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			<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
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			<H3>fix phonon command
			</H3>
			<P><B>Syntax:</B>
			</P>
			<PRE>fix ID group-ID phonon N Noutput Nwait map_file prefix keyword values ...
			</PRE>
			<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command

			<LI>phonon = style name of this fix command

			<LI>N = measure the Green's function every this many timesteps

			<LI>Noutput = output the dynamical matrix every this many measurements

			<LI>Nwait = wait this many timesteps before measuring

			<LI>map_file = file containing the mapping info between atom ID and the lattice indices

			<LI>prefix = prefix for output files

			<LI>one or none keyword/value pairs may be appended

			<LI>keyword = <I>sysdim</I> or <I>nasr</I>

			<PRE> <I>sysdim</I> value = d
			d = dimension of the system, usually the same as the MD model dimension
			<I>nasr</I> value = n
			n = number of iterations to enforce the acoustic sum rule
			</PRE>

			</UL>
			<P><B>Examples:</B>
			</P>
			<PRE>fix 1 all phonon 20 5000 200000 map.in LJ1D sysdim 1
			fix 1 all phonon 20 5000 200000 map.in EAM3D
			</PRE>
			<P><B>Description:</B>
			</P>
			<P>Calculate the dynamical matrix from molecular dynamics simulations
			based on fluctuation-dissipation theory for a group of atoms.
			</P>
			<P>Consider a crystal with <I>N</I> unit cells in three dimensions labelled <I>l
			= (l<sub>1</sub>,l<sub>2</sub>,l<sub>3</sub>)</I> where <I>l<sub>i</sub></I>
			are integers. Each unit cell is defined by three linearly independent
			vectors <B>a</B><sub>1</sub>, <B>a</B><sub>2</sub>, <B>a</B><sub>3</sub> forming a
			parallelipiped, containing <I>K</I> basis atoms labelled <I>k</I>.
			</P>
			<P>Based on fluctuation-dissipation theory, the force constant
			coefficients of the system in reciprocal space are given by
			(<A HREF = "#Campana">Campañá</A> , <A HREF = "#Kong">Kong</A>)
			<center><b>Φ</b><sub>kα,k'β</sub>(<b>q</b>) =
			k<sub>B</sub>T
			<b>G</b><sup>-1</sup><sub>kα,k'β</sub>(<b>q</b>),</center>
			</P>
			<P>where <B>G</B> is the Green's functions coefficients given by
			</P>
			<center><b>G</b><sub>kα,k'β</sub>(<b>q</b>) =
			<<b>u</b><sub>kα</sub>(<b>q</b>)<b>u</b><sub>k'β</sub><sup>*</sup>(<b>q</b>)>,</center>

			<P>where <...> denotes the ensemble average, and
			<center><B>u</B><sub>kα</sub>(<b>q</b>) = ∑<sub>l</sub>
			<b>u</b><sub>lkα</sub> exp(i<B>qr</B><sub>l</sub>)</center>
			</P>
			<P>is the α component of the atomic displacement for the <I>k</I>th atom
			in the unit cell in reciprocal space at <B>q</B>. In practice, the Green's
			functions coefficients can also be measured according to the following
			formula,
			</P>
			<center><b>G</b><sub>kα,k'β</sub>(<b>q</b>) =
			<<b>R</b><sub>kα</sub>(<b>q</b>)<b>R</b><sup>*</sup><sub>k'β</sub>(<b>q</b>)>
			- <<b>R</b>><sub>kα</sub>(<b>q</b>)<<b>R</b>><sup>*</sup><sub>k'β</sub>(<b>q</b>),
			</center>

			<P>where <B>R</B> is the instantaneous positions of atoms, and <<B>R</B>> is the
			averaged atomic positions. It gives essentially the same results as
			the displacement method and is easier to implement in an MD code.
			</P>
			<P>Once the force constant matrix is known, the dynamical matrix <B>D</B> can
			then be obtained by
			</P>
			<center><b>D</b><sub>kα, k'β</sub>(<b>q</b>) = (m<sub>k</sub>m<sub>k'</sub>)<sup>-1/2</sup> <b>Φ</b><sub>kα,k'β</sub>(<b>q</b>)</center>

			<P>whose eigenvalues are exactly the phonon frequencies at <B>q</B>.
			</P>
			<P>This fix uses positions of atoms in the specified group and calculates
			two-point correlations. To achieve this. the positions of the atoms
			are examined every <I>Nevery</I> steps and are Fourier-transformed into
			reciprocal space, where the averaging process and correlation
			computation is then done. After every <I>Noutput</I> measurements, the
			matrix <B>G</B>(<B>q</B>) is calculated and inverted to obtain the elastic
			stiffness coefficients. The dynamical matrices are then constructed
			and written to <I>prefix</I>.bin.timestep files in binary format and to the
			file <I>prefix</I>.log for each wavevector <B>q</B>.
			</P>
			<P>A detailed description of this method can be found in
			(<A HREF = "#Kong2011">Kong2011</A>).
			</P>
			<P>The <I>sysdim</I> keyword is optional. If specified with a value smaller
			than the dimensionality of the LAMMPS simulation, its value is used
			for the dynamical matrix calculation. For example, using LAMMPS ot
			model a 2D or 3D system, the phonon dispersion of a 1D atomic chain
			can be computed using <I>sysdim</I> = 1.
			</P>
			<P>The <I>nasr</I> keyword is optional. An iterative procedure is employed to
			enforce the acoustic sum rule on Φ at Γ, and the number
			provided by keyword <I>nasr</I> gives the total number of iterations. For a
			system whose unit cell has only one atom, <I>nasr</I> = 1 is sufficient;
			for other systems, <I>nasr</I> = 10 is typically sufficient.
			</P>
			<P>The <I>map_file</I> contains the mapping information between the lattice
			indices and the atom IDs, which tells the code which atom sits at
			which lattice point; the lattice indices start from 0. An auxiliary
			code, <A HREF = "http://code.google.com/p/latgen">latgen</A>, can be employed to
			generate the compatible map file for various crystals.
			</P>
			<P>The calculated dynamical matrix elements are written out in
			<A HREF = "units.html">energy/distance^2/mass</A> units. The coordinates for <I>q</I>
			points in the log file is in the units of the basis vectors of the
			corresponding reciprocal lattice.
			</P>
			<P><B>Restart, fix_modify, output, run start/stop, minimize info:</B>
			</P>
			<P>No information about this fix is written to <A HREF = "restart.html">binary restart
			files</A>.
			</P>
			<P>The <A HREF = "fix_modify.html">fix_modify</A> <I>temp</I> 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.
			</P>
			<P>No global scalar or vector or per-atom quantities are stored by this
			fix for access by various <A HREF = "Section_howto.html#4_15">output commands</A>.
			</P>
			<P>Instead, this fix outputs its initialization information (including
			mapping information) and the calculated dynamical matrices to the file
			<I>prefix</I>.log, with the specified <I>prefix</I>. The dynamical matrices are
			also written to files <I>prefix</I>.bin.timestep in binary format. These
			can be read by the post-processing tool in tools/phonon to compute the
			phonon density of states and/or phonon dispersion curves.
			</P>
			<P>No parameter of this fix can be used with the <I>start/stop</I> keywords
			of the <A HREF = "run.html">run</A> command.
			</P>
			<P>This fix is not invoked during <A HREF = "minimize.html">energy minimization</A>.
			</P>
			<P><B>Restrictions:</B>
			</P>
			<P>This fix assumes a crystalline system with periodical lattice. The
			temperature of the system should not exceed the melting temperature to
			keep the system in its solid state.
			</P>
			<P>This fix is part of the USER-PHONON package. It is only enabled if
			LAMMPS was built with that package. See the <A HREF = "Section_start.html#start_3">Making
			LAMMPS</A> section for more info.
			</P>
			<P>This fix requires LAMMPS be built with an FFT library. See the
			<A HREF = "Section_start.html#start_2">Making LAMMPS</A> section for more info.
			</P>
			<P><B>Related commands:</B>
			</P>
			<P><A HREF = "compute_msd.html">compute msd</A>
			</P>
			<P><B>Default:</B>
			</P>
			<P>The option defaults are sysdim = the same dimemsion as specified by
			the <A HREF = "dimension">dimension</A> command, and nasr = 20.
			</P>
			<HR>

			<A NAME = "Campana"></A>

			<P><B>(Campañá)</B> C. Campañá and
			M. H. Müser, <I>Practical Green's function approach to the
			simulation of elastic semi-infinite solids</I>, <A HREF = "http://dx.doi.org/10.1103/PhysRevB.74.075420">Phys. Rev. B <B>74</B>,
			075420 (2006)</A>
			</P>
			<A NAME = "Kong"></A>

			<P><B>(Kong)</B> L.T. Kong, G. Bartels, C. Campañá,
			C. Denniston, and Martin H. Müser, <I>Implementation of Green's
			function molecular dynamics: An extension to LAMMPS</I>, <A HREF = "http://dx.doi.org/10.1016/j.cpc.2008.12.035">Computer
			Physics Communications <B>180</B>(6):1004-1010
			(2009).</A>
			</P>
			<P>L.T. Kong, C. Denniston, and Martin H. Müser,
			<I>An improved version of the Green's function molecular dynamics
			method</I>, <A HREF = "http://dx.doi.org/10.1016/j.cpc.2010.10.006">Computer Physics Communications <B>182</B>(2):540-541
			(2011).</A>
			</P>
			<A NAME = "Kong2011"></A>

			<P><B>(Kong2011)</B> L.T. Kong, <I>Phonon dispersion measured directly from
			molecular dynamics simulations</I>, <A HREF = "http://dx.doi.org/10.1016/j.cpc.2011.04.019">Computer Physics Communications
			<B>182</B>(10):2201-2207,
			(2011).</A>
			</P>
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