Phonons Debye frequency

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Compared with the momentum of impinging atoms or ions, we may safely neglect the momentum transferred by the absorbed photons and thus we can neglect direct knock-on effects in photochemistry. The strong interaction between photons and the electronic system of the crystal leads to an excitation of the electrons by photon absorption as the primary effect. This excitation causes either the formation of a localized exciton or an (e +h ) defect pair. Non-localized electron defects can be described by planar waves which may be scattered, trapped, etc. Their behavior has been explained with the electron theory of solids [A.H. Wilson (1953)]. Electrons which are trapped by their interaction with impurities or which are self-trapped by interaction with phonons may be localized for a long time (in terms of the reciprocal Debye frequency) before they leave their potential minimum in a hopping type of process activated by thermal fluctuations. 

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,... 

Subtraction of the resolution function reveals the vibrational contribution S (v) to the excitation probability. PHOENIX adjusts the subtraction weight to achieve the best match to the one-phonon contribution to the vibrational signal near Eq expected for a Debye frequency distribution (D (v) a v y The Fourier-log algorithm then yields the dominant first-order vibrational contribution... 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

The properties of a one-dimensional Fermi gas model with two characteristic energies, two bandwidth cut-offs, is studied. The direct electron-electron coupling and the phonon mediated effective coupling are cut off at energies Ea and <0,, respectively, where E is the bandwidth of the electron energy band and to is the Debye frequency. The model is treated in the framework of renormalization group approach. It is shown that this model can be mapped on the usual one cut-off model and the results obtained for that model can be applied. 

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]. 

AcAc, acetylacetonate EPR, electron paramagnetic resonance DPM, dipivaloylmethane Tc, Correlation time for molecular tumbling A/x, concentration of spins X (per unit volume) D, mutual translational self-diffusion coefficient of the molecules containing A and X a, distance of closest approach of A and X ye, magnetogyric ratio for the electron C, spin-rotation interaction constant (assumed to be isotropic) Ashielding anisotropy <7 <7j ) coo, Debye frequency 0d, the corresponding Debye temperature Fa, spin-phonon coupling constant. 

A natural choice for to is Tuof, where cud is the Debye frequency this is an approximate upper limit of the frequency of phonons in the solid (see chapter 6), so with both ek and ek within a shell of this thickness above the Fermi level, their difference is likely to be smaller than the phonon energy cuq and hence their interaction, given by Eq. (8.5), attractive, as assumed in the Cooper model. Cooper also showed that the radius R of the bound electron pair is... 

This relation provides an explanation for the isotope effect in its simplest form from our discussion of phonons in chapter 6, we can take the Debye frequency... 

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