Phonon curve

Miscellaneous Solids. Anderson and Walmsley (1965) described the far infrared spectra of ammonia, H S, and their deuterated analogs. The far infrared spectra of H S and D2S have also been reported by Taimasalu and Robinson (1965). Giguere and Chapados (1966) reported the far infrared spectra of solid and DjO. All lattice modes predicted to be active were observed. Tubino and Zerbi (1970) studied the phonon curves and frequency distributions for a hydrogen-bonded system the -form of solid formic acid and of its deuterated isotopes. The low-energy phonons arising from the hydrogen bond were found to be strongly dispersed. 

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 .

I2H2O as a function of the reciprocal temperature. The points are data obtained from fits of the Mdssbauer spectra (Fig. 6.6). The broken curve is a fit to the Einstein model for a Raman process. The dotted curve corresponds to a contribution from a direct process due to interactions between the electronic spins and low-energy phonons associated with critical fluctuations near the phase transition temperature. (Reprinted with permission from [32] copyright 1979 by the Institute of Physics)... 

Fig. 2.2. Two generation models of coherent optical phonons, (a), (c), (e) impulsive stimulated Raman scattering (ISRS). (b), (d), (f) displacive excitation of coherent phonons (DECP). Graphs (e) and (f) display the time evolution of the driving force (grey areas) and that of the displacement (solid, curves) for ISRS and DECP, respectively...

Fig. 2.8. Left oscillatory part of the reflectivity change of Bi (0001) surface at 8K (open circles). Fit to the double damped harmonic function (solid curve) shows that the Aig and Eg components (broken and dotted curves) are a sine and a cosine functions of time, respectively. Right pump polarization dependence of the amplitudes of coherent Aig and Eg phonons of Bi (0001). Adapted from [25]...

Fig. 2.17. Left transient reflectivity change of Zn and Cd at 7K. Inset shows the imaginary part of the dielectric function of Zn, Cd, and Mg. Right amplitude of the coherent E2g phonon of Zn as a function of temperature. Solid curve in the right panel represents the fit to np. From [56]...

Fig. 3.12. Selective enhancement of coherent Aig phonon of Bi-Bi vibration in Bi0. 3iSbo.69 mixed crystal at room temperature, fa c] reflectivity change m time domain with single, double and triple pulses. Dotted curves represent SHG profile of the pump pulses, (d-f) corresponding FT spectra. From [33]...

Figure 5. Model spectra of a naked neutron star. The emitted spectrum with electron-phonon damping accounted for and Tsurf = 106 K. Left panel uniform surface temperature right panel meridional temperature variation. The dashed line is the blackbody at Tsurf and the dash-dotted line the blackbody which best-fits the calculated spectrum in the 0.1-2 keV range. The two models shown in each panel are computed for a dipole field Bp = 5 x 1013 G (upper solid curve) and Bp = 3 x 1013 G (lower solid curve). The spectra are at the star surface and no red-shift correction has been applied. From Turolla, Zane and Drake (2004).

The band structure of bulk silicon, with possible optical transitions for (c) absorption and (d) emission of a photon, together with (e) the dispersion curves of phonon branches, is shown on the right. After [Kol5],... 

Figure 5.10 The configurational coordinate diagram for the ABe center oscillating as a breathing mode. The broken curves are parabolas within the approximation of the harmonic oscillator. The horizontal full lines are phonon states.

The discrete energy levels sketched as horizontal lines on each potential curve of Figure 5.10 are consistent with the quantized energy levels (phonon levels) of a harmonic oscillator. For each harmonic oscillator at frequency 12, the permitted phonon energies are given by... 

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... 

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. 

---