Phonon annihilation

In the gas phase, the asymmetric CO stretch lifetime is 1.28 0.1 ns. The solvent can provide an alternative relaxation pathway that requires single phonon excitation (or phonon annihilation) (102) at 150 cm-1. Some support for this picture is provided by the results shown in Fig. 8. When Ar is the solvent at 3 mol/L, a single exponential decay is observed with a lifetime that is the same as the zero density lifetime, within experimental error. While Ar is effective at relaxing the low-frequency modes of W(CO)6, as discussed in conjunction with Fig. 8, it has no affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off at "-60 cm-1 (108). If the role of the solvent is to open a relaxation pathway involving intermolecular interactions that require the deposition of 150 cm-1 into the solvent, then in Ar the process would require the excitation of three phonons. A three-phonon process would be much less probable than single phonon processes that may occur in the polyatomic solvents. In this picture, the differences in the actual lifetimes measured in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in the phonon DOS at 150 cm-1 or to the magnitude of the coupling matrix elements. 

Here we use the convention that co > 0 for phonon annihilation and < 0 for phonon creation. For the differential reflection coefficient, or the fraction of incident atoms which are scattered into final solid angle dQf with energy Ef to Ef + dEf, the result from perturbation theory—as, for example, from the distorted wave Bom approximation [39, 41]—is that... 

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

Here we have introduce he phonon frequencies UJj (q) and the polarization vectors ej (0,j). n( w) is the Bose ractor, phonon creation (energy loss) correspond to 03. phonon annihilation (energy gain) corresponds to (q)>0. V(q,z) is the 2-dimensional Fourier transform of the potential and Q now is considered in the extended zone scheme. The Debye Waller factor 2W is given by ... 

Introducing the electron and phonon annihilation and creation operators, Oa, al bj, b], we can rewrite the Hamiltonian in the notation of second quantization after a canonical transformation... 

A development of synchrotron radiation facility made possible to perform the nuclear resonant scattering with synchrotron radiation. Elastic scattering is identical, in principle, to the Mossbauer resonance by y photons from radioactive nuclei. From the inelastic scattering one can observe the scattering involved phonon annihilation and creation in solid. Nuclear resonant scattering with synchrotron radiation will briefly described in final part of this chapter. 

Fig. 10. Ag(100)-c(2x2) Cl. Extended zone plot of the phonon energies measured by HATOF [83Lam,84Lam]. Negative frequencies correspond to phonon annihilation. The dashed lines indieate the bulk band edge, while the solid lines are the surface modes and S4 as calculated by Castiel et al. [76Cas] for bare Ag(001) surface. Along...

The scattering conditions can be discussed in a modified Ewald construction similar to that used for LEED (Chapter 3.2.1). Here, it has to be modified for the helium atom energy gain or loss due to phonon annihilation or creation, respectively, and... 

Figure C3.5.2. VER transitions involved in the decay of vibration Q by cubic and quartic anhannonic coupling (from [M])- Transitions involving discrete vibrations are represented by arrows. Transitions involving phonons (continuous energy states) are represented by wiggly arrows. In (a), the transition denoted (i) is the ladder down-conversion process, where D is annihilated and a lower-energy vibration cu and a phonon co are created.

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

Several types of spin-lattice relaxation processes have been described in the literature [31]. Here a brief overview of some of the most important ones is given. The simplest spin-lattice process is the direct process in which a spin transition is accompanied by the creation or annihilation of a single phonon such that the electronic spin transition energy, A, is exchanged by the phonon energy, hcoq. Using the Debye model for the phonon spectrum, one finds for k T A that... 

At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy hcOq is annihilated and a phonon with energy HcOr is created. The energy difference TicOq — ha>r is taken up in a transition of the electronic spin. In the Debye approximation for the phonon spectrum, this gives rise to a relaxation rate given by... 

Nuclear absorption of incident X-rays (from the synchrotron beam) occurs elastically, provided their energy, y, coincides precisely with the energy of the nuclear transition, Eq, of the Mossbauer isotope (elastic or zero-phonon peak at = E m Fig. 9.34). Nuclear absorption may also proceed inelasticaUy, by creation or annihilation of a phonon. This process causes inelastic sidebands in the energy spectrum around the central elastic peak (Fig. 9.34) and is termed nuclear inelastic scattering (NIS). 

Fig. 9.34 Monitoring of inelastic excitations by nuclear resonant scattering. The sidebands of the excitation probability densities for phonon creation, S(E), and for annihilation, S —E), are related by the Boltzmann factor, i.e., S(—E) = S E) tTvp —Elk T). This imbalance, known as detailed balance, is an intrinsic feature of each NIS spectrum and allows the determination of the temperature T at which the spectrum was recorded...

If the dephasing time of the coherent phonons depend critically on the carrier density, photo-injection of carriers with the second pump pulse can annihilate them partially or completely, depending on its fluence but not on its relative timing. Such incoherent control was demonstrated for the LO phonons of GaAs [37],... 

This formula describes the exchange of a single phonon of wavevector Q, frequency co(0 ) and polarization e(Q,j). n is the Bose factor for annihilation (—) or creation (+) of a phonon, respectively, i.e. the phonon occupation number. 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Inelastic scattering of radiation in solids is typified by the Raman effect, which involves the creation or annihilation of phonons or magnons. If a single phonon is involved, the scattering event is referred to as the first-order Raman effect in second-order Raman effect two phonons are involved. The polarizability associated with a phonon mode can be represented as a power series of the phonon amplitude, u, as follows ... 

Conversely, it might be possible for energy exchange from B to A to take place. This could happen if phonons were annihilated in the exchange process the major requirement then being that AE KT, so that again energy is conserved. 

For paramagnetic spin systems, there are two major processes of relaxation (55). One relaxation mode involves spin-flipping accompanied by lattice phonon creation and/or annihilation (spin-lattice relaxation), and the other mode is due to the mutual flipping of neighboring spins such that equilibrium between the spins is maintained (spin-spin relaxation). For the former mode of relaxation, th decreases with increasing temperature, and the latter relaxation mode, while in certain cases temperature dependent, becomes more important (th decreases) as the concentration of spins increases. 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

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