Crystallization potential

Figure B3.2.4. A schematic illustration of an energy-independent augmented plane wave basis fimction used in the LAPW method. The black sine fimction represents the plane wave, the localized oscillations represent the augmentation of the fimction inside the atomic spheres used for the solution of the Sclirodinger equation. The nuclei are represented by filled black circles. In the lower part of the picture, the crystal potential is sketched.

It is assumed from the outset that the crystal potential may be written as a sum of two parts ... 

Smart, DJ. and Humphreys, C.J. (1978) The crystal potential in electron diffraction and in band theory, Inst. Phys. Confs. Ser., 41, 145-149. 

The steady-state wave function V(r) describing electrons with energy E moving in a crystal potential V(r) obeys the Schrodinger equation ... 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

The relation between an HREM image and the projected crystal potential is quite complex if the crystal is thick. To obtain an image which can be directly interpreted in terms of projected potential, crystals have to be well aligned, thin enough to be close to weak-phase-objects and the defocus value for the objective lens should be optimal, i.e. at the Scherzer defocus. 

The crystal potential for L-alanine calculated with Eq. (8.34) is shown in Fig. 8.1(a). The term Ospherical.atom(r) can be evaluated in direct space by the methods described in the following section. The term 0(0) for the independent-atom model [not exactly equal to the true 0(0)] was evaluated by a summation of the IAM potential over the unit cell. 

In both sexes, CCM-OJ provided an alkali load that significantly increased urinary pH compared to basal levels and versus milk consumption, and also increased urinary citrate excretion versus basal levels. An elevated urine pH and citrate level are generally considered to reduce Ca oxalate supersaturation and crystallization potential (Odvina, 2006). However, in this study the relative supersaturation measurement for Ca oxalate was not different between the CCM-OJ and milk treatment groups, or between either treatment and the basal levels. Although the alkalizing effect of milk was less than that of CCM-OJ, it also induced a higher urinary pH compared to basal levels (p <. 01 and p <. 05 in women and men, respectively). 

The phase contrast is produced by the phase modulation of the incident electron wave when it is transmitted through the sample crystal potential V(x, y). The propagation of a plane electron wave traversing through a thin sample is thus treated as a weak (scattering) phase object. The wavefunction at the exit... 

Dipolar ions like CN and OH can be incorporated into solids like NaCl and KCl. Several small dopant ions like Cu and Li ions get stabilized in off-centre positions (slightly away from the lattice positions) in host lattices like KCl, giving rise to dipoles. These dipoles, which are present in the field of the crystal potential, are both polarizable and orientable in an external field, hence the name paraelectric impurities. Molecular ions like SJ, SeJ, Nf and O J can also be incorporated into alkali halides. Their optical spectra and relaxation behaviour are of diagnostic value in studying the host lattices. These impurities are characterized by an electric dipole vector and an elastic dipole tensor. The dipole moments and the orientation direction of a variety of paraelectric impurities have been studied in recent years. The reorientation movements may be classical or involve quantum-mechanical tunnelling. 

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

Table 9.1 shows that the tunneling splitting A decreases exponentially with increasing lattice parameter a0. In the point-ion approximation in a cubic crystal potential, one would expect a decrease of the potential... 

Single crystal (Box 2.1) Search of crystal potential hypersurfaces... 

Here Sr is the overlap integral between two adjacent carbon layers (a-a ) (see Figure 6.10) and is defined similar to Equation 6.2. The difference between a and P atoms and the terms SE and SXE in the crystal potentials are neglected in Equation 6.12. From Equation 6.12, the energies of the n-bands are derived... 

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