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Figure 22
Density distribution of the two-scale octadecagonal quasicrystal: the black line is the local free-energy density from equation (57)[link]. Here q = k18 ≃ 1.970 and [{\tilde \rho}_1] = [{\tilde \rho}_q] = 1/13. Note that the purple-colored octadecagonal density distribution extends into the positive values of [\phi \,\gt] 2, where the free-energy density is negative, making its overall free energy [{\cal F}] ≃ −1.223 × 10−3 [\lt] 0. On the other hand, the density distribution of the single-scale hexagonal structure, which is plotted here for reference, cannot extend beyond ϕ = 2 without running into the barrier at [\phi \,\lt] −1, which would force its free energy to become positive. This approach for forcing the quasicrystal structure to be the minimum free-energy state succeeds theoretically for all 6n-fold quasicrystals, where n [\ge] 2, although they become increasingly fragile.

IUCrJ
Volume 5| Part 3| May 2018| Pages 247-268
ISSN: 2052-2525
https://doi.org/10.1107/S2052252518001161