Boltzmann function

The second step is the molecular dynamics (MD) calculation that is based on the solution of the Newtonian equations of motion. An arbitrary starting conformation is chosen and the atoms in the molecule can move under the restriction of a certain force field using the thermal energy, distributed via Boltzmann functions to the atoms in the molecule in a stochastic manner. The aim is to find the conformation with minimal energy when the experimental distances and sometimes simultaneously the bond angles as derived from vicinal or direct coupling constants are used as constraints. 

A useful means for compiling or comparing the results of a large number of experiments is to fit the observed populations to some distribution function. A reasonable first try for thermally activated processes would seem to be a Boltzmann function, where the population in a particular rotational level (J)... 

Fig. 1-3. Probability density of electron energy distribution, fli), state density, D(t), and occupied electron density. Die) fit), in an allowed energy band much higher than the Fermi level in solid semiconductors, where the Boltzmann function is applicable.

Since the band gap is relatively narrow in semiconductors, a few electrons in the fully occupied valence band are thermally excited up to the conduction band leaving positive holes (vacant electrons) in the valence band. The concentration, ni, of thermally exdted electron-hole pairs is given, to a first approximation, by the Boltzmann function as shown in Eqn. 2-6 ... 

The concentration of electrons, n, in the conduction band of n-type semiconductors and the hole concentration, p, in the valence band of p-type semiconductors are given by Eqn. 2-7 and Eqn. 2-10, respectively. The concentration of ionized donors, IVd-, and the concentration of ionized acceptors, iVx-, are derived by using the Fermi function approximated by the Boltzmann function as shown in Eqns. 2-18 and 2-19, respectively ... 

Fig. 2-20. Electron state density and ranges of Fermi energy where electron occupation probability in the conduction band of an electron ensemble of low electron density (e.g., semiconductor) follows Boltzmann function (Y i)or Fermi function (y > 1) y = electron activity coeffident ET =transition level from Y 4= 1 to Y > 1 0(t) = electron energy state density CB = conduction band. [From Rosenberg, I960.]...

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

Doping a p-type semiconductor generates fixed acceptor sites with a density Na, and an equal number of mobile carriers with an opposite charge h+, whose distribution is controlled by the local value of the potential T>(x), following the Boltzmann function so that the mobile charge distribution is given by ... 

All of the silicon detectors can be cooled to reduce the thermal noise that produces a background under all the induced signals. The thermal noise is created by random fluctuations that promote an electron across the bandgap into the conduction band resulting in an electron-hole pair. The number of promoted electrons will be proportional to a Boltzmann function containing the bandgap A and the temperature T ... 

Due to the above-mentioned weak coupling and to the inequality (2), the distribution of translational, rotational and, to an approximation also of vibrational degrees of freedom — p(t), p(r) and p(v) respectively — can be satisfactorily approximated by the Boltzmann functions exp(—j3KeK) with the corresponding temperatures... 

From the Boltzmann function, the relative population of the upper state, N+, and the lower state, AT, energy levels at thermal equilibrium is given by... 

In Equation 4.2, Vtest is the test potential and Vrev is the reversal potential. In our system, Vrev for K+ was calculated using the Nernst equation as -78 mV at +20°C and this is the value we used in our calculations. The K+ conductances were plotted as a function of test potential (Figure 4.5E) and the data were described by the following Boltzmann function (Equation 4.3),... 

A very similar result is obtained if we consider a very dilute solution of both A and B in solvent S, all of which species are made of spherical molecules that have nearly equal volumes. Then the concentration of A-B neighbors in such a solution is given by the product of the concentration of A molecules A a, the number of near-neighbor sites, Z, the mole fraction of B in the solution riB, and the Boltzmann function ... 

Fig. 2. The three dimensional velocity distribution of all the atoms after impact of a cold cluster of 125 Ar atoms at a surface at a velocity of 25 km/s (= 0.25 A/fs). Shown for comparison is a Meixwell-Boltzmann functional form for the same mean energy. Even at this high velocity of impact, the velocity distribution after the impact is essentially isotropic. See Sec. 4.

Figure 2 compares the distribution after the impact, computed from molecular dynamics simulations, to the Maocwell-Boltzmann functional form. In Sec. 4, we discuss the reasons why the initially directed energy is so rapidly thermalized. 

Electrons in semiconductors may be regarded as low-density particle ensembles such that their occupancy in the valence and conduction bands may be approximated by the Boltzmann function [14, 15] ... 

The intrinsic constant expressions written with Boltzmann function terms in Eqs. (10.35) and (10.36), correspond to hypothetical protonation and deprotonation reactions written... 

MINTEQA2 treats electrostatic terms (Boltzmann function terms) as if they were separate components. Explain. 

It should be noted that the observed distribution functions might not be thermal. In fact, for a large number of product state distributions from uni- and bi-molecular reactions, one observes significant deviations form the Boltzmann functions, which reflect particular state-to-state chemical reaction dynamics. 

The Fermi-Dirac function can be approximated by the Boltzmann function if Ep - E) is much greater than kpT. The Fermi energy Ep determines n and p, the concentrations of... 

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