Lattice force constants

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

The Raman and infrared spectra for C70 are much more complicated than for Cfio because of the lower symmetry and the large number of Raman-active modes (53) and infrared active modes (31) out of a total of 122 possible vibrational mode frequencies. Nevertheless, well-resolved infrared spectra [88, 103] and Raman spectra have been observed [95, 103, 104]. Using polarization studies and a force constant model calculation [103, 105], an attempt has been made to assign mode symmetries to all the intramolecular modes. Making use of a force constant model based on Ceo and a small perturbation to account for the weakening of the force constants for the belt atoms around the equator, reasonable consistency between the model calculation and the experimentally determined lattice modes [103, 105] has been achieved. 

At the same time, many lattice dynamics models have been constructed from force-constant models or ab-initio methods. Recently, the technique of molecular dynamics (MD) simulation has been widely used" " to study vibrations, surface melting, roughening and disordering. In particular, it has been demonstrated " " " that the presence of adatoms modifies drastically the vibrational properties of surfaces. Lately, the dynamical properties of Cu adatoms on Cu(lOO) " and Cu(lll) faces have been calculated using MD simulations and a many-body potential based on the tight-binding (TB) second-moment aproximation (SMA). " ... 

Lattice vibrations are calculated by applying the second order perturbation theory approach of Varma and Weber , thereby combining first principles short range force constants with the electron-phonon coupling matrix arising from a tight-binding theory. 

Here eK, gk are the force constants for the pure solute K, which can be determined from measurements of its second virial coefficient, and q, oq are similar, but as yet unknown, constants characteristic for the / -hydroquinone lattice. 

Figure 2. A schematic of the free energy density of an aperiodic lattice as a function of the effective Einstein oscillator force constant a (a is also an inverse square of the locahzation length used as input in the density functional of the liquid). Specifically, the curves shown characterize the system near the dynamical transition at Ta, when a secondary, metastable minimum in F a) begins to appear as the temperature is lowered. Taken from Ref. [47] with permission.

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

Figure 8.8 One-dimensional diatomic chain with lattice parameter 2a and force constant K.

In Eq. (6.1) 1 is the unit operator, there is one state /> associated with each lattice site, and l and l A A = 1, 2, 3, 4) label a molecule and its nearest neighbors in the tetrahedral lattice. Weare and Alben show also that the theorem remains valid when small distortions away from tetrahedrality exist, hence it can be used to describe a random amorphous solid derived from a tetrahedral parent lattice. Basically, the density of states of the amorphous solid is a somewhat washed out version of that of the parent lattice. The general shape of the frequency spectrum is not much altered by the inclusion of a non zero bond-bending force constant provided the ratio of it to the bond stretching force constant is small relative to unity. 

Our development has assumed temperature independent force constants. In real liquids, however, there is a small temperature dependence of frequencies and force constants due to anharmonicities, lattice expansion, etc. The incorporation of these effects into the theory is treated in later sections. 

June et al. (12) used TST as an alternative method to investigate Xe diffusion in silicalite. Interactions between the zeolite oxygen atoms and the Xe atoms were modeled with a 6-12 Lennard-Jones function, with potential parameters similar to those used in previous MD simulations (11). Simulations were performed with both a rigid and a flexible zeolite lattice, and those that included flexibility of the zeolite framework employed a harmonic term to describe the motion of the zeolite atoms, with a force constant and bond length data taken from previous simulations (26). 

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