Bravais parameters

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

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). 

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

The dye molecules are positioned at sites along the linear channels. The length of a site is equal to a number ns times the length of c, so that one dye molecule fits into one site. Thus ns is the number of unit cells that form a site we name the ns-site. The parameter ns depends on the size of the dye molecules and on the length of the primitive unit cell. As an example, a dye with a length of 1.5 nm in zeolite L requires two primitive unit cells, therefore ns = 2 and the sites are called 2-site. The sites form a new (pseudo) Bravais lattice with the primitive vectors a, b, and ns c in favorable cases. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The 14 Bravais lattices are enumerated in Table 9-4 as the following types primitive (P, R), side-centered (C), face-centered (F), and body-centered (7). The numbering of the Bravais lattices in Table 9-4 corresponds to that in Figure 9-20. The lattice parameters are also enumerated in the table. In addition, the distribution of lattice types among the crystal systems is shown. 

Nitride Homogeneity range 1 — x at 1700 K Structure (Bravais lattice) Lattice parameter (room temp.) (nm) Density (gem ) Color Microhardness (room temp.) (GPa) Melting point (K) Heat conduct. (room temp.) (Wm- K- ) Thermal exp. coefficient at 1000K(10 K- ) comp. (1 — x) Electr. resistance (room temp.) ( xl2 cm) Supercond. transition temp. Tc (K) comp. (1 — x)... 

Salt Bravais lattice Bravais la paramete a ttic sOO c Mols.y unit cell Mol. Vol. (A.3)... 

Regardless of which indexing method was employed, the resulting unit cell (especially when it is triclinic) shall be reduced using either Delaunay-Ito or Niggli method in order to enable the comparison of different solutions and to facilitate database and literature searches. Furthermore, the relationships between reduced unit cell parameters must be used to properly determine the Bravais lattice. The Niggli-reduced cell is considered standard and therefore, is preferable. 

A pattern is made of a cubic substance with unhltered chromium radiation. The observed sin 0 values and intensities are 0.265(m), 0.321(vs), 0.528(w), 0.638(s), 0.793(s), and 0.958(vs). Index these lines and state which are due to Ka and which to Kfi radiation. Determine the Bravais lattice and lattice parameter. Identify the substance by reference to Appendix 5. 

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

This approach consists of correlating the position of the peaks with each other. In this method, we consider a Bravais lattice and set approximate values for the cell parameters. The changes in peak positions are made by modifying these cell parameters. It is easy to understand, in this case, that the position of each peak strongly depends on the positions of the other peaks. 

An orthorhombic body centred Bravais lattice has lattice parameters ... 

Conventionally, the superconductors that we treat are labelled with four digits in a set that correspond, in order, to the stoichiometric coefficients of Tl, Ba, Cu and Ca. Figure 17-5 shows the known structures for m=. In these structures the Bravais lattice is primitive P, and the number of sheets that occur in one unit cell corresponds to the sum of four stoichiometric coefficients 4, 6, 8, 10 and 12 for n=l, 2, 3, 5 and 5, in order. The structures are tetragonal P4/mmm and the value of cell parameter a, 3.85A, corresponds to twice the length of the Cu—O bond. From member n to n+l, cell parameter c increases by about... 

There are several known approaches to classification of individual crystals in accordance with their symmetry and crystallochemislry. The particles which form a crystal are distributed in certain points in space. These points are separated by certain distances (translations) equal to each other in any chosen direction in the crystal. Crystal lattice is a diagram that describes the location of particles (individual or groups) in a crystal. The lattice parameters are three non-coplanar translations that form the crystal lattice. Three basic translations form the unit cell of a crystal. August Bravais (184S) has shown that all possible crystal lattice structures belong to one or another of fourteen lattice types (Bravais lattices). The Bravais lattices, both primitive and non-primitive, are the contents of Table 3. 

The relative or absolute value of these different parameters allows us to define seven base lattices or Bravais lattices, which are ... 

Figure 3. Pressure dependence of structural parameters for the Pbnm phase, a) and b) show the internal ones, while c), d) and e) show the rescaled Bravais lattice parameters a/a , b/b , and c/co, where ao, bo, Co are zero pressure values. The insets show experimental results of Ross and Hazen [28] (o) and ours ( ).

C2/c pyroxenes contain 40 atoms per cell and are described by 18 free parameters. The Bravais lattice is monoclinic with a/I angle around... 

The lattice parameters a, c and values of coordinates u, v for some crystals are given in fable 2. More details may be found in papers by Aldred (1984), Mdll and Schafer (1971), Lohmuller et al. (1973), Feuss and Kallel (1972), and Radhakrishna et al. (1981). The Bravais lattice in this case is also a body-centered tetragonal, the first two coordination spheres of a rare-earfli ion consist of two fours of oxygen ions. Rare-earth ions are distributed with almost the same density as in tetrafluorides. 

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