I’m trying to understand the formulation of ECIs for the cluster expansion of a 3 body system using the following reference, but I’m having trouble following the proof.
Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the Alloy Theoretic Automated Toolkit
https://arxiv.org/abs/0906.1608
I have some example data that I’m using to help me understand, where I compare the ECIs generated for a 2-body Al,Ti system and a 3-body Al,Ti,Si system using the following lat.in file.
3.1 3.1 5.062 90 90 120
1 0 0
0 1 0
0 0 1
0 0 0 Al,Ti, Si
0.6666666 0.3333333 0.5 Al,Ti,Si
This results in the following clusinfo.out files for the 2 and 3 body systems respectively.
2 3.099886 6 0.050458
2 3.100000 6 0.049634
3 3.100000 12 -0.004061
3 3.100000 2 -0.006092
3 3.100000 2 0.000000
2 3.099886 6 -0.002474
2 3.099886 12 -0.023477
2 3.099886 6 0.133232
2 3.100000 6 0.002404
2 3.100000 12 -0.039656
2 3.100000 6 0.105034
2 4.383981 6 0.009612
2 4.383981 12 -0.009351
2 4.383981 6 0.019565
2 5.062000 2 -0.019693
2 5.062000 4 -0.018041
2 5.062000 2 0.066512
2 5.369292 12 0.000000
2 5.369292 24 0.000000
2 5.369292 12 0.000000
2 5.369358 6 0.020870
2 5.369358 6 0.000000
2 5.369358 6 0.000000
2 5.369358 6 0.000000
Focusing on just the first line in the Al-Ti system
2 3.099886 6 0.050458
Appears related to these corresponding lines in the Al-Ti-Si system
2 3.099886 6 -0.002474
2 3.099886 12 -0.023477
2 3.099886 6 0.133232
My question is, what are the associated ‘spin’ variables for these ECIs? And why does the second line have twice the symmetry of the other values?