Triangular planar lattice

Figure 2.4 The four principal frustrated lattices triangular planar (TP), kagome,...

The triangular planar (D3h symmetry) CO/ molecular ion with 24 electrons (AB324-type) in CaC03 is easily ionized by radiation to electron and hole centres self-trapped in the lattice or an oxygen vacancy type C02 molecular ion at the anon site. Molecular orbital schemes based on the general scheme of AB3 molecules with 25,24 and 23 electrons for atoms A (B, C, Si, N, P, As and S) and B (O) characterize their specific -factor. Hence, the anisotropic -factor of these radicals estimated from the powder spectrum has been to identify the radical species.1... 

Figure 2.39 A view of the B (or A) sites in the pyrochlore lattice as a stacking of alternating kagome and triangular planar layers normal to the <111> direction. Reprinted with permission from Gardner et al., 2010 [6]. Copyright (2010) American Physical Society...

Fig. 2.16. Planar orientational structures of nonpolar molecules on a triangular lattice.

Fig. 2.20. Allowed planar molecular orientations in the representation of double (upper row) and ordinary (lower row) azimuthal angles two rigidly fixed molecular orientations (n = 2) for a square lattice (a) and for a triangular lattice (b), and four discrete orientations (n = 4) on a square lattice (c).

It is noteworthy that the critical value given by Eq. (2.5.11) is exactly half as large as for two quadrupole orientations on a square lattice (cf. Eq. (2.5.7)). This is a vindication of the inference that a halved critical temperature results from a corresponding change in the orientation space dimensionality. Thus, one might with good reason anticipate that the critical temperature for a triangular lattice of quadruples with arbitrary planar orientations should also be approximately half as... 

Figure 59 The magnetic structure of Gd2Ti207 at 50 mK showing the Kagome type planes separated by paramagnetic Gd + ions on a planar triangular lattice. (Reprinted with permission from J.M.D. Champion, A.S. Wills, T. Fennell, S.T. Bramwell, J.S. Gardner, M.A. Green, Phys.Rev., 2001, B64, R140407. 2001 by the American Physical Society)...

Fig. 24. Herringbone (a) and pinwiiccl (b) orientational ordering of uniaxial diatomic molecules on a triangular lattice. The heavy bars represent planar rotators and the circles denote vacancies, p is the degeneracy of the ordering. From Mourilsen (1985).

To proceed, consider two finite planar networks, a regular Euclidean triangular lattice (interior valence v — 6, but with boundary defect sites of valence v = A and v = 2) of dimension d — 2, and a fractal lattice... 

The evolution of S t) was studied for J = 1 lattices, d = 2 square-planar and triangular lattices, and the Sierpinski gasket. In each case, it was observed that after an initial interval of time (which is of the order of one), S(t) grows linearly with in t). For a finite lattice, on a time scale which depends on the size of the lattice, the evolution curve deviates from strictly linear behavior. The linear regime persists longer the larger the size of the lattice. 

Displayed in Figure 4.24 are the results of calculations on a linear lattice of 21 sites, a 21 x 21 square-planar lattice, a triangular lattice of 361 sites. 

Presented in Figures 4.32, 4.33, and 4.34 for the choice of initial conditions wherein reactant trajectories are initiated from all possible nontrapping sites of the system (the entire lattice case) are results calculated for (n) versus N for one-, two-, and three-layer lattices. In each figure, two profiles are displayed the uppermost one corresponds to results calculated for an ensemble based on dimension d — 2 planar lattices of hexagonal symmetry (v = 3) while the lower curve is based on J = 2 planar lattices of triangular symmetry (v = 6). The ratios of reaction efficiency, ( (II))/( ( ))... 

Figure 6. Schematic picture of the six (2V3 x Ji)R30° hetringbone ground states of the anisotropic-planar-rotor model on a triangular lattice (2.5). Small and large dots represent empty and occupied sites, respectively. (Adapted from Fig. 1 of Ref. 60.)...

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