The Lattice

Within the lattice so alled transition probabilities X(z, rjz, f) are defined, which are normalized such that 1 = 532 = 2-1, 2, z + tY i =r-t. r, r+tX(z, rjz, r ). These transition probabilities obey the internal balance equation L(z, r) k(z, rjz, r ) = L(z, r ) A,(z, f z, r). For very large values of r the curvature of the lattice rings vanishes and the transition probabilities converge to constant values X. 

These transition probabilities are used in the site fraction of any property X that depends on the coordinates (z, r) as  

The site averages, used systematically in the computation of the interactions as well as the chain statistics, account for the local curvature in the coordinate system. 

To convert volume fractions into molar concentrations, a choice is needed for the size of a lattice site h. Here we have used b = 0.5 nm, which is a compromise between the size of water molecules and surfactant segments. For a given molecule this conversion depends on the molar volume (v), which we will express in units b, and using the quantity N as the number of segments in a molecule (Table 5.1). For example, to convert the volume fraction for the ions one needs to multiply by roughly 10. For the surfactant the conversion factor is dose to unity. 

With this coordinate system in place, we can now focus on the computation of the volume fractions and the segment potentials. 

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The crystal structure determines not only the arrangement of atoms in the lattice but also the external form of the crystal. 

Kapustinskii equation For an ionic crystal composed of cations and anions, of respective charge and z, which behave as hard spheres, the lattice energy (U) may be obtained from the expression... 

Surface Stresses and Edge Energies. Some surface tension values, that is, values of the surface stress t, are included in Table VII-2. These are obtained by applying Eq. Vll-5 to the appropriate lattice sums. The calculation is very sensitive to the form of the lattice potential. Earlier calculations have given widely different results, including negative r [43, 51, 52]. 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

The density of dislocations is usually stated in terms of the number of dislocation lines intersecting unit area in the crystal it ranges from 10 cm for good crystals to 10 cm" in cold-worked metals. Thus, dislocations are separated by 10 -10 A, or every crystal grain larger than about 100 A will have dislocations on its surface one surface atom in a thousand is apt to be near a dislocation. By elastic theory, the increased potential energy of the lattice near... 

The oriented overgrowth of a crystalline phase on the surface of a substrate that is also crystalline is called epitaxial growth [104]. Usually it is required that the lattices of the two crystalline phases match, and it can be a rather complicated process [105]. Some new applications enlist amorphous substrates or grow new phases on a surface with a rather poor lattice match. 

Knowing the lattice is usually not sufficient to reconstruct the crystal structure. A knowledge of the vectors (a, b, c) does not specify the positions of the atoms within the unit cell. The positions of the atoms withm the unit cell is given by a set of vectors i., = 1, 2, 3... u where n is the number of atoms in the unit cell. The set of vectors, x., is called the basis. For simple elemental structures, the unit cell may contain only one atom. The lattice sites in this case can be chosen to correspond to the atomic sites, and no basis exists. 

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

Because (k) = (k + G), a knowledge of (k) within a given volume called the Brillouin zone is sufficient to detennine (k) for all k. In one dimension, G = Imld where d is the lattice spacing between atoms. In this case, E k) is known once k is detennined for -%ld < k < %ld. (For example, m the Kronig-Peimey model (fignre Al.3.6). d = a + b and/rwas defined only to within a vector 2nl a + b).) In tlnee dimensions, this subspace can result in complex polyhedrons for the Brillouin zone. 

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

The linear dependence of C witii temperahire agrees well with experiment, but the pre-factor can differ by a factor of two or more from the free electron value. The origin of the difference is thought to arise from several factors the electrons are not tndy free, they interact with each other and with the crystal lattice, and the dynamical behaviour the electrons interacting witii the lattice results in an effective mass which differs from the free electron mass. For example, as the electron moves tlirough tiie lattice, the lattice can distort and exert a dragging force. 

An alternative fomuilation of the nearest-neighbour Ising model is to consider the number of up f T land down [i] spins, the numbers of nearest-neighbour pairs of spins IT 11- U fl- IT Hand their distribution over the lattice sites. Not all of the spin densities are independent since... 

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

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