Commensurate sublattices

When adsorbed on the atomically highly corrugated, anisotropic Cu(llO) surface N2 orders commensurably in a complex seven-sublattice pinwheel structure... 

Consider, for example, the case of the quasi-one-dimensional organic metal TTT2I3+5. This material often exhibits a set of disordered iodine sublattices either commensurate or incommensurate with the main lattice. Lowe-Ma et al. [85] have analyzed the intensity of the corresponding set of diffuse reciprocal layers and postulated that, in their samples, part of the It ions are substituted by I2 and I- moieties. In fact, the intensity of the zero layer is not negligible and no three-dimensional ordering of iodine chains is observed at low temperature. However, TTT2I3+8 crystals are often characterized by a varying amount of positional disorder [136] of iodine columns rather than by a chemical disorder. 

It is important to remark here that the periodicity of the cation sublattice in the chain direction z just coincides with 2kF periodicity in the case of TEA(TCNQ)2, and with 4kF periodicity in the case of MEM(TCNQ)2. This fact would suffice by itself to account, in terms of electron-cation interaction and commensurability effect, for the intrinsic chain tetramerization of TEA(TCNQ)2 and for the intrinsic chain dimerization of MEM(TCNQ)2, in particular for the residual dimerization still above 335 K. However, the cation subsystem certainly has a more direct, steric influence than through only the electron-ion interaction discussed above on the overall structural properties of the organic salts. 

The systems exhibiting the 1x2 structure are then also expected to show commensurate and incommensurate phases, quite similar to those observed for the ANNNI model [75]. In the lattice gas systems the presence of incommensurate phases is restricted to the situations in which the substrate lattice can be divided into a certain number of equivalent interpenetrating sublattices and the ordered state corresponds to the preferential occupation of one of those sublattices. Incommensurability is manifested by the presence of regions with different occupied sublattices and the formation of walls between the domains of commensurate phase. In the case of the discussed here systems exhibiting 1x2 ordered phase we have two sublattices, since particles occupy alternate rows. Figure 6 shows examples of equilibrium configurations demonstrating the formation of incommensurate structure when the ordered 1x2 phase is heated up. 

Fig.7.3. Calculated pressure P, bulk modulus B, and sublattice magnetisation m, as functions of atomic radius for commensurate anti ferromagnetic chromium [7.19]. The filled circles show the measured equation of state, and the filled squares indicate the measured magnetisation, upper panel, and the measured bulk modulus, lower panel... 

Figure 5. Schematic picture of (a) the ideal (2 x 1) two-sublattice in-plane herringbone structure and (b) the ideal four-sublattice pinwheel structure within the (->/3 x /3)/ 30° lattice which is commensurate on the honeycomb lattice of the (0001) basal plane of graphite. The principal axes of the ellipses correspond to the 95% electronic charge density contour given in Table I.

Figure 29. Distribution function of the cosine of the relative angle between two N2 molecules i and j in the (->/3 X J3)R30° commensurate phase on graphite obtained from Monte Carlo simulations for either first, second, and third nearest-neighbor pairs (y). Solid line is for pairs ( ) ) which belong to the same sublattice, and dashed line is for the intersublattice pairs. The herringbone transition is located in this model at around 25 K. (Adapted from Fig. 4 of Ref. 183.)...

Uniaxially compressed phases and the commensurate reference stmcture were furthermore examined [342] by molecular dynamics simulations along the lines of the work reported in Refs. 232 and 340 (see Section III.D.l), except for small alterations in the potential models. The 96 molecules were put into a rectangular cell which was uniaxially compressed by 5 % perpendicular to a glide line of the herringbone sublattice stmcture that is, the center-of-mass lattice is contracted toward the glide line this compression allows the same periodic boundary conditions to be effective for both adsorbate and graphite lattices. It should be noted, however, that even this does not ensure a simulation of the tme equilibrium situation because every solid accommodates even in equilibrium a certain number of vacancies and interstitials. In simulations with a constant number of particles the net number of such defects is acmally constrained to some constant value, which is not necessarily the correct equilibrium value [338, 339]. Two temperatures well below and above the orientational disordering transition at 15 K and... 

It is found [138] that the increase of the corrugation due to the inclusion of axially symmetric (experimentally determined bulk) quadrupole moments located at the carbon sites [361] which model the aspheiical charge distribution in the graphite substrate [see (3.9) and (3.10) in Section III.D.l] stabilizes the commensurate herringbone structure. This structure is head-tail-ordered as in Ref. 17 (see Fig. 53a or Fig. 54Z>, where the molecular axes have a systematic out-of-plane tilt) the unit cell is deformed because of the displacement of the molecular centers on the two sublattices. The Brillouin-zone-center frequency gap in the phonon spectrum is estimated [138] to amount to about 10 K in the ground state,... 

Mobility of the ions implies that their association with the PAc chains is weak (or at least not strongly directed). As the weak or undirected bonding limit of the two sublattices is approached, there is no reason to expect the periodicity of the ions along the channel to be compatible with that of the polymer. The resulting sublattices would be incommensurate, and as found in the simulations, they might exhibit a composition and dopant ion periodicities that need not be described by PAKl, PAK2, or any other commensurate structure. 

---