Phonon concept

It is interesting to note that the simple Morse potential model, when employed with appropriate values for the parameters a and D (a = 2.3 x 1010 m 1, D = 5.6 x 10 19 J as derived from spectroscopic and thermochemical data), gives fb = 6.4 nN and eb = 20%, which are quite comparable to the results obtained with the more sophisticated theoretical techniques [89]. The best experimental data determined on highly oriented UHMWPE fibers give values which are significantly lower than the theoretical estimates (fb 2 nN, b = 4%), the differences are generally explained by the presence of faults in the bulk sample [72, 90] or by the phonon concept of thermomechanical strength [15]. 

Several calculations applying the "frozen phonon" concept to different substances h ve accumulated in the past 3-4 years. Be d l the worJcQon Si already mentioned in Section 1, phonons in Ge and A1 were studied as well, with the aid of the norm-conserving pseudopotentials. Ref. 41 demonstrates the use of forces, instead of energies, in frozen phonon calculations. (This will still be explained in Section 5.1.) In the context of static calculations (pha g diagrams), S. Froyen evaluated also the TO(r) frequency in GaAs. (7.84 THz), and energies of the frozen phonons at r and X in NaCl. ... 

The value of the deformed polymer consists of two parts the free volume of non-deformed polymer and the dilatational free volume, which is the consequence of the polymer density change in the deformation process [35]. The first component of the decrease in due to introduction of the epoxy polymer in HOPE (that is illustrated by a reduction in the permeability to gas) will require compensation at the expense of the second component increase, which should result in growth in the yield strain y [35]. This assumption is confirmed experimentally (see Figure 8.9). Within the frameworks of the yielding phonon concept [4, 19] this means that not only the parameters characterising the initial non-deformed polymer structure should be taken into account, but also its degree of modification in the deformation process, which can be realised with the help of the Griinesen parameter y. In the first approximation the first factor can be characterised with... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

In a general concept of a symmetry-restricted anharmonic theory Krumhansl relates the phonon anomalies to the electron band topology. The latter is directly determined by the competition of nearest neighbour interactions which in turn can be a function of stress, composition and temperature Nagasawa, Yoshida Makita simulated the <110> ... 

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

Such a concept of quasi-Fermi level is valid under the condition that the time constant for the establishment of thermal equilibrium of electrons or holes in the conduction or valence band (the redprocal of the rate of establishing equilibrium between electrons and phonons) is much smaller than the time constant for the recombination of exdted electron-hole pairs (the redprocal d the recombination... 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

The concept of phonons describing vibrations in extended solids is a core topic in solid-state physics. Two classic books for this field are ... 

According to the concept of the displacive-type ferroelectric phase transition [10], an increase in the dielectric constant corresponds directly to the softening of the IR-active transverse phonon. When the crystal can be regarded as an assembly of the vibrators of normal coordinates, the soft phonon... 

The concept of a mobility edge has proved useful in the description of the nondegenerate gas of electrons in the conduction band of non-crystalline semiconductors. Here recent theoretical work (see Dersch and Thomas 1985, Dersch et al. 1987, Mott 1988, Overhof and Thomas 1989) has emphasized that, since even at zero temperature an electron can jump downwards with the emission of a phonon, the localized states always have a finite lifetime x and so are broadened with width AE fi/x. This allows non-activated hopping from one such state to another, the states are delocalized by phonons. In this book we discuss only degenerate electron gases here neither the Fermi energy at T=0 nor the mobility edge is broadened by interaction with phonons or by electron-electron interaction this will be shown in Chapter 2. 

We shall in this book use the concept of a degenerate gas of small—or at any rate heavy—polarons. Clearly we should not expect these to be formed unless the number of carriers is considerably less than the number of sites. We also remark, as mentioned earlier, that in all metals, at temperatures less than B phonons lead to a certain mass enhancement, of order less than 2. A treatment is given by Ashcroft and Mermin (1976, p. 520). This affects the thermopower some results for an amorphous alloy (Ca AlJ from Naugle (1984) are shown in Fig. 2.2. A theoretical treatment of the range between this situation and the polaron gas has not yet been given. 

Now, the non-adiabatic electron transitions is examined only when electron matrix element Fif is small (see the criterion (10) and (10a)). It is the criterion of applicability of the perturbation theory on F f, but it is not the criterion of applicability of the concept of non-adiabatic transition between two crossing diabatic terms. As it is known (see, for example, ref. [5]) the true image of terms is changed on taking into account the interaction V. Denote two terms without inter-term interaction as E[(R) and E (R), where R is the generalized nuclear coordinate. If the crystal phonons (or the outer-sphere variables in a polar medium) only participate in the transition, then E[(R) and E (R) are the parabolic terms independent of the value of shift of... 

Finally, the structure and the energy diagram of bipolarons is given in Fig. 1.12. Bipolarons are double charged carriers where a strong interaction with the lattice (electron-phonon interaction) can lead to a stabilization of two charges despite the Coulomb repulsion. A more detailed and complete discussion of quasiparticles, their generation, their occurrence and extended concepts can be found in the literature [50-52],... 

Soliton — Solitons (solitary waves) are neutral or charged quasiparticles which were introduced in solid state physics in order to describe the electron-phonon coupling. In one-dimensional chainlike structures there is a strong coupling of the electronic states to conformational excitations (solitons), therefore, the concept of soliton (-> polaron, - bipolaron) became an essential tool to explain the behavior of - conducting polymers. While in traditional three-dimensional -> semiconductors due to their rigid structure the conventional concept of - electrons and -> holes as dominant excitations is considered, in the case of polymers the dominant electronic excitations are inherently coupled to chain distortions [i]. 

The high-field saturation of the carrier velocity can have various origins, e.g. a finite bandwidth of a non-parabolic transporting (here valence) bands, or the emission of optical phonons. It is believed that the high-field saturation of the drift carrier velocity in the crystal directions where the band model concept can be applied is due to the first one. Then [420],... 

One should realize that the anisotropy of the electronic properties (in the t s) imparts a similar anisotropy to the electron-acoustic-phonon interaction. Details and refinements of these concepts can be found in Ref. 22. 

Thermally-induced network vibrations broaden the absorption edge and shift the band gap of semiconductors. The thermal disorder couples to the optical transition through the deformation potential, which describes how the electronic energy varies with the displacement of the atoms. The bond strain in an amorphous material is also a displacement of atoms from their ideal position, and can be described by a similar approach. The description of static disorder in terms of frozen phonons is a helpful concept which goes back 20 years. Amorphous materials, of course, also have the additional disordering of the real phonon vibrations. 

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