\(\renewcommand{\AA}{\text{Å}}\)

# compute centro/atom command

## Syntax

    compute ID group-ID centro/atom lattice keyword value ...

- ID, group-ID are documented in compute command

- centro/atom = style name of this compute command

- lattice = fcc or bcc or N = # of neighbors per atom to include

- zero or more keyword/value pairs may be appended

- keyword = axes

      axes value = no or yes
        no = do not calculate 3 symmetry axes
        yes = calculate 3 symmetry axes

## Examples

    compute 1 all centro/atom fcc

    compute 1 all centro/atom 8

## Description

Define a computation that calculates the centro-symmetry parameter for each atom in the group, for either FCC or BCC lattices, depending on the choice of the lattice argument. In solid-state systems the centro-symmetry parameter is a useful measure of the local lattice disorder around an atom and can be used to characterize whether the atom is part of a perfect lattice, a local defect (e.g. a dislocation or stacking fault), or at a surface.

The value of the centro-symmetry parameter will be 0.0 for atoms not in the specified compute group.

This parameter is computed using the following formula from (Kelchner)

\[CS = \sum_{i = 1}^{N/2} | \vec{R}_i + \vec{R}_{i+N/2} |^2\]

where the \(N\) nearest neighbors of each atom are identified and \(\vec{R}_i\) and \(\vec{R}_{i+N/2}\) are vectors from the central atom to a particular pair of nearest neighbors. There are \(N (N-1)/2\) possible neighbor pairs that can contribute to this formula. The quantity in the sum is computed for each, and the \(N/2\) smallest are used. This will typically be for pairs of atoms in symmetrically opposite positions with respect to the central atom; hence the \(i+N/2\) notation.

\(N\) is an input parameter, which should be set to correspond to the number of nearest neighbors in the underlying lattice of atoms. If the keyword fcc or bcc is used, N is set to 12 and 8 respectively. More generally, N can be set to a positive, even integer.

For an atom on a lattice site, surrounded by atoms on a perfect lattice, the centro-symmetry parameter will be 0. It will be near 0 for small thermal perturbations of a perfect lattice. If a point defect exists, the symmetry is broken, and the parameter will be a larger positive value. An atom at a surface will have a large positive parameter. If the atom does not have \(N\) neighbors (within the potential cutoff), then its centro-symmetry parameter is set to 0.0.

If the keyword axes has the setting yes, then this compute also estimates three symmetry axes for each atom’s local neighborhood. The first two of these are the vectors joining the two pairs of neighbor atoms with smallest contributions to the centrosymmetry parameter, i.e. the two most symmetric pairs of atoms. The third vector is normal to the first two by the right-hand rule. All three vectors are normalized to unit length. For FCC crystals, the first two vectors will lie along a \(\langle110\rangle\) direction, while the third vector will lie along either a \(\langle100\rangle\) or \(\langle111\rangle\) direction. For HCP crystals, the first two vectors will lie along \(\langle1000\rangle\) directions, while the third vector will lie along \(\langle0001\rangle\). This provides a simple way to measure local orientation in HCP structures. In general, the axes keyword can be used to estimate the orientation of symmetry axes in the neighborhood of any atom.

Only atoms within the cutoff of the pairwise neighbor list are considered as possible neighbors. Atoms not in the compute group are included in the \(N\) neighbors used in this calculation.

The neighbor list needed to compute this quantity is constructed each time the calculation is performed (e.g., each time a snapshot of atoms is dumped). Thus it can be inefficient to compute/dump this quantity too frequently or to have multiple compute/dump commands, each with a centro/atom style.

## Output info

By default, this compute calculates the centrosymmetry value for each atom as a per-atom vector, which can be accessed by any command that uses per-atom values from a compute as input. See the Howto output page for an overview of LAMMPS output options.

If the axes keyword setting is yes, then a per-atom array is calculated. The first column is the centrosymmetry parameter. The next three columns are the x, y, and z components of the first symmetry axis, followed by the second, and third symmetry axes in columns 5–7 and 8–10.

The centrosymmetry values are unitless values \(\ge 0.0\). Their magnitude depends on the lattice style due to the number of contributing neighbor pairs in the summation in the formula above. And it depends on the local defects surrounding the central atom, as described above. For the axes yes case, the vector components are also unitless, since they represent spatial directions.

Here are typical centro-symmetry values, from a nanoindentation simulation into gold (FCC). These were provided by Jon Zimmerman (Sandia):

    Bulk lattice = 0
    Dislocation core ~ 1.0 (0.5 to 1.25)
    Stacking faults ~ 5.0 (4.0 to 6.0)
    Free surface ~ 23.0

These values are not normalized by the square of the lattice parameter. If they were, normalized values would be:

    Bulk lattice = 0
    Dislocation core ~ 0.06 (0.03 to 0.075)
    Stacking faults ~ 0.3 (0.24 to 0.36)
    Free surface ~ 1.38

For BCC materials, the values for dislocation cores and free surfaces would be somewhat different, due to their being only 8 neighbors instead of 12.

## Restrictions

none

## Related commands

compute cna/atom

## Default

The default value for the optional keyword is axes = no.

(Kelchner) Kelchner, Plimpton, Hamilton, Phys Rev B, 58, 11085 (1998).