Debye approximation

Fig. 2.3. Schematic variation of absorption coefficient as a function of the frequency of absorbed radiation experimental curve (1) and theoretical curves in Debye approximation (2) and impact approximation (3) (Rocard formula).

In the Ising-type model, the change of molecular volume AV due to the LS<->HS transformation leads to a change of phonon frequencies of the lattice. The effect may be treated within the Debye approximation which requires that the interaction parameters and J2 are replaced by J and J 2 where ... 

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... 

The resonance width for low-frequency inodes rjq averaged over wave vectors is given in the Debye approximation as follows 143... 

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 experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

The recoilless fraction, /, has been calculated (13) for monotomic lattices using the Debye approximation. When the specific heat Debye temperatures of the alkali iodides are inserted in the Debye-Waller factor, a large variation of f follows (from 0.79 in Lil to 0.15/xCsI). It is not... 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

In any crystal, the low-frequency acoustic modes dominate at low temperatures, so that the approximation that w is proportional to q becomes increasingly valid as is evident from Fig. 2.2. In particular, the T dependence of the specific heat at very low temperatures is well predicted by the Debye approximation. 

Before turning to the applications of the Debye approximation, we should elaborate more fully on a point that was glossed over. This is the assumption —made at the outset, but explicated in going from Equation (58) to Equation (59) —that the scattering behavior of each scattering element is independent of what happens elsewhere in the particle. The approximation that the phase difference between scattered waves depends only on their location in the particle and is independent of any material property of the particle is valid as long as... 

Mie wrote the scattering and absorption cross sections as power series in the size parameter 0, restricting the series to the first few terms. This truncation of the series restricts the Mie theory to particles with dimensions less than the wavelength of light but, unlike the Rayleigh and Debye approximations, applies to absorbing and nonabsorbing particles. 

Here /jn(f) is the intensity of the incident radiation and 0 is the phase of the interferometer in the dark. The functions N(< >) and M(< >) relate the intensities of the transmitted and intracavity fields to that of the incident light. The function 7ref (0 corresponds to the intensity of radiation from an additional source, which is very likely to be present in a real device to control the operating point. This description is valid in a plane-wave approximation, provided that we neglect transverse effects and the intracavity buildup time in comparison with the characteristic relaxation time of nonlinear response in the system. It has been shown that the Debye approximation holds for many OB systems with different mechanisms of nonlinearity. 

Double Debye Approximation for Complex Permittivity of Heavy Water... 

In accordance with Ref. 54 both for liquid H20 and D20 the double Debye approximation is applicable in the frequency range up to 2THz (i.e., up to 70cm-1) and in the temperature range from 273 K to 303 K. 

G. H. Meeten, Conservative dichroism in the Rayleigh-Gans-Debye approximation, J. Coll. Inter. Sci., 84,235 (1981). 

Figure 3. The scattering intensity, Cj, per particle as a function of particle diameter according to the exact Mie theory (solid line) and the Rayleigh Debye approximation (dotted line).

For small colloidal particles (a < 1 /mi), with spherically symmetric distribution of scattering material inside its volume, the field amplitude is given by the Rayleigh-Gans-Debye approximation [38]... 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

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