Standing waves, electronic

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Figure Bl.19.6. Constant current 50 mn x 50 mn image of a Cu(l 11) surface held at 4 K. Tliree monatomic steps and numerous point defects are visible. Spatial oscillations (electronic standing waves) with a...

Crommie M F, Lutz C P and Eigler D M 1993 Imaging standing waves in a two-dimensional electron gas Nature 363 524... 

Fig. 3.17 The two possible sets of standing waves at the Brillouin zone boundary. Standing wave A concentrates electron density at the nuclei, whereas wave B concentrates electron density between the nuclei. Wave A thus has a lower energy than wave B.

The picture of the electron in an orbit as a standing wave does, however, pose the important question of where the electron, regarded as a particle, is. We shall consider the answer to this for the case of an electron travelling with constant velocity in a direction x. The de Broglie picture of this is of a wave with a specific wavelength travelling in the x direction as in Figure 1.4(a), and it is clear that we cannot specify where the electron is. 

Figure 1.3 (a) A standing wave for an electron in an orbit with n = 6. (b) A travelling wave, resulting... 

The construction of a TXRF system, including X-ray source, energy-dispersive detector and pulse-processing electronics, is similar to that of conventional XRF. The geometrical arrangement must also enable total reflection of a monochromatic primary beam. The totally reflected beam interferes with the incident primary beam. This interference causes the formation of standing waves above the surface of a homogeneous sample, as depicted in Fig. 4.1, or within a multiple-layered sample. Part of the primary beam fades away in an evanescent wave field in the bulk or substrate [4.28],... 

In recent years, scanning tunneling electron microscopes have produced pictures of electron waves, an example of which appears in Figure 7-15c. Here, two atoms on an otherwise smooth metal surface act like the floats in Figure 7-15a. and cause the electrons in the metal to set up a standing wave pattern. 

A three-dimensional (3D) piece of metal can be considered as a crystal of infinite extension in the directions x, y and z with standing waves with the wave numbers k, ky and k, each being occupied with two electrons as a maximum. In a piece of bulk metal the energy differences Sk y are so small that A->0, identical with quasi-free continuously distributed electrons. Since the energy of free electrons varies with the square of the wave numbers, its dependence on k describes a parabola. Figure 4a shows these relations. 

We have already seen (p. 2) that the individual electrons of an atom can be symbolised by wave functions, and some physical analogy can be drawn between the behaviour of such a wave-like electron and the standing waves that can be generated in a string fastened at both ends—the electron in a (one-dimensional) box analogy. The first three possible modes of vibration will thus be (Fig. 12.1) ... 

According to De Broglie an electron in a Bohr orbit is associated with a standing wave. To avoid self destruction by interference an integral number of wavelengts are required to span the orbit of radius r, which implies n — 2nr, or nh/2n = pr, which is the Bohr condition. As a physical argument the wave conjecture is less plausible, but not indefensible. One possible interpretation considers the superposition of several waves rather than a single monochromatic wave to simulate the behaviour of a particle. 

The electron behaves as a standing wave with an integral number of half wavelengths fitting into the one-dimensional box, with boundary conditions... 

In the quantum-mechanical description of atoms and molecules, electrons have characteristics of waves as well as particles. In the familiar case of the hydrogen atom, the orbitals Is, 2s, 2p,... describe the different possible standing wave patterns of electron distribution, for a single electron moving in the potential field of a proton. The motion of the electrons in any atom or molecule is described as fully as possibly by a set of wave functions associated with the ground and excited states. 

A standing wave (SW) microwave linear accelerator consists of a linear array of resonant cavities that are energized by a common source of microwave power. These cavities are nearly isolated by webs with small-diameter apertures, and the high-energy electron beam passes through these apertures. However, they are coupled through intermediate cavities, which stabilize the microwave phase relationship between the accelerating cavities. 

By 1926, just in time for Davidson and Germer s 1927 experiment, Schrodinger put into mathematical form an idea due to de Broglie (1924). It was that the sometimes wavelike character of electrons could be the basis of the quantum states. The waves had to fit into the space available (e.g., the distance between two nuclei in a solid), and it was this need to fit and make a standing wave that made only certain states—certain wavelengths (or energies)—possible. 

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