Boltzmann distribution coupling

In this case, a spin A that was coupled to the a orientation of the B spin may end up, after the exchange, coupled to either a or (3. Because of the Boltzmann distribution, the amounts of a and P orientation are each... 

For all known cases of iron-sulfur proteins, J > 0, meaning that the system is antiferromagnetically coupled through the Fe-S-Fe moiety. Equation (4) produces a series of levels, each characterized by a total spin S, with an associated energy, which are populated according to the Boltzmann distribution. Note that for each S level there is in principle an electron relaxation time. For most purposes it is convenient to refer to an effective relaxation time for the whole cluster. 

These equations are valid in the absence of degeneracy. Like Eqs. (10) they are characterized by the fact that elements with the same difference of indices m — m are coupled among themselves. Their equilibrium solution is a Boltzmann distribution. 

Expansion of the Boltzmann Distribution in the Dipolar Coupling Parameter... 

Figure 4. Series of IR femtosecond/picosecond laser pulses for the sequence of vibrational transitions 0H(u = 0)- -OH(u = 5)- 0H(u = 10) for the model OH coupled to a thermal heat bath cf. Fig. 1 and Table I. The notations are as in Fig. 3. The laser pulses compete against dissipation. After the pulses, the coupled system relaxes toward a Boltzmann distribution (T = 300 K).

Table 2.1. Intensities of CH multiplets in Coupled 13C NMR Spectra with Boltzmann Distribution of Nuclei (a) and after Selective Polarization Transfer (b).

The kinetic theory of gases, coupled with the Boltzmann distribution, lets us predict a wide variety of the macroscopic properties of gases, and the agreement with experiment is excellent—even though we are making extremely crude approximations to the microscopic structure of the individual molecules. Molecules are much more complicated than tiny billiard balls, yet at standard temperature and pressure, the answers we get from the simplest theories are good to within a few percent. 

Similar modelling has been performed for both of these systems, based on the cube model. Following Hand and Harris, the molecular motion was coupled to the surface oscillator via a rigid shift of the Z-coordinate in the PES, i.e. V(Z, r,. .., y) = V(Z — y, r...), where y is the oscillator coordinate. For the H2/Pd system [80], six molecular degrees-of-freedom were included in a classical treatment, while four molecular degrees-of-freedom were included in a quantum solution for the H2/Cu system [81, 82]. In the classical calculations, the surface temperature dependence was introduced by sampling the surface vibration from a Boltzmann distribution. In quantum calculations, this is not possible, and many calculations were required, each in a different initial surface oscillator state. The results... 

FIGURE 9.9 Energy levels, populations, and, 3C spectra in the SPT experiment. Top Energy levels and excess populations in (a) the normal Boltzmann distribution and (b) the distribution after a selective pulse on line vx has inverted the populations of the aa and /3a levels. Bottom , 3C spectra (coupled and decoupled) obtained with a) Boltzmann distribution (b) altered population distribution (c) altered distribution with phase inversion of line v3. 

---