\(\renewcommand{\AA}{\text{Å}}\)

# fix phonon command

## Syntax

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

- ID, group-ID are documented in fix command

- phonon = style name of this fix command

- N = measure the Green’s function every this many timesteps

- Noutput = output the dynamical matrix every this many measurements

- Nwait = wait this many timesteps before measuring

- map_file = file or GAMMA

      file is the file that contains the mapping info between atom ID and the lattice indices.

      GAMMA flags to treate the whole simulation box as a unit cell, so that the mapping
      info can be generated internally. In this case, dynamical matrix at only the gamma-point
      will/can be evaluated.

- prefix = prefix for output files

- one or none keyword/value pairs may be appended

- keyword = sysdim or nasr

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

## Examples

    fix 1 all phonon 20 5000 200000 map.in LJ1D sysdim 1
    fix 1 all phonon 20 5000 200000 map.in EAM3D
    fix 1 all phonon 10 5000 500000 GAMMA EAM0D nasr 100

## Description

Calculate 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 labeled \(l = (l_1, l_2, l_3)\) where \(l_i\) are integers. Each unit cell is defined by three linearly independent vectors \(\mathbf{a}_1\), \(\mathbf{a}_2\), \(\mathbf{a}_3\) forming a parallelepiped, containing \(K\) basis atoms labeled \(k\).

Based on fluctuation-dissipation theory, the force constant coefficients of the system in reciprocal space are given by (Campana , Kong)

\[\mathbf{\Phi}_{k\alpha,k^\prime \beta}(\mathbf{q}) = k_B T \mathbf{G}^{-1}_{k\alpha,k^\prime \beta}(\mathbf{q})\]

where \(\mathbf{G}\) is the Green’s functions coefficients given by

\[\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) = \left< \mathbf{u}_{k\alpha}(\mathbf{q}) \bullet \mathbf{u}_{k^\prime \beta}^*(\mathbf{q}) \right>\]

where \(\left< \ldots \right>\) denotes the ensemble average, and

\[\mathbf{u}_{k\alpha}(\mathbf{q}) = \sum_l \mathbf{u}_{l k \alpha} \exp{(i\mathbf{qr}_l)}\]

is the \(\alpha\) component of the atomic displacement for the \(k\) th atom in the unit cell in reciprocal space at \(\mathbf{q}\). In practice, the Green’s functions coefficients can also be measured according to the following formula,

\[\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) = \left< \mathbf{R}_{k \alpha}(\mathbf{q}) \bullet \mathbf{R}^*_{k^\prime \beta}(\mathbf{q}) \right> - \left<\mathbf{R}\right>_{k \alpha}(\mathbf{q}) \bullet \left<\mathbf{R}\right>^*_{k^\prime \beta}(\mathbf{q})\]

where \(\mathbf{R}\) is the instantaneous positions of atoms, and \(\left<\mathbf{R}\right>\) is the averaged atomic positions. It gives essentially the same results as the displacement method and is easier to implement in an MD code.

Once the force constant matrix is known, the dynamical matrix \(\mathbf{D}\) can then be obtained by

\[\mathbf{D}_{k\alpha, k^\prime\beta}(\mathbf{q}) = (m_k m_{k^\prime})^{-\frac{1}{2}} \mathbf{\Phi}_{k \alpha, k^\prime \beta}(\mathbf{q})\]

whose eigenvalues are exactly the phonon frequencies at \(\mathbf{q}\).

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 Nevery steps and are Fourier-transformed into reciprocal space, where the averaging process and correlation computation is then done. After every Noutput measurements, the matrix \(\mathbf{G}(\mathbf{q})\) is calculated and inverted to obtain the elastic stiffness coefficients. The dynamical matrices are then constructed and written to prefix.bin.timestep files in binary format and to the file prefix.log for each wave-vector \(\mathbf{q}\).

A detailed description of this method can be found in (Kong2011).

The sysdim 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 to model a 2D or 3D system, the phonon dispersion of a 1D atomic chain can be computed using sysdim = 1.

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 a system whose unit cell has only one atom, nasr = 1 is sufficient; for other systems, nasr = 10 is typically sufficient.

The map_file 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, latgen, can be employed to generate the compatible map file for various crystals.

In case one simulates a non-periodic system, where the whole simulation box is treated as a unit cell, one can set map_file as GAMMA, so that the mapping info will be generated internally and a file is not needed. In this case, the dynamical matrix at only the gamma-point will/can be evaluated. Please keep in mind that fix-phonon is designed for cyrstals, it will be inefficient and even degrade the performance of LAMMPS in case the unit cell is too large.

The calculated dynamical matrix elements are written out in energy/distance^2/mass units. The coordinates for q points in the log file is in the units 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.

The fix_modify 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.

Instead, this fix outputs its initialization information (including mapping information) and the calculated dynamical matrices to the file prefix.log, with the specified prefix. The dynamical matrices are also written to files prefix.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.

No parameter of this fix can be used with the start/stop keywords of the run command.

This fix is not invoked during energy minimization.

## Restrictions

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.

This fix is part of the PHONON package. It is only enabled if LAMMPS was built with that package. This fix also requires LAMMPS to be built with 3d-FFT support which is included in the KSPACE package. See the Build package page for more info.

## Related commands

compute msd, dynamical_matrix

## Default

The option defaults are sysdim = the same dimension as specified by the dimension command, and nasr = 20.

(Campana) C. Campana and M. H. Muser, Practical Green’s function approach to the simulation of elastic semi-infinite solids, Phys. Rev. B [74], 075420 (2006)

(Kong) L.T. Kong, G. Bartels, C. Campana, C. Denniston, and Martin H. Muser, Implementation of Green’s function molecular dynamics: An extension to LAMMPS, Computer Physics Communications [180](6):1004-1010 (2009).

L.T. Kong, C. Denniston, and Martin H. Muser, An improved version of the Green’s function molecular dynamics method, Computer Physics Communications [182](2):540-541 (2011).

(Kong2011) L.T. Kong, Phonon dispersion measured directly from molecular dynamics simulations, Computer Physics Communications [182](10):2201-2207, (2011).