Crystal lattice energy, description

The estimate via the simple relationship Eq. (5.207) is in approximate agreement with the experimental data. Thus, for silver halides ApH° UMad/ (cf. Eq. 5.8). Computer experiments for fluids yield crystal lattice energy, the dielectric permittivity of the crystal enters the equation for the quasi lattice energy of the defect lattice. In contrast to the intrinsic concentration at low temperatures (see Eq. (5.85)) the implicit relationship of x for say AgCl is... 

In molecular crystals, the relative importance of the electrostatic, repulsive, and van de Waals interactions is strongly dependent on the nature of the molecule. Nevertheless, in many studies the lattice energy of molecular crystals is simply evaluated with the exp-6 model of Eq. (9.45), which in principle accounts for the van der Waals and repulsive interaction only. As underlined by Desiraju (1989), this formalism may give an approximate description, but it ignores many structure-defining interactions which are electrostatic in nature. The electrostatic interactions have a much more complex angular dependence than the pairwise atom-atom potential functions, and are thus important in defining the structure that actually occurs. 

The electronic structure of a solid metal or semiconductor is described by the band theory that considers the possible energy states of delocalized electrons in the crystal lattice. An apparent difficulty for the application of band theory to solid state catalysis is that the theory describes the situation in an infinitely extended lattice whereas the catalytic process is located on an external crystal surface where the lattice ends. In attempting to develop a correlation between catalytic surface processes and the bulk electronic properties of catalysts as described by the band theory, the approach taken in the following pages will be to assume a correlation between bulk and surface electronic properties. For example, it is assumed that lack of electrons in the bulk results in empty orbitals in the surface conversely, excess electrons in the bulk should result in occupied orbitals in the surface (7). This principle gains strong support from the consistency of the description thus achieved. In the following, the principle will be applied to supported catalysts. 

For application to nonmolecular solids, the bond description is similar but certain modifications are needed. First, the covalent energy must be multiplied by the equivalent number n of two electron covalent bonds per formula unit that must be broken for atomization. The evaluation of n will be discussed in detail presently. Second, the ionic energy must be evaluated as the potential energy over the entire crystal, corrected for the repulsions among adjacent electronic spheres. This is done by using the Born-Mayer equation for lattice energy, multiplying this expression by an empirical constant, a, which is 1 for the halides and less than 1 for the chal-cides, as follows ... 

Section (2) develops a theoretical account of plastic deformation and energy dissipation at the atomic or molecular level. The AFM observations show that plastic deformation of shocked or impacted crystals can significantly deform both the crystal lattice and its molecular components. These molecular and sub-molecular scale processes require a quantum mechanical description and necessarily involve the lattice and molecular potentials of the deforming crystals. A deformed lattice potential is developed which when combined with a quantum mechanical account of plastic flow in crystalline solids will be shown to give reasonably complete and accurate descriptions of the plastic flow and initiation properties of damaged and deformed explosive crystals. The deformed lattice potential allows, for the first time, the damaged state of the crystal lattice to be taken into account when determining crystal response to shock or impact. 

A description of the present state of this theory will be presented in the following discussion [cf. Balandin 8-11)]. The multiplet theory deals with numerical values of bond lengths and bond energies, as well as with the geometrical form of reacting molecules and the crystal lattices of catalysts. This allows fairly definite results to be obtained for many reactions on an atomic level. It is this point that singles out the multiplet theory from a number of other theories on catalysis. 

Chapter 4 deals with the crystal lattice. Here, we discuss a basic concept of reciprocal lattice in detail and present the Wigner-Seitz cell and the Brillouin zone. These notions are commonly used in any description of the energy of electrons in solids. 

The exponent n is an important quantity in the empirical description of the interatomic interactions. This quantity uniquely determines the dependence of energy on distance, E r) Series of the type of (15.18) have been calculated and tabulated (see [7]). For n < 3 these series diverge. As n ooA approaches the number of nearest neighbors, which is 12 for the face-centered crystal lattice. 

More complicated is the description of the melting of copolymers with Ml or partial isomorphism or isodimorphism (Sect. 5.1.10). The changes in concentration in the mixed crystals as well as the melt must be considered and the change of the lattice energy with cocrystalhzation must be known. The simplest description assumes that there is no difference in concentration between melt and crystal, and there is a linear change of the heat of fusion, AH, with concentration. This leads to T = [ 1 - XB(AHj/AHu)]Tm°, where AH is the heat of fusion per repeating unit of the homopolymer. 

That this is the correct description for neutron slowing down in zirconium hydride has been experimentally verified [5 21], and the quantum hv has been measured to be 0.13 ev. Due to the finite mass of the zirconium atoms, it is not quite correct to treat the motion of the hydrogen atom as if it were in a fixed potential well. It is necessary to include the motion of the zirconium atoms in the ordinary vibrations of the crystal lattice. This introduces a Doppler broadening of the energy-transfer cross section so that energy transfers with a certain spread in the neighborhood of multiples of hv are allowed. To treat this broadening in detail, it is necessary to treat... 

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