\(\renewcommand{\AA}{\text{Å}}\)

# lattice command¶

## Syntax¶

```
lattice style scale keyword values ...
```

style =

*none*or*sc*or*bcc*or*fcc*or*hcp*or*diamond*or*sq*or*sq2*or*hex*or*custom*scale = scale factor between lattice and simulation box

scale = reduced density rho* (for LJ units) scale = lattice constant in distance units (for all other units)

zero or more keyword/value pairs may be appended

keyword =

*origin*or*orient*or*spacing*or*a1*or*a2*or*a3*or*basis**origin*values = x y z x,y,z = fractions of a unit cell (0 <= x,y,z < 1)*orient*values = dim i j k dim =*x*or*y*or*z*i,j,k = integer lattice directions*spacing*values = dx dy dz dx,dy,dz = lattice spacings in the x,y,z box directions*a1*,*a2*,*a3*values = x y z x,y,z = primitive vector components that define unit cell*basis*values = x y z x,y,z = fractional coords of a basis atom (0 <= x,y,z < 1)

## Examples¶

```
lattice fcc 3.52
lattice hex 0.85
lattice sq 0.8 origin 0.0 0.5 0.0 orient x 1 1 0 orient y -1 1 0
lattice custom 3.52 a1 1.0 0.0 0.0 a2 0.5 1.0 0.0 a3 0.0 0.0 0.5 &
basis 0.0 0.0 0.0 basis 0.5 0.5 0.5
lattice none 2.0
```

## Description¶

Define a lattice for use by other commands. In LAMMPS, a lattice is simply a set of points in space, determined by a unit cell with basis atoms, that is replicated infinitely in all dimensions. The arguments of the lattice command can be used to define a wide variety of crystallographic lattices.

A lattice is used by LAMMPS in two ways. First, the create_atoms command creates atoms on the lattice points inside the simulation box. Note that the create_atoms command allows different atom types to be assigned to different basis atoms of the lattice. Second, the lattice spacing in the x,y,z dimensions implied by the lattice, can be used by other commands as distance units (e.g. create_box, region and velocity), which are often convenient to use when the underlying problem geometry is atoms on a lattice.

The lattice style must be consistent with the dimension of the
simulation - see the dimension command. Styles *sc*
or *bcc* or *fcc* or *hcp* or *diamond* are for 3d problems. Styles
*sq* or *sq2* or *hex* are for 2d problems. Style *custom* can be
used for either 2d or 3d problems.

A lattice consists of a unit cell, a set of basis atoms within that cell, and a set of transformation parameters (scale, origin, orient) that map the unit cell into the simulation box. The vectors a1,a2,a3 are the edge vectors of the unit cell. This is the nomenclature for “primitive” vectors in solid-state crystallography, but in LAMMPS the unit cell they determine does not have to be a “primitive cell” of minimum volume.

Note that the lattice command can be used multiple times in an input script. Each time it is invoked, the lattice attributes are re-defined and are used for all subsequent commands (that use lattice attributes). For example, a sequence of lattice, region, and create_atoms commands can be repeated multiple times to build a poly-crystalline model with different geometric regions populated with atoms in different lattice orientations.

A lattice of style *none* does not define a unit cell and basis set,
so it cannot be used with the create_atoms
command. However it does define a lattice spacing via the specified
scale parameter. As explained above the lattice spacings in x,y,z can
be used by other commands as distance units. No additional
keyword/value pairs can be specified for the *none* style. By
default, a “lattice none 1.0” is defined, which means the lattice
spacing is the same as one distance unit, as defined by the
units command.

Lattices of style *sc*, *fcc*, *bcc*, and *diamond* are 3d lattices
that define a cubic unit cell with edge length = 1.0. This means a1 =
1 0 0, a2 = 0 1 0, and a3 = 0 0 1. Style *hcp* has a1 = 1 0 0, a2 = 0
sqrt(3) 0, and a3 = 0 0 sqrt(8/3). The placement of the basis atoms
within the unit cell are described in any solid-state physics text. A
*sc* lattice has 1 basis atom at the lower-left-bottom corner of the
cube. A *bcc* lattice has 2 basis atoms, one at the corner and one at
the center of the cube. A *fcc* lattice has 4 basis atoms, one at the
corner and 3 at the cube face centers. A *hcp* lattice has 4 basis
atoms, two in the z = 0 plane and 2 in the z = 0.5 plane. A *diamond*
lattice has 8 basis atoms.

Lattices of style *sq* and *sq2* are 2d lattices that define a square
unit cell with edge length = 1.0. This means a1 = 1 0 0 and a2 = 0 1
0. A *sq* lattice has 1 basis atom at the lower-left corner of the
square. A *sq2* lattice has 2 basis atoms, one at the corner and one
at the center of the square. A *hex* style is also a 2d lattice, but
the unit cell is rectangular, with a1 = 1 0 0 and a2 = 0 sqrt(3) 0.
It has 2 basis atoms, one at the corner and one at the center of the
rectangle.

A lattice of style *custom* allows you to specify a1, a2, a3, and a
list of basis atoms to put in the unit cell. By default, a1 and a2
and a3 are 3 orthogonal unit vectors (edges of a unit cube). But you
can specify them to be of any length and non-orthogonal to each other,
so that they describe a tilted parallelepiped. Via the *basis*
keyword you add atoms, one at a time, to the unit cell. Its arguments
are fractional coordinates (0.0 <= x,y,z < 1.0). The position vector
x of a basis atom within the unit cell is thus a linear combination of
the unit cell’s 3 edge vectors, i.e. x = bx a1 + by a2 + bz a3,
where bx,by,bz are the 3 values specified for the *basis* keyword.

This sub-section discusses the arguments that determine how the idealized unit cell is transformed into a lattice of points within the simulation box.

The *scale* argument determines how the size of the unit cell will be
scaled when mapping it into the simulation box. I.e. it determines a
multiplicative factor to apply to the unit cell, to convert it to a
lattice of the desired size and distance units in the simulation box.
The meaning of the *scale* argument depends on the units
being used in your simulation.

For all unit styles except *lj*, the scale argument is specified in
the distance units defined by the unit style. For example, in *real*
or *metal* units, if the unit cell is a unit cube with edge length
1.0, specifying scale = 3.52 would create a cubic lattice with a
spacing of 3.52 Angstroms. In *cgs* units, the spacing would be 3.52
cm.

For unit style *lj*, the scale argument is the Lennard-Jones reduced
density, typically written as rho*. LAMMPS converts this value into
the multiplicative factor via the formula “factor^dim = rho/rho*”,
where rho = N/V with V = the volume of the lattice unit cell and N =
the number of basis atoms in the unit cell (described below), and dim
= 2 or 3 for the dimensionality of the simulation. Effectively, this
means that if LJ particles of size sigma = 1.0 are used in the
simulation, the lattice of particles will be at the desired reduced
density.

The *origin* option specifies how the unit cell will be shifted or
translated when mapping it into the simulation box. The x,y,z values
are fractional values (0.0 <= x,y,z < 1.0) meaning shift the lattice
by a fraction of the lattice spacing in each dimension. The meaning
of “lattice spacing” is discussed below.

The *orient* option specifies how the unit cell will be rotated when
mapping it into the simulation box. The *dim* argument is one of the
3 coordinate axes in the simulation box. The other 3 arguments are
the crystallographic direction in the lattice that you want to orient
along that axis, specified as integers. E.g. “orient x 2 1 0” means
the x-axis in the simulation box will be the [210] lattice
direction, and similarly for y and z. The 3 lattice directions you
specify do not have to be unit vectors, but they must be mutually
orthogonal and obey the right-hand rule, i.e. (X cross Y) points in
the Z direction.

Note

The preceding paragraph describing lattice directions is only
valid for orthogonal cubic unit cells (or square in 2d). If you are
using a *hcp* or *hex* lattice or the more general lattice style
*custom* with non-orthogonal a1,a2,a3 vectors, then you should think
of the 3 *orient* vectors as creating a 3x3 rotation matrix which is
applied to a1,a2,a3 to rotate the original unit cell to a new
orientation in the simulation box.

Several LAMMPS commands have the option to use distance units that are inferred from “lattice spacings” in the x,y,z box directions. E.g. the region command can create a block of size 10x20x20, where 10 means 10 lattice spacings in the x direction.

Note

Though they are called lattice spacings, all the commands that
have a “units lattice” option, simply use the 3 values as scale
factors on the distance units defined by the units
command. Thus if you do not like the lattice spacings computed by
LAMMPS (e.g. for a non-orthogonal or rotated unit cell), you can
define the 3 values to be whatever you wish, via the *spacing* option.

If the *spacing* option is not specified, the lattice spacings are
computed by LAMMPS in the following way. A unit cell of the lattice
is mapped into the simulation box (scaled and rotated), so that it now
has (perhaps) a modified size and orientation. The lattice spacing in
X is defined as the difference between the min/max extent of the x
coordinates of the 8 corner points of the modified unit cell (4 in
2d). Similarly, the Y and Z lattice spacings are defined as the
difference in the min/max of the y and z coordinates.

Note that if the unit cell is orthogonal with axis-aligned edges (no
rotation via the *orient* keyword), then the lattice spacings in each
dimension are simply the scale factor (described above) multiplied by
the length of a1,a2,a3. Thus a *hex* style lattice with a scale
factor of 3.0 Angstroms, would have a lattice spacing of 3.0 in x and
3*sqrt(3.0) in y.

Note

For non-orthogonal unit cells and/or when a rotation is applied
via the *orient* keyword, then the lattice spacings computed by LAMMPS
are typically less intuitive. In particular, in these cases, there is
no guarantee that a particular lattice spacing is an integer multiple
of the periodicity of the lattice in that direction. Thus, if you
create an orthogonal periodic simulation box whose size in a dimension
is a multiple of the lattice spacing, and then fill it with atoms via
the create_atoms command, you will NOT necessarily
create a periodic system. I.e. atoms may overlap incorrectly at the
faces of the simulation box.

The *spacing* option sets the 3 lattice spacings directly. All must
be non-zero (use 1.0 for dz in a 2d simulation). The specified values
are multiplied by the multiplicative factor described above that is
associated with the scale factor. Thus a spacing of 1.0 means one
unit cell edge length independent of the scale factor. As mentioned
above, this option can be useful if the spacings LAMMPS computes are
inconvenient to use in subsequent commands, which can be the case for
non-orthogonal or rotated lattices.

Note that whenever the lattice command is used, the values of the lattice spacings LAMMPS calculates are printed out. Thus their effect in commands that use the spacings should be decipherable.

Example commands for generating a Wurtzite crystal. The lattice constants approximate those of CdSe. The \(\sqrt{3}\times 1\) orthorhombic supercell is used with the x, y, and z directions oriented along \([\bar{1}\bar{2}30]\), \([10\bar{1}0]\), and \([0001]\), respectively.

```
variable a equal 4.34
variable b equal $a*sqrt(3.0)
variable c equal $a*sqrt(8.0/3.0)
variable third equal 1.0/3.0
variable five6 equal 5.0/6.0
lattice custom 1.0 &
a1 $b 0.0 0.0 &
a2 0.0 $a 0.0 &
a3 0.0 0.0 $c &
basis 0.0 0.0 0.0 &
basis 0.5 0.5 0.0 &
basis ${third} 0.0 0.5 &
basis ${five6} 0.5 0.5 &
basis 0.0 0.0 0.625 &
basis 0.5 0.5 0.625 &
basis ${third} 0.0 0.125 &
basis ${five6} 0.5 0.125
region myreg block 0 1 0 1 0 1
create_box 2 myreg
create_atoms 1 box &
basis 5 2 &
basis 6 2 &
basis 7 2 &
basis 8 2
```

## Restrictions¶

The *a1,a2,a3,basis* keywords can only be used with style *custom*.

## Default¶

```
lattice none 1.0
```

For other lattice styles, the option defaults are origin = 0.0 0.0 0.0, orient = x 1 0 0, orient = y 0 1 0, orient = z 0 0 1, a1 = 1 0 0, a2 = 0 1 0, and a3 = 0 0 1.