Solid momentum conservation

Returning to the momentum equations, the third term on the right-hand side of Eqs. (145) and (146) contains the gas-solid momentum-exchange models Likewise, the fourth term on the right-hand side of Eqs. (146) contains the solid-solid momentum-exchange model fsm . Note that because solid-solid interactions conserve momentum, the latter must be defined such that... 

There is a condition of momentum conservation for photons and electrons which must also be satisfied in the photoemission process. For band electrons, for which the Bloch wavefunctions are characterized by the wavenumber k (proportional to the momentum p of the electron), the momentum conservation condition is important to determine the angular distribution of the photoemitted electrons. Angular J esolved FhotoEmission spectroscopy (ARPES), schematized in Fig. 2, is potentially able to provide, and has been used to obtain, the E(fc) dispersion curves for solids. 

Here the and -f signs correspond to those Raman processes in which an elementary excitation is created or annihilated (Stokes and anti-Stokes processes, respectively). In crystalline solids, the quasi-momentum conservation gives the following relation between the wave vectors k, of the incident light, kj of the scattered light, and q of the elementary excitation ... 

The prediction of such limits rests on the integrations in eqs. 4.64 and 4.63, and these in turn rest on the assumption that momentum conservation in the electrode is not a constraint. If it were, then only electrons and holes very near the Fermi level could be involved in interfacial ET and HT, because replacement of an electron by a hole at the Fermi level does not change the electronic momentum of the solid, whereas the removal of a low-momentum electron of energy well below the Fermi level, which creates a hot (high-momentum) hole, does. If this were a real constraint, then ET rate coefficients at metal electrodes should behave like ET rate constants for homogeneous ET, and show an inverted region past the Marcus maximum rather than tending to a limit. 

In order to obtain the energy band dispersion from UPS experiments, we need to use the momentum conservation role as well as the energy conservation role upon photoelectron emission. A three-step model is generally adopted for the photoelectron spectroscopy process, which consists of an optical dipole excitation in the solid, followed by transport to the surface and emission to the vacuum [37, 38]. General assumptions are as follows (i) both the energy and momentum of the electrons are conserved during the optical transition, (ii) the momentum component parallel to the surface is conserved while the electron escapes through the surface, and (iii) the final continuum state in the solid is a parabolic free-electron-like band in a constant inner potential Vq,... 

In the continuum (Euler-Euler)-type formulation, the gas, liquid, and solid phases are assumed to be continuum and the volume-averaged mass and momentum equations (see Table 6.10) are solved for each phase separately to predict the pressure, phase holdup, and phase velocity distributions. As a result of time and volume averaging, additional terms appear in the momentum conservation equations. These additional terms need closure models and such unclosed terms are highlighted in Table 6.10. 

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