Triangular lattice structure monolayers

Fig. 24. Contour plot of the structure factor (the kinematic LEED intensity) of a x y/i monolayer in a triangular lattice gas with nearest-neighbor repulsion, at a temperature k TIi = 0.355 (about 5% above T ) and a chemical potential // = 1.5 (0c = 0.336 at the transition temperature.) Contour increments are in a (common) logarithmic scale separated by 0.1, starting with 3.2 at the outermost contour. Center of the surface Brillouin zon is to the left k, and k the radial and azimuthal components of kH, are in units of nlXla, a being the lattice spacing. Data are based on averages over 2x10 Monte Carlo steps per site. (From Bartelt et...

Elastic neutron diffraction was first performed (analyzer in Fig. 1 set to zero energy transfer) to establish the structure of the monolayer at low temperature. Three Bragg reflections were observed which could be indexed by a triangular lattice having a nearest-neighbor distance about 10% smaller than required for a 3 X /3 R30° commensurate structure (every third carbon hexagon in the graphite basal plane occupied). 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

Figure 15 (left) Schematic view of a modulated lattice of rods characteristic of a spatially finite bilayer structure, (right) Model lineshape calculations for a powder-averaged diffraction profile resulting from an A-B stacking sequence for (a) monolayer, (b) bilayer, (c) trilayer thick film, (d) illustrates the diffraction profile for an A-B-C trilayer. A triangular, commensurate, close-packed structure was used in each case. The in-plane lattice spacing was the same in all cases. The interplane distance was held fixed between all layers ... 

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