Boltzmann occupancy factor

In applying this formula to the determination of the peak absorption coefficient, it is customary to combine with it the rotational partition function contribution kT/hB giving the term hB J/kT, (ref 3, p. 117). This equates to hvpd/IkT in linear and symmetric top molecules and replaces the fimn term in Equation 1.21, with becoming fj. This latter term takes into account the Boltzmann occupancy factor for that state fj = exp — [57(7 + )/kT]... 

The quantities averaged on the right here are the Boltzmann factor for the binding energy, as is usual for the PDT, but multiplied by the indicator function for the event that there are zero occupants of the defined inner shell. 

In Fig. 63 the occupation of state 1 is equal to Nly the occupation of state 2 is equal to N2 and the total occupation is equal to N1+N2=N. The viscoelastic and plastic shear strain is proportional to the decrease of the occupation of state 1 or proportional to the increase of the occupation of state 2. Without external stress the probability for transition from state 1 to state 2 (v ) is proportional to the Boltzmann factor expI-Uf T)-1], and for the inverse transition 2—>1 (v) the probability is proportional to N2exp[-U(kBT) 1]. 

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... 

A notable measure of the intermolecular forces is the maximum frequency v of the lattice vibrations (optical phonons). In a typical organic molecular crystal, it is of the order of 3.5 THz in Si, in contrast, it is 14THz. Thus the difference in the Boltzmann factors exp(-hv/feT) for the thermal occupation of phonon states, which plays a decisive role in many solid-state properties, is already great when comparing organic and inorganic solids at room temperature, and it becomes very much greater at low temperatures (Table 1.2). 

This expression differ from the Shannon entropy in two basic aspects first, it has the multiplying Boltzmann factor, a signature of thermodynamic phenomena, and second, the logarithm is taken in the natural basis. The probability pk is the occupation number for the energy level Ek, or its population. At thermal equilibrium at temperature T, the probabilities can be obtained from the principle of maximization of the entropy [12] ... 

Figure D.l. Average occupation numbers in the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions, for = 10 (left) and f = 100 (right), on an energy scale in which fXfo = p-be = 1 for the FD and BE distributions. The factor nl(2nmfl in the Maxwell-Boltzmann distribution of the ideal gas is equal to unity.

By simplifying the occupation probabilities for a nondegenerate semiconductor, which means replacing them with their Boltzmann factors, we get [38]... 

A mechanism of spin-lattice and spin-spin relaxations was also worked out at the same time. The point is that the relative levels occupancy N" N, between which a transition occurs, is defined by the Boltzmann factor. The probability of transition between two levels is greater the larger the distance between them and, consequently, the more N" N differs from 1. In y-gamma resonance, the energy dilference is of the order of 10 eV, so the upper level is always nearly free and resonance absorption of quantum is nearly always possible. In NMR resonance, the difference E"—E is extraordinarily small and factor N"/N is close to 1. Under such conditions, the occupancy of both levels is the same and the probability of transition should be very small and, basically, paramagnetic resonance must not be observed. However, it turns out that another mechanism of removing the excitation from the upper level exists. 

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