Diamond lattice transforms

Three different unit cells conveniently describe the cubic crystal structure of diamond. This structure is shared by silicon and germanium. Closely related is the zinc blende structure, which has two different elements in its unit cell, and lacks the inversion symmetry of the diamond structure. The zinc blende structure is not only seen in zinc sulphide, but also cubic silicon carbide, boron nitride, and several other compounds.

In all cases there is a two-atom primitive cell, an eight-atom cubic cell, and also a six-atom cell which describes the crystal in a hexagonal setting. This is often used when comparing the cubic form of silicon carbide with the hexagonal forms.

So recorded here are the transform matrices for moving between these lattices.

From 2 atom

  0.00  2.73 2.73
  2.73  0.00 2.73
  2.73  2.73 0.00

To 6 atom: -x='(-1,1,0)(-1,0,1)(1,1,1)'
To 8 atom: -x='(-1,1,1)(1,-1,1)(1,1,-1)'

From 6 atom

  2.73  -2.73   0.00
  2.73   0.00  -2.73
  5.46   5.46   5.46
%endblock LATTICE_CART

(As this keeps the same orientation in space as the other cells, the c axis will be in the x+y+z Cartesian direction as here, for that is where the 3 axis lies.)

To 2 atom: -x='(-1/3,-1/3,1/3)(2/3,-1/3,1/3)(-1/3,2/3,1/3)'
To 8 atom: -x='(2/3,2/3,1/3)(-4/3,2/3,1/3)(2/3,-4/3,1/3)'
To 6 atom with c along z: -a

Note that versions of c2x prior to 2.40d do not accept fractions in the argument to -x, so the above will need converting to decimals.

The six atom cell with c along z will be written as:

  3.8608030   0.0000000   0.0000000
  1.9304015   3.3435535   0.0000000
  0.0000000   0.0000000   9.4569974
%endblock LATTICE_CART

Use -a -15 for extra precision in the output.

From 8 atom

  5.46   0.00   0.00
  0.00   5.46   0.00
  0.00   0.00   5.46
%endblock LATTICE_CART

To 2 atom: -x='(0,0.5,0.5)(0.5,0,0.5)(0.5,0.5,0)'
To 6 atom: -x='(.5,-.5,0)(.5,0,-.5)(1,1,1)'