Crystals, lattice vibrations

Electrons of still lower energy have been called subvibrational (Mozumder and Magee, 1967). These electrons are hot (epithermal) and must still lose energy to become thermal with energy (3/2)kBT — 0.0375 eV at T = 300 K. Subvibrational electrons are characterized not by forbiddenness of intramolecular vibrational excitation, but by their low cross section. Three avenues of energy loss of subvibrational electrons have been considered (1) elastic collision, (2) excitation of rotation (free or hindered), and (3) excitation of inter-molecular vibration (including, in crystals, lattice vibrations). 

Crystal lattice packing, 12 249-250 Crystal lattice vibrations, 14 236 Crystalline adsorbents, 1 586, 589. See also Molecular sieves Zeolites for gas separation, 1 631 properties and applications, l 588t Crystalline alkali silicates, atomic structure of, 22 454-455 Crystalline cellulose, 5 373-379 Crystalline epoxy resins, 10 373-374 Crystalline flake graphite, 12 793 manufacture and processing of, 12 781-784... 

The total entropy of the system is the sum of contributions from spins and from the crystal lattice vibrations (the latter being characterized by the temperature T of the crystal) ... 

Vibrational Spectroscopy [Infrared (mid-IR, NIR), Raman]. In contrast to X-ray powder diffraction, which probes the orderly arrangement of molecules in the crystal lattice, vibration spectroscopy probes differences in the influence of the solid state on the molecular spectroscopy. As a result, there is often a severe overlap of the majority of the spectra for different forms of the pharmaceutical. Sometimes complete resolution of the vibrational modes of a particular functional group suffices to differentiate the solid-state form and allows direct quantification. In other instances, particularly with near-infrared (NIR) spectroscopy, the overlap of spectral features results in the need to rely on more sophisticated approaches for quantification. Of the spectroscopic methods which have been shown to be useful for quantitative analysis, vibrational (mid-IR absorption, Raman scattering, and NIR) spectroscopy is perhaps the most amenable to routine, on-line, off-line, and quality-control quantitation. 

The change in the intensity with temperature is calculated with the temperature factor. This change is produced by the crystal lattice vibrations, that is, the scattering atoms or ions vibrate around their standard positions as was previously explained (see Section 1.4) consequently, as the crystal temperature increases, the intensity of the Bragg-reflected beams decreases without affecting the peak positions [25], Debye and Waller were the first to study the effect of thermal vibration on the intensities of the diffraction maxima. They showed that thermal vibrations do not break up the coherent diffraction this effect merely reduces the intensity of the peaks by an exponential correction factor, named the temperature factor, D(0) [2,26], given by... 

With dielectric permeability for the region of crystal lattice vibrations of InSe (e1 = 6.8) and GaSe crystals (e1 = 6.2) [17] it is not difficult to show that appearance of a crystal layer between two H2 molecular sheets lead to screening of their interaction. It results in increase of parameter Ch2 in a matrix of a layered crystal, which becomes practically conterminous with parameter CGaSe and C rlSc in the given crystals. 

A magnetic relaxation process is always involved in energy exchange, and the exchange of energy between the spin centers is much faster (by several orders) than that between spin and the crystal lattice. The energy exchange between spin centers is called spin-spin relaxation and that between spin and the crystal lattice vibrations is called spin-lattice relaxation. 

Starr, T. L. and Williams, D. E. (1977 ). Coulombic nonbonded interatomic potential functions derived from crystal-lattice vibrational frequencies in hydrocarbons. Acta Crystallogr A, 33, 771-6. [153]... 

The orbital triplet states (transforming as the Ti or Ty irreducible representations) interact with crystal lattice vibrations, and this interaction affects also the SO splitting of these states. From the computational point of view, the effects of interaction between the electronic levels arising from the orbital triplet and surrounding ions can be modeled easier if each orbital triplet would be considered individually, without explicit treatment of mixture with electronic states from other... 

Fig. 1 General scheme of the origin of the 3d ion energy levels with different interactions considered. From the left to the right (a) the LS terms (Coulomb interaction between the 3d electrons) (b) crystal field splitting of the LS terms (c) SO splitting of the crystal field energy levels (d) Ham quenching of the SO splitting (effect of the crystal lattice vibrations)...

Here P(i) is linear momentum conjugated to the distortion coordinate 2(0-In the theory of the JT effect, the linear-coupling case E b described by the Hamiltonian (7) is the easiest one. Its matrix part includes just diagonal matrices. As distinguished from this simple case, the general JT case is a tough problem of complex dynamics of electrons coupled to crystal lattice vibrations. The... 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

The stability of stagnant motions is a significant problem not only in the theoretical subject but also in the practical measurement. For instance, in the experiments of quartz oscillators the 1 // spectral fluctuations are frequently observed, which is considered to be good examples for the nonstationary motions generated in the Hamiltonian dynamics of crystal lattice vibrations [6-8],... 

T. L. Starr and D. E. Williams, Acta Crystallogr. Ser. A, A106, 771 (1977). Coulombic Non-bonded Interatomic Potential Functions Derived from Crystal-Lattice Vibrational Frequencies in Hydrocarbons. 

For a solid with a full energy band and an empty band close above it in energy, what do you expect to happen as the crystal lattice vibration energy is transferred to the electrons ... 

For gases at normal pressures and temperatures (STP) impact broadening is also negligible when compared to broadening in condensed systems. However, if collisions could be approximated by crystal lattice vibrations in condensed systems, half-widths according to interruption broadening could range to 1500 cm 1. 

In the macroscopic approach, when studying crystal lattice vibrations, one deals with the mechanical displacements u x,t) of ions. In the range of large wavelengths, u x,t) is the relative displacement of two atoms constituting a unit cell. Relative displacements of atoms of various species leads to a change in the lattice polarizability with the polarization vector P x, t), which can be expressed as... 

The dispersion relation contains the most important information concerning vibration normal modes in a crystal. Lattice vibrations can be measured experimentally by means of classical vibration spectroscopic techniques (infrared and Raman) or neutron inelastic scattering. However, only the latter technique allows one to measure the full spectrum in a range of k vectors, whereas with infrared and Raman spectroscopy, only lattice vibrations at T can be detected. This limitation for measuring phonon dispersions is serious, becuase neutron scattering experiments are demanding. 

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals. 

In a crystal lattice, vibrations of different frequencies appear simultaneously, forming the vibrational spectrum. This spectrum is characterized by a function of distribution of vibration by the frequencies g(v). The physical meaning of the function g(i ) is that being multiplied by a small interval dv,it gives the number of vibrations... 

The method of neutron spectroscopy is the most efficient tool for study frequencies of the crystal lattice vibrations. This method is based on the scattering of the low-energy, so-called heat neutrons by the nuclei of solids. The wavelength of the normal vibrations and of the heat neutrons are values of the same order as the energies are. As a result of the interaction of the low-energy neutrons with solids, quanta of normal vibrations of the crystal lattice (phonons) are created or, conversely, annihilated. The collision neutron-phonon changes the state of the neutron essentially and this change can be detected experimentally. 

In dielectric VV paramagnets, nuclear spins are coupled to crystal lattice vibrations much more weakly than in intermetallic compounds therefore it is difficult to cool the crystal lattice. However, very low temperatures of nuclear spin systems have been obtained Tm in TmV04, 0.1 mK TminTmP04,0.4 mK (Suzuki et al. 1981) Ho in H0VO4, 1 mK (Suzuki et al. 1978). 

Electron-Phonon Interaction Interplay between electrons and crystal-lattice vibrations that accounts for both ordinary resistance and the zero resistance of superconductivity. 

Phonons Based on an analogy between crystal lattice vibrations and those of an electromagnetic field, these particles of quantized vibrational energy were used by physicists to facilitate calculations of thermal and electrical conduction in solids. 

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