Phonons wavevectors

What happens if you try to image an object with periodicity outside this spatial frequency range The spatial frequency will be displaced by an integral multiple of 2n/a so as to lie in the range that the framestore can handle. There is an almost exact analogy with phonon wavevectors in a crystal lattice or, if you prefer, with why stage-coach wheels appear to go backwards in movies. The effect is illustrated in Fig. 3.8 with a feature with a periodic structure of spatial frequency Kj. It will be stored in the framestore with a spatial frequency dz JCj — 27T K/ap, with ng = 1 in the example here. This will appear as a periodic structure in the image that bears no apparent relationship to the object. 

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form... 

In the above approximation excitons and optical phonons act as independent and uncorrelated quasiparticles. In contrast to the case of weak resonant coupling, the vibronic states are characterized by a set of wavevectors, corresponding to the excitonic wavevector and to the phonon wavevectors, and not by one value of the wavevector. Thus by excitation by light of these multiparticle states the conservation laws for the wavevector and for the energy have the form (note that in the ground state excitons and molecule oscillations are absent because of the low temperature T)... 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

Figure 43. TOP spectra transformed to energy transfer distributions for a Kr monolayer on Pt(lll) at four different angles. With decreasing incident angle, larger phonon wavevectors are probed. One can see that the two phonons which lie very close in energy in panel a merge together in panels b and c with increasing wavevector, and that only one phonon remains in the spec-tmm after a certain wavevector is reached (panel d). (This figure has been reproduced from Fig. 2 of Ref. 129, with permission.)...

Fig. 5.7 A scattering diagram in reciprocal space. The scattering vector Q is the difference between the wavevectors k/ of the incident wave and k rof the scattered wave. Q is an observable of inelastic neutron scattering. It determines the value of the phonon wavevector K Q=K-Cuptoa lattice vector C in the reciprocal lattice.

Phonons once propagating in a crystal system undergo various other scattering interactions. Such scattering events, which cause a change in the phonon wavevector or phase, occur at crystal boundaries and as a result of interactions with lattice imperfections or with conduction electrons. It is possible experimentally to limit interactions with surfaces and electrons the latter by concentrating on insulators and semiconductors tjith low carrier concentrations. 

For the interpretation of the observed band progressions, the vibrational frequencies of a molecular chain with fixed ends can be classified in terms of the phonon wavevector k ... 

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