Scattering by a free electron

The second subregion corresponds to large to and q and is related to close collisions (the knock-on). At very large transferred momenta (qa0> 1) the inelastic scattering by a molecule (an atom) is actually the elastic scattering by a free electron with the cross section given by Rutherford formula. In this case the function /(to, q) can be presented analytically as a delta function ... 

At the point P, the wave scattered by a free electron is polarized in the plane Eq/s. Its amplitude is proportional to the component of the acceleration of the electron perpendicular to s,... 

A photon (2 = 1 pm) is scattered by a free electron at an angle 9 = 90°. What part of its energy W does the photon transmit to the electron ... 

The first factor in square brackets represents the Thomson cross-section for scattering from a free electron. The second square bracket describes the atomic arrangement of electrons through the atomic form factor, F, and incoherent scatter function, S. Finally, the last square bracket contains the factor s(x), the molecular interference function that describes the modification to the atomic scattering cross-section induced by the spatial arrangement of atoms in their molecules. 

By definition, the atomic scattering factor /(x) is given in terms of the amplitude scattered by a single electron at the lattice point. It is useful, however, to have the scattered amplitude/I in terms of the incident amplitude Aq. From classical electromagnetic theory, it follows that if a wave of amplitude Aq is incident on a free electron, the amplitude A of the radiation emitted in the forward direction, at a distance R (meters) from the electron, is given by... 

Fig. 3.9. Scattering of a polarized electromagnetic wave of amplitude Eq and intensity Iq by a free electron of charge -e and mass m. The accelerated electron emits a secondary wave E( >) of intensity /( >)...

In earlier discussions, it was noted that the motion of an electron in a perfect crystal can be represented by a free electron with an effective mass m somewhat different than the real mass of an electron. In this model, once the effective mass has been determined, the atoms of the perfect crystal can be discarded and the electron viewed as moving within free space. If the crystal is not perfect, however, those deviations from perfection remain after the perfect crystal lattice has been discarded and act as scattering sites within the free space seen by the electron in the crystal. 

Example 3. The mean free path of electrons scattered by a crystal lattice is known to iavolve temperature 9, energy E, the elastic constant C, the Planck s constant the Boltzmann constant and the electron mass M. (see, for example, (25)). The problem is to derive a general equation among these variables. 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

Inelastic photon scattering processes are also possible. In 1928, the Indian scientist C. V. Raman (who won the Nobel Prize in 1930) demonstrated a type of inelastic scattering that had already been predicted by A. Smekal in 1923. This type of scattering gave rise to a new type of spectroscopy, Raman spectroscopy, in which the light is inelastically scattered by a substance. This effect is in some ways similar to the Compton effect, which occurs as a result of the inelastic scattering of electromagnetic radiation by free electrons. 

The damping term in a metal is due to the scattering suffered by the free electrons with atoms and electrons in the solid, which produces the electrical resistivity. 

The Fermi surface plays an important role in the theory of metals. It is defined by the reciprocal-space wavevectors of the electrons with largest kinetic energy, and is the highest occupied molecular orbital (HOMO) in molecular orbital theory. For a free electron gas, the Fermi surface is spherical, that is, the kinetic energy of the electrons is only dependent on the magnitude, not on the direction of the wavevector. In a free electron gas the electrons are completely delocalized and will not contribute to the intensity of the Bragg reflections. As a result, an accurate scale factor may not be obtainable from a least-squares refinement with neutral atom scattering factors. 

More recent work has shown, however, that an exponential decay of the screening potential as in (21) is not correct, and that round any scattering centre the charge density falls off as r 3 cos 2kFr. This we shall now show, by introducing the phase shifts t/, defined as follows (cf. (13)). Consider the wave functions Fx of a free electron in the field of an impurity. These behave at large distances from the impurity as (Mott and Massey 1965)... 

From the gas phase scattering data we conclude that a plane wave state for the electron in liquid helium lies at positive energy relative to the vacuum level in agreement with Sommers electron injection experiment. We proceed now to a semiquantitative treatment of free electron states in liquids characterized by a positive scattering length, which will be used to estimate the energy of interaction of a free electron with liquid helium (18). 

Fig. 10.16. Absorption of X-rays as a function of photon energy h by a free atom and by an atom in a lattice. The spherical electron wave from the central atom is scattered back by the neighbouring atoms, which leads to interference, which is constructive or destructive depending on the wavelength of the electron wave (or the kinetic energy of the electron) and the distance between the atoms. As a result, the X-ray absorption probability is modulated and the spectrum shows fine structure which represents the EXAFS spectrum [37],...

If the time-dependent wave function of electrons Vl(t) is known, the current density approach outlined in Section 2 can be used directly to calculate the energy loss and also to analyze the spatial distribution of related effects. This is the case when the energy loss to a free electron is considered. Assuming that the electron is initially at rest, we can describe its state in the projectile frame as a state of scattering in the projectile Coulomb field. The wave function in the laboratory frame is then given by the expression... 

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