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]. 

Our renormalization procedure is internally consistent in that the physical value of the tunneling amplitude depends on the scaling variable—the bare coupling Aq—only logarithmically. This bare coupling must scale with the only quantum scale in the problem—the Debye frequency, as pointed out in the first section. 

Such an approximate description of acoustic vibrations is referred to as the Debye approximation and the limiting frequency coo is called the Debye frequency. The... 

The simplest model to describe lattice vibrations is the Einstein model, in which all atoms vibrate as harmonic oscillators with one frequency. A more realistic model is the Debye model. Also in this case the atoms vibrate as harmonic oscillators, but now with a distribution of frequencies which is proportional to o and extends to a maximum called the Debye frequency, (Oq. It is customary to express this frequency as a temperature, the Debye temperature, defined by... 

TABLE IV. Properties of the water monomer in gas phase and in solution. Distances in A, angles in degrees, dipole moment, H, in Debyes, frequencies in cm 1. 

An observation of motion of single atoms and single atomic clusters with STEM was reported by Isaacson et al,192 They observed atomic jumps of single uranium atoms on a very thin carbon film of —15 A thickness or less. Coupled motion of two to three atoms could also be seen. As the temperature of the thin film could not be controlled, no Arrhenius plot could be obtained. Instead, the Debye frequency , kTIh, was used to calculate the activation energy of surface diffusion, as is also sometimes done in field ion microscopy. That the atomic jumps were not induced by electron bombardment was checked by observing the atomic hopping frequencies as a function of the electron beam intensity. 

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. 

We shall assume that only the levels with < lod where lod > A is the Debye frequency are paired. Assuming also that the density of states v is... 

Figure 6.19. Relaxation rate tor Raman transitions (R) and two-photon emission (T) in a two-level system as a function of temperature. The value of asymmetry is 0, 4, and 32 cm 1 as indicated. The tunneling matrix element is 0.25 cm 1 and the Debye frequency is 80 cm 1. (From Silbey and Trommsdorf [1990].)...

Debye frequency of the surface vibrations corresponds better to the observed value of Tc. The results for D(T) leave no doubt that surface diffusion of hydrogen isotopes is quantal. However, the weak dependence of Dc on mass cannot be rationalized within the framework of any one-dimensional model for the tunneling. 

Figure 15.8 A model of temperature-dependent relaxation time spectrum for PMN. G (r, T) is the number of polar regions having a relaxation time r, T/ is the freezing temperature, and td is the inverse Debye frequency (from [16]).

The more recent theories of chemical conversions [59-61] take into account the fact that the process of overcoming the activation barrier involves a cooperative change of more than one degree of freedom for the starting reagents subsystem. For the surface processes this is expected to lead to a need for considering the dynamics of the solid atom motion and, at least, the model should include information on Debye frequencies for its atoms (see, e.g., Ref. [62]). An additional inconvenience of the models for the elementary surface processes is associated with the fact that the frequencies of the surface atom oscillations differ from those inside the solid. Consideration of the multiphonon contributions to the probabilities that the elementary process can take place results in a significant modification of its rate constant up to the complete disappearance of the activation form of the temperature dependence [63,64]. 

If the lattice is assumed to have a Debye frequency spectrum with o>m = kOnw/hy then... 

From Eq. (3.1) we see that the Debye frequency iw varies proportionally to the velocity of elastic waves, divided by the cube root of the volume, and from Eq. (3.9) wc see that the velocity of either longitudinal or transverse waves varies inversely as the square root of the compressibility times the density, if we assume that Poisson s ratio is independent of the volume. As we shall see later, this assumption can hardly be... 

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