Momentum-conservation rule

In materials with a band structure such as that sketched in Figure 4.8(b), the bottom point in the conduction band has a quite different wave vector from that of the top point in the valence band. These are called indirect-gap materials. Transitions at the gap photon energy are not allowed by the rule given in Equation (4.29), but they are still possible with the participation of lattice phonons. These transitions are called indirect transitions. The momentum conservation rule for indirect transitions can be written as... 

Thus, the chemical interconversion for equal electronic parity channels has four separated aspects i) activation via molding of reactants ii) population of TS rovibrational quantum states iii) population of reactants molded into configurations covered by the TS, and iv) relaxation towards products in their ground states. All such changes are submitted to energy and angular momentum conservation rules. 

The electronic quantum state ofthe pair H,ls> H+>= in> remains invariant at all distances. The electron transfer will not take place in a direct manner because the electronic parity is equal for both channels. The interconversion process requires aTS with parity -1. Among the states available to a system decomposable in one electron and two protons (or proton deuterium, etc) there are the hydrogen molecule ion species. The first electronic excited state (leu) ofthe molecular ion H2+ provides an "intermediate" (Q-state) for the interconversion once angular momentum conservation rules are fulfilled. The state (lau) is found above the in> and out> states leading to resonance in the cross section. This state may either relax to the (lrg) state yielding the hydrogen molecule ion and emitting a photon as this state is 2.8eV below dissociation, or it may take the product channels. This is a FC-like process. The reaction (27) is a prototype of electron transfer (ET). Thus, for any ET reaction whose in> and out> asymptotic electronic states share the same parity, the actual interconversion would require the mediation of a TS. 

Relaxation of the momentum conservation rule is another important change when the crystallite size decreases to 1-2 nm. This can be understood in terms of Heisenberg s uncertainty principle Ap Ax > h. In a large single crystal of... 

Figure 3.9 Energy and momentum conservation rules for measurements of valence band dispersion EbW (a) and the momentum conservation upon photoelectron escape to vacuum (b).

Figure 19 Schematic representation of the Raman effect (a) principle of Raman scattering, if/ is the scattering angle (b) vector representation of the momentum conservation rule.

With regard to Raman scattering or infrared absorption, one of the major tasks consists of assigning the experimentally observed lines to the different irreducible representations (species) predicted by a previous group theory analysis (see Sec. V.F) based on the supposedly known space group of the molecule or crystal. The species are labeled by the irreducible representations of the point group at the center of the first Brillouin zone because of the momentum conservation rule evoked earlier. 

With UPS the density of states (DOS) is measured, more precisely a convolution of the densities of unoccupied and occupied states (as a consequence of the final state sitting in unoccupied parts of the DOS of the studied material, obeying momentum conservation rules) which is therefore referred to as Joint Density of States (JDOS) [37]. In order to better understand these effects (and the difference between monochromatic and tuneable light sources), one can describe the flux of photo electrons in a specific direction R by using Fermi s golden mle. From this results, that the number N of the emitted photo electrons is due to the selection rule for the momentum Afc = 0, proportional to the combined DOS of the concerned bands. It is given in good approximation as [60, 86] ... 

The phonon confinement model [23] attributes the redshift of the asymmetric Raman line to relaxation of the -vector selection rule for the excitation of the Raman active phonons due to their locahzation. The relaxation of the momentum conservation rule arises from the finite crystalline size and the diameter distribution of the nanosolid in the films. When the size is decreased, the rule of momentum conservation will be relaxed and the Raman active modes will not be limited at the center of the Brillouin zone [21]. The large surface-to-volume ratio of a nanodot strongly affects the optical properties mainly due to introducing surface polarization states [28]. 

Because of the momentum conservation rule during the light scattering process, first-order Raman scattering is caused by phonons at the Brillouin zone center. The Raman scattering efficiency S with polarization detection is given by ... 

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