Phonon dispersion, curves

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

Figure 3 Phonon dispersion curves obtained by inelastic neutron scattering revealing precursor behaviour prior to the 14M transformation in Ni-AI. The dip at q = 1/6 [110] (a) deepens upon cooling and (b) shifts under an external load .

Kinematics of surface phonon He spectroscopy. The thick lines correspond to scan curves of a 18 meV He beam. The thin lines display the Rayleigh phonon dispersion curve of Pt(lll) along the f M azimuth. 

Fig. 10. (a) He time-of-flight spectrum taken from a LiF(001) surface along the < 100) azimuth at an incident angle Si = 64.2°. The primary beam energy was 19.2 meV. (After Ref 25.). (b) Measured Rayleigh phonon dispersion curve of LiFfOOl) < 100), including a scan curve (dashed) for the kinematical conditions in (a). (After Ref. 25.)... 

Fig. 23. Calculated and measured surface phonon dispersion curves of the (111) surfaces of the noble metals Cu, Ag and Au. (After Ref. 45.)...

Fig, 26. Experimental dispersion curve of the Kr monolayer and measured line width broadening As of the Kr creation phonon peaks. The solid line in the dispersion plot is the clean Pt(lll) Rayleigh phonon dispersion curve and the dashed line the longitudinal phonon bulk band edge of the Pt(l 11) substrate, both in the r Mn azimuth which is coincident with the r Kk, azimuth. 

Detailed electronic energy-band calculations have revealed the existence of appropriate surface states near the Fermi energy, indicative of an electronically driven surface instability. Angle-resolved photoemission studies, however, showed that the Fermi surface is very curved and the nesting is far from perfect. Recently Wang and Weber have calculated the surface phonon dispersion curve of the unreconstructed clean W(100) surface based on the first principles energy-band calculations of Mattheis and Hamann. ... 

It is of extreme interest to note that the calculated Tm for Nb and Ta are lower than the observed and conversely, the calculated Tm for Mo and W are higher than the observed. This fact is rather puzzling in view of the fact that all four elements have a common crystal structure of bcc. Furthermore, the four elements, Nb (sometime called Cb) Mo, Ta and W all have 4d, 5d atomic orbital and they all form continuous solid solutions with one another. Therefore, metallurgically, the four elements are totally compatible with one another which contradicts the sharp differences in their observed vs. calculated melting temperatures. To understand these differences we first look at the differences (or similarities) in their phonon dispersion curves (Fig. 1) along the three principal axes. A distinct difference is observed between the dispersion curves for Mo and W, on one hand, and those for Nb and Ta on the other in the [ 0] and [i 00] directions. 

Ph. Ghosez, X. Gonze and J.-P. Michenaud, "Ab Initio phonon dispersion curves and interatomic force constants of barium titanate," accepted for publication in Ferroelectrics. 

Both A - and Ei-modes are Raman and IR active. The two nonpolar E2-modes E and E are Raman active only. The Bi-modes are IR and Raman inactive (silent modes). Phonon dispersion curves of wurtzite-structure and rocksalt-structure ZnO throughout the Brillouin Zone were reported in [106-108]. For crystals with wurtzite crystal structure, pure longitudinal or... 

AIN exists in two types the hexagonal (wurtzite structure) and the cubic (zincblende structure). The former is more stable, and has been investigated in more detail. The wurtzitic AIN has two formula units per unit cell (4 atoms per cell) and 9 optical branches to the phonon dispersion curves [1] ... 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals/ Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions. 

Stevenson, Ed., Plenum Press, New York, 1966, Vol. 6 of Scottish Universities Summer School, Chapter 2, pp. 53—72. Theory of Phonon Dispersion Curves. 

The periodicity of the multilayer provides an additional scattering of the phonons and introduces an artificial Brillouin zone. This leads to a folding of the phonon dispersion curves at wave vectors n/L, where L is the multilayer spacing. 

Fig. 9.29. The folded phonon dispersion curves for a multilayer structure. Raman scattering occurs at the intersection with the light wave vector k. ...

Fig. 9.29 illustrates the intersection of the light wave vector and phonon dispersion curves and predicts pairs of Raman lines. The Raman shift is 10-100 cm for layer widths of order 100 A. 

The parameters of the ion-ion interaction potentials in ZnO crystals are listed in Table 1. Ihe values of the parameters were obtained by theoretical fits to the measured elastic and dielectric constants, the lattice constants, the lattice energy (see Table 2), and the phonon dispersion curves. 

As an example. Fig. 3 plots the phonon dispersion curves for three highly S5mimetric directions in the Brillouin zone of the perfect ZnO crystal. Comparison of the theoretical and experimental frequencies shows good agreement for the acoustic branches. The densities of phonon states of the perfect ZnO crystal calculated by integrating over the Brillouin zone are displayed in Fig. 4. Comparison of the results of our calculation and a calcu-... 

Figure 3. Phonon dispersion curves for ZnO crystals. The points denote experimental values. ...

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. 

In addition to calculating energies, it is also possible to calculate routinely a range of crystal properties, including the lattice stability, the elastic and dielectric and piezoelectric constants, and the phonon dispersion curves. The techniques used, which are quite standard, require knowledge of both first and second derivatives of the energy with respect to the atomic coordinates. Indeed it is useful to describe two quantities first the vector, g, whose components g are defined as ... 

The expressions used in calculating the properties referred to above from these derivatives are discussed in greater detail in Reference 9. For more detailed discussions of the calculation of phonon dispersion curves from the second derivative or dynamical matrix W, the reader should consult References 41 and 42. Finally, we note that by the term lattice stability we refer to the equilibrium conditions both for the atoms within the unit cell, and for the unit cell as a whole. The former are available from the gradient vector g, while the latter are described in terms of the six components ei- ee, which define the strain matrix e, where... 

The single-phonon coherent inelastic processes described above are often used for measurements of phonon dispersion curves with single crystal samples (Figure 6). These studies are not trivial, and as a result, it is often the case that measurements will only focus on a small subset of all phonon modes (particularly the lower energy modes). Examples of measurements of relatively complete sets of dispersion curves of minerals include calcite, sapphire, and quartz. It should be appreciated that the measurements... 

Figure 7 Phonon dispersion curves for face-centered cubic Cu at 296 K. (Reproduced with permission Svensson, Brockhouse and Rowe 1967, American Physical Society)...

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