Lattice planes description

It may be recalled that an alternative description for a crystal stmcture can be made in terms of sets of lattice planes, which intersect the unit cell axes at ua, VU2, and was-The reciprocals of the coefficients are transformed to the smallest three integers having the same ratios, h, k, and I, which are used to denote the plane (hkl). Of course, the lattice planes may or may not coincide with the layers of atoms. Any such set of planes is completely specihed by the interplanar spacing, dhU7 and the unit vector normal to the set, since the former is given by the projection of, for example, u ui onto n kh that is dhki = u ui- n ki- The reciprocal lattice vector is defined as ... 

Figure 1.7 Geometrical description of a lattice plane in terms of real space basis vectors.

A description of the minerals occurring most commonly in soils will be given in this section. Before this can be done, however, some basic concepts used to describe ionic sohds need to be presented. The reader with no background in mineralogy, and unfamiliar with terms such as unit cells, lattice points, and lattice planes, should read the appendix of this chapter first. 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

Diffraction by a crystal can be described as a superposition of the spherical wave contributions from all atoms. As already discussed, the description can be greatly simplified if we compare the crystal to a three-dimensional diffraction grating. The process can then be viewed as reflection at the lattice planes, followed by interference. This is depicted in Figure 10.27, which shows two planes separated by a distance d. The incident beam is reflected, partly from the top row and partly from the second row. For constructive interference to occur, the path difference 8 must be a multiple of the wavelength 2, as expressed by Bragg s Law ... 

Epitaxial crystal growth methods such as molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) have advanced to the point that active regions of essentially arbitrary thicknesses can be prepared (see Thin films, film deposition techniques). Most semiconductors used for lasers are cubic crystals where the lattice constant, the dimension of the cube, is equal to two atomic plane distances. When the thickness of this layer is reduced to dimensions on the order of 0.01 )J.m, between 20 and 30 atomic plane distances, quantum mechanics is needed for an accurate description of the confined carrier energies (11). Such layers are called quantum wells and the lasers containing such layers in their active regions are known as quantum well lasers (12). 

Of conrse, many reflections are possible from a regnlar lattice, and if the set of vectors snch as h is represented then we can graphically visualise all the reflecting planes and consequent reflected directions in the crystal. We have dednced virtually all the rules already, but we formalise the description below. 

The input data for the model consist of the description of the lattice deformation and the choice of the slip system in the lattice-invariant shear. The model has successfully predicted the observed geometrical features of many martensitic transformations. The observed and calculated habit planes generally have high indices that result from the condition that they be macroscopically invariant. 

The aim of this work is to demonstrate that the above-mentioned unusual properties of cuprates can be interpreted in the framework of the t-J model of a Cu-O plane which is a common structure element of these crystals. The model was shown to describe correctly the low-energy part of the spectrum of the realistic extended Hubbard model [4], To take proper account of strong electron correlations inherent in moderately doped cuprate perovskites the description in terms of Hubbard operators and Mori s projection operator technique [5] are used. The self-energy equations for hole and spin Green s functions obtained in this approach are self-consistently solved for the ranges of hole concentrations 0 < x < 0.16 and temperatures 2 K< T <1200 K. Lattices with 20x20 sites and larger are used. 

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