Phonons as Quantized Lattice Vibrations

We can thus make an important deduction from our studies, namely that associated defects hove definite effects iqKm the eneigy hand structure of solids. It is not surprising, then, that reactivity of solids depends considerably upon the degree of purity of the solid itself. 

There are certain equations which can be used to describe the observed properties of the fundamental unit of quantized energy, the photon. It can be shown that the vectors, F M, of a photon (these vectors are 

Secondly, if one uses monochromatic radiation, l.e.- a laser, one can obtain evidence of Stokes and anti-Stokes scattering of phonons, i.e.-lattice vibrations, as shown in the following  

You will note that the frequency shift is quantized. 

That is, the interaction of a photon with lattice vibrations (phonons) is a quantum unit which is either positive or negative in regard to frequency. 

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Charge carriers in a semiconductor are always in random thermal motion with an average thermal speed, given by the equipartion relation of classical thermodynamics as m v /2 = 3KT/2. As a result of this random thermal motion, carriers diffuse from regions of higher concentration. Applying an electric field superposes a drift of carriers on this random thermal motion. Carriers are accelerated by the electric field but lose momentum to collisions with impurities or phonons, ie, quantized lattice vibrations. This results in a drift speed, which is proportional to the electric field = p E where E is the electric field in volts per cm and is the electron s mobility in units of cm /Vs. 

It becomes apparent that we can predict, and calculate the expected number of phonon modes present in a given lattice, by knowing the types and numbers of atoms present. We have now shown how phonons have been defined as quantized lattice vibrations. It should be clear that phonon energy is completely defined by Quantum Mechanics and that if we wish to add or subtract energy during a photon interaction (such as an excitation process in a phosphor), the amount that can be added is a direct function of the type of atoms (ions) comprising the solid and its structure. 

The transition arises dynamically through the interaction between the electrons and the quantized lattice vibrations of the solid, phonons (437). The phonons, in a manner similar to electrons, are assigned a wavevector q = 2jt/A where k is the wavelength of the lattice vibration. There is an energy associated with each phonon of wavevector q, as indicated schematically in Fig. 12. The actual dispersion relation is a function of the mass of the atoms in a... 

Phonons are quasiparticles, which are quantized lattice vibrations of all atoms in a solid material. Oscillating properties of the individual atoms in nonequivalent positions in a compound, however, are not necessarily equivalent. The dynamics of certain atoms in a compound influence characteristics such as the vibration of the impurity or doped atoms in metals and the rare-earth atom oscillations in filled skutterudite antimonides. Therefore, the ability to measure the element-specific phonon density of states is an advantageous feature of the method based on nuclear resonant inelastic scattering. Element-specific studies on the atomic motions in filled skutterudites have been performed (Long et al. 2005 Wille et al. 2007 Tsutsui et al. 2008). 

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

The optical spectral region consists of internal vibrations (discussed in Section 1.13) and lattice vibrations (external). The fundamental modes of vibration that show infrared and/or Raman activities are located in the center Brillouin zone where k = 0, and for a diatomic linear lattice, are the longwave limit. The lattice (external) modes are weak in energy and are found at lower frequencies (far infrared region). These modes are further classified as translations and rotations (or librations), and occur in ionic or molecular crystals. Acoustical and optical modes are often termed phonon modes because they involve wave motions in a crystal lattice chain (as demonstrated in Fig. l-38b) that are quantized in energy. 

The rate of the relaxation process is often expressed as a relaxation time, t, that gives a measure how fast the higher spin level returns to the lower. The lattice vibrations of a crystalline solid are quantized phonons with energy ha) and mechanisms of the following types may occur ... 

Phonons are quantized vibrational waves, just as photons are quantized electromagnetic waves. In each case the energy of the quasi-particle is given by the famous Planck formula, E — hv, where v is the firequency of the light, in the case of the photon, or the frequency of the vibration, in the case of the phonon. Vibrational waves in a periodic one-dimensional lattice such as an ordered linear or helical polymer are periodic both in time and in space. Thus they possess both a frequency and a wave length, A. 

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. 

C. Heavy crystalline moderators. For crystalline materials, the dynamics of the atomic motions is well represented in terms of the quantized, simple-harmonic vibrations of the lattice. These excitations are commonly known as phonons, and are of considerable interest to the solid-state physicist. Since the materials of interest as reactor moderators will occur in polycrystalline form, the use of the incoherent approximation to determine the cross... 

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