The force-constant model

The displacements S will be taken to be always much smaller in magnitude than the bond distances, consistent with the assumption of small deviations from equilibrium, in which case the harmonic approximation makes sense. Using this fact, we can expand in powers of the small quantity s — Sj /fe, which gives to lowest order 

By similar arguments we can obtain an expression in terms of the variables S , for the bond bending energy, defined to be 

For the left-hand side of this last equation, we find 

Combining the two contributions, Eqs. (6.26) and (6.30), we obtain for the total potential energy 

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Table I summarizes some of the results of the dynamical calculations for adsorbed butane. The calculated surface vibratory mode frequencies are in reasonable agreement with the observed spectrum, lying in the range 50-125 cm"1 with the rocking mode about the chain axis having the highest frequency followed by the closely spaced bouncing and orthogonal rocking modes. Although there is some variation depending on the force-constant model used, the calculated frequencies are within 30 cm of the experimental values.

Figure 6.1. Schematic representation of bond stretching and bond bending in the force-constant model. The filled circles denote the positions of atoms in the ideal configuration and the open circles represent their displaced positions Si, Sj are the atomic displacements, rij is the original bond length, r -j the distorted bond length, and AOij the change in orientation of the bond (the angular distortion).

We next apply the force-constant model to a simple example that includes all the essential features needed to demonstrate the behavior of phonons. Our example is based on a system that is periodic in one dimension (with lattice vector ai = (a/V5)[x -I- y]), but exists in a two-dimensional space, and has two atoms per unit cellatpositionsti = a/2 )k, ti = —(a/2V2)x. This type of atomic arrangement... 

Figure6.4. ToprthephononspectrumofSialongthehigh-symmetrydirectionsL — T — X, calculated within the force-constant model with -s/Kr/Msi = 8.828 THz, = 2.245...

The force-constant model Table 6.2. Phonon modes in Si. 

Frequencies of four high-symmetry modes are given (in THz) at the center (F) and the boundary (X) of the Brillouin Zone, as obtained by theoretical calculations and by experimental measurements. DFT results are from Refs. [65, 67]. The force-constant model is based on nearest neighbor interactions only, with values of Kr and ko chosen to reproduce exactly the experimental frequencies marked by asterisks. The atomic displacements corresponding to these modes are shown in Fig. 6.4. 

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). 

Of course, the guesses above aren t really guesses. They are predicated on many years of Raman and other spectroscopic experience and calculations that are the reverse of the calculation we descr ibed. In spectroscopic studies, one normally calculates the force constants from the stretching frequencies in modeling, one... 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a stmcture to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical energy, but a classical mechanical one). The sum of the components is called the force field energy, or steric energy, which also routinely includes the electrostatic energy components. Typically, the steric energy is expressed as... 

This force equation can now be used to find the force in model systems such as that of an ideal dielectric sphere (relative dielectric constant Ko) in an ideal perfectly insulating dielectric fluid (relative dielectric constant K ). The force can now be written as... 

The thirty-two silent modes of Coo have been studied by various techniques [7], the most fruitful being higher-order Raman and infra-red spectroscopy. Because of the molecular nature of solid Cqq, the higher-order spectra are relatively sharp. Thus overtone and combination modes can be resolved, and with the help of a force constant model for the vibrational modes, various observed molecular frequencies can be identified with specific vibrational modes. Using this strategy, the 32 silent intramolecular modes of Ceo have been determined [101, 102]. 

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). " ... 

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

For a given a the force constant ko can be chosen in a way that the displacement d of the Drude particle remains much smaller than the interatomic distance. This guarantees that the resulting induced dipole jl, is almost equivalent to a point dipole. In the Drude polarizable model the only relevant parameter is the combination q /ko which defines the atomic polarizability, a. It is... 

Isotope superlattices of nonpolar semiconductors gave an insight on how the coherent optical phonon wavepackets are created [49]. High-order coherent confined optical phonons were observed in 70Ge/74Ge isotope superlattices. Comparison with the calculated spectrum based on a planar force-constant model and a bond polarizability approach indicated that the coherent phonon amplitudes are determined solely by the degree of the atomic displacement, and that only the Raman active odd-number-order modes are observable. 

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