Boltzmann equilibrium

Figure 4.1 (a) Populations at Boltzmann equilibrium before application of a radio-... 

Now if S is strongly irradiated, then it is saturated and S is no longer at its Boltzmann equilibrium. Therefore it cannot maintain the Boltzmann equilibrium of spins 7, and the intensity of the 7 signal is changed. Equalizing S populations produces a proportional change in 7 populations such that equation 3.41 can be written in which r IS is called the nuclear Overhauser enhancement (NOE) factor. 

Even at the highest fields, the NMR experiment would not be practicable if mechanisms did not exist to restore the Boltzmann equilibrium that is perturbed as the result of the absorption of electromagnetic radiation in making an NMR measurement. These mechanisms are known by the general term of relaxation and are not confined to NMR spectroscopy. Because of the small magnitude of the Boltzmann excess in the NMR experiment, relaxation is more critical and more important in NMR than in other forms of spectroscopy. 

Fig. 1. MM2 energies of 2-methylperhydropyrido[2,l-b][l,3]oxazine in chloroform. The top number indicates the energy (kcal mol 1) relative to the most stable species. The bottom numbers indicate the percentage of the Boltzmann equilibrium distribution for all species.

At this point, one simply remarks that at long times (t >r") the relation (4.155) yields the Boltzmann equilibrium distribution. 

Assuming a Boltzmann equilibrium distribution for the initial condition of the oscillator, that is,... 

A quantity relevant to the discussion of the relaxation in the presence of a sink is the broadness of the probability density function of the initial condition. The following case where the system is made reactive (knr = 0) at times t 0, but is unreactive (/cnr = 0) for t < 0—corresponding to physical situations—can be solved easily. Indeed, the initial distribution p0(x0) equals the Boltzmann equilibrium distribution [see also Eq. (4.157)], that is,... 

Fig. 22 shows the results of photometry of plates similar to that illustrated in Fig. 21. The relative intensities of suitable transitions were determined from the asymptotic limit at long time delays when the system attains equilibrium. (These resemble, but are not identical to, the relative/ values because of the usual instrumental effects which depend on line width.) The time variation of the relative concentrations is shown in Fig. 23 the upper four levels attain Boltzmann equilibrium amongst themselves after 100 /isec, to form a coupled (by collision) system overpopulated with respect to the 5DA state. The equilibration of the upper four levels causes the initial rise (Fig. 22) in the population of Fe(a5D3). Thus relaxation amongst the sub-levels is formally similar to vibrational relaxation in most polyatomic molecules, in which excitation to the first vibrational level is the rate determining step. In both cases, this result is due to the translational overlap term, for example, in the simple form of equation (14) of Section 3.

These equations have been analyzed in some detail by Lucht and Laurendeau (2) and Chan and Daily (3). When the laser is turned on, molecules in the state are pumped to the E state. Since the laser selectively pumps from one rotational sublevel to another, the other rotational sublevels in the E state can be populated only by rotational relaxation. If quenching, or electronic de-excitation, is fast compared to rotational relaxation, then only the laser coupled state will be populated. On the other hand, if rotational relaxation is fast then all the rotational states will be populated and in Boltzmann equilibrium. 

Gorban et al. in their works (Gorban, 2007 Gorban et al., 2001,2006) seems to be more comprehensive for our discussion. The works unfolded the idea of the Ehrenfests (1959) on the isolated system tending toward the Boltzmann equilibrium trajectory as a result of "agitations."... 

During normal acquisition time of approximately 1 s, these nuclei cannot relax between pulses the spin system becomes saturated and the signal-to-noise ratio decreases. A common solution for slow relaxation is a recovery delay after signal acquisition. The pulse delay should be of the order of four to five times the longest Tj in the sample, allowing all nuclei to return to the Boltzmann equilibrium state before being flipped by the next pulse. 

Our NMR theory is almost complete, but there is one more thing to consider before we set about designing a spectrometer. We indicated previously that at equilibrium in the absence of an external magnetic field, all nuclear spin states are degenerate and, therefore, of equal probability and population. Then, when immersed in a magnetic field, the spin states establish a new (Boltzmann) equilibrium distribution with a slight excess of nuclei in the lower energy state. 

Take a moment to look back at Figure 2.10. It shows how the magnitude of the z-component net magnetization vector M, when displaced from its Boltzmann equilibrium value, always recovers exponentially back to this equilibrium value at a rate determined by Th the spin-lattice (longitudinal) relaxation time. In this case, we will label this value Mq, the equilibrium magnitude of M in the magnetic field. Since the magnitude of M at any time (M,) is solely dependent on the difference between the populations of the up and down spin states, we can recast Eq. (2.9) in the form... 

Fig. 17 Energy as a function of a T-T distance and b T-T-T angle used in the simulation procedure (calculated as smoothing spline fits to Boltzmann equilibrium interpretations of the histogrammed data taken from 32 representative zeolite crystal structures). Only the central portions are shown, c The contribution to the energy sum for the merging of two symmetry-related atoms merging is only permitted when the two atoms are at less than a defined minimum distance [84], Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

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