The wave nature of electrons

The second idea related to the LEED technique is, as its name indicates, the diffraction phenomenon. With the wave nature of the electrons established, information on the interaction of a beam of light with matter can be extrapolated to understand the interaction of a beam of electrons with a crystal. In this sense, the corresponding relationship is that of a monochromatic beam of light—that with a single frequency and single wavelength—with that of a beam of electrons of fixed energy. 

ELECTRON MICROSCOPE. The concepts that eventually led to the development of electron microscopes came out or the discovery of the wave nature of the electron in 1924. The effective wavelength of the electrons varies with accelerating voltage and is less than 1 A >. = /< 150/ V) A. This short wavelength makes possible far better resolution and higher magnification in the electron microscope as compared with the optical microscope. 

Wavefunctions of electrons in atoms are called atomic orbitals. The name was chosen to suggest something less definite than an orbit of an electron around a nucleus and to take into account the wave nature of the electron. The mathematical expressions for atomic orbitals—which are obtained as solutions of the Schrodinger equation—are more complicated than the sine functions for the particle in a box, but their essential features are quite simple. Moreover, we must never lose sight of their interpretation, that the square of a wavefunction tells us the probability density of an electron at each point. To visualize this probability density, we can think of a cloud centered on the nucleus. The density of the cloud at each point represents the probability of finding an electron there. Denser regions of the cloud therefore represent locations where the electron is more likely to be found. 

The wave nature of the electron was discovered in diffraction experiments and since that time electrons have been widely used in structural studies both of crystals and of gases. A convenient potential is about 40 kV and this gives a wavelength of about 0-06 A. This is the right order of magnitude for diffraction studies of molecular structure. 

For a potential difference of 150 V, or 60,000 V we have therefore X = 1 A or 0.05 A respectively, which agrees with the experiments carried out by Davisson and Germer with nickel single crystals or by Thomson with gold foil. With the observed angles of deviation the wave length could be calculated from the lattice spacings in the same way as with X-rays. We can therefore look upon (4 ) also as the purely experimental result of these experiments from which the wave nature of the electron and the correctness of equation (4 ) appear experimentally. 

The squared wave functions are the probability densities and show the difference between classical and quantum mechanical behavior. Classical mechanics predicts that the electron has equal probability of being at any point in the box. The wave nature of the electron gives it the extremes of high and low probability at different locations in the box. 

At large distances from the nucleus, the electron density, or probability of finding the electron, falls off rapidly. The 2s orbital also has a nodal surface, a surface with zero electron density, in this case a sphere with r = 2uq where the probability is zero. Nodes appear naturally as a result of the wave nature of the electron they occur in the functions that result from solving the wave equation for 4. A node is a surface where the wave function is zero as it changes sign (as at r = 2aQ, in the 2s orbital) this requires that = 0, and the probability of finding the electron at that point is also zero. 

This chapter introduces the electronic structure of the atom, from the early shell structure of the Bohr theory, using the single principal quantum n, through the wave nature of the electron, the Schrodinger wave equation, and the need for the four quantum numbers, n, /, m, and to describe the occurrence of the s, p, d and / orbitals. The evidence for this more complicated shell structure is seen in the photoelectronic spectra of the elements this justifies the one electron orbital description of the atom and from which the s-, p-, d- and/- block structure of the Periodic Table is developed. 

However, although the Bohr theory, involving a single quantum number n, was adequate to explain the line spectrum of the hydrogen atom with a single valence electron (Figures 2.4 and 2.5, respectively), it was inadequate to explain, in detail, the line spectrum of elements with more than one electron. To do this, it was found necessary to introduce the idea of three further quantum numbers, in addition to the principal quantum number, n. These arise from the wave nature of the electron. 

Footnote The Wave Nature of the Electron. So far the electron has been considered as a particle, with clearly quantised energy levels, that can be precisely measured, as in the emission lines of the spectrum of hydrogen. Because the electron is so small and light, the accuracy with which it can be measured is very uncertain. This is associated with the Heisenberg Uncertainty Principle, which states that it is impossible to determine both the position and momentum of an electron simultaneously , i.e. Ax Ap = hl2it, where Ax is the uncertainty in measuring the position of the electron and Ap is the uncertainty in measuring the momentum (p = mass X velocity) of the electron. The two uncertainties bear an inverse relationship to each other. Consequently, if the position of the... 

BohrPT Bohr Model of the Atom - Four Quantum Numbers -Electron Configuration of the Atom - Electron Shells - Shapes of Orbitals - Wave Nature of the Electron - Wave Functions, Radial and... 

To conclude this section, there is one other phenomenon we should like to discuss, viz. the Raman effect. Let it be mentioned beforehand, however, that this is not a revolutionary discovery, like, for example, the discovery of the wave nature of the electron, but an effect which was predicted by the quantum theory (Smekal (1923), Kramers-Heisenberg) some years before it was found experimentally, though it can also be explained within the framework of classical physics (Cabannes (1928), Rocard, Placzek) its great importance rests rather on the facility with which it can be applied to the study of molecules, and on the colossal amount of material relating to it which has been accumulated so quickly. The effect was discovered simultaneously (1928) by Raman in India, and by Landsberg and Mandelstam in Russia. They found that scattered light contains, in addition to the frequency of the incident light, a series of other frequencies. 

The QSE arise from the wave nature of the electrons, and the ensuing discrete energy-level structure of the individual clusters. An important point here is that,... 

The wave nature of the electron and the physical implications thereof were discussed recently in some detail (Boeyens, 2010). As in the theory of general relativity it is accepted that an empty universe is featureless and flat, but that curvature of space-time causes wavelike distortion of the vacuum. The equivalent of an infinite plane wave in flat space develops interference effects, like wave packets, in curved space, interpreted as units of mass and energy. 

De Broglie s hypothesis and Heisenberg s uncertainty principle set the stage for a new and more broadly applicable theory of atomic structure. In this approach, any attempt to define precisely the instantaneous location and momentum of the electron is abandoned. The wave nature of the electron is recognized, and its behavior is described in terms appropriate to waves. The result is a model that precisely describes the energy of the electron while describing its location not precisely but rather in terms of probabilities. 

Open problems in writing the basic organic chonistry textbook include the selection of concepts for the representation of the material, but also the level of the explanation of the complex phenomena such as reaction mechanisms or the electron structure. Here I propose the compromises. First compromise is related to the mode of the systematization of the contents, which can traditionally be based either on the classes of compounds, or on the classes of reactions. Here, the main chapter titles contain the reaction types, but the subtitles involve the compound classes. The electronic effects as well as the nature of the chemical bond is described by using the quasi-classical approach starting with the wave nature of the electron, and building the molecular orbitals from the linear combination of the atomic orbitals on the principle of the qualitative MO model. Hybridization is avoided because all the phenomena on this level can be simply explained by non-hybridized molecular orbitals. 

In the kinetic field there is another way in which the small mass of the proton may be important. It is well known that the behaviour of electrons cannot be accounted for in terms of a particulate model but that it is necessary to take into account the wave nature of the electron on the other hand, it is usually supposed that the motion of nuclei can be described with sufficient accuracy by the laws of classical mechanics. This is undoubtedly true for most nuclei, but calculation shows that the proton may, on account of its small mass, show considerable deviations from classical behaviour. This phenomenon is often described as the tunnel effect and should be detectable experimentally, especially by a detailed analysis of kinetic isotope effects. At present the experimental evidence is meagre, but the problem is an interesting one and will be treated in some detail. 

Quantum Mechanical Tunneling If the material of interest is an insulator, classical physics predicts that electrons incident on the metal/insulator interface will be reflected if the energy of an electron is less than the interfacial potential barrier of the interface. The electron cannot penetrate the barrier, and its passage from one electrode to the other is inhibited. However, quantum mechanics predicts that an electron will overcome this barrier due to the wave nature of the electrons. The electron wavefunction decays rapidly with depth of penetration from the electrode/insulator interface, and for barriers of microscopic thickness, the wavefunction is essentially zero at the opposite interface [18]. This means that the probability of finding the electron is essentially zero at the other electrode. However, if the barrier is very thin (<5 nm), the wavefunction has a nonzero value at the other electrode, allowing... 

Electromagnetic Radiation 4-12 The Photoelectric Effect 4-13 Atomic Spectra and the Bohr Atom 4-14 The Wave Nature of the Electron 4-15 The Quantum Mechanical Picture of the Atom... 

Quantum-mechanical tunneling For barrier thickness of 1-4 nm, the wave-nature of the electrons enables penetration through thin barriers. This is especially true for heavily doped semiconductors. 

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