Force Constant Models

An example of a force constant treatment for A1 is shown in Fig.3.18 only three nearest-neighbour force constants have been used whose values have been determined from the three elastic constants. To obtain better agreement, it is necessary to use a large number of force constants extending to at least fifth neighbours [4.62]. 

The results for lead (Fig.4.25) showed the presence in this metal (and in other polyvalent metals having a large electron-phonon interaction) of interionic forces of long range and complex nature. 

From the above it is clear that for metals such as Pb and Nb where strong electron-phonon interaction is present, an analysis in terms of phenomenological force constants is not meaningful because the number of force constants is very large and lack physical interpretation. 

2 Coulomb Interaction in the Uniform-Background Lattice Model 

Fairly plausible results were obtained for the TA-modes, but the LA-mode has the characteristies of an optic branch in that it approaches a nonvanishing frequency (O) for q 0 given by 

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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... 

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. 

The model of Jayanthi etal. overcomes the phenomenological force-constant models and thus avoids the large number of hypothetical force constants, sometimes used in these calculations. Jayanthi et al. calculate the charge density in each unit cell by an expansion over many-body interactions, which arise from the coupling of the electronic deformations to... 

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.

We can in fact fit the speed of sound and the zone-boundary mode to a simple nearest-neighbor force-constant model for the Weber system, just as we did in Table 9-2 for the tetrahedral solids. We obtain force constants... 

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

IR and Raman spectra of M2Mn207, where M = In or Tl, were analysed using a short-range force constant model.255... 

Figure 6. Dispersion relation for an isobaric crystal, where Mq = M. When /j = /2 for this force constant model, the diatomic crystal is equivalent to a monatomic crystal with lattice spacing a 2 or Brillouin zone boundary at 2ir/a. Hence the optical branch (solid line) appears as the portion of the acoustic branch from via to 2ir/a (dashed line) folded back to the zone center from the real Brillouin zone boundary (dotted vertical line).

Figure 31. Surface phonon dispersion for Cu(lll). The open circles are from HAS experiments, and the open triangles are from EELS experiments. The surface modes shown as solid lines and bulk band boundaries are based on a simple force constant model. The X and Y designations indicate the polarizations of the corresponding modes as identified in the reduced zone diagram in the inset. (Reproduced from Fig. 3 in Ref. 99, with permission.)...

Figure 32. Surface phonon dispersion for Nb(OOl). The data are the solid points which were taken at 900 K. Panels a and b correspond to slab dynamics calculations with two different force constant models the calculation in panel b uses the force constants from the bulk phonon fits. The solid lines represent the surface phonons and resonances polarized mainly longitudinally (or parallel), the lines with long dashes represent phonons polarized mainly perpendicularly, and those with short dashes are shear horizontal. (Reproduced from Fig. 6 of Ref. 107, with permission.)...

TOF measurements of the dispersion in the normal phase agree by and large with a force constant model based on the neutron scattering data [114]. However, at temperatures below the normal-incommensurate phase transi-... 

Born-Model Calculations.- A very much more complex model is that due to Catlow and co-workers who have developed a methodology whereby structure prediction takes place based upon Born s model for ionic solids. The interactions which are considered, are, for the most part, non-bonded interactions, and thus can be considered as a potential model and not a force constant model. 

We also studied crystalline-ropes, optimizing the intertube distance, in conparison with other theoretical studies using the pseudopotential LCAO approach (31), tight-binding methods (32,33), or an enqiirical force-constant model (34), as well as with eiqierimental data whenever available. 

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