Crystal constant neighbor distance

Crystal Nearest-Neighbor Distance (A) Rate Constants (O Effective Tunneling Splitting (cm 1) 4>o... 

In ionic crystals with d = nearest neighbor distance, the ions repulse each other strongly when d becomes smaller than the equilibrium value d0. This can be described by an inverse power function, +l/dn, where n is a power of order, 9. As for the electrostatic attractions, these repulsions must be summed over the N molecules of the crystal structure, yielding another constant, D. The energy, < > per molecule (ion pair) is then ... 

Figure 6 The perfect (111) surface of an fee crystal. AH atoms shown are considered as surface atoms. The surface sites form a triangular lattice of lattice constant o/i/3, where a is the nearest neighbor distance of the fee crystal. Atoni at site A is allowed to evaporate, while site B is ready to host a condensation event. Also shown is the cubic cell of a sublattice which defines the parallel updating algorithm.

In this expression r is some combination of elastic constants ( 1q11 erg/cm ), b the magnitude of the Burger s vector of the dislocation, a the nearest neighbor distance in the crystal and... 

Silicon is a semiconductor with an intrinsic conductivity of 4.3 x 10" Q" cm and a band gap of I.I2eV at 300K. It has a diamond crystal structure characteristic of the elements with four covalently bonded atoms. As shown in Fig. 2.1, the lattice constant, a, is 5.43 A for the diamond lattice of silicon crystal structure. The distance between the nearest two neighbors is V3a/4, that is, 2.35 A, and the radius of the silicon atom is 1.18 A if a hard sphere model is used. Some physical parameters of silicon are listed in Table 2.1. 

It is generally agreed that thermally induced vibrations of atoms in solids play a major role in melting [2.144]. The simple vibrational model of Linde-mann predicts a lattice instability when the root-mean-square amplitude of the thermal vibrations reaches a certain fraction / of the next neighbor distances. However, the Lindemann constant/varies considerably for different substances because lattice anharmonicity and soft modes are not considered, thus limiting the predictive power of such a law. Furthermore, Born proposed the collapse of the crystal lattice to occur when one of the effective elastic shear moduli vanishes [2.138], Experimentally, it is found instead that the shear modulus as a function of dilatation is not reduced to zero at Tm and would vanish at temperatures far above Tm for a wide range of different substances [2.145]... 

The silver halides, with the exception of one crystal modification of Agl, have cubic crystal structures. Crystal structure data, relevant lattice properties and low temperature dielectric constants, are collected in Table 3. AgF, AgCl, and AgBr all have the NaCl rocksalt structure in which there are four silver halide pairs per nonprimitive unit cell with cation-anion nearest neighbor distances equal to one-half a lattice constant. 

Here Me is the Madelung constant, a the nearest-neighbor distance in crystal, qr the charge ascribed to the lattice ions beyond the fragment under consideration, L numbers atoms in the fragment considered. 

TABLE X. Crystal Constants (E = cohesive energy, d = nearest-neighbor distance between lattice points)... 

The carbon atoms in a methane crystal occupy the lattice points of a face-centered cubic structure. At temperatures below 20.4° K the molecules do not freely rotate and seem to be parallel to each other with all the CH bonds directed towards the body diagonals. The distance between two nearest-neighbor carbon atoms will be called "nearest-neighbor distance and denoted by d (so that the lattice constant is V2d),... 

In the solid state, the Ceo molecules crystallize into a cubic structure with a lattice constant of 14.17A, a nearest neighbor Ceo-Ceo distance of 10.02A [41], and a mass density of 1.72 g/cm (corresponding to 1.44 Ceo... 

In Eq. (2.1), Uo represents the energy of the crystal at the absolute zero as a function of the distance r between nearest neighboring atoms, where to represents the value of this distance in equilibrium, when U0 has its minimum value. The quantity L is the energy required to break up the metal into atoms at the absolute zero, or the latent heat of vaporization at the absolute zero, a quantity that can be found from experimental measurements of vapor pressure, as we saw in Chap. XI. The quantity a is an empirical constant. We shall determine L from the experimental value of the vapor pressure, ro from the experimental density at the absolute zero, and a from the compressibility. To compare with our formulation of Chap. XIII, we expand Eq. (2.1) in Taylor s series, finding... 

I lGURK 20-2 The body-centered cubic structure. The central atom sits at the center of a cube formed by k,r. its eight nearest neighbors, shaded to distinguish them, though every atom and its environment (in the extended crystal) is identical. The six second neighbors lie a distance 15 percent further away. We construct a Bloch sum with wave number in the z-direction, giving phase factors shown for atoms in each plane of constant z. 

NaCl crystal. Let a denote the distance between atoms 1 and 2. Then the length of the primitive cell is 2a. We consider waves that propagate in the direction shown by the arrow, and assume that each plane interacts only with its neighboring planes. If the force constants (f) are identical between all these neighboring planes, we have... 

We move on now to consider a diatomic crystal where each unit cell has two different atoms G and H with masses Mq and Mfj. Again, the onedimensional crystal has N atoms of each type with lattice spacing a, and within each cell the atoms are separated by distance b. Furthermore, we take force constants /i and for the binding of the atoms within each cell and between nearest-neighbor atoms outside the cells, respectively, as illustrated in Fig. 3. By analogy to Eq. (2) we expect the form of the wave to be... 

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