Reflection of electrons

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

A similar argument can be made for electronic features such as electron density, polarization and polarizability. These are critically dependent on the ionization state of the molecule, but the conformahonal state is also highly influential. One highly approximate yet useful reflection of electron density is afforded by the polar surface area (PSA), a measure of the extent of polar (hydrophilic) regions on a molecular surface (see Chapter 5). 

Figure 1. Various configurations of an optical field for laser-induced electron optics (a) focusing of electrons by the TEM oi mode configuration of the laser field (b) reflection of electrons by the evanescent laser field (c) reflective focusing of electrons by the curved evanescent laser field.

Let us consider the possibility of reflection of electrons by an evanescent laser wave formed due to total internal reflection of femtosecond laser pulses from a dielectric-vacuum interface [4] (Fig. lb). Such a laser field was considered elsewhere [7, 8] to effect the mirror reflection of atoms (references to the latest works on the mirror reflection of atoms can be found in Refs. 9 and 10). The light intensity distribution in the evanescent wave in the vacuum may be represented in the form [11]... 

The character of reflection of electrons from the evanescent wave strongly depends on the relationship between the duration r of the laser pulse and the time of flight of an electron through the laser wave, rtr. It may be shown that when the laser pulse duration is much longer than the characteristic transit time rtr, the character of reflection of the electrons is close to the mirror. Where the relationship between these times is reversed, the mirrorlike character of reflection is disturbed. Let us make some simple estimates of the laser field and electron beam parameters with which the reflection of electrons is possible. 

The possibility of reflection of electrons by an evanescent wave formed upon the total internal reflection of femtosecond light pulses from a dielectric-vacuum interface is quite realistic. The duration of the reflected electron pulses may be as long as 100 fs. In the case of electrons reflecting from a curved evanescent wave, one can simultaneously control the duration of the reflected electron pulse and affect its focusing (Fig. lc). Of course, one can imagine many other schemes for controlling the motion of electrons, as is now the case with resonant laser radiation of moderate intensity [9, 10]. In other words, one can think of the possibility of developing femtosecond laser-induced electron optics. Such ultrashort electron pulses may possibly find application in studies into the molecular dynamics of chemical reactions [1,2]. 

A is a reflection of electronic factors due to the added electron weakening the metal—metal multiple bond slightly and hence shifting the metal atom separation toward the high side of observed quadruple bond values, a compounding of this effect would be anticipated in the more reduced Re2 Cl4(PEt3)4 case. Such is not observed. The separation actually decreases to... 

Reflection of Electron-Transfer Step in Reaction Stereochemistry... 

In the SC state the excess current Iexc (1), which is due to the Andreev reflection of electron quasiparticles from the N — S boundary inaW-c — S contact (c stands for constriction ), can be written as... 

T o incorporate the reflections of electrons from the p+- i interface into the theory, consider the effect of the electron mirror on the photoconductivity of the i layer. It is assumed that the electrons are totally reflected back into the i layer when the internal electric fields are highest at Jd —> 0. The electrons are not reflected and back-diffusion exists when the internal fields are low at JT = 0 and = Jp. With the assumption that the electrons carry most of the current and that /d /, where /d is the electron diffusion length and / is the length of the i layer, the resistance of the i layer is... 

We have measured with great accuracy the reflectivity of electron doped Pr2 sCe, Cu() at various Ce doping levels. An optical conductivity spectral weight analysis shows that a partial gap opens at low temperatures for Ce concentrations up to x = 0.15. A spin density wave model reproduces satisfactorily the data. 

We think we understand the regular reflection of light and x-rays—and we should understand the reflections of electrons as well if electrons were... 

Bragg reflection of electrons in liquid alloys. Phys. Letters (1967). 

The next three subsections address the not-so-transparent concept of how and why bands form in solids. Three approaches are discussed. The first is a simple qualitative model. The second is slightly more quantitative and sheds some light on the relationship between the properties of the atoms making up a solid and its band gap. The last model is included because it is physically the most tangible and because it relates the formation of bands to the total internal reflection of electrons by the periodically arranged atoms. 

In the case of metals their UV-VIS properties can be described by quantized plasma vibrations, so-called plasmons. Plasmons can be excited by inelastic scattering of electrons or reflection of electrons or photons on thin metal layers. More details are available in the literature [13]. 

Electrons, however, do have one advantage. Because they are charged they can be focused by magnetic lenses to form an image. The mechanism of diffraction as an electron beam passes through a thin flake of solid allows defects such as dislocations to be imaged with a resolution close to atomic dimensions. Similarly, diffraction (reflection) of electrons from surfaces of thick solids allows surface details to be recorded, also with a resolution close to atomic scales. Thus although electron diffraction is not widely used in structure determination it is used as an important tool in the exploration of the microstructures and nanostructures of solids. 

Show that the four functions of (13.90) have the indicated eigenvalues with respeet to a (tJ xz) reflection of electronic coordinates. Start by showing that this reflection converts to —[Pg.456]

The first two functions in (13.89) are symmetric. They therefore go with the antisymmetric singlet spin function (11.60). Clearly, these two spatial functions have different energies. The last two functions in (13.89) are antisymmetric and hence are the spatial factors in the wave functions of the two 2 terms. The four functions in (13.89) are found to have eigenvalue +1 or -1 with respect to reflection of electronic coordinates in the xz cr symmetry plane containing the molecular (z) axis (Prob. 13.30). The superscripts + and - refer to this eigenvalue. 

Kapitza, P., and Dirac, P. A. M. (1933). Reflection of electrons by standing light waves. Proceedings of Cambridge Philosophical Society, 29, 297-300. 

Most readers are probably familiar with the Bragg formula for x-ray diffraction (XRD), 2d sin 6 = nX, and the selection rules that tell us that the sum of the Miller indices for body-centered cubic (bcc) crystals must be even and the indices for face-centered cubic (fee) crystals must be all even or all odd in order to produce a diffraction peak. The intent of this chapter is to give a more general derivation of the Laue conditions leading to Bragg reflections of electrons and phonons that take place in three-dimensional (3D) crystals. These reflections from lattice planes within the crystal are fimdamental to the imderstand-ing of the thermal and electronic properties of materials. In the process, the reciprocal lattice will be introduced which will be widely used in subsequent chapters to describe how materials behave thermally, electronically, magnetically, and photonically. 

One can visualize the distribution of momenta in momentum space in the absence of an applied electric field as a sphere with radius pp centered about the origin as shown in Figure 18.3. This we call the Fermi sphere and its surface is called the Fermi surface. Later, we show that the Fermi surface is distorted by Bragg reflections of electrons near boundaries of the Brillouin zones. The electrons below the Fermi energy are in filled states and cannot respond directly to the applied electric field. Only those electrons at the Fermi energy can be accelerated by the applied field to a velocity Vd + vp. However, this creates new states at the Fermi surface in which electrons just below the Fermi level can move into, and so on. The net result is that each electron is able to increase its velocity by Vd. 

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