lammps/doc/fix_phonon.txt

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:link(ld,Manual.html)
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fix phonon command :h3

[Syntax:]

fix ID group-ID phonon N Noutput Nwait map_file prefix keyword values ... :pre

ID, group-ID are documented in "fix"_fix.html command :ulb,l
phonon = style name of this fix command :l
N = measure the Green's function every this many timesteps :l
Noutput = output the dynamical matrix every this many measurements :l
Nwait = wait this many timesteps before measuring :l
map_file = file containing the mapping info between atom ID and the lattice indices :l
prefix = prefix for output files :l
one or none keyword/value pairs may be appended :l
keyword = {sysdim} or {nasr} :l
  {sysdim} value = d
    d = dimension of the system, usually is the same as the MD dimension.
  {nasr} value = n
    n = number of iterations to enforce the acoustic sum rule. :pre
:ule

[Examples:]

fix 1 all phonon 20 5000 200000 map.in LJ1D sysdim 1
fix 1 all phonon 20 5000 200000 map.in EAM3D :pre

[Description:]

To measure the dynamical matrix from molecular dynamics simulations
based on fluctuation-dissipation theory for a group of atoms.

Consider a crystal with {N} unit cells in three dimensions labelled {l
= (l<sub>1</sub>,l<sub>2</sub>,l<sub>3</sub>)} where {l<sub>i</sub>}
are integers.  Each unit cell is defined by three linearly independent
vectors [a]<sub>1</sub>, [a]<sub>2</sub>, [a]<sub>3</sub> forming a
parallelipiped, containing {K} basis atoms labelled {k}.

Based on fluctuation-dissipation theory, the force costant
coefficients of the system in the reciprocal space could be given by
("Campa&ntilde;&aacute;"_#campana , "Kong"_#kong)
<center><b>&Phi;</b><sub>k&alpha;,k'&beta;</sub>(<b>q</b>) =
k<sub>B</sub>T
<b>G</b><sup>-1</sup><sub>k&alpha;,k'&beta;</sub>(<b>q</b>),</center>

where [G] is the Green's functions coefficients given by

<center><b>G</b><sub>k&alpha;,k'&beta;</sub>(<b>q</b>) =
<<b>u</b><sub>k&alpha;</sub>(<b>q</b>)&#149;<b>u</b><sub>k'&beta;</sub><sup>*</sup>(<b>q</b>)>,</center>

where <...> denotes the enssemble average and
<center>[u]<sub>k&alpha;</sub>(<b>q</b>) = &sum;<sub>l</sub>
<b>u</b><sub>lk&alpha;</sub> exp(i[qr]<sub>l</sub>)</center>

is the &alpha; component of the atomic displacement for the {k}th atom
in the unit cell in the reciprocal space at [q]. In practice, the
Green's functions coefficients can also be measured according to the
following formula,

<center><b>G</b><sub>k&alpha;,k'&beta;</sub>(<b>q</b>) =
<<b>R</b><sub>k&alpha;</sub>(<b>q</b>)&#149;<b>R</b><sup>*</sup><sub>k'&beta;</sub>(<b>q</b>)>
- <<b>R</b>><sub>k&alpha;</sub>(<b>q</b>)&#149;<<b>R</b>><sup>*</sup><sub>k'&beta;</sub>(<b>q</b>),
</center>

where [R] is the instantaneous positions of atoms, and <[R]> is the
averaged atomic positions. It gives essentially the same
results as the displacement method and is easier to implement into an MD code.

Once the force constant matrix is known, the dynamical matrix [D]
could then be obtained by

<center><b>D</b><sub>k&alpha;, k'&beta;</sub>(<b>q</b>) = (m<sub>k</sub>m<sub>k'</sub>)<sup>-1/2</sup> <b>&Phi;</b><sub>k&alpha;,k'&beta;</sub>(<b>q</b>)</center>

whose eigenvalues are exactly the phonon frequencies at [q].

This fix observes the positions of atoms in the corresponding group
and the two-point correlations. To achieve this the positions of the
atoms are examined every Nevery steps and are Fourier-transformed
(with "FFT3d wrapper"_kspace_style.html from kspace package of LAMMPS)
into reciprocal space, in which the averaging process and correlation
computation is then done. After every Noutput measurements, the matrix
[G]([q]) is calculated and inverted to get the elastic stiffness
coefficients. The dynamical matrices are then constructed and written
to file "prefix".bin."timestep" in binary format and to file
"prefix".log for each wavevector [q].

A detailed description of this method can be found in ("Kong2011"_#kong2011).

The {sysdim} keyword is optional, if specified and its value is
smaller than the dimensionality of the molecular dynamics simulation,
its value will be taken as the dimensionality of the system under
study. For example, in LAMMPS, one can simulate a 2D or 3D system,
while if one wants to study the phonon dispersion of a 1D atomic
chain, one can set the {sysdim} keyword to be {1} and the code will
assume a 1D system is studied.

The {nasr} keyword is optional. An iterative procedure is employed to
enforce the acoustic sum rule on &Phi; at &Gamma;, and the number
provided by keyword {nasr} gives the total number of iterations. For
system whose unit cell has only one atom, {nasr} = 1 is sufficient;
while for other systems, {nasr} = 10 should be enough.

The {map_file} contains the mapping information between the lattice
indices and the atom IDs, which simply tells the code which atom sits
at which lattice point; the lattice indices start from 0. An auxiliary
code, "latgen"_http://code.google.com/p/latgen, can be employed to
generate the compatible map file for various crystals.

The unit of the measured dynamical matrix is
"energy/distance^2/mass"_units.html.  The coordinates for {q} points
in the log file is in unit of the basis vectors of the corresponding
reciprocal lattice.

[Restart, fix_modify, output, run start/stop, minimize info:]

No information about this fix is written to "binary restart
files"_restart.html. 

The "fix_modify"_fix_modify.html {temp} option
is supported by this fix. You can use it to change the 
temperature compute from thermo_temp to the one that reflects the true
temperature of atoms in the group.

No global scalar or vector or per-atom
quantities are stored by this fix for access by various "output
commands"_Section_howto.html#4_15.
Instead, this fix outputs the initialization information (in which the
mapping information can be found) and the calculated
dynamical matrices into file "prefix".log, where "prefix" will be
replaced by the prefix given; besides, the
dynamical matrices will also be written to file
"prefix".bin."timestep" in binary format, which could be
used by a postprocessing code {phana} to get the phonon density of states
and/or phonon dispersion curves.
At the end of run, these information will also be written.

No parameter of this fix can be used with the {start/stop} keywords
of the "run"_run.html command.

This fix is not invoked during "energy minimization"_minimize.html.

[Restrictions:]

This fix assumes a crystalline system with periodical lattice. The temperature
of the system should not exceed the melting temperature to keep in solid state.

[Related commands:]

"compute msd"_compute_msd.html

"fix gfc"_http://code.google.com/p/fix-gfmd

[Note:] Compiling of this fix along with LAMMPS requires that the
"FFT3d wrappers of kspace"_kspace_style.html of LAMMPS is included
while compiling.

[Default:]

The option defaults are {sysdim} = domain->dimension, {nasr} = 20.

:line

:link(campana)
[(Campa&ntilde;&aacute;)] C. Campa&ntilde;&aacute; and
M. H. M&uuml;ser, {Practical Green's function approach to the
simulation of elastic semi-infinite solids}, "Phys. Rev. B [74],
075420 (2006)"_http://dx.doi.org/10.1103/PhysRevB.74.075420

:link(kong)
[(Kong)] L.T. Kong, G. Bartels, C. Campa&ntilde;&aacute;,
C. Denniston, and Martin H. M&uuml;ser, {Implementation of Green's
function molecular dynamics: An extension to LAMMPS}, "Computer
Physics Communications [180](6):1004-1010
(2009)."_http://dx.doi.org/10.1016/j.cpc.2008.12.035

L.T. Kong, C. Denniston, and Martin H. M&uuml;ser,
{An improved version of the Green's function molecular dynamics
method}, "Computer Physics Communications [182](2):540-541
(2011)."_http://dx.doi.org/10.1016/j.cpc.2010.10.006

:link(kong2011)
[(Kong2011)] L.T. Kong, {Phonon dispersion measured directly from
molecular dynamics simulations}, "Computer Physics Communications
[182](10):2201-2207,
(2011)."_http://dx.doi.org/10.1016/j.cpc.2011.04.019