this
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fix. You can use it to change the temperature compute from thermo_temp
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to the one that reflects the true temperature of atoms in the group.
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No global scalar or vector or per-atom quantities are stored by this
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fix for access by various :doc:`output commands <Howto_output>`.
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Instead, this fix outputs its initialization information (including
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mapping information) and the calculated dynamical matrices to the file
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*prefix*\ .log, with the specified *prefix*\ . The dynamical matrices are
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also written to files *prefix*\ .bin.timestep in binary format. These
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can be read by the post-processing tool in tools/phonon to compute the
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phonon density of states and/or phonon dispersion curves.
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No parameter of this fix can be used with the *start/stop* keywords
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of the :doc:`run <run>` command.
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This fix is not invoked during :doc:`energy minimization <minimize>`.
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Restrictions
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""""""""""""
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This fix assumes a crystalline system with periodical lattice. The
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temperature of the system should not exceed the melting temperature to
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keep the system in its solid state.
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This fix is part of the PHONON package. It is only enabled if
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LAMMPS was built with that package. See the :doc:`Build package <Build_package>` page for more info.
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This fix requires LAMMPS be built with an FFT library. See the :doc:`Build settings <Build_settings>` page for details.
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Related commands
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""""""""""""""""
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:doc:`compute msd <compute_msd>`,
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:doc:`dynamical_matrix <dynamical_matrix>`
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Default
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"""""""
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The option defaults are sysdim = the same dimension as specified by
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the :doc:`dimension <dimension>` command, and nasr = 20.
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----------
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.. _Campana:
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**(Campana)** C. Campana and
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M. H. Muser, *Practical Green's function approach to the
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simulation of elastic semi-infinite solids*, `Phys. Rev. B [74], 075420 (2006) <https://doi.org/10.1103/PhysRevB.74.075420>`_
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.. _Kong:
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**(Kong)** L.T. Kong, G. Bartels, C. Campana,
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C. Denniston, and Martin H. Muser, *Implementation of Green's
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function molecular dynamics: An extension to LAMMPS*, `Computer Physics Communications [180](6):1004-1010 (2009). <https://doi.org/10.1016/j.cpc.2008.12.035>`_
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L.T. Kong, C. Denniston, and Martin H. Muser,
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*An improved version of the Green's function molecular dynamics
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method*, `Computer Physics Communications [182](2):540-541 (2011). <https://doi.org/10.1016/j.cpc.2010.10.006>`_
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.. _Kong2011:
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**(Kong2011)** L.T. Kong, *Phonon dispersion measured directly from
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molecular dynamics simulations*, `Computer Physics Communications [182](10):2201-2207, (2011). <https://doi.org/10.1016/j.cpc.2011.04.019>`_
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