Phonon dispersion Equations

Consider a monatomic lattice. This maybe a metal or other element. Our approach is to apply Hooke s Law for an elastic vibrator, in which the 

We can rewrite these equations in the form of a traveling wave, i.e.- 

Now consider a cubic monatomic lattice. We can use a plane net to show the position of each atom. and the displacement can be shown as crosses, so as to show how each atom is displaced from a given equilibrium point. This is illustrated in 5.4.9. on the next page. 

We have illustrated two types of vibration, one in which the atoms vibrate together in an equal distance (shown as a longitudinal vibration) and the other where the atoms vibrate an unequal distance, relative to one another (here shown as a transverse vibration). Note however, that the atoms vibrate in phase, regardless of the direction and the distance. We can write equations using these terms by considering u as a coordinate and p as a displacement, in terms of the total force required. Ft, and the individual forces on each atom, Fi, as a form of Hookes Law  

What this means is that a standing wave will occur for nearest neighbor planes of the reciprocal lattice. Since all atoms in this example are equivalent, we can replace 2 FI by a constant, C., as follows  

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I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

By assuming harmonic forces and periodic boundary conditions, we can obtain a normal mode distribution function of the nuclear displacements at absolute zero temperature (under normal circumstances). The problem is then reduced to a classic system of coupled oscillators. The displacements of the coupled nuclei are the resultants of a series of monochromatic waves (the normal modes). The number of normal vibrational modes is determined by the number of degrees of freedom of the system (i.e. 3N, where N is the number of nuclei). Under these conditions the one-phonon dispersion relation can be evaluated and the DOS is obtained. Hence, the measured scattering intensities of equations (10) and (11) can be reconstructed. 

Using the Green function method and some decoupling approximations corresponding to the RPA and the Zeeeman reduced splitting smaller than Debye phonon quantum, it is possible to get the dispersion equation... 

Note that this equation is like a dispersion equation (which we have already discussed). Phonon relaxation rates follow these equations only at low temperatures, i.e.- 4,2 °K. 

Theoretical calculations of surface phonon dispersion have been carried out in two ways. One method is to use a Green s function technique which treats the surface as a perturbation of the bulk periodicity in the z-direction [34, 35]. The other is a slab dynamics calculation in which the crystal is represented by a slab of typically 15-30 layers thick, and periodic boundary conditions are employed to treat interactions outside the unit cell as the equations of motion for each atom are solved [28, 33, 35, 37]. In the latter both the bulk and the surface modes are found and the surface localized modes are identified by the decay of the vibrational amplitudes into the bulk in the former the surface modes can be obtained directly. When the frequency of a surface mode lies within a bulk band of the same symmetry, then hybridization can take place. In this event the mode can no longer be regarded as strictly surface localized and is referred to as a surface resonance [24]. Figure 8, adapted from Benedek and Toennies [24], shows how the bulk and surface modes develop as more and more layers are taken in a slab dynamics calculation. 

The elastic constants determine the acoustic phonon dispersion for long wavelengths according to the equations of motion in the continuum limit ... 

The self-consistent calculations were performed on superc lls with m = 4 to 8 (quadrupled to octupled) and displacements lu ranging from 0.01 to 0.02 The Results from different supercells are consistent within <0.004 x 10 dyn/cm and the force constants given in Tab. 5.1 are those obtained on the smallest of the supercells tried (which we expect to limit the roundoff errors and thus to be more reliable) they will be discussed later. Writing down the equations of motion for the linear chains shown schematically in Fig. 5.3.1 leads to 2 x 2 secular equations, which have as solutions the phonon dispersion shown in Fig. 5.3.3 before reaching this point, however, two problems require an adequate treatment anharmonlclty and spatial extent of forces. They are the topics of the next two Subsections. 

Comparing the vibrational branches and electronic bands calculations we note that in the former case the equations for different k values are solved independently while in the latter case the self-consistent calculation is necessary due to the BZ summation in the HP or KS Hamiltonian (see Chapters 4 and 7). Once the phonon dispersion in a crystal is known, thermodynamic functions can be calculated on the basis of statistical mechanics equations. As an example, the Helmholtz free energy, F, can be obtained as ... 

Fig. 26. Gruneisen analysis for UPtj. The upper plot shows the temperature-dependent Gruneisen parameter Q T) determined from C(T ) and ii T) using eq. (1). The equation S2 T) = S2 C, ICT) + Q C JCt) is then solved for C (T) and CpiiCT) under the assumption Q =73 and flp, =2.35. These are shown in the middle plot (Cf solid circles Cp dotted curve) and compared to the theoretical prediction for a Kondo doublet with 7 k = 16K (solid lines) and to the lattice-specific heat deduced from phonon dispersion curves (dashed line). The bottom plot shows the Kondo and phonon contributions to the thermal expansion deduced from the same analysis. Data are from Franse et al. (1989).

The phonon dispersion curves of the linear chain (or, in the three-dimensional case, of a crystal) can then be obtained by the well-known eigenvalue equation of the GF method, ... 

Previously we had only considered the optical modes at fc = 0 because the large difference between the momentum of a photon and a phonon would not permit energy and momentum to be conserved otherwise. Obviously, k carmot be exactly zero, so let us explore the region of small fc by developing the dispersion relation couple with phonons. Rewriting Equation 23.34,... 

Solving this equation for o) and plotting o> versus fc provides a dispersion curve for NaCl as shown in Figure 24.5. Notice that the curve has two branches the upper branch is the phonon dispersion relation for the longitudinal mode and approaches >l at fc = 0 while the lower branch is the dispersion curve for the transverse mode, which approaches wj as fc increases. No frequencies can propagate between < x and o>ij which causes an energy gap in this region. 

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... 

The electromagnetic fields of the right- and left-propagating polaritons, respectively, follow the wave equations with the speeds and damping rates of the different frequency components dispersed according to the frequency- and wavevector-dependent complex refractive index n = v/e(k, oj). A typical example of the dispersion of these modes is shown in Fig. 1 for the case of a real permittivity e. The term Ao(r,t) represents the envelope of the wavepacket on the phonon-polariton coordinate A. Note that this phonon-polariton coordinate is a linear combination of ionic and electromagnetic displacements, which both contribute to the polarization... 

The spectrum of polaritons can be found by means of Maxwell s macroscopic equations (see Ch. 4), provided that the dielectric tensor of the medium (44) is assumed to be known. Without going into details, we emphasize here that always a gap appears in the polariton spectrum (here we ignore spatial dispersion) in the region of the fundamental dipole-active vibration (C-phonon, exciton, etc.). At present, there is a sufficiently detailed theory for RSL by phonon-polaritons, taking many phonon bands into consideration. With this theory the RSL cross-section can be calculated for various scattering angles provided that the dielectric tensor of the crystal is known, as well as the dependence of the polarizability of the crystal on the displacement of the lattice sites and the electric field generated by this displacement (45). 

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