Phonon elastic waves

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... 

The quanta of the elastic wave energy are called phonons The themral average number of phonons in an elastic wave of frequency or is given, just as in the case of photons, by... 

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. 

There are differences between photons and phonons while the total number of photons in a cavity is infinite, the number of elastic modes m a finite solid is finite and equals 3N if there are N atoms in a three-dimensional solid. Furthennore, an elastic wave has tliree possible polarizations, two transverse and one longimdinal, in contrast to only... 

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... 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

For an elastic wave in a medium that is both isotropic and continuous the acoustic phonons have three different wave-polarisations, one longitudinal, with the atomic displacements in the direction of the wave propagation and two with transverse polarisation, the atomic displacements perpendicular to the propagation vector. In a crystal, the transverse modes are not necessarily degenerate except in specific symmetry directions and, because the atoms are located in discrete positions, the velocity of propagation will depend on its direction. 

In electrically insulating solids, heat is transferred in the form of elastic waves or phonons [1], Anything that affects the propagation of the phonons through the solid affects the thermal conductivity of the solid. In a pure crystalline ceramic, the intrinsic thermal conductivity is limited by the energy dissipated during phonon-phonon collisions or so-called Umklapp processes [15], Commonly, the intrinsic thermal conductivity of solids is described by (5). 

Phonon excitations are dependent on the condition that the temporal width of the excitation pulses, Tp, is short compared to the single acoustic oscillation periods (1 /Tp> 1 /Ta)- Hence the interaction term "impulsive" is incorporated into the name ISLS. Material modes where Xp > T A will not be efficiently excited. The acoustic wavelength and wave vectors describing the two coherently excited elastic waves are (A,, k ) where... 

The lattice vibration causes elastic waves in crystals. The quantum of die lattice vibrational energy is called phonon, in analogy with the photon of the electromagnetic wave. 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

The line width y is related to the damping of elastic waves, whieh results from anharmonic interaetions of the acoustie phonons with other phonons, relaxation... 

Debye, Petrus (Peter) Josephus Wilhelmus (1884-1966) was born in the Netherlands and became a naturalized American citizen in 1946. In 1912 he proposed the idea of quantized elastic waves, called phonons. From 1940 to 1952 Debye was Professor of Chemistry at Cornell University. He won the Nobel Prize in Chemistry in 1936. 

The usual polymers do not conduct electricity. Consequently, heat cannot be transferred by electrons in these polymers. Heat must be mostly transported by elastic waves (phonons in the corpuscular picture). The distance at which the intensity has decreased to ie is known as the free path length. This free path length is comparatively independent of temperature for glasses, amorphous polymers, and liquids and is about 0.7 nm. From this., it can be concluded that the weak decrease in thermal conductivity observed for amorphous polymers below the glass-transition temperature is essentially due to a decrease in the heat capacity with temperature (see Figures 10-26 and 10-4). 

Debye s theory of the specific heats of solids depends on the existence of a high number of standing, high frequency, elastic waves that are associated with thermal lattice vibrations. Central to his approach is the proposal that in a solid the phonon spectral density (p) increases continuously, and with a direct dependence on the square of the frequency (cutoff frequency (Q, at about 10 Hz) above which the phonon density vanishes for a solid continuum containing N atoms in a sample of volume V, the proportionality constant is 6 V/v, where V is the velocity of propagation. At a typical nuclear... 

In this chapter we examine the energetic contributions of the lattice vibrations. These are the most important, nonchemical, thermal mcdtations and involve motion of the nuclei. Lattice vibrations are quantized. In the same way as photons are, as the respective quasi-particles, equivalent to electromagnetic waves, there are quasiparticles allocated to these elastic waves termed phonons. 

P. Debye modified the Einstein s model by introduction of inter-atomic forces in a crystal model. This is equivalent as to take phonons into account (refer to Section 9.3.1). To each elastic wave (phonon) the Bom-Karman atomic chain was attracted spread out in a three-dimensional array (Figure 9.13 and 9.15). As a result of reflection from external crystal borders, standing waves with various values co and k (refer to Section 2.9.2 and 2.9.3) are formed. There is the certain relationship between the wavelength of standing waves X and the size of the crystal L expressed by the eqn (2.9.8). Phase speed of running... 

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