Phonons Debye model

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Low-temperature behaviour. In the Debye model, when T upper limit, can be approximately replaced by co, die integral over v then has a value 7t /15 and the total phonon energy reduces to... 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

The temperature dependence of sod is related to that of the recoil-free fraction /(T) = Qxp[— x )Ey / Hc) ], where (x ) is the mean square displacement (2.14). Both quantities, (x ) and can be derived from the Debye model for the energy distribution of phonons in a solid (see Sect. 2.4). The second-order Doppler shift is thereby given as [20]... 

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... 

When the displacements of the nuclei are considered in terms of the phonon modes of the crystal, the reorganization energy can be expressed in terms of the relative shifts of the equilibrium positions of acoustic vibrations of acoustic vibrations. Using the Debye model and assuming that the friction is independent of frequency, the reorganization energy is proportional to the friction coefficient, Et = AI2>7Ty]0 2DQ (see Section 2.3). 

There are two different temperature regimes of diffusive behavior they are analogous to those described by Holstein [1959] for polaron motion. At the lowest temperatures, coherent motion takes place in which the lattice oscillations are not excited transitions in which the phonon occupation numbers are not changed are dominant. The Frank-Condon factor is described by (2.51), and for the resonant case one has in the Debye model ... 

Fig. 1. Temperature dependence of the homogeneous width y (a) and position 8 (b) (in u>d units) of a zero-phonon line in the Debye model for different values of the interaction parameter wcx/w indicated in the right-side boxes. The instability limit corresponds to wcr/w = 1.

Zimmermann and Konig211) introduce the phonon contribution of the lattice by a Debye model with an interpolated Debye temperature... 

N is Avogadro s number and is the Debye temperature) and whose intercept at T = 0 is y. In the Debye model, the slope has a dependency (actually, r" for an n-dimen-sional solid) owing to the lattice or phonon contribution to the heat capacity. Of course, the heat capacity normally measured is Cp, the heat capacity at constant pressure. However, for solids the difference between Cp and Cy is typically less than 1 percent at low temperatures and thus can be neglected. 

The fact that a quantum oscillator of frequency u> does not interact effectively with a bath of temperature smaller than hu>/kg implies that if the low temperature behavior of the solid heat capacity is associated with vibrational motions, it must be related to the low frequency phonon modes. The Debye model combines this observation with two additional physical ideas One is the fact that the low frequency (long wavelength) limit of the dispersion relation must be... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

These concepts may be quantified as follows. A quantum of lattice vibration is termed a phonon, and the mean deviation of an atom from its lattice position is the mean-square displacement (u ). Phonons are detected by vibrational spectroscopy by absorption peaks below 500 cm . According to the Debye model, atoms vibrate as harmonic oscillators with a distribution of frequencies, the highest of which is COD- then the Debye temperature 9d is defined as... 

The temperature dependence of the resistivity of amorphous non-magnetic metals has been studied by many authors. The most popular theory is still Ziman s theory for the resistivity of liquid metals and several authors have extended this theory to include a dynamic structure factor. Cote and Meisel (1979) used an isotropic Debye spectrum for the phonons to calculate the dynamic structure factor. They obtained a quadratic dependence of the resistivity on temperature at low temperatures. Frobose and Jackie (1977) used both a Debye model and a model of uncorrelated Einstein oscillators in conjunction with a dynamic structure factor to analyse the resistivity. They found that the second model leads to a satisfactory fit for the temperature dependence of the resistivity of CuSn alloys for T 10 K. Ohkawa and Yosida (1977) predict a T ... 

This has been used for two-level tunnelling systems in insulating glasses. The coupling coefficient Fip from the phonon deformation potential should be independent of T and A, because the density of phonon modes in the Debye model is proportional to co up to the maximum frequency cod and this co-dependence counteracts the smaller overlap for larger A. The electron rate Re may therefore dominate the total rate at small values of A, while Rip may be faster for large A up to the Debye energy k T. ... 

Other processes where two phonons are emitted are also possible and they are important for large AE. The coupling must be treated as an adjustable parameter, but the A and T dependencies of the numerical integral (6.7) can be calculated. However, the Debye model for the phonons is of course uncertain for large a>. 

The Einstein model gives a good qualitative agreement with the real behavior of solids, but the quantitative agreement is poor (Cranshaw et al. 1985). A more realistic representation of a solid is given by the Debye model. The model describes the lattice vibration of solids as a superposition of independent vibrational modes (i.e., collective wave motion of the lattice, associated with phonons ) with different frequencies. The (normalized) density function p(co) of the vibrational frequencies is monotonically increasing up to a characteristic maximum of cod, where it abruptly drops to zero (Kittel 1968) ... 

The heat transfer via the solid backbone of aerogels depends on the backbone stmcture and connectivity (Chap. 21), and its chemical composition. For a given temperature gradient within an aerogel, heat is transferred by diffusing phonons via the chains of the aerogel backbone, where the mean free path of the phonons is far below the dimensions of the mostly amorphous, dielectric primary particles. Within the primary particles, the thermal conductivity is a property of the backbone material and is described in terms of the phonon diffusion model by Debye [7], i.e., ... 

Dulong-Petit limit can be reproduced. In the Debye model, lattice vibrations are approximated as a continuous elastic body considering only acoustic modes, and therefore, phonon spectrum is treated linearly. In... 

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