Lattice vibrations structure factors

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. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

In all properties studied with pseudopotenlial theory, the first step is the evaluation of the structure factors. For simplicity, let us consider a metallic crystal with a single ion per primitive cell -either a body-centered or face-centered cubic structure. We must specify the ion positions in the presence of a lattice vibration, as we did in Section 9-D for covalent solids. There, however, we were able to work with the linear force equations and could give displacements in complex form. Here the energy must be computed, and that requires terms quadratic in the displacements. It is easier to keep everything straight if we specify displacements as real. Fora lattice vibration of wave number k, we write the displacement of the ion with equilibrium position r, as... 

A lattice vibration of wave number k reduces the structure factor at the lattice wave numbers (solid dots), but... 

Finally, a note on disorder of the membrane stacks and on attempts to correct for it in the analysis of diffraction data. Generally, two kinds of disorder are being discussed in crystal structure Disorder of the first kind refers to displacements of the structural elements (for example the one-dimensional unit cell of a membrane stack) from the ideal positions prescribed by the periodic lattice. The effect on the diffraction pattern is indistinguishable from that of thermal vibrations and may, therefore, be expressed as a Debye-Waller temperature factor so that the structure factor, expressed as a cosine series, includes a Gaussian terra, according to... 

It therefore follows that the total intensity of a reflection depends on how far apart the various lattices are, i.e. on the positions and identities of the atoms in the cell. The square root of the total intensity of a reflection from a particular set of planes with Miller indices h, k and I is called a structure amplitude or structure factor, Ffj i. It depends not only on the fractional coordinates of the atoms x, y and z and their scattering powers,/, but also on parameters that describe the vibrational movement of the atoms in the lattice. Every vibrational movement leads to partial destructive interference and therefore a decrease of intensity of the reflected beam. Assuming that each atom i moves isotropically, with a mean-square amplitude of vibration (/, we can define quantities A and B by Eqs 10.8 and 10.9. 

The standard procrystal model assumes that the electron distribution in the crystal is very nearly equal to a superposition of previously known rigid atomic density distributions, which are smeared by harmonic lattice vibrations. The structure factor F(h, k, 1) with integer Miller indices h, k, I for N atoms per unit cell, located at the positions then becomes... 

An EXRAFS smdy [15] revealed that the covalent bond in the Tellurium MC (0.2792 nm) is shorter and stronger than the bond (0.2835 nm) in the triagonal Te (t-Te) bulk structure. The Debye-Waller factor (square of the mean amplitude of lattice vibration) of the Te chain is larger than that of the bulk, but the thermal evolution of the Debye-Waller factor is slower than that of the bulk, which suggests the Te-Te bond in the chain is stronger than it is in the bulk, see Fig. 25.1a. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

The vibration bands relative to phosphate groups in the apatite structure differ from the normal modes of the P04 isolated ion, due to distortions of the PO4 tetrahedra in the apatite lattice and vibrational coupling [4]. Therefore, site-group and factor-group analyses were applied [15,16,18,21] to elucidate the vibrational spectra observed (Fig. 5) and band assignments of infrared (IR) and Raman bands have been given (Table 3). 

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