Phonon dispersion: diagram

Figure 3.5 Phonon dispersion diagram for a complete unit cell with degrees of freedom. From Kieffer (1985). Reprinted with permission of The Mineralogical Society of America.

For Nal, the large group of points in the center of the bulk band in FX near the F point are probably due to phonon-assisted bound state resonances which were also found for NaCl and for LiF [58, 61, 63]. In the case of NaCl, the bound state energies had been determined by other scattering experiments [75, 76] so that the peaks in the TOF spectra due to bulk phonon resonances could be reliably removed from the phonon dispersion diagram in Fig. 24. For Nal the values of the bound states still need to be established. 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

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.)...

Inelastic processes are now possible whenever the curve (2.95) (o(AK) crosses a phonon dispersion curve (see Fig. 2.13). Such plots are called zone diagrams and are essential for the correct interpretation of time-of-flight (TOF) spectra of atom-surface inelastic scattering processes [117]. 

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.

Fig. 5.2 A schematic energy diagram J2(K) of the internal and the external molecular vibrations in molecular crystals. Q is the frequency, hS2 the energy and K is the magnitude of the wavevector in a particular direction, e.g. in the direction a. (C = 0 is the centre and K = itja the boundary of the Brillouin zone, with the lattice constant a. P is the usual notation for the centre of the Brillouin zone. MSi is a low-frequency internal molecular oscillation with a small or vanishing dispersion const.). MSi is a high-frequency internal molecular oscillation. All together, there are 3N-6 internal modes N is the number of atoms per molecule. OP is an optical phonon in which whole molecules are excited to carry out translational or libration oscillations whose frequencies are...

Fig. 9. Magnon dispersion for Tb in the ferromagnetic phase along the three principal axis directions. Symbols along the top of the diagram label the directions in the conventional notation shown in fig. 4. Dashed curves labelled PH are [longitudinal acoustic (LA), transverse acoustic (TA), and transverse optic (TO)] phonon branches which interact with the magnons. (After Mackintosh and Bjerrum-Moller 1972.)...

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