Degeneracy factors

In this equation as well as in the succeeding discussions, we have suppressed, for notational simplicity, the pemuitation or degeneracy factor of two, required for SFG. 

There is a degeneracy factor of two associated with a n orbital compared with the nondegeneracy of a (7 orbital, so that it might be expected that the integrated intensity of the second band system would be twice that of each of the other two. However, although the second band system is the most intense, other factors affect the relative intensities so that they are only an approximate guide to orbital degeneracies. 

The sum in the denominator relates to the quantum states. The formula is often written in terms of energy levels rather than quantum states in the case that some of the energy levels are degenerate, with degeneracy factors gi then the formula can be modified to refer to energy-level populations directly ... 

The partition function may alternatively be written as a sum over all distinct energy levels, times a degeneracy factor g,-. 

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. 

Clearly, Bj f embodies the final-level degeneracy factor gf, the perturbation matrix elements, and the 2n factor in the earlier expression for Ri f. The spontaneous rate of transition from the excited to the lower level is found to be independent of photon intensity, because it deals with a process that does not require collision with a photon to occur, and is usually denoted A f. The rate of photon-stimulated upward transitions from state f to state i (gi Rf i = gi Rif in the present case) is also proportional to g(Cdf,i), so it is written by convention as ... 

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... 

The exponential energy factor in Eq. (H) gives decreasing populations with increasing 7, but the degeneracy factor (27 + 1) works in the opposite direction. As a result, rotational populations increase initially with increasing 7, reach a peak, and subsequently decrease. 

In his derivation, Frieden uses the concept of a number-count set nm, each member representing the number of photons counted in a spectral interval. The total number of photons m= x nm is taken as known to be N. In terms of frequencies vm, the values of the object spectrum are given by om = nmhvm, where h is Planck s constant. The number of normal modes or degrees of freedom available for occupation by photons of frequency vw is labeled zm. The Bose-Einstein degeneracy factor... 

Principle (17) now consists of just the Bose-Einstein degeneracy factor, exactly Kikuchi and Soffer s form (1977). Also, as these authors showed (see also Section IX.B), in the case (11a) of sparsely occupied df it becomes Jaynes s maximum-entropy form (39) (Jaynes, 1968). Hence, both the Kikuchi-Soffer and Jaynes estimators are special cases of the ML approach, corresponding to the prior knowledge that the unknown spectrum is equal-energy white with the highest conviction. 

Here Nc is the density of states in the conduction band, g the level degeneracy factor, n the carrier concentration in the band, A the activation energy of the level, Boltzmann s constant, and T the temperature. Now, in general, except at fairly low temperatures, the occupancy for shallow levels (with/ = /s) will be small, i.e., fs 1, and consequently... 

Thus, the bound-state degeneracy factor, = 0/ 1 falls out in a very natural way. It arises simply because of the assumption that the state with na + 2 electrons is at too high an energy to be stable (see Look, 1981, for more detail). 

This equation has precisely the same form as its donor analog, Eq. (B36), except for the inversion of the degeneracy factor in AV. 

We next consider an acceptor center near the valence band. Here the states can sometimes be described in terms of a deep s-like state (not important in the problem), and three p-like states (p+ = px + ip, P- = px — ip, and pz). Suppose that these p-like states are degenerate (or nearly so, within a few kT), so that their six electrons (three spin-up, three spin-down) can be considered equivalent. Consider the acceptor formed by Cd on a Ga site in GaAs. This site has seven electrons, two in the deep s-like state and five in the p-like states. Therefore, the unoccupied state has degeneracy gA0 = 6 /5 l = 6, and the occupied state (after accepting one electron) has gA1 = 6 /6 = 1. Thus, in Eq. (B41), the degeneracy factor is gAJgA0 = b... 

In this expression, the degeneracy factor gN represents the number of ways the TV molecules may be placed on M sites, and is given by the combinatorial formula (see Equation (3.52) in Section 3.4a.2 or Equation (2.46) in Section 2.6a) ... 

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