Free-electron scale

Figure 12-5. Kcprcscmauun of Uie calculated injcciiou curretu on a 111 j vs scale. Tlic dashed line indicates tile slopes predicted by Fowler Nordheiin tunneling theory lor A=0.8eV assuming that the effective mass equals the free electron mass.

However, time-resolved X-ray diffraction remains a young science. It is still impossible, or is at least very difficult, to attain time scales below to a picosecond. General characteristics of subpicosecond X-ray diffraction and absorption are hardly understood. To progress in this direction, free electron laser X-ray sources are actually under construction subject to heavy financial constraints. Nevertheless, this field is exceptionally promising. Working therein is a challenge for everybody ... 

The Vacuum Reference The first reference in the double-reference method enables the surface potential of the metal slab to be related to the vacuum scale. This relationship is determined by calculating the workfunction of the model metal/water/adsorbate interface, including a few layers of water molecules. The workfunction, — < ermi. is then used to calibrate the system Fermi level to an electrochemical reference electrode. It is convenient to choose the normal hydrogen electrode (NHE), as it has been experimentally and theoretically determined that the NHE potential is —4.8 V with respect to the free electron in a vacuum [Wagner, 1993]. We therefore apply the relationship... 

The lowest level of the conduction band for metals Vc and the highest level of the valence band for semiconductors is very often used as a reference point for the energies of the Fermi levels in this book, however, the energy of a free electron at rest in a vacuum will be used as the reference point for the scale of the Fermi levels (cf. Fig. 3.2.). 

Fortunately, in favorable cases enhancement mechanisms operate which increase the signal from the interface by a factor of 105 — 106, so that spectra of good quality can be observed - hence the name surface-enhanced Raman spectroscopy (SERS). However, these mechanisms seem to operate only on metals with broad free-electron-like bands, in particular on the sp metals copper, silver and gold. Furthermore, the electrodes must be roughened on a microscopic scale. These conditions severely limit the applicability of Raman spectroscopy to electrochemical interfaces. Nevertheless, SERS is a fascinating phenomenon, and though not universally applicable, it can yield valuable information on many interesting systems, and its usefulness is expected to increase as instrumentation and preparation techniques improve. 

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. 

The major advantage of time-resolved X-ray techniques, as compared to optical spectroscopy, is that their wavelength X as well as the pulse duration r can be chosen to fit the atomic scales. This is not the case for optical spectroscopy, where the wavelength X exceeds interatomic distances by three orders of magnitude at least. Unfortunately, X-ray techniques also have their drawbacks. They require large scale instruments such as the synchrotron. Even much larger instruments based on free electron lasers are actually under construction. The... 

The calculated extinction spectrum of a polydispersion of small aluminum spheres (mean radius 0.01 jam, fractional standard deviation 0.15) is shown in Fig. 11.4 both scales are logarithmic. In some ways spectral extinction by metallic particles is less interesting than that by insulating particles, such as those discussed in the preceding two sections. The free-electron contribution to absorption in metals, which dominates other absorption bands, extends from radio to far-ultraviolet frequencies. Hence, extinction features in the transparent region of insulating particles, such as ripple and interference structure, are suppressed in metallic particles because of their inherent opacity. But extinction by metallic particles is not without its interesting aspects. 

Electron transfer is a fast reaction ( 10-12s) and obeys the Franck-Condon Principle of energy conservation. To describe the transfer of electron between an electrolyte in solution and a semiconductor electrode, the energy levels of both the systems at electrode-electrolyte interface must be described in terms of a common energy scale. The absolute scale of redox potential is defined with reference to free electron in vacuum where E=0. The energy levels of an electron donor and an electron acceptor are directly related to the gas phase electronic work function of the donor and to the electron affinity of the acceptor respectively. In solution, the energetics of donor-acceptor property can be described as in Figure 9.6. 

Phenol radical cations exist only in strong acidic solutions (pKa -1) [1, 2]. However, in non-polar media phenol radical cations with lifetimes up to some hundred nanoseconds were obtained by pulse radiolysis [3], The free electron transfer from phenols (ArOH) to primary parent solvent radical cations (RX +) (1) resulted in the parallel and synchroneous generation of phenol radical cations as well as phenoxyl radicals in equal amounts, caused by an extremely rapid electron jump in the time scale of molecule oscillations since the rotation of the hydroxyl groups around the C - OH is strongly connected with pulsations in the electron distribution of the highest molecular orbitals [4-6]. 

Figure 8.24 Term diagram for XPS, AES, and EDX. The vacuum energy Evac defines the zero point of the energy scale. The binding energy of electrons Eb, the Fermi energy Ef and the kinetic energy of free electrons Ekin are indicated.

---