Lattice frequencies

However, the obscure choice of frequencies in the visible and UV regions in the original calculations may have been guided by a desire to fit experimental heats. In fact, the Debye rotational and translational crystal frequencies relate to sublimation energies of the lattice, and, together with internal molecular vibrations, can be used to calculate thermodynamic functions (16). An indirect connection between maximum lattice frequencies (vm) and heats of formation may hold because the former is inversely related to interatomic dimensions (see Section IV,D,1) ... 

Safford and Naumann (128) have shown that the time-of-flight spectra for 4.6M solutions of KF, KC1, CsCl, NaCl, and LiCl show peaks in the inelastic scattering region which coincide both in frequency and shape with the ice-like (structured) frequencies of pure water. Also, solutions of KSCN, KI, KBr, and NaC104 have lattice frequencies where they are found for water although in these cases apparently with less resolution and less intensity. Even an 18.5-M solution of KSCN showed a similar behavior. We take this to suggest that elements of water structure remain in these solutions (as discussed elsewhere in this paper, where we noted that the thermal anomalies occur at approximately the same temperatures, even for relatively concentrated solutions, as where they occur in pure water see also Ref. 103). 

The above discussion of Equations 2 and 3 has been predicted on the assumption of harmonic frequencies for all 3N modes. More realistically, these are at best described as slightly anharmonic frequencies which we approximate with an effective harmonic force field. For lattice frequencies in particular, anharmonicity is expected to be important here it arises both from the anharmonic curvature in the potential and from the expansion of the lattice on warming. Consequently, the force constants used to describe the lattice modes become temperature dependent. The approach amounts to a simple extension of the ideas at the basis of the pseudoharmonic theory of solid lattices (2, 3) to the condensed phases which interest us. One phenomenological result of such anharmonicity is that Equation 3 now takes the form ... 

Figure 6. Dependence of lattice frequencies on lattice aluminium content for HMFI zeolites. Reproduced with permission from reference 24.

The examples discussed here are the simplest possible and real systems are usually more complicated. Moreover, we have used simplified models for example, v is the characteristic frequency of the interstitial ion or an ion adjacent to a vacant lattice position it will be diflerent from the characteristic lattice frequency and will not be the same for Schottky and Frenkel defects. However, the general principles of the transfer of matter through a crystalline solid are as they have been given here. 

At each lattice frequency there is a phonon density that is characteristic of the material considered. The phonon spectral density vanishes above a characteristic cut off frequency that is typically 1013 Hz for many solids. The cut off point of the phonon density is some orders of magnitude greater than the Larmor frequencies of nuclei currently found in NMR spectroscopy, and so there is a significant phonon density at these NMR frequencies to influence nuclear transitions. 

A further consequence of the choice (4.31) is that the phonon-like frequencies in (4.23) can be obtained from an algebraic equation involving only the pure lattice frequencies corh and the coupling constants krh. In effect f2rk are solutions of... 

The adjustable parameters appearing in Equation (12.6) must be carefully optimized against experimental data. These are either geometric (cell dimensions, molecular position and orientation) or energetic (sublimation enthalpies, lattice frequencies) in nature. Calculation of geometric features depends primarily on a proper balance of the attractive and repulsive parts of the potentials, not on their absolute magnitudes. Calculation of lattice energy (to be compared with sublimation enthalpies) depends primarily on the well depths of individual atom-atom interactions. Finally, lattice vibrations depend critically on the second derivative of the potential and are usually the most difficult to calculate correctly. 

Lattice Frequencies [19] (in cm ) at the F Point of the Two-Dimensional Brillouin Zone for Orientationally Ordered N2 Monolayer on Graphite flora Time-Dependent Hartree Lattice Dynamics Including Up to Quartic Displacement Terms, the Spherical Expanded Ab Initio N2-N2 Potential [23], and an Empirical N2-Graphite Potential [326] ... 

The thermodynamics of crystalline substances (including polymers) have been considered either based on the Griineisen parameter, yo, or deriving the statistical thermodynamic lattice theory of solid polymers. The Griineisen dimensionless parameter was originally defined as a density gradient of the crystalline lattice frequency, v [Warfield et al., 1983] ... 

Two general forms have been used for the pair potential v. The first was introduced by Walmsley and Pople (1964) in their treatment of the <1 = 0 lattice frequencies of solid COj. It consists of a 6-12 Lennard-Jones term (between molecular centers) and an orientation-dependent term in the form of a quadrupole-quadrupole interaetion. The seeond form, which has found wide application, consists of a sum over atom-atom interactions, summed over the nonbonded atoms of the two molecules. This type of pair potential function was developed for organic molecules and was used to account for the crystal structures of these systems. Kitaigorodskii (1966) determined the parameters for such potentials in this way. Dows (1962) first applied such a potential to the calculation of the librational lattice modes frequencies of solid ethylene using hydrogen-hydrogen repulsion terms as given by de Boer (1942). The usual form of this type of potential contains 6-exponential atom-atom terms ... 

As has been shown in Section IIC.2, the harmonic approximation is expected to be applicable to the treatment of the librational motions of COj. Additional support has been obtained for this recently (Suzuki and Schnepp, 1971). It is therefore possible to judge the validity of intermolecular pair potentials by comparison of calculated lattice frequencies with experiment. Table V summarizes available experimental and theoretical results for the q = 0 modes. It is again seen here, as was the case for a-Ng, that the translational frequencies can be closely reproduced from a gas phase Lennard-Jones potential (Walmsley and... 

The usual procedure for calculating the lattice frequencies is to replace the classical equation (Born and Huang, 1954),... 

This effective potential has been used to obtain lattice frequencies, using (3,15) and (3.17) (de Wette, Nosanow, and Werthainer, 1967 Klump, Schnepp, and Nosanow, 1970). In addition the ground-state energy must be minimized, thus obtaining an additional dependence of the effective force constants on the correlation. This minimization gives... 

As pointed out by Koehler (1968), even the solution obtained from these modified force constants is not the most meaningful one, because it is obtained from the harmonic part only of the wavefunction (3.22). He shows that the complete dispersion curves are not obtained from the force constant matrix alone, but first it has to be transformed by a unitary matrix diagonalizing the matrix of second derivatives of the correlation function/ . For q O, however, the frequencies coincide and (3.27) is sufficient for the calculation of lattice frequencies. 

When in a molecular solid all the atoms of a certain type are replaced by a different isotope, the frequencies of the vibrational excitations are simply shifted. So, for instance, when ethene crystals are grown from rather than C2H4, all the translational phonon frequencies decrease by a factor of J/32/28, the librations about the C = C axis by a factor of j/2 and the librations about the other axes by different but fixed amounts. The actual shifts found in the lattice frequencies can be used to determine the nature of the lattice modes. 

Provided that the temperature is high enough for the relation hVi kT to be satisfied, even for the highest lattice frequencies, the Helmholtz free energy of the crystal is given by equation (13 46) ... 

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