Acoustic phonon, dispersion

Collins, D.R., W.G. Stirling, C.R.A. Catlow, and G. Rowbotham. 1993. Determination of acoustic phonon dispersion curves in layer silicates by inelastic neutron scattering and computer simulation techniques. Phys. Chem. Miner. 19 520-527. 

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

The excitation spectrum is determined from the poles of D (q, energy states excitations). The latter enter via the frequency dependent susceptibility q V, interaction between both types of modes leads to a hybridization (Elliott et al., 1972). Due to the elastic scattering processes within a CEF-energy level there are 5 =o-contributions to the quadrupole susceptibility (see eq. 17.108). It was shown that these processes have to be included in the expressions for the elastic constants and therefore contribute to a Jahn-Teller phase transition. However they do not show up in the acoustic phonon dispersion which one would measure... 

Fig. 54. The longitudinal acoustic phonon dispersion relations along the [100] axis of a- and y-Ce (Stassis 1.0 1988). The splitting of the dispersion curve for a-Ce is the result of mode mixing.

Figure 3 Acoustic phonon dispersion curves and phonon density of states of NbC and ZrC (21). Depressed parts, as indicated by arrows, are formed on the dispersion curve of NbC, which moves the state density of the compound to the low-energy side.

The quite another temperature dependence of the rate constant at helium temperatures is resulted in the case when the principal contribution to dispersion a in formula (25a) gives the acoustic phonons. Their frequencies lie in the interval [0, lud], where tuD is Debye s frequency. Even if hin0 kT, it exists always in the range of such low frequencies that haxkT. It is these phonons that give the contribution depending on the temperature in the dispersion a [15], One assumes that the displacements of the equilibrium positions of phonon modes Sqs do not depend on frequency. Then, the calculation of the rate constant gives at low temperatures, hcou>kT,... 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

Figure 2.20. Right part The polariton dispersion at a few tens of reciprocal centimeters below the bottom of the excitonic band, vs the wave vector, or the refractive index n = ck/w (notice the logarithmic scale). The arrows indicate transitions with creation of one acoustical phonon, with linear dispersion in k (with a sound velocity of 2000 m/s). For the transitions T, Tt, T3 the final momentum is negligible compared to the initial momentum, and the unidimensional picture suffices. For the transitions between T3 and the point A, the direction of the final wave vectors should be taken into account. Left part The density of states m( ) (2.141) of the polaritons in the same energy region. This diagram explains why the transitions T, will be much slower than the transitions around T3 and the point A. The very rapid increase of m( ) at a few reciprocal centimeters below E0 shows the effect of the thermal barrier.

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-... 

Apart from acoustic phonons, which account for heat transport in insulating media, propagation of vibrational energy is usually not considered in crystals, as the dispersion of optical modes is normally very small over the Brillouin zone. However, there is an important class of optical vibrations in crystals for which spatial propagation can be the dominant property at optically accessible wave vectors. This class is identical with that of infrared active modes and its members are known as phonon-polaritons. ... 

Here T is the temperature of acoustic phonons (thermostat), T is the temperature of optical phonons (284), the anharmonicity constant k is much less than 1, and flq is the frequency of acoustic phonons it is possible to assume ilq = (10 2 to 10 1)flDebye. The coupling between the optical and acoustic phonons is strongest near f>Debye 1 and because of this, for sufficiently large anharmonicity, k > 10 2 even at T = T", the last exponential multiplier can be approximated by the exponent below, with the dispersion being neglected ... 

Figure 3. Schematic Illustration of dispersion curves of an acoustic phonon, a band electron, and the SWAP. The Incident phonon -q Is scattered as q2. (Reproduced with permission from reference 5. Copyright 1985 Nljhoff.)...

Michael A. Stroscio, Mitra Dutta, Slavalor Rufo, and Jianyong Yang, Dispersion and damping of acoustic phonons in quantum dots, IEEE Transactions on Nanotechnology, 3, 32-36 (2004). 

Fig. 10. (a) A ID lattice of rigid diatomic molecules. The dispersion curve for acoustic phonons that result from translations runs from zero frequency to a cut-off termed the Debye frequency. The vibron has no dispersion. Adding flexibility to the molecules introduces dispersion in the vibron state and narrows the gap between vibrons and phonon as shown at right, (b) A 3D lattice of flexible naphthalene molecules. The 12 phonons overlap significantly with the two or three lowest energy vibrations, termed doorway modes. Doorway modes are coupled to both phonons and higher frequency vibrations associated with bond breaking. Adapted from ref. [91]. 

Now we have two (2) phonon dispersion curves, a so-called optical branch and a lower energy acoustical branch. The standing waves are better understood in terms of the actual displacement the atoms undergo ... 

NUMBER OF BRANCHES OF PHONON DISPERSION For y - atoms Acoustical = 3... 

Figure 7. Bulk phonon dispersion curves for KBr and RbCl in their <100> and <111> high-symmetry directions. Both crystals have fee lattices and rocksalt structures. Note that the transverse branches, labeled TA (transverse acoustic) and TO (transverse optical), are doubly degenerate in these directions. (Adapted from Fig. 3 of Ref. 32.)...

The effects of relaxation on the calculated surface phonon dispersion in Rbl have apparently been verified, particularly by the observation of a surface optical mode which lies above the bulk phonon optical bands. Except for the mysterious acoustic band mode in Rbl, the Shell model calculations have generally been quite accurate in predicting surface vibrational mode energies in both high-symmetry directions of the alkali halide (001) surfaces. 

Figure 33. Surface phonon dispersion for W(OOl) in the FM portion of the SBZ showing the measured Rayleigh wave (R) and longitudinal (L) modes. The data in the upper panel were obtained at 1200 K, while in the lower panel the data shown by open circles were obtained at 500 K and those represented by closed circles were obtained at 300 K. The edges of the transverse acoustic (TA) and longitudinal acoustic (LA) bulk bands are given by the hatched lines. The vertical lines in the lower panel denote the widths in the energy transfer distributions of these points. (Reproduced from Figs. 10 and 13 of Ref. 110, with permission.)...

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