# Triclinic (non-orthogonal) simulation boxes

By default, LAMMPS uses an orthogonal simulation box to encompass the
particles. The [boundary](boundary) command sets the boundary conditions
of the box (periodic, non-periodic, etc). The orthogonal box has its
\"origin\" at (xlo,ylo,zlo) and is defined by 3 edge vectors starting
from the origin given by **a** = (xhi-xlo,0,0); **b** = (0,yhi-ylo,0);
**c** = (0,0,zhi-zlo). The 6 parameters (xlo,xhi,ylo,yhi,zlo,zhi) are
defined at the time the simulation box is created, e.g. by the
[create_box](create_box) or [read_data](read_data) or
[read_restart](read_restart) commands. Additionally, LAMMPS defines box
size parameters lx,ly,lz where lx = xhi-xlo, and similarly in the y and
z dimensions. The 6 parameters, as well as lx,ly,lz, can be output via
the [thermo_style custom](thermo_style) command.

LAMMPS also allows simulations to be performed in triclinic
(non-orthogonal) simulation boxes shaped as a parallelepiped with
triclinic symmetry. The parallelepiped has its \"origin\" at
(xlo,ylo,zlo) and is defined by 3 edge vectors starting from the origin
given by **a** = (xhi-xlo,0,0); **b** = (xy,yhi-ylo,0); **c** =
(xz,yz,zhi-zlo). *xy,xz,yz* can be 0.0 or positive or negative values
and are called \"tilt factors\" because they are the amount of
displacement applied to faces of an originally orthogonal box to
transform it into the parallelepiped. In LAMMPS the triclinic simulation
box edge vectors **a**, **b**, and **c** cannot be arbitrary vectors. As
indicated, **a** must lie on the positive x axis. **b** must lie in the
xy plane, with strictly positive y component. **c** may have any
orientation with strictly positive z component. The requirement that
**a**, **b**, and **c** have strictly positive x, y, and z components,
respectively, ensures that **a**, **b**, and **c** form a complete
right-handed basis. These restrictions impose no loss of generality,
since it is possible to rotate/invert any set of 3 crystal basis vectors
so that they conform to the restrictions.

For example, assume that the 3 vectors **A**,\*\*B\*\*,\*\*C\*\* are the
edge vectors of a general parallelepiped, where there is no restriction
on **A**,\*\*B\*\*,\*\*C\*\* other than they form a complete
right-handed basis i.e. **A** x **B** . **C** \> 0. The equivalent
LAMMPS **a**,\*\*b\*\*,\*\*c\*\* are a linear rotation of **A**, **B**,
and **C** and can be computed as follows:

$$\begin{aligned}
\begin{pmatrix} \mathbf{a}  & \mathbf{b}  & \mathbf{c} \end{pmatrix} = &
\begin{pmatrix}
  a_x & b_x & c_x \\
  0   & b_y & c_y \\
  0   & 0   & c_z \\
\end{pmatrix} \\
a_x = & A \\
b_x = & \mathbf{B} \cdot \mathbf{\hat{A}} \quad = \quad B \cos{\gamma} \\
b_y = & |\mathbf{\hat{A}} \times \mathbf{B}| \quad = \quad B \sin{\gamma} \quad =  \quad  \sqrt{B^2 - {b_x}^2} \\
c_x = & \mathbf{C} \cdot \mathbf{\hat{A}} \quad = \quad C \cos{\beta} \\
c_y = & \mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})} \times \mathbf{\hat{A}} \quad = \quad \frac{\mathbf{B} \cdot \mathbf{C} - b_x c_x}{b_y} \\
c_z = & |\mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})}|\quad = \quad \sqrt{C^2 - {c_x}^2 - {c_y}^2}
\end{aligned}$$

where A = \| **A** \| indicates the scalar length of **A**. The hat
symbol (\^) indicates the corresponding unit vector. $\beta$ and
$\gamma$ are angles between the vectors described below. Note that by
construction, **a**, **b**, and **c** have strictly positive x, y, and z
components, respectively. If it should happen that **A**, **B**, and
**C** form a left-handed basis, then the above equations are not valid
for **c**. In this case, it is necessary to first apply an inversion.
This can be achieved by interchanging two basis vectors or by changing
the sign of one of them.

For consistency, the same rotation/inversion applied to the basis
vectors must also be applied to atom positions, velocities, and any
other vector quantities. This can be conveniently achieved by first
converting to fractional coordinates in the old basis and then
converting to distance coordinates in the new basis. The transformation
is given by the following equation:

$$\begin{aligned}
\mathbf{x} = & \begin{pmatrix} \mathbf{a}  & \mathbf{b}  & \mathbf{c} \end{pmatrix} \cdot \frac{1}{V}
  \begin{pmatrix}
    \mathbf{B \times C}  \\
    \mathbf{C \times A}  \\
    \mathbf{A \times B}
  \end{pmatrix} \cdot \mathbf{X}
\end{aligned}$$

where *V* is the volume of the box, **X** is the original vector
quantity and **x** is the vector in the LAMMPS basis.

There is no requirement that a triclinic box be periodic in any
dimension, though it typically should be in at least the second
dimension of the tilt (y in xy) if you want to enforce a shift in
periodic boundary conditions across that boundary. Some commands that
work with triclinic boxes, e.g. the [fix deform](fix_deform) and [fix
npt](fix_nh) commands, require periodicity or non-shrink-wrap boundary
conditions in specific dimensions. See the command doc pages for
details.

The 9 parameters (xlo,xhi,ylo,yhi,zlo,zhi,xy,xz,yz) are defined at the
time the simulation box is created. This happens in one of 3 ways. If
the [create_box](create_box) command is used with a region of style
*prism*, then a triclinic box is setup. See the [region](region) command
for details. If the [read_data](read_data) command is used to define the
simulation box, and the header of the data file contains a line with the
\"xy xz yz\" keyword, then a triclinic box is setup. See the
[read_data](read_data) command for details. Finally, if the
[read_restart](read_restart) command reads a restart file which was
written from a simulation using a triclinic box, then a triclinic box
will be setup for the restarted simulation.

Note that you can define a triclinic box with all 3 tilt factors = 0.0,
so that it is initially orthogonal. This is necessary if the box will
become non-orthogonal, e.g. due to the [fix npt](fix_nh) or [fix
deform](fix_deform) commands. Alternatively, you can use the
[change_box](change_box) command to convert a simulation box from
orthogonal to triclinic and vice versa.

As with orthogonal boxes, LAMMPS defines triclinic box size parameters
lx,ly,lz where lx = xhi-xlo, and similarly in the y and z dimensions.
The 9 parameters, as well as lx,ly,lz, can be output via the
[thermo_style custom](thermo_style) command.

To avoid extremely tilted boxes (which would be computationally
inefficient), LAMMPS normally requires that no tilt factor can skew the
box more than half the distance of the parallel box length, which is the
first dimension in the tilt factor (x for xz). This is required both
when the simulation box is created, e.g. via the
[create_box](create_box) or [read_data](read_data) commands, as well as
when the box shape changes dynamically during a simulation, e.g. via the
[fix deform](fix_deform) or [fix npt](fix_nh) commands.

For example, if xlo = 2 and xhi = 12, then the x box length is 10 and
the xy tilt factor must be between -5 and 5. Similarly, both xz and yz
must be between -(xhi-xlo)/2 and +(yhi-ylo)/2. Note that this is not a
limitation, since if the maximum tilt factor is 5 (as in this example),
then configurations with tilt = \..., -15, -5, 5, 15, 25, \... are
geometrically all equivalent. If the box tilt exceeds this limit during
a dynamics run (e.g. via the [fix deform](fix_deform) command), then the
box is \"flipped\" to an equivalent shape with a tilt factor within the
bounds, so the run can continue. See the [fix deform](fix_deform) page
for further details.

One exception to this rule is if the first dimension in the tilt factor
(x for xy) is non-periodic. In that case, the limits on the tilt factor
are not enforced, since flipping the box in that dimension does not
change the atom positions due to non-periodicity. In this mode, if you
tilt the system to extreme angles, the simulation will simply become
inefficient, due to the highly skewed simulation box.

Box flips that may occur using the [fix deform](fix_deform) or [fix
npt](fix_nh) commands can be turned off using the *flip no* option with
either of the commands.

Note that if a simulation box has a large tilt factor, LAMMPS will run
less efficiently, due to the large volume of communication needed to
acquire ghost atoms around a processor\'s irregular-shaped subdomain.
For extreme values of tilt, LAMMPS may also lose atoms and generate an
error.

Triclinic crystal structures are often defined using three lattice
constants *a*, *b*, and *c*, and three angles $\alpha$, $\beta$, and
$\gamma$. Note that in this nomenclature, the a, b, and c lattice
constants are the scalar lengths of the edge vectors **a**, **b**, and
**c** defined above. The relationship between these 6 quantities (a, b,
c, $\alpha$, $\beta$, $\gamma$) and the LAMMPS box sizes (lx,ly,lz) =
(xhi-xlo,yhi-ylo,zhi-zlo) and tilt factors (xy,xz,yz) is as follows:

$$\begin{aligned}
a   = & {\rm lx} \\
b^2 = &  {\rm ly}^2 +  {\rm xy}^2 \\
c^2 = &  {\rm lz}^2 +  {\rm xz}^2 +  {\rm yz}^2 \\
\cos{\alpha} = & \frac{{\rm xy}*{\rm xz} + {\rm ly}*{\rm yz}}{b*c} \\
\cos{\beta}  = & \frac{\rm xz}{c} \\
\cos{\gamma} = & \frac{\rm xy}{b} \\
\end{aligned}$$

The inverse relationship can be written as follows:

$$\begin{aligned}
{\rm lx}   = & a \\
{\rm xy}   = & b \cos{\gamma}  \\
{\rm xz}   = & c \cos{\beta}\\
{\rm ly}^2 = & b^2 - {\rm xy}^2 \\
{\rm yz}   = & \frac{b*c \cos{\alpha} - {\rm xy}*{\rm xz}}{\rm ly} \\
{\rm lz}^2 = & c^2 - {\rm xz}^2 - {\rm yz}^2 \\
\end{aligned}$$

The values of *a*, *b*, *c*, $\alpha$ , $\beta$, and $\gamma$ can be
printed out or accessed by computes using the [thermo_style
custom](thermo_style) keywords *cella*, *cellb*, *cellc*, *cellalpha*,
*cellbeta*, *cellgamma*, respectively.

As discussed on the [dump](dump) command doc page, when the BOX BOUNDS
for a snapshot is written to a dump file for a triclinic box, an
orthogonal bounding box which encloses the triclinic simulation box is
output, along with the 3 tilt factors (xy, xz, yz) of the triclinic box,
formatted as follows:

    ITEM: BOX BOUNDS xy xz yz
    xlo_bound xhi_bound xy
    ylo_bound yhi_bound xz
    zlo_bound zhi_bound yz

This bounding box is convenient for many visualization programs and is
calculated from the 9 triclinic box parameters
(xlo,xhi,ylo,yhi,zlo,zhi,xy,xz,yz) as follows:

    xlo_bound = xlo + MIN(0.0,xy,xz,xy+xz)
    xhi_bound = xhi + MAX(0.0,xy,xz,xy+xz)
    ylo_bound = ylo + MIN(0.0,yz)
    yhi_bound = yhi + MAX(0.0,yz)
    zlo_bound = zlo
    zhi_bound = zhi

These formulas can be inverted if you need to convert the bounding box
back into the triclinic box parameters, e.g. xlo = xlo_bound
-MIN(0.0,xy,xz,xy+xz).

One use of triclinic simulation boxes is to model solid-state crystals
with triclinic symmetry. The [lattice](lattice) command can be used with
non-orthogonal basis vectors to define a lattice that will tile a
triclinic simulation box via the [create_atoms](create_atoms) command.

A second use is to run Parrinello-Rahman dynamics via the [fix
npt](fix_nh) command, which will adjust the xy, xz, yz tilt factors to
compensate for off-diagonal components of the pressure tensor. The
analog for an [energy minimization](minimize) is the [fix
box/relax](fix_box_relax) command.

A third use is to shear a bulk solid to study the response of the
material. The [fix deform](fix_deform) command can be used for this
purpose. It allows dynamic control of the xy, xz, yz tilt factors as a
simulation runs. This is discussed in the next section on
non-equilibrium MD (NEMD) simulations.