Problem in tranforming Si unit cell

[Sharat Chandra]

Hi

I want to transform the standard Si unit cell to another one with the (111) plane oriented in c direction. For that I have worked out the transforms as given below. The Si structure is in P1 symmetry.

Si lattice vectors:
 a = (100), b = (010), c = (001), 8 atoms in the unit cell.

Transformations to cell with (111) plane perpendicular to c-axis:

 a' = a - c, b' = b - c, c' = a + b + c.
So the transformation matrix to be fed in the 'Option' panel in the 'Unit Cell' tab is
 a' = (1 0 -1), b' = (0 1 -1) and c' = (1 1 1).

This gives the proper transformation with 24 atoms in the unit cell.

Lattice parameters
   a        b        c       alpha    beta     gamma
 7.68130  7.68130  9.40763  90.0000  90.0000  60.0000
Unit-cell volume = 480.707187 Å^3

Now if I want to transform this unit cell to another one with all angles = 90 degrees, the transformations are:

 a" = a' = a - c, b" = -a'/2 + b' = -a/2 + b -c/2, c" = c' = b - c.
So the transformation matrix to be fed in the 'Option' panel in the 'Unit Cell' tab is
 a" = (1 0 -1), b" = (-1/2 1 -1/2) and c" = (1 1 1).

The problem is if I feed this matrix directly in the Si crystal, it gives me a unit cell with 24 atoms, but with all the angles != 90 degrees:

Lattice parameters
   a        b        c       alpha    beta     gamma
 8.14725  7.68130  8.14725  76.3670  83.6206  76.3670
Unit-cell volume = 480.707221 Å^3

But if I export the 60 degree structure in the first step to a cif file and then enter the following transformations in the Option panel after reopening it in VESTA:

 a" = a' = (1 0 0), b" = -a'/2 + b' = (-1/2 1 0) and c" = c' = (0 0 1)
 (just the transforms in terms of a', b' and c'),

I get a proper structure with 24 atoms and with all angles = 90 degrees and the (111) plane oriented in the c direction.

Lattice parameters
   a        b        c       alpha    beta     gamma
 6.65220  7.68130  9.40763  90.0000  90.0000  90.0000
Unit-cell volume = 480.706880 Å^3

I am not able to understand what is the problem and what is the mistake I am making. Can someone please clarify.

Regards

Sharat Chandra

[Philip Howie]

Dear Sharat,

If you look at the right side of the 'Option' window, you'll see that the transformation matrix, P, postmultiplies the original lattice vectors.  The matrix you want is then the transpose of the one you have, i.e.

     (1  -1/2  1)
     (0   1     1)
     (1  -1/2  1)

If you apply this to the original Si structure, you'll obtain the correct structure with alpha = beta = gamma = 90 °.

Hope this helps,
Philip

[Sharat Chandra]

Dear Philip

Thanks a lot!

Of course it solves the problem. I do know now that I am probably blind on the right side vision :)

Regards

Sharat