Lattice constant distance, bond-length

AIN, GaN and InN crystallise in the wurtzite structure which is characterised by lattice parameters a and c, as well as by u-value (u = b/c, where b is a bond-length in the c-direction). For the ideal wurtzite structure, c/a = 1.633 and u = 0.375. In contrast to the cubic sphalerite structure, the wurtzite structure offers two possibilities to deviate from the ideal arrangement, by changing the c/a ratio and by changing the u value. Such deviations are often observed in wurtzite-type structures [1] but there exists a strong correlation between the c/a ratio and the u parameter if c/a decreases, then u increases in such a way that the four tetrahedral distances remain nearly constant and the tetrahedral angles are distorted [2]. The bond lengths would be equal if ... 

The regular orbit displayed in Figure 2.7, is the geometry on the unit sphere such that the bond length , the Euclidean distance between adjacent vertices, is constant. This restriction is not necessary from a symmetry viewpoint it may be relaxed subject only to the requirement that the local four, three and two-fold symmetries are maintained. One important example of such a relaxation occurs for the regular orbit of the Oh Crystallographic point group. In the simplest model crystal of Oh point symmetry, the primitive cubic array, for example, as in cubium, lattice points are distributed as dictated by the lattice vector Rmnp such that... 

From Fig. Ic one notices that in the NaTl structure both the Na and Tl atoms form diamond-like sublattices (Fig. Id). The diamond structure arises from the NaTl structure by leaving vacancies V in one sublattice (Table 1). Therefore in the B32 structure (Fig. Ic) the bond lengths d(A-A), d(A-B) and d(B-B) are equal and the atoms A and B must have equal sizes to get a spacefilling distribution . It follows that the atomic radii in B32 type compounds rB32 should be half of the distance to the nearest neighbours. In Table 2 the lattice constants for the binary Zintl phases are listed. The resulting atomic... 

Several approaches have been used for determining functional forms for the pair sum Eq. [12]. Once the Hamiltonian matrix elements had been specified, Chadi, for example, used a near-neighbor harmonic interaction for covalent materials where the force constants and minimum energy distances were fit to bulk moduli and lattice constants, respectively. This expression was then used to predict energies and bond lengths for surfaces and related structures. More recently. Ho and coworkers have fit the pair sum to the universal binding energy relation. This reproduces not only lattice constant and bulk modulus, but also ensures reasonable nonlinear interatomic interactions that account for properties like thermal expansion. 

The unit cell of ThCr2Si2-type is displayed in fig. 1. This type of crystal structure consists of tetrahedra composed of X atoms with a transition metal inside. The X-X distances are usually close to the sum of covalent radii of X atoms, similarly as are the T-X contacts. Thus strong chemical interactions are expected within the layers composed of tetrahedra which in turns, consist of 4X atoms. The bond lengths are critically dependent on the magnitude of the z parameter and the da ratio (a, c are the lattice constants). 

Figure 1.2 Schematic representation of a wurtzitic ZnO structure with lattice constants a in the basal plane and c in the basal direction, u parameter, which is expressed as the bond length or the nearest-neighbor distance b divided by c (0.375 in ideal costal), a and p (109.47° in ideal costal) bond angles, and three types of second-nearest-neighbor distances b, b 2, and b 3.

An initial network configuration was set up by randomly choosing sizes for all sites and bonds, while observing the imposed twofold size distribution. Sites were placed at the nodes of the cubic lattice and bonds in between the nodes. Node to node distance was constant (i.e. equal to the diameter of the largest site) the length of each bond was adjusted to a value enough to connect its two neighbouring sites. This particular selection of bond-site distance defined the porosity of the network. 

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