A same cluster is considered twice

Alloy Theoretic Automated Toolkit (ATAT)
1

Dear Axel,

First of all, let me thanks again for the recent correction you made on "xtalutil.c++",
that fixed a bug which made "corrdump" able to find now all clusters of rank 4 (quadruplet) in the C15-phase.

I am back again with this C15-phase, where you can find the 'lat.in" of the full-random primitive cell below:


4.9497 4.9497 4.9497 60 60 60
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0
     0.375000000         0.375000000         0.375000000 Cu,Mg
     0.625000000         0.625000000         0.625000000 Cu,Mg
     0.000000000         0.000000000         0.000000000 Cu,Mg
     0.500000000         0.000000000         0.000000000 Cu,Mg
     0.000000000         0.500000000         0.000000000 Cu,Mg
     0.000000000         0.000000000         0.500000000 Cu,Mg

As you know,
AB2-C15 is a FCC cell with two non-equivalent sites,
MgCu2 its prototype, SG: Fd-3m (227) with 2 origin descriptions.
Here, we choose A=Cu,Mg on 8a (1/8,1/8,1/8) and B=Cu,Mg on 16d (0,0,0),
with origin 1 + 1/8 shift description, cubic cell parameter, a=7.

By asking the command :

corrdump -l=lat.in -nop -clus -3=3

2 triplets, with a min distance of 2.90202 are found: one with a multiplicity 24 and the other with 12.
If we see the detail of these clusters in "cluster.out", we can see that, in fact, this 2 clusters are precisely the same: The triplet A-B-B:


24
2.90202
3
1.00000 1.00000 0.50000 0 0
1.00000 0.50000 1.00000 0 0
0.62500 0.62500 0.62500 0 0

12
2.90202
3
1.00000 1.00000 0.50000 0 0
1.00000 1.00000 0.00000 0 0
0.62500 0.62500 0.62500 0 0

You may answer that (1.00000 0.50000 1.00000) are different than (1.00000 1.00000 0.00000),
since an atom of the center of the edge of the cell is different that in vertices position,
but looking carefully the structure, the "B-16d" site positions are exactly equivalent in term of coordination, distances and global environment.

Please check on the crystal structure attached.
Pair distances are equal and angles too:
phi(Cu1-Mg2-Cu2) = 50.4788(0) deg. phi(Cu4-Mg2-Cu2) = 50.4788(0) deg.

[b]My questions are :

  1. How it is possible that the same cluster A-B-B is considered twice?[/b]
  2. By doing CEM calculations on 60 structures with CV<0.035, I found that the associated ECI are of opposite sign! How identical crystallographic clusters could have opposite sign ECI energy?

Thus I think there is a small error in the crystal detection for complex structures, as the C15 is.
Let me know if the problem is clear enough and if I could help you in a way to solve this bug.

Many Many thanks in advance,
JC.

2

BTW, first thank you for your earlier bug report, which did help me fix a bug.
However, this time I believe the code actually does give the right output.
The key is that two cluster may be, taken in isolation, geometrically identical, but nevertheless be distinct when they are embedded in the whole crystal because their surroundings differ. Specifically, if one applies a symmetry operation that maps one cluster onto the other, the rest of the crystal is not mapped onto itself.
In this case, you can see it in the following way (it took me a while to find a simple construction to prove it independently of my code’s routines). If you compute the normal to the planes containing each respective triplet, you get (in cartesian coordinates):
[-1 1 4] for one cluster
[1 0 -1] for the other
But this crystal has a cubic symmetry (spacegroup 227) and there is not cubic symmetry operation that will map the top vector on the bottom one, which indicates that the crystal would not be mapped onto itself if you performed any symmetry operation that map one cluster onto the other.
Do let me know if I made a mistake in my calculations.
And thanks again for exceptionally detailed and helpful bug report even if there turns to be not bug.

3

Thanks Alex,
This is clear to me now.

After analyzing the closest atom to the normal of both side of triangular plans,
the two triplet clusters are in fact different.

The first cluster presents one Cu and one Mg atoms,
forming two tetrahedrons with the common face of the triplet,
… whereas the second cluster presents tetrahedrons with two Mg neighbors.

I did not expected that the environment of the triplets would be different since the crystallographic position of sites is identical.

My apologies for taking your time to finding an easily proof to that.
That helps :wink:

Best regards,
mcjc