Solids lattice vibrations

Cyc/o-Undecasulfur Su was first prepared in 1982 and vibrational spectra served to identify this orthorhombic allotrope as a new phase of elemental sulfur [160]. Later, the molecular and crystal structures were determined by X-ray diffraction [161, 162]. The Sn molecules are of C2 symmetry but occupy sites of Cl symmetry. The vibrational spectra show signals for the SS stretching modes between 410 and 480 cm and the bending, torsion and lattice vibrations below 290 cm [160, 162]. For a detailed list of wavenumbers, see [160]. The vibrational spectra of solid Sn are shown in Fig. 23. 

The variation in the lattice vibration of the solid products was examined by utilizing the FT-IR technique at successive DGC process times and the results are presented in Fig. 5. The absorption bands at 550 cm and 450 cm" are assigned to the vibration of the MFI-type zeolite and the internal vibration of tetrahedral inorganic atoms. The band 960 cm" has been assigned to the 0-Si stretching vibration associated with the incorporation of titanium species into silica lattice [4], This indicates that the amorphous wall of Ti-MCM-41 was transformed into the TS-1 structure. 

The Mossbauer effect can only be detected in the solid state because the absorption and emission events must occur without energy losses due to recoil effects. The fraction of the absorption and emission events without exchange of recoil energy is called the recoilless fraction, f. It depends on temperature and on the energy of the lattice vibrations /is high for a rigid lattice, but low for surface atoms. 

ALL CHANGES IN PHASE involve a release or absorption of calories. One reason for this is that each solid has its own heat capacity. That is, there is a characteristic heat content for each material which depends upon the atoms composing the solid, the nature of the lattice vibrations within it, and its structure. The total heat content, or enthalpy, of each solid is defined by ... 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

While the examples in Scheme 7.16 hinted at the practicality of the solid state photodecarbonylation of ketones, the factors controlling this reaction remained unknown until very recently. As a starting point to understand and predict the photochemical behavior of ketones in terms of their molecular structures, we recall that most of the thermal (kinetic) energy of crystals is in the form of lattice vibrations. 

The heat transfer in a solid is due both to lattice vibrations (phonons) and to conduction electrons. Experiments show that in reasonably pure metals, nearly all the heat is carried by the electrons. In impure metals, alloys and semiconductors, however, an appreciable... 

When the temperature is such that hv kT, neither of the limiting cases described earlier can be used. For many solids, the frequency of lattice vibration is on the order of 1013 Hz, so that the temperature at which the value of the heat capacity deviates substantially from 3R is above 300 to 400 K. For a series of vibrational energy levels that are multiples of some fundamental frequency, the energies are 0, hv, 2hv, 3hi/, etc. For these levels, the populations of the states (n0, nu n2, etc.) will be in the ratio 1 e hl T e Jh,/Ikr e etc. The total number of vibrational states possible for N atoms is 3N... 

Control of the lattice vibrations in crystals has so far been achieved only through classical interference. Optical control in solids is far more complicated than in atoms and molecules, especially because of the strong interaction between the phononic and electronic subsystems. Nevertheless, we expect that the rapidly developing pulse-shaping techniques will further stimulate pioneering studies on optical control of coherent phonons. 

The intensity of the Mossbauer effect is determined by the recoil-free fraction, or /factor, which can be considered as a kind of efficiency. It is determined by the lattice vibrations of the solid to which the nucleus belongs, the mass of the nucleus and the photon energy, Ea and is given by ... 

The term exp(-2k2c ) in (6-9) accounts for the disorder of the solid. Static disorder arises if atoms of the same coordination shell have slightly different distances to the central atom. Amorphous solids, for instance, possess large static disorder. Dynamic disorder, on the other hand, is caused by lattice vibrations of the atoms, as explained in Appendix 1. Dynamic disorder becomes much less important at lower temperatures, and it is therefore an important advantage to measure spectra at cryogenic temperatures, especially if a sample consists of highly dispersed particles. The same argument holds in X-ray and electron diffraction, as well as in Mossbauer spectroscopy. 

Vibrations in molecules or in solid lattices are excited by the absorption of photons (infrared spectroscopy), or by the scattering of photons (Raman spectroscopy), electrons (electron energy loss spectroscopy) or neutrons (inelastic neutron scattering). If the vibration is excited by the interaction of the bond with a wave... 

All solid state ionic conduction proceeds by means of hops between well-defined lattice sites. Ions spend most of their time on specific sites where their only movement is that of small oscillations at lattice vibrational frequencies (10 Hz). Occasionally, ions can hop into adjacent... 

Herman F. (1959). Lattice vibrational spectrum of germanium. J. Phys. Chem. Solids, 8 405-418. 

The temperature ranges in which these simple behaviours are approximated depend on the vibrational frequencies of the molecules involved in the reaction. For the calculation of a partition function ratio for a pair of isotopic molecules, the vibrational frequencies of each molecule must be known. When solid materials are considered, the evaluation of partition function ratios becomes even more complicated, because it is necessary to consider not only the independent internal vibrations of each molecule, but also the lattice vibrations. 

Notions of high - and low -frequency limiting behavior depend on one s point of view, and the notation reflects this what is low frequency to an ultraviolet spectroscopist may be high frequency to an infrared spectroscopist. For insulating solids the value of c in the near infrared is often denoted as c0 by ultraviolet spectroscopists it refers to frequencies low compared with certain oscillators—electrons in this example—which may, however, be high compared with lattice vibrational frequencies. Consequently, this same limiting value is denoted as by infrared workers. 

A simple application of the multiple-oscillator theory is to fit measured reflectance data for MgO in the Reststrahlen region. In Section 9.1 we considered the electronic excitations of MgO, whereas we now turn our attention to its lattice vibrations. A glance at the far-infrared reflectance spectrum of MgO in Fig. 9.7 shows that it does not completely exhibit one-oscillator behavior there is an additional shoulder on the high-frequency side of the main reflectance peak, which signals a weaker, but still appreciable, second oscillator. The solid curves in Fig. 9.7 show the results of a two-oscillator calculation using (9.25) the reflectance data were taken from Jasperse et al. (1966), who give the following parameters for MgO at 295°K ... 

The extinction curves for magnesium oxide particles (Fig. 11.2) and aluminum particles (Fig. 11.4) show the dominance of surface modes. The strong extinction by MgO particles near 0.07 eV( - 17 ju.m) is a surface mode associated with lattice vibrations. Even more striking is the extinction feature in aluminum that dominates the ultraviolet region near 8 eV no corresponding feature exists in the bulk solid. Magnesium oxide and aluminum particles will be treated in more detail, both theoretically and experimentally, in this chapter. 

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. 

McCombie, C. W., 1970. Modification of lattice vibrations by impurities, in Far-Infrared Properties of Solids, S. S. Mitra and S. Nudelman (Eds.), Plenum, New York, pp. 297-359. 

Microcrystals exhibit properties distinctly different from those of bulk solids. The fractional change in lattice spacing has been found to increase with decreasing particle size in FejOj. Magnetic hyperfine fields in a-FejOj and FejO are lower in the microcrystalline phase compared to those of the bulk crystalline phases. The tetra-gonality (i.e. the departure of the axial ratio from unity) of ferroelectric BaTiOj decreases with decrease in particle size in PZT, the low-frequency dielectric constant decreases and the Curie temperature increases with decreasing particle size. The small particle size in microcrystals cannot apparently sustain low-frequency lattice vibrations. 

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