Waves, interference

Diffraction is based on wave interference, whether the wave is an electromagnetic wave (optical, x-ray, etc), or a quantum mechanical wave associated with a particle (electron, neutron, atom, etc), or any other kind of wave. To obtain infonnation about atomic positions, one exploits the interference between different scattering trajectories among atoms in a solid or at a surface, since this interference is very sensitive to differences in patii lengths and hence to relative atomic positions (see chapter B1.9). 

From the above descriptions, it becomes apparent that one can include a wide variety of teclmiques under the label diffraction methods . Table Bl.21.1 lists many techniques used for surface stmctural detemiination, and specifies which can be considered diffraction methods due to their use of wave interference (table Bl.21.1 also explains many teclmique acronyms commonly used in surface science). The diffraction methods range from the classic case of XRD and the analogous case of FEED to much more subtle cases like XAFS (listed as both SEXAFS (surface extended XAFS) and NEXAFS (near-edge XAFS) in the table). 

Table 81.21.1. Surface stmctural detemiination methods. The second colunni indicates whether a technique can be considered a diffraction method, in the sense of relying on wave interference. Also shown are statistics of surface stmctural detemiinations, extracted from the Surface Stmcture Database [14], up to 1997. Counted here are only detailed and complete stmctural determinations, in which typically the experiment is simulated computationally and atomic positions are fitted to experiment. (Some stmctural detemiinations are perfomied by combining two or more methods those are counted more than once in this table, so that the colunnis add up to more than the actual 1113 stmctural detemiinations included in the database.)...

FIGURE 1.19 In this illustration, the peaks of the waves of electromagnetic radiation are represented by orange lines. When radiation coming from the left (the vertical lines) passes through a pair of closely spaced slits, circ ular waves are generated at each slit. These waves interfere with each other. Where they interfere constructively (as indicated by the positions of the dotted lines), a bright line is seen on the screen behind the slits where the interference is destructive, the screen is dark. 

Examples of wave patterns, (a) Floats produce standing water waves. (Z>) X rays generate wave interference patterns, (c) Protruding atoms on a metal surface generate standing electron waves. 

The last two CGC in Eq. (12) evidently dictate that rather different partial wave interference contributions are made to each of the angular parameters. This will impact on the dynamical information conveyed by each one. Equally important, the phase subexpression... 

Aji/Ak. At the points of maximum amplitude, the two original plane waves interfere constructively. At the nodes in Figure 1.2(a), the two original plane waves interfere destructively and cancel each other out. 

Figure 2.74 Schematic representation of the electron wave interference effects giving rise to the Kronig fine structure on X-ray absorption edges (sec text).

Harigaya K, Kobayashi Y, Kawatsu N et al (2004) Tuning magnetism and novel electronic wave interference patterns in nanographite materials. Physica E Low Dimens Syst Nanos-truct 22 708-711... 

In LEED, electrons of well-defined (but variable) energy and direction of propagation diffract off a crystal surface. Usually only the elastically diffracted electrons are considered and we shall do so here as well. The electrons are scattered mainly by the individual atom cores of the surface and produce, because of the quantum-mechanical wave nature of electrons, wave interferences that depend strongly on the relative atomic positions of the surface under examination. 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

The primary goal of quantum control is the manipulation of dynamics phenomena. To practically meet this goal, the closed loop procedure in Fig. 2 was suggested to circumvent the lack of detailed quantitative information about the Hamiltonians of most realistic systems[2]. This paucity of quantitative Hamiltonian information is an especially serious matter, as it is expected that delicate quantum wave interference will be required to obtain the highest degree of control. 

Recent years have seen a flurry of activity in both the theoretical and experimental aspects of control over molecular processes [1] (see also S. A. Rice, Perspectives on the Control of Quantum Many-Body Dynamics Application to Chemical Reactions, this volume). Most of the emphasis has been on the use of optical fields as a means for control, although other approaches can be envisioned in special circumstances [2]. The key underlying principle of the overall subject is the achievement of control through the manipulation of quantum wave interferences [1, 3], although full control will surely not be lost in the incoherent regime. 

The LCAO-MO in Eq. 1 has a lower energy than either of the atomic orbitals used in its construction. The two atomic orbitals are like waves centered on different nuclei. Between the nuclei, the waves interfere constructively with each other in the sense that the total amplitude of the wavefunction is increased where they overlap (Fig. 3.29). The increased amplitude in the internuclear region means that there is a greater probability of finding the electron in the orbital in locations where it can interact with both nuclei. Because an electron that occupies the molecular orbital is attracted to both nuclei, it has a lower energy than when it is confined to an atomic orbital on one atom. A combination of atomic... 

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