Ratio degeneracy

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

B) Compare the expected and observed largest-to-smallest intensity ratios. Accidental degeneracies (or am/ line-width dependence) may cause the observed ratio to be bigger than expected. It is rarer (but not unknown) to find a smaller ratio than expected. 

The reorientation of the B—H complex at 100 K complicates the analysis of the stress splitting data. The ratios of the intensities of the stress split components were extrapolated to zero stress to determine the site degeneracies for each stress orientation and hence to deduce the symmetry of the complex (Herrero and Stutzmann, 1988b). A unique configuration could not be found to fit the data for all stress directions it was suggested that the configuration of the complex must depend upon the applied stress. For the [110] stress direction it was proposed that the H is displaced from the trigonal axis in the direction away from the C site, while for [100] stress the H is supposed to be displaced toward the C site. 

Because of charge neutrality, ye- t] B and is consequently negligible. That the same thing holds for yVe (and for yV/tI) is a postulate, but a very plausible one because otherwise we would have neutrino or antineutrino degeneracy with LV 2> B. The upshot is that the chemical potentials of neutrons and protons are equal and so from Eq. (2.43) one has a simple Boltzmann-type equilibrium ratio (n/p)eq = e-fa-mptflkT = g-1.29M eV/kT (4.36)... 

As stated earlier, in the state of thermal equilibrium at room temperature, dihydrogen (H2) contains 25.1% parahydrogen (nuclear singlet state) and 74.9% orthohydrogen (nuclear triplet state) [19]. This behavior reflects the three-fold degeneracy of the triplet state and the almost equal population of the energy levels, as demanded by the Maxwell-Boltzmann distribution. At lower temperatures, different ratios prevail (Fig. 12.5) due to the different symmetry of the singlet and the triplet state [19]. 

Strictly speaking, these equations should also contain pre-exponential factors related to the degeneracy ratios of the states considered (SS), This is, however, irrelevant for what follows. 

The triple degeneracy of the LUMO of Cgo was confirmed experimentally in several steps between 1990 and 1992 with the detection of Cgo and Cgo [27], Cgo - [28], Cgo - [29], Cgo - [30], and finally Cgo [31]. Owing to limitations in the solvent potential window, the elusive hexaanion species was only detected when the experiment was carried out under vacuum, at low temperature (—lO C), and using a 0.1 M tetra-n-butylammonium hexafluorophosphate (TBAPFg) electrolyte solution in a solvent mixture consisting of toluene/acetonitrile (PhMe/MeCN) in a 4 to 1 ratio. Under these conditions, using cyclic voltammetry (CV)... 

The high-temperature transport data has been well rationalized 0 on the basis of a diffusional model with AH = 0.16 eV for all x, which is an agreement with localized Fe " -ion configurations responsible for a cooperative Jahn-Teller distortion below Tn. The ratio (1 - c)/c obtained from Eq. (15) for the Seebeck coefficient, with a spin-degeneracy factor P = 2, gives the factor c(l — c) entering the conductivity expression... 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

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