Lattice Transformations in CrystalMaker

CrystalMaker lets you take an existing unit cell and transform its lattice type (e.g., to convert a Face-Centred cell to a Primitive cell), project the cell onto a lattice plane, or to define a much larger, "Supercell".

Getting Started

Choose the Transform > Unit Cell command to display a submenu with transformation options.

General Transformation
Here, you can enter a transformation matrix, [m_ij], that relates the new crystallographic unit cell (basis) vectors, to those of the existing cell. Thus,


				[a']    [m₁₁ m₁₂ m₁₃] [a]

				[b']       = [m₂₁ m₂₂ m₂₃] [b]

				[c']    [m₃₁ m₃₂ m₃₃] [c]

Where {a, b, c} are the unit cell vectors of your existing cell (i.e., the existing basis vectors), and {a', b', c'} are the unit cell vectors for the new unit cell (i.e., your new basis vectors).

Projection onto a Lattice Plane
Here you can specify the Miller Indices of a lattice plane, onto which the unit cell will be projected. CrystalMaker attempts to create a Primitive cell (where possible), with the x- and y crystallographic axes lying in the plane, and the z-axis directed out of the plane.

This type of transformation is very useful in surface studies, as it makes it easy to display precise ranges of atoms parallel to a given surface. Note, however, that your origin lattice plane is now parallel to (001) in the new transformed crystal setting.

Supercell
If you have an existing crystal structure and you wish to use a larger unit cell, you can specify the relative size of the new unit cell, with respect to the old cell. This can be useful when investigating order/disorder relations and/or creating a structure as a prelude to simulating "superlattice" reflexions.

Example: Convert a C-face centred cell to a primitive cell.

Our existing cell has unit cell vectors {a, b, c}, with lattice points at the corners of the unit cell, and at {1/2, 1/2, 0}. We'll transform this cell to a Primitive cell with axial vectors: {a', b', c'}. These will be chosen such that the vector from the origin to {1/2, 1/2, 0} defines our new x-axis, and the vector to {-1/2, 1/2, 0} defines our new y-axis, with the z-axis remaining unchanged, i.e.,

a' = a/2 + b/2
b' = -a/2 + b/2
c' = c

Our transformation matrix is therefore:

[	0.5	0.5	0	]
[	-0.5	0.5	0	]
[	0	0	1	]

Note: When transforming a unit cell, CrystalMaker will discard any spacegroup symmetry, generating new sites, as required to preserve the structure. The fractional coordinates of sites in the structure will be transformed to match the new basis vectors.