CE with CS for a tetragonal alloy

Alloy Theoretic Automated Toolkit (ATAT)
1

Hallo everyone,

I am trying to produce a CE with CS for a tetragonal alloy.
The description and illustration of the issue is in the attachments, because I could not write formula in the forum.
My three questions are also in the attachment.

Thank you for reading.

With best regards,

maria

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2

I have trouble reading your pictures (they are too large to be viewed). Can you send them as a zip file we can download?

3

Yes, of course!
The zip-file contains the original pdf-file and the same as pictures.
[attachment deleted]
Thank you for reading.

With best regards,

maria

4

These are nontrivial questions! Here are my suggestions:

  1. The component of the direction vector u for the harmonics should be in cartesian coordinates (not multiples of 1/a or 1/c). But the direction of the corresponding planes should be calculated with the and c lattice parameters artificially set to some conventional value than does not change throughout your calculations. (Perhaps set a to the average a parameters of the two pure phases and similarly for c). So u represents the direction of the planes in the lattice before it has relaxed, because this is how the Ising model is defined (with a fixed a and c even if the relaxed a and c change).
  2. The space group I42d (actually you probably mean I(-4)2d ) does not have inversion symmetry so, in principle, gencs answer’s is correct in making [111] and [-1 -1 -1] inequivalent. But the physical system will always contain both planes at the same time, so the harmonic should be symmetrized with respect to inversion. You can force that by running gencs with the -s option to read a set of generators for the point group from the sym.in file. You would add the inversion symmetry to the actual generators of the point group. Example (I am not sure this is correct for your system):
4

-1 0 0
0 -1 0
0 0 1
0 0 0

1 0 0
0 -1 0
0 0 -1
0 0 0

-1 0 0
0 1 0
0 0 -1
0 0 0

-1 0 0
0 -1 0
0 0 -1
0 0 0
  1. Usually, you would include or exclude harmonics of the same order at the same time because they represent similar angular accuracies.
5

Thank you for your answers. They are very helpful!
But I would like to continue to discuss about the question/answer 1.

To represent the planes in the cartesian coordinate system, it is possible to take the vectors b_1=(c_m,0,0), b_2=(0,c_m,0) and b_3=(0,0,a_m) and multiply them with the miller indices of the plane (and normalize). c_m and a_m are here the conventional values of the lattice parameters.
I have already tested if the harmonic functions change when I put different values for a_m and c_m. And they do not change, e.g. by stretching/compression in the z-direction when the ration between c_m and a_m changes.
That is why I came to the idea to take the miller indices and propose the vector
u=1/Sq(3)*(1,1,1) for the plane [111] even when the direction in the cartesian coordinate system is not the same.
In my opinion it makes sense to look at the program gencs. It seems to take only the symmetries and work in a dimensionless system, like the miller indices.
Am I right?

With best regards,

maria

6

Just to clarify: the vector "u" has to be normalized to unit length (and be dimensionless) in the expression of the harmonics. The meaning of this unit vector depends on the c/a ratio you use when specifying a miller plane. You need to pick a reference c/a ratio and keep it the same every time you use the harmonic expression.
So u does not have to be the actual Cartesian direction, but you have to pick a convention on how you scale each coordinate and stick to it throughout.
(BTW, the csfit code does not handle systems other than cubic at this point.)

7

Thank you for your answers. Now this is much clearer for me. :)).
Am I right with my last statement? I am really not sure about this.

With best regards,

maria

8

You are right that the kspacecs.c++ is rather general except for the hard-coded kubic harmonics.
The main reason I didn’t go beyond this is because the fitting part (csfit) was much more difficult to implement generally (due to the possible complex low-symmetry relaxations), so that was my hurdle. Now, if you are willing and have the time, it would be really fantastic to implement this "right".