sqsgen under the hood

Pair Short-Range-Order

What does sqsgen actually compute?

sqsgenerator computes so-called Warren-Cowley short-range-order (WC-SRO) parameters, to quantify the “randomness” of an atomistic model. Thus consider a binary alloy with just \(A\) and \(B\) atoms, then the SRO parameter is defined as

(1)\[ \alpha_{AB} = 1 - \dfrac{N_{AB}}{NMx_Ax_B} \]

Where \(N_{AB}\) the number of \(A-B\) bonds, \(N\) the number of atoms in the cell, \(x_A\) (\(x_B\)) the mole fraction of \(A\) (\(B\)) and \(M\) the number of neighbors on the lattice (e. g \(M^{fcc}=12\)). Thus it compares the number of \(A-B\) bonds in a given (numerator) configuration with that of statistical ideal solid solution (denominator). Thus the following cases can occur

  • \(\alpha_{AB} < 0\): system tends to be ordered

  • \(\alpha_{AB} > 0\): system tends to be clustered

  • \(\alpha_{AB} \approx 0\): system tends to be disordered

Therefore, sqsgenerator tries to optimize a given system such that the SRO approaches zero.

Can sqsgenerator handle also second-nearest neighbors?

Yes it actually can! Indeed, on a fcc lattice each atom exhibits \(M^{fcc}_1=12\) nearest neighbors, but there are also \(M^{fcc}_2=6\) second nearest neighbors and even \(M^{fcc}_3=24\) third nearest neighbors. However, it is very easy to generalize the Warren-Cowley SRO to more “coordination shells” which, therefore becomes a vector

(2)\[ \alpha^k_{AB} = 1 - \dfrac{N_{AB}^k}{NM^kx_Ax_B}. \]

But what about optimizing multi-component alloys?

Similarly, the generalization to multi-component systems is a straight as for the the coordination shells. Consider a system of \(\{A,B,C\}\) atoms. Now we have to also consider \(A-C\) and \(B-C\) bonds, thus resulting in 6 SRO parameters for each coordination shells (also \(A-A\), \(B-B\) and \(C-C\) bonds are considered), thus the initial parameter becomes a three dimensional object

(3)\[ \alpha^i_{\xi\eta} = 1 - \dfrac{N^i_{\xi\eta}}{NM^ix_{\xi}x_{\eta}} \]

where again \(M^{i}\) is the number of sites in the \(i\)-th coordination shell, \(x_\xi\) and \(x_\eta\) are the mole-fraction of the corresponding species and \(N^i_{\xi\eta}\) the number of \(\xi-\eta\)bonds”/pairs in the coordination shell.

Putting it all together!

This is a quite large number of parameters to optimize. In order to squeeze those together we produce a single scalar value by simply summing over those parameters. Therefore we define our “objective-function” as for a certain atomic configuration \(\sigma\) as,

(4)\[ \mathcal{O}(\sigma) = \sum_{k}w^i\sum_{\xi,\eta} p_{\xi\eta} | \tilde{\alpha}_{\xi\eta}-\alpha^i_{\xi\eta} | \]

where \(w^i\) are the coordination shell_weights, to take into account the decreasing influence of shells which are farther away. \(\tilde{\alpha}_{\xi\eta}\) are the target_objective parameters. Those can be used to optimize pairs between different species differently. \(p_{\xi\eta}\) are pair_weights and allows the user to weight pair types differently.

However for the sake of computational efficiency we merge \(p_{\xi\eta}\) and \(w^i\) into \(\tilde{p}_{\xi\eta}^i\) where

(5)\[ \tilde{p}_{\xi\eta}^i = w^i p_{\xi\eta} \]

Moreover, the \(\dfrac{1}{NM^i x_\xi x_\eta}\) term does not depend on a specific configuration \(\sigma\) therefore it can be computed before the actual optimization process. Hence we gather those and name them “prefactors\(f_{\xi\eta}^i\) where

(6)\[ f_{\xi\eta}^i = (NM^ix_\xi x_\eta)^{-1} \]

such that Eq. (3) can be factored to

(7)\[ \alpha^i_{\xi\eta} = 1 - f_{\xi\eta}^i N^i_{\xi\eta} \]

Employing the above definition the objective function \(\mathcal{O}(\sigma)\) can be written as

(8)\[\begin{split} \mathcal{O}(\sigma) &= \sum_{i,\xi,\eta} \tilde{p}_{\xi\eta}^i | \tilde{\alpha}_{\xi\eta}-\alpha^i_{\xi\eta} | \\ &=\sum_{i,\xi,\eta} \underbrace{\tilde{p}_{\xi\eta}^i}_{\texttt{pair_weights}} | \overbrace{\tilde{\alpha}_{\xi\eta}}^{\texttt{target_objective}}- \overbrace{(1 - \underbrace{f^i_{\xi\eta}}_{\texttt{prefactors}}N^i_{\xi\eta})}^{\alpha^i_{\xi\eta}} | \end{split}\]

sqsgenerator tries different atomic configurations \(\sigma\) by “randomly shuffling” or “systematically” (in lexicographical order) to filter out those configuration which minimize \(\mathcal{O}(\sigma)\).

Note

In the technical part of the documentation we refer to \(w^i\) as the shell_weights, \(\tilde{\alpha}_{\xi\eta}\) as the target_objective, \(\tilde{p}^i_{\xi\eta}\) as the pair_weights and \(f_{\xi\eta}^i\) as the prefactors

Note

The mapping of an atomic configuration \(\sigma\) on a scalar values via \(\mathcal{O}(\sigma)\) is non-injective, thus you might end up with quite a lot of configurations, which are “equal” in the here presented formalism.

Triplet-based Short-Range-Order

The objective-function above is a non-injective function, meaning, that one can end up with a lot of structures which are the same in the formalism presented above, therefore it might be beneficial to further investigate those structures, by not counting pairs, but rather triplets.

the Pair-Shell-Matrix

To be able to generalize Warren-Cowley parameters to triplets we introduce a “pair-shell-matrix\(\mathbf{c}\) of dimensions (\(N\times N\)) where \(N\) is the number of atoms in the system under consideration. Each entry \(c_{ij}\) denotes the “coordination-shell” between lattice site \(i\) and lattice site \(j\).

Number of triplets for a tuple of coordination shell

Similarly as above, when we generalized Warren-Cowley SRO parameter to an arbitrary amount of coordination shells, we can do the same for triplets. However, a pair of lattice sites is uniquely characterized by a single distance/coordination, we need three distances/coordination-shells to properly describe a triplet of lattice positions. E. g all triplets that can be formed from first, second and third nearest neighbors will be identified by \(\{1,2,3\}\). Let \(i\),\(j\) and \(k\) be coordination shells, then \(\{i,j,k\}\)​ represents the set of a tuples that can be generated upon permuting the sequence. In other words \(\{i,j,k\} = \{(i,j,k), (i,k,j),\ldots\}\). Furthermore let \(\widetilde{M}\) be the set of all lattice positions with a sphere with radius \(\max(i,j,k)\).

It follows that the number of triplets that can be formed \(T^{\left\{i,j,k\right\}}\) reads as

(9)\[ T^{\left\{i,j,k\right\}} = \left|\left\{ (\xi,\eta,\zeta) | (c_{\xi\eta}, c_{\eta\zeta},c_{\xi\zeta}) \in \left\{i,j,k\right\} \right\} \right|_{\forall \xi,\eta,\zeta \in \widetilde{M}} \]

The number of all triplets are computed from the input structure before the iteration itself starts.

(10)\[ \beta^{ijk}_{\xi\eta\zeta} = 1 - \dfrac{N^{ijk}_{\xi\eta\zeta}}{\frac{1}{|\{i,j,k\}|}NT^{\left\{i,j,k\right\}}x_\xi x_\eta x_\zeta} \]