<p>where the <em>N</em> nearest neighbors of each atom are identified and Ri and
Ri+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.</p>
<p><em>N</em> 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 <em>fcc</em> or <em>bcc</em> is used, <em>N</em> is set to 12 and 8
respectively. More generally, <em>N</em> can be set to a positive, even
integer.</p>
<p>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 <em>N</em> neighbors (within
the potential cutoff), then its centro-symmetry parameter is set to
0.0.</p>
<p>If the keyword <em>axes</em> has the setting <em>yes</em>, 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 <110> direction, while the third vector will lie
along either a <100> or <111> direction. For HCP crystals, the first
two vectors will lie along <1000> directions, while the third vector
will lie along <0001>. This provides a simple way to measure local
orientation in HCP structures. In general, the <em>axes</em> keyword can be
used to estimate the orientation of symmetry axes in the neighborhood
of any atom.</p>
<p>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 <em>N</em> neighbors used in this calculation.</p>
<p>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
<em>centro/atom</em> style.</p>
<p><strong>Output info:</strong></p>
<p>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 <a class="reference internal" href="Section_howto.html#howto-15"><span class="std std-ref">Section_howto 15</span></a> for an overview of LAMMPS output
options.</p>
<p>If the <em>axes</em> keyword setting is <em>yes</em>, 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.</p>
<p>The centrosymmetry values are unitless values >= 0.0. Their magnitude
depends on the lattice style due to the number of contibuting 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 <em>axes yes</em> case, the vector components are also unitless, since
they represent spatial directions.</p>
<p>Here are typical centro-symmetry values, from a a nanoindentation
simulation into gold (FCC). These were provided by Jon Zimmerman
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