Equilibrium unit cell

The decomposition of magnesium hydroxide has been evaluated as a process of nucleation and growth of MgO crystals within the brucite matrix. The formation of a defect layer of hydroxide structure, which suddenly recrystallizes to the cubic MgO structure when the fracture stress is exceeded in the defect layer, has been postulated (Freund et al., 1975 Guilliat et al., 1970 Garn et al., 1978 Lpnvik, 1978). The MgO crystals formed have an expanded cubic lattice that upon increasing calcination temperature gradually decreases in size until the equilibrium unit cell dimension is reached. 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

Johansson and coworkers [182-184] have analyzed polyacrylamide gel structure via several different approaches. They developed an analytical model of the gel structure using a single cylindrical unit cell coupled with a distribution of unit cells. They considered the distribution of unit cells to be of several types, including (1) Ogston distribution, (2) Gaussian distribution of chains, and (3) a fractal network of pores [182-184]. They [183] used the equilibrium partition coefficient... 

Interesting is a comparison of the volumes occupied by individual complexes in solution and in the solid state. The partial molal volumes can be obtained from precise measurements of the solution densities of the complexes as a function of concentration [177]. These values may be subsequently compared with the unit cell volumes per complex molecule derived from the crystal structure. For Fe[HB(pz)3]2, the apparent molal volume in tetrahydrofuran solution was determined as 340.9 em mol Taking into account that the complex in solution forms an equilibrium between 86% LS and 14% HS isomers and employing the volume difference between the two spin states AF° = 23.6 cm mol S the volume of the LS isomer was calculated as 337.6 cm mol This value agrees closely with the volume of 337.3 cm mol for the completely LS complex in solid Fe[HB(pz)3]2 [105]. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

Primary crystallization occurs when chain segments from a molten polymer that is below its equilibrium melting temperature deposit themselves on the growing face of a crystallite or a nucleus. Primary crystal growth takes place in the "a and b directions, relative to the unit cell, as shown schematically in Fig. 7.8. Inevitably, either the a or b direction of growth is thermodynamically favored and lamellae tend to grow faster in one direction than the other. The crystallite thickness, i.e., the c dimension of the crystallite, remains constant for a given crystallization temperature. Crystallite thickness is proportional to the crystallization temperature. 

The structure of a vapor-quenched alloy may be either crystalline, in which the periodicity of the unit cell is repeated within the crystallites, or amorphous, in which there is no translational periodicity even over a distance of several lattice spacings. Mader (64) has given the following criteria for the formation of an amorphous structure the equilibrium diagram must show limited terminal solubilities of the two components, and a size difference of greater than 10% should exist between the component atoms. A ball model simulation experiment has been used to illustrate the effects of size difference and rate of deposition on the structure of quench-cooled alloy films (68). Concentrated alloys of Cu-Ag (35-65%... 

The remark just made suggests that a natural place to begin our discussion of equilibrium equations is with the occupation of different charge states. Let a hydrogen in charge state i(i = +, 0, or - ) have possible minimum-energy positions in each unit cell, of volume O0, of the silicon lattice. (O0 contains two Si atoms, so our equations below will be applicable also to zincblende-type semiconductors.) To account for spin degener-ancies, vibrational excitations, etc., let us define the partition function... 

Sigma (a) bonds Sigma bonds have the orbital overlap on a line drawn between the two nuclei, simple cubic unit cell The simple cubic unit cell has particles located at the corners of a simple cube, single displacement (replacement) reactions Single displacement reactions are reactions in which atoms of an element replace the atoms of another element in a compound, solid A solid is a state of matter that has both a definite shape and a definite volume, solubility product constant (/ p) The solubility product constant is the equilibrium constant associated with sparingly soluble salts and is the product of the ionic concentrations, each one raised to the power of the coefficient in the balanced chemical equation, solute The solute is the component of the solution that is there in smallest amount, solution A solution is defined as a homogeneous mixture composed of solvent and one or more solutes. 

Ion-exchange equilibrium In most experiments described here, monovalent cationic dyes have been used. D/ and M/ denote the dye cation and the alkali metal cation in solution. Z stands for zeolite and Y describes the cation concentration inside the zeolite. For monovalent cations and dyes which occupy two unit cells in zeolite L (e.g., Py" or Ox+), we must use (Mr ),] to describe the state of a given... 

Lattice Points positions in a unit cell occupied by atom, molecules, or ions Law of Definite Proportion law that states that different samples of the same compound always contain elemental mass percentages that are constant Law of Mass Action mathematical expression based on the ratio between products and reactants at equilibrium, an equation to determine the equilibrium rate constant Law of Multiple Proportions law that states when two elements combine to form more than one compound that the mass of one element compared to the fixed mass of... 

Figure 7.12 Dependence of the hydrogen equilibrium pressure on the unit-cell volume of various LnNij-type compounds (Ln = rare earth). Open circles, LnCoj closed circles, LnNij open triangles, LaCo, (Following Buschow van...

---