Atomic displacements, periodicity

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... 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

Raman spectra as a function of temperature are shown in Fig. 21.6b for the C2B4S2 SL. Other superlattices exhibit similar temperature evolution of Raman spectra. These data were used to determine Tc using the same approach as described in the previous section, based on the fact that cubic centrosymmetric perovskite-type crystals have no first-order Raman active modes in the paraelectric phase. The temperature evolution of Raman spectra has indicated that all SLs remain in the tetragonal ferroelectric phase with out-of-plane polarization in the entire temperature range below T. The Tc determination is illustrated in Fig. 21.7 for three of the SLs studied SIBICI, S2B4C2, and S1B3C1. Again, the normalized intensities of the TO2 and TO4 phonon peaks (marked by arrows in Fig. 21.6b) were used. In the three-component SLs studied, a structural asymmetry is introduced by the presence of the three different layers, BaTiOs, SrTiOs, and CaTiOs, in each period. Therefore, the phonon peaks should not disappear from the spectra completely upon transition to the paraelectric phase at T. Raman intensity should rather drop to some small but non-zero value. However, this inversion symmetry breakdown appears to have a small effect in terms of atomic displacement patterns associated with phonons, and this residual above-Tc Raman intensity appears too small to be detected. Therefore, the observed temperature evolution of Raman intensities shows a behavior similar to that of symmetric two-component superlattices. 

Peaks occur in a difference map in positions in the unit cell where the model did not include enough electron density valleys appear in places where the model contained too much electron density. This information may be used to obtain more precise atomic positions, atomic displacement parameters, or atomic numbers. For example, in the last category, the identities of atoms (carbon or nitrogen) in a tricyclic molecule were established by setting all atoms to one type (carbon in this case) in the structure factor calculation. A difference map was calculated with the calculated phases and examined for excess electron density at atomic positions (Table 9.2). It was found to be possible to distinguish between nitrogen (seven electrons) and carbon (six electrons), even though these atoms are adjacent in the Periodic Table. 

Diamond has been studied by coherent INS [16]. Fig. 11.7 shows the dispersion curves calculated by periodic DFT these are in excellent with the experimental data. Fig 11.8 shows a comparison of the INS spectrum derived from the dispersion curves and the TFXA experimental spectrum. Examination of the atomic displacements shows that the spectrum can be approximately described as C-C stretching modes above 1000 cm and deformation modes below 1000 cm. The agreement is good except for the features at 154 and 331 cm. These are not a failure of the calculation but are a consequence of the use of graphite for the analysing crystal ( 3.4.2.3) and will be discussed in 11.2.2. 

The atomic displacement u(r) produces an excess of energy (eV) that surpasses the mean thermal energy (meV) leading to a change in the equilibrium position from r to r. Thus, in the dynamic matrix we have to add the periodic perturbation ... 

Before solving the equations of motion, the periodic nature of the solid must be taken into account by including the dependence of the atomic displacements on the wavevector q, and hence the displacement is modified as follows... 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

The first two-periodic all-electron HP LCAO calculations of the rutile relaxed surfaces, made in [779], gave atomic displacements of surface atoms that did not differ significantly from the later results of DPT-PW investigations. Purther periodic LCAO studies of Ti02 bare surfaces have been made in [777,799,800]. Por studies of H2O adsorption on Ti02 the single-slab periodic HP-LCAO and DPT-LCAO methods were first applied in [790] and compared with PW-DPT results to test various methods with cyclic- and embedded-cluster calculations and resolve discrepancies between the methods. 

FIGURE 9.8. Atomic displacements of normal modes for selected frequencies in periodic rrafw-polyacetylene. The numbers indicate the frequency and are given in cm . 

A neutron is captured by a nucleus, producing an excited compound nucleus with a mass of (M2 + 1) which is unstable and decays over a very short time period ( 10 s), resulting in the emission of neutrons, protons and y photons or fission products. The nuclear recoil caused by the emission of the decay products can lead to atom displacement, just as in the previous case. In contrast to PKAs, these atoms are termed recoil atoms [44]. 

What are the atoms really doing during a lattice vibration First of all, a real lattice vibration must be a complex combination of atomic displacements in the three directions of space, but still its nature and physical meaning do not change from those of the simple one-dimensional example given above. Molecules in crystals are very constrained, so that oscillations are restricted to relatively small displacements from equilibrium. This is the reason why the harmonic assumption can be successfully applied in lattice dynamics. We are now dealing with a dynamic, time-dependent phenomenon, in which atomic displacements are periodic in time (equation 2.19). The illustrations in Fig. 6.2 and 6.6 differ in one crucial point the picture of atomic orbital combinations is static, while vibrational modes have an additional phase term that depends on time, and describes the periodic oscillation of the nuclei around the... 

A crystal structure consists of a periodic pattern of atomic nuclei and a continuous (also periodic) distribution of electron density. As X-rays are scattered by electrons, it is possible to extract structure factors from the reflection intensities, and, by their Fourier transform, to calculate the electron density map, the peaks of which correspond to the centres of atoms. We can then approximate the structure by placing at these points isolated atoms of corresponding elements with ideal spherical symmetry. The positions (coordinates) of these atoms and their displacement parameters (see Atomic displacements below), can be refined by the least-squares technique so as to achieve the best... 

The periodic boundary conditions require that the atomic displacements for atoms separated by a translation N. a., or a sum of such translations, must be the same... 

Now, we show in Appendix A that for the diatomic linear chain, the eigenvalues and the atomic displacements are periodic functions of t = Z-rrm/a. The extension of the results to the general three-dimensional case is straightforward and gives... 

A) Periodicity of Elgenfrequencies and Atomic Displacements in Reciprocal Space... 

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