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Table 2 Total number of possible atom combinations for different substitution levels x and different supercell sizes of the \(\hbox {A}_\mathrm{x} \hbox {Pb}_\mathrm{1-x}\hbox {Te}\) system

From: Supercell program: a combinatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals

N 8 16 24 32 64
Cell, \(a\times b \times c\) \(1\times 1 \times 1\) \(1\times 1 \times 2\) \(1\times 1 \times 3\) \(1\times 2 \times 2\) \(2\times 2 \times 2\)
Symmetry operations 192 128 192 256 1536
\(x=\frac{1}{16}\) N/A N/A N/A 16 (1) 496 (5)
\(x=\frac{1}{8}\) N/A 8 (1) N/A 120 (5) 35,960 (71)
\(x=\frac{1}{4}\) 4 (1) 28 (4) 220 (9) 1820 (33) 10,518,300 (8043)
\(x=\frac{1}{2}\) 6 (1) 70 (8) 924 (34) 12,870 (153) 601,080,390 (404,582)
  1. N is the total number of atoms in the supercell. The number of unique (non-symmetric) combinations are given in parenthesis. The total number of combinations depends only on N, whereas the number of unique combinations can depend on the supercell formula \(a\times b \times c\). The number of combinations for substitutions \(1-x\) is equal to the number of combinations for \(x\) and are consequently not shown