Rectangular lattice parameter

In the general case of arbitrary two-dimensional Bravais lattices (not rectangular and rhombic), the ground state, depending on the lattice parameters (x0 and y0 in Fig. 2.13), is characterized by ferroelectric (0.25 < x0 <0.5) or stratified bisublattice antiferroelectric ordering (0 < x0 < 0.25). 

The probability Y(p) of connecting set formation was calculated as a ratio of connecting set number to the number of all possible configurations. The Probability function Y(p) for a three-dimensional rectangular lattice d = 3, 1 = 2 was calculated by a method similar to that of Ref. 62 and Y(p) of Eq. (243). Each Mi bond from the fractal random set Yl/ifp) possesses Hall properties (a, fik) namely ohmic conductivity and a Hall parameter. 

A rectangular primitive plane lattice has lattice parameters ... 

Fig. 14.1. The crystal structure of YBa2Cu307 x. The lattice parameters are a = 0.383 nm, b — 0.388 nm and c — 1.17 nm. The lattice parameters depend on the oxygen content [e.g. 14.11]. In the following figures, the structure is represented by a rectangular box shown to the right.

In contrast to the hexagonal-shaped P j -phase, another new phase, called the -phase, is also found in the products of diffusion synthesis [7]. The X i-phase is a rectangular plate-shaped crystal. The determination of the detailed structure is still in progress. Preliminary results show that the lattice parameters of the X -phase are similar to that of the 0-phase [3,4] with Z = 4. However, the new phase is characterized by its peculiar transport behavior (see below). Table 1 shows the structure parameters of the P - and phases, in comparison with the other isomers. It can be seen that the reduced unit cell volume in the X j-phase is a little larger than that of the 0-phase, indicating either the existence of excess iodine atoms, or the misarrangement of the iodine chain. In the following, we will present experimental results for the two new phases. 

In a crystal, the spacing of the lattice points is an important parameter to define its stmcture. In crystal stracture theory one normally uses the so-called Miller indices hkt) to characterize the lattice plane. These indices are the reciprocal of intersection distances (any resulting fractions are removed by multiplying with an appropriate factor). For example, in a 2D rectangular lattice, such as the one shown in Figure 26.1, the planes passing through the lattice points are represented by the smallest intersection... 

If the fibres are uniaxial, there is still some symmetry in the material, and the number of parameters needed to describe the elastic behaviour is smaller than 21, the value for a triclinic lattice. If the fibres are directed, but their positions in space are irregular or arranged on a hexagonal lattice, the material is transversally isotropic i. e., its properties are the same in all directions perpendicular to the fibre direction. In this case, there are five independent elastic constants (see section 2.4.6). If the fibres are uniaxial and arranged on a rectangular lattice, the material is orthotropic, and the number of independent elastic components is nine (see section 2.4.5). 

Consider, for example, crystals with face-centered cubic Bravais lattices. For the (001), (110) and (111) sections the plane lattices are square, rectangular and hexagonal, respectively. The basic translation vectors of the direct and reciprocal lattices for these three cases are given in Table 11.2 (ai and 02 are given in units a/2, Bi in units 27r/a, where a is the cubic lattice parameter). Note that for a cubic lattice the planes (100), (010) and (001) are equivalent. The equivalence takes place also for (110), (101), and (oil) planes. We see that the vectors Bi(i = 1,2) are now not the translation vectors of the three-dimensional reciprocal lattice. Therefore, the boundaries of BZ-2 do not coincide with those of BZ-3. 

---