Fermi selection rules

See also Fermions Fermi Resonance Fermi Selection Rules Fermi Surface and Ferminm. 

FERMI SELECTION RULES. A set of selection rules for bem decay where g(E) dE is the number of slates in the energy range E to E + dE. 

Fermi selection rules and Gamow-Teller (GT) selection rules are alternative sets of rules for allowed beta transitions both are currently believed to be valid, so that a transition allowed according to either set is actually allowed. 

In a case where the transition of an energy state is from 0 to 1 in any one of the vibrational states (vi,v2,v3,. ..), the transition is considered as fundamental and is allowed by selection rules. When a transition is from the ground state to v — 2,3,. .., and all others are zero, it is known as an overtone. Transitions from the ground state to a state for which Vj = 1 and vj = 1 simultaneously are known as combination bands. Other combinations, such as v — 1, Vj = 1, v = 1, or v, — 2, v7 — 1, etc., are also possible. In the strictest form, overtones and combinations are not allowed, however they do appear (weaker than fundamentals) due to anharmonicity or Fermi resonance. 

Although the occupied orbitals are of main importance, since they are directly involved in the formation of the chemical bond, the unoccupied states also provide complementary information. In X-ray absorption spectroscopy (XAS), often denoted Near Edge X-ray Absorption Fine Structure (NEXAFS), we excite a core electron to the empty states above the Fermi level [3,4,13]. There is a close connection between XES and XAS where the former gives information on the occupied orbitals while the latter relates to the character and symmetry of the unoccupied levels. Both are governed by the dipole selection rule and the localized character of the core orbitals allows a simple atom-specific projection of the electronic structure the major difference is in the final states. In XAS the empty electronic states are probed... 

Figure 24. Effects of temperature and isotopic substitution on the spectral densities of crystalline adipic acid in the absence of Fermi resonance. Comparison between theoiy (Eq. (279)) (thick Line) and experiment [101] (grayed spectra). Left column calculations using the breaking of the IR selection rule (r)° = 0). Right column same calculations but without the breaking of the IR selection rule (r 0 = 0).

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

The metal X-edge fine structure corresponds to the transition of a Is electron of the absorbing atom to the unoccupied levels of p symmetry situated just beyond the Fermi level, and to any such hybrid levels in the conduction band which may have a p admixture (23,24,116,199). The height of the absorption edge is related to the number of p electrons lacking. These transitions obey all selection rules, and in first-row transition elements the 4p orbitals are unoccupied the 4p-5p distances are about 12 eV and the distance ratio 4p-5p 5p-6p 6p-7p 4 2 1. With the broadening of the higher levels, 5p and 6p absorption often overlap. 

The electronic transitions probed by x-ray absorption spectroscopy involve the excitation of a core electron into either unoccupied bound electron states near the Fermi level of the material or at higher energies into the continuum of states producing a photoelectron. These electronic excitations must obey spectroscopic selection rules and thus can provide information about the symmetry of an atom s environment, its oxidation state, and sometimes, with the assistance of comprehensive theoretical calculations, details about the geometry of ligands and other nearby atoms. This information is derived from excitations into bound states and low lying resonances above the Fermi level and is referred to as the x-ray absorption near edge structure (XANES). 

These selection rules can be related to spin-orbit coupling with the help of the Fermi golden ride. (Cf. Section 5.2.3.) The values k j 10 s for naphthalene and kgj 10 s " for I-bromonaphthalene (Birks, 1970), the difference of which can be explained through the heavy atom effect, also indicate clearly the influence of spin-orbit coupling. The fact that the rate... 

Due to the intermediate core-excited state in the XES spectroscopic process, there are additional selection rules for molecules with an inversion center which allows one to distinguish the symmetry of both the occupied and unoccupied orbitals [41]. Due to the two-step character of the XES measurements with initial absorption (excitation) and subsequent emission (de-excitation) steps, we find selection rules for molecules with inversion symmetry that require the same characteristics of initial and final states. New states below the Eermi level can then be identified and attributed to the population of the n orbital through the back-donation process while above the Fermi level the presence of states of 7i-character confirm the 7i-donation process [40, 42]. 

The rate of a general, proximity-induced, two-photon absorption process can be calculated by combining the Fermi Golden rule, Eq. (5.6), with Eq. (5.13). For each of the four specific cases to be studied, the selection rules normally dictate that only two of the four terms in Eq. (5.13) arise for any given mechanism, thus reducing the number of terms in the rate equation to just four. A brief derivation of the rate of absorption for each case is presented below. 

As is well known, the selection rules allow RSL by polaritons only in crystals without a center of inversion. This is precisely the kind of crystal in which Fermi resonance with polaritons (to be discussed below) was found to be the physical phenomenon in which the special features of the biphonon spectrum were most evident. 

Let a molecule have two vibrational modes with frequencies uia and uii,. If the second-order resonance condition 2uia uif, is fulfilled, then the huif, transition in the infrared spectrum can split if the interaction is allowed by symmetry of molecule into two lines of comparable intensity. The second line cannot be explained as a result of the interaction of light with the vibrational a mode because the transition with excitation of two hwa quanta is forbidden due to the well-known n — n 1 selection rule for a harmonic oscillator. Fermi explained (7) this experimental observation as a result of a nonlinear resonance interaction of two vibrational modes with each other. Since that time the notion of Fermi resonance has been generalized to processes with participation of different types of quanta (e.g. + iv2 u>3, lo + iv2 — UJ3 — 0J4, and so on) and to elec-... 

The additional enhancement provided by coadsorbed halide ions on a colloidal silver surface has been pointed out [401]. As an explanation, morphological changes effected by the strongly adsorbed anions have been invoked. The photo-driven CT that is assumed to proceed from filled metal states near the Fermi level of the metal to the first and second excited state in the case of pyrazine adsorbed on polycrystalline gold has been invoked as the cause for the breakdown of Raman selection rules upon adsorption (i.e. the activation of originally Raman-forbidden vibrational modes) [314]. 

In XAS the incident X-rays traverse through the sample. X-rays of appropriate energy are absorbed by the atoms and, as a result, a core electron is excited to an unoccupied state above the Fermi energy, Ep. The decrease in the transmitted X-ray intensity is measured. However, in this spectroscopy selection rules are rigorously obeyed and all symmetries must be considered to interpret the spectral features. The schematic diagram, in the one-electron approximation, of the X-ray excitation of a core state is shown in fig. 1. 

It has been shown that Fermi resonances between chemical bond vibrational levels and van der Waals modes can dramatically reduce vibrational predissociation lifetimes. The selection rule becomes altered because the definitions of the quantum numbers become blurred by the Fermi resonances. This is illustrated in the recent study of Tiller, Peet and Clary and shown in Fig. 5. Here 5 0,0,0> mixes with nearby 17 0,0,10> whose vibrational predissociation lifetime is calculated to be two orders of magnitude shorter than the prepared state. 

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