Crystal polarizability

Usually the dependence of the nonlinear polarizability tensors on the wavevec-tors k, k jk", etc. is neglected. In this approximation in crystals with an inversion center the third-rank tensor (u>, uiu>"), which changes sign by inversion, vanishes and nonlinear processes involving three photons are absent. Explicit expressions for the nonlinear polarizability tensors, similar to the tensor e u , k), can be obtained within a microscopic theory. Above we presented various methods for calculation of the tensor k). The important point was that to obtain this tensor it was necessary to establish the crystal polarizability in the linear approximation with respect to the electric field. The nonlinear polarizability tensors can be obtained in a similar way, but now the crystal polarizability must be determined by taking into account higher order terms with respect to the electric field (see (16)). 

Expressions for the force constant, i.r. absorption frequency, Debye temperature, cohesive energy, and atomization energy of alkali-metal halide crystals have been obtained. Gaussian and modified Gaussian interatomic functions were used as a basis the potential parameters were evaluated, using molecular force constants and interatomic distances. A linear dependence between spectroscopically determined values of crystal ionicity and crystal parameters (e.g. interatomic distances, atomic vibrations) has been observed. Such a correlation permits quantitative prediction of coefficients of thermal expansion and amplitude of thermal vibrations of the atoms. The temperature dependence (295—773 K) of the atomic vibrations for NaF, NaCl, KCl, and KBr has been determined, and molecular dynamics calculations have been performed on Lil and NaCl. Empirical values for free ion polarizabilities of alkali-metal, alkaline-earth-metal, and halide ions have been obtained from static crystal polarizabilities the results for the cations are in agreement with recent experimental and theoretical work. 

Mahan GD (1980) Polarizability of ions in crystals. Solid State Ionics 1 29-45 Fowler PW, Pyper NC (1985) In-crystal ionic polarizabilities derived by combining experimental and ab initio results. Proc Roy Soc London A 398 377-393 Fowler PW, Madden PA (1985) In-crystal polarizability of CP . J Phys Chem 89 2581-2585 Pyper NC, Pike CG, Edwards PP (1992) The polarizabilities of species present in ionic-solutions. Mol Phys 76 353-372... 

A related advantage of studying crystalline matter is that one can have synnnetry-related operations that greatly expedite the discussion of a chemical bond. For example, in an elemental crystal of diamond, all the chemical bonds are equivalent. There are no tenninating bonds and the characterization of one bond is sufficient to understand die entire system. If one were to know the binding energy or polarizability associated with one bond, then properties of the diamond crystal associated with all the bonds could be extracted. In contrast, molecular systems often contain different bonds and always have atoms at the boundary between the molecule and the vacuum. 

The induction energy is inlierently non-additive. In fact, the non-additivity is displayed elegantly in a distributed polarizability approach [28]. Non-additive induction energies have been found to stabilize what appear to be highly improbable crystal structures of the alkalme earth halides [57]. 

The fluoride ion is the least polarizable anion. It is small, having a diameter of 0.136 nm, 0.045 nm smaller than the chloride ion. The isoelectronic E and ions are the only anions of comparable size to many cations. These anions are about the same size as K" and Ba " and smaller than Rb" and Cs". The small size of E allows for high coordination numbers and leads to different crystal forms and solubiUties, and higher bond energies than are evidenced by the other haUdes. Bonds between fluorine and other elements are strong whereas the fluorine—fluorine bond is much weaker, 158.8 kj/mol (37.95 kcal/mol), than the chlorine—chlorine bond which is 242.58 kJ/mol (57.98 kcal/mol). This bond weakness relative to the second-row elements is also seen ia 0-0 and N—N single bonds and results from electronic repulsion. 

In order for dipole—dipole and dipole-iaduced dipole iateractioas to be effective, the molecule must coataia polar groups and/or be highly polarizable. Ease of electronic distortion is favored by the presence of aromatic groups and double or triple bonds. These groups frequently are found ia the molecular stmcture of Hquid crystal compouads. The most common nematogenic and smectogenic molecules are of the type shown ia Table 2. 

Cu crystallizes in the fee and its melting point is 1356 K. The experimental data for single-crystal Cu/H20 interfaces are also controversial. 567 570,572 57X The first studies with Cu(l 11), Cu(100), and Cu(l 10) in surface-inactive electrolyte solutions (NaF, Na2S04) show a capacitance minimum at E less negative than the positive limit of ideal polarizability of Cu electrodes (Table 11). depends on the method of surface... 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

By making the approximation of setting matrix B to zero and A to Ao, the resolvent matrix becomes diagonal, every coupling between the crystal orbitals disappears and the polarizability reads ... 

The electrochemisty of ITIES is developing mainly on the basis of the studies of the water-nitrobenzene and water-1,2-dichloroethane interfaces. The polarizability ranges of these interfaces in the presence of typical electrolytes (Scheme 13) are about 0.30 V. Extension of these ranges has been achieved using other organic ions or/and solvents [2,8]. For example, TBA ions may be substituted by tetraphenylarsonium crystal violet cations and TPhB ions by dicarbollyl cobaltate (III) anions [1,2]. 

Both theoretical and experimental data (in the solid, liquid, and gas phases) prove that the tendency of halocarbons to work as XB donors decreases in the order I > Br > Cl [66-68]. Clearly, polarizability and not electronegativity plays a key role. 3-Halo-cyanoacetylene works as self-complementary module and the N X distance is beautifully consistent with the scale reported above, being 2.932, 2.978 and 2.984 A in the iodo, bromo and chloro derivatives, respectively [69,70]. The same trend is observed when a phenyl, rather than a triple bond, spaces the donor and acceptor sites. The N Br distance in 4-bromobenzonitrile is longer than in the 4-iodo derivative [71,72] and no XB is present in the chloro and fluoro analogues, wherein molecules are pinned by N H and X- H short contacts [73]. PFCs have a very poor tendency, if any, to work as XB donors [74-77] and no crystal engineering can be based on such tectons. However, F2 is a quite strong XB donor and several adducts have been described in the gas phase [11,18] (see also the chapter by Legon in this volume). 

Cao J, Berne BJ (1993) Theory of polarizable liquid crystals optical birefringence. J Chem Phys 99(3) 2213-2220... 

Figure 3.9 C44 elastic moduli vs. reciprocal polarizabilities for prototype alkali halide crystals.

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