TP lattice

The square (tp) lattice, (Figure 3.5g), has, as principle symmetry element, a tetrad rotation axis through the lattice point at the unit cell origin, which necessitates a tetrad through each lattice point. This generates additional diads at the centre of each unit cell side, and another tetrad at the cell... 

In building up these patterns, it will become apparent that point groups with a tetrad axis can only be combined with a square (tp) lattice, and those with triad or hexad axes can only be combined with a hexagonal (hp) lattice. 

Figure 2.5 The noncollinear 120° magnetic structure for a TP lattice. The two forms are degenerate chiral pairs. Reprinted with permission from Gaulin, 1994 [3]. Copyright (1994) Springer Science + Business Media...

Figure 2.8 A representation of the RVB spin liquid state on a TP lattice. The ground state is a superposition of states in which each spin is AF coupled to another to form 5 = 0 dimers...

While the TP lattice is realised in many inorganic materials, the transition metal di- and trihalides such as CrX3 or NiXy, where X is a Cl, Br or I, come to mind, for most of these the nn, intraplanar exchange is F and frustration does not play a role. In part this is due to the predominance of intraplanar 90° M-X-M superexchange which is often F. The salts,VBr2 and VCL, do appear to have intraplanar AF... 

Barium titanate is usually produced by the soHd-state reaction of barium carbonate and titanium dioxide. Dielectric and pie2oelectric properties of BaTiO can be affected by stoichiometry, micro stmcture, and additive ions that can enter into soHd solution. In the perovskite lattice, substitutions of Pb ", Sr ", Ca ", and Cd " can be made for part of the barium ions, maintaining the ferroelectric characteristics. Similarly, the TP" ion can partially be replaced with Sn +, Zr +, Ce +, and Th +. The possibihties for forming solution alloys in all these stmctures offer a range of compositions, which present a... 

The sequence g(r-i)w+2i igrN-t-i , is similarly defined between Tp and Tr-n lattices, by successive additions of randomly selected edges linking vertices a dis-... 

As before, the time dependence of R(t) is chosen to satisfy the boundary conditions R(0) = Ri,R Tp) = Rp anddR 0)/dt = dR(Tp)/dt = 0, so that the additional phase vanishes at the initial and the final times. A comparison of Eqs. (3.20) and (3.39) reveals a remarkable difference between the acceleration possible in a lattice system and that in a continuous system, in that there is a lower limit to Tp in a lattice system, whereas there is no such limits for a continuous system. Thus, for a lattice system, there is also a higher limit to dR/dt that depends on cp , because the maximum phase difference allowed between the sites is limited to 2jt. 

The specimen intensity transform X is a type of convolution product of the particle intensity transform Ip and the particle orientation density function ( 1,2). The procedure that we have used to simulate Ip involves firstly the calculation of the intensity transform for an infinite particle, with appropriate allowances for random fluctuations in atomic positions and for matrix scattering. A mapping of Xp is then carried out which includes the effects of finite particle dimensions and of intraparticle lattice disorder, if this is present. A mapping of Is is then obtained from Tp by incorporating the effects of imperfect particle orientation. 

For particular lattices with two molecules per unit cell, with simple structure and nearest-neighbor interactions (linear alternating chains, square 2D lattices, or simple cubic lattices of the NaCl type), the absorption occurs for two values of q q = 0 (the b component) and q = Q (the a component) see Section II.B.l.b. In these simple lattices the environment of the site is symmetric and the complete inversion in P, TP, (2.77) is not necessary, because, in the coupling of P, to P, only the completely symmetric state S> intervenes ... 

One of the initial motivations for pressure studies was to suppress the CDW transitions in TTF-TCNQ and its derivatives and thereby stabilize a metallic, and possibly superconducting, state at low temperatures [2]. Experiments on TTF-TCNQ and TSeF-TCNQ [27] showed an increase in the CDW or Peierls transition temperatures (Tp) with pressure, as shown in Fig. 12 [80], Later work on materials such as HMTTF-TCNQ showed that the transitions could be suppressed by pressure, but a true metallic state was not obtained up to about 30 kbar [81]. Instead, the ground state was very reminiscent of the semimetallic behavior observed for HMTSF-TCNQ, as shown by the resistivity data in Fig. 13. One possible mechanism for the formation of a semimetallic state is that, as proposed by Weger [82], it arises simply from hybridization of donor and acceptor wave functions. However, diffuse x-ray scattering lines [34] and reasonably sharp conductivity anomalies are often observed, so in many cases incommensurate lattice distortions must play a role. In other words, a semimetallic state can also arise when the Q vector of the CDW does not destroy the whole Fermi surface (FS) but leaves small pockets of holes and electrons. Such a situation is particularly likely in two-chain materials, where the direction of Q is determined not just by the FS nesting properties but by the Coulomb interaction between CDWs on the two chains [10]. 

The sudden increase in Tp of TTF-TCNQ near 19 kbar and associated hysteresis in the transition [80] lead to the suggestion that in this pressure region (Fig. 12), the charge transfer is 1, and 2fcF is commensurate with the crystalline lattice. This was later confirmed directly by neutron scattering on deuterated samples for which it occurred at somewhat lower pressures [84]. 

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