Boltzmann equilibrium nuclear relaxation

An interesting observation is that the decay of the excited planar form is clearly non exponential, while the decay of the twisted state is exponential, at least for "long" times, i.e. after the solvent has relaxed. Thus the deactivation of the twisted state is a "classical" reaction, because it occurs from a species which has had time enough to equilibrate with its surrounding. The twisted state corresponds to a local minimum in a potential-energy-versus nuclear-and-solvent-coordinates diagram. By contrast, the deactivation of the planar excited state is an example of ultrafast reaction for which the Boltzmann equilibrium is not reached (Note that the conversion to the twisted form involves almost no activation energy). 

Several processes, both within the molecule (intramolecular) and between molecnles (inter-molecular), contribute to spin-lattice relaxation. The principal contributor is magnetic dipole-dipole interaction. The spin of an excited nucleus interacts with the spins of other magnetic nuclei that are in the same molecule or in nearby molecules. These interactions can indnce nuclear spin transitions and exchanges. Eventually, the system relaxes back to the Boltzmann equilibrium. This mechanism is especially effective if there are hydrogen atoms nearby. For carbon nuclei, relaxation is fastest if hydrogen atoms are directly bonded, as in CH, CH2, and CH3 groups. Spin-lattice relaxation is also most effective in larger molecules, which tumble (rotate) slowly, and it is very inefficient in small molecules, which tumble faster. 

If the relaxation of spin I is significantly affected by the motions of a different spin, S, then in general any deviation from Boltzmann equilibrium of the S spins will also render the I spin populations non-Boltzmann. The resulting change in the intensity of resonance I is called a nuclear Overhauser enhancement (NOE) when spins I and S both belong to nuclei. The nuclear Overhauser enhancement factor is conventionally ri, so that 1 -l- tjjs is defined as the intensity of the I resonance when the S spin populations are equalized, divided by the intensity of the I resonance when the... 

Relaxation refers to all processes which regenerate the Boltzmann distribution of nuclear spins on their precession states and the resulting equilibrium magnetisation along the static magnetic field. Relaxation also destroys the transverse magnetisation arising from phase coherenee of nuelear spins built up upon NMR excitation. 

Spin-lattice relaxation is the steady (exponential) build-up or regeneration of the Boltzmann distribution (equilibrium magnetisation) of nuelear spins in the static magnetic field. The lattice is the molecular environment of the nuclear spin with whieh energy is exchanged. 

Relaxation is an inherent property of all nuclear spins. There are two predominant types of relaxation processes in NMR of liquids. These relaxation processes are denoted by the longitudinal (Ti) and transverse (T2) relaxation time constants. When a sample is excited from its thermal equihbrium with an RF pulse, its tendency is to relax back to its Boltzmann distribution. The amount of time to re-equilibrate is typically on the order of seconds to minutes. T, and T2 relaxation processes operate simultaneously. The recovery of magnetization to the equilibrium state along the z-axis is longitudinal or the 7 relaxation time. The loss of coherence of the ensemble of excited spins (uniform distribution) in the x-, y-plane following the completion of a pulse is transverse or T2... 

The physical basis of current MRI methods has its origin in the fact that, in a strong magnetic field, the nuclear spins of water protons in different tissues relax back to equilibrium at different rates, when subject to perturbation from the resting Boltzmann distribution by the application of a short radio frequency (rf) pulse. For the most common type of spin-echo imaging, return to equilibrium takes place in accord with equation 1 and is governed by two time constants T and T2, the longitudinal and transverse relaxation times, respectively. 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

Figure 9. Excitation and relaxation in a population of spins, (a) Before pulse, (b) Induction of phase coherence along y by Hi, and consequent tipping of macroscopic magnetization, M. (c) Dephasing of nuclear magnetic moments by spin-spin relaxation, i.e., M,. = 0. (d) Re-establishment of the Boltzmann distribution (Afj is at its equilibrium value)(a = d).

In all the experiments mentioned above, the Overhauser effect has been observed by irradiating the e.s.r. signal of the dissolved free radicals. However the essential conditions for production of an Overhauser effect are that the populations of the electron spin Zeeman levels should depart from their thermal equilibrium value and that, as the electron spins relax and attempt to restore the Boltzmann distribution among their levels, they should interact with the nuclear spins present in solution. 

A well-known and important phenomenon in the area of nuclear-spin resonance (NMR) in gases, liquids, or solid samples is dynamic nuclear-spin polarisation (DNP) (see e.g. [M6]). This term refers to deviations of the nuclear magnetisation from its thermal-equilibrium value, thus a deviation from the Boltzmann distribution of the populations of the nuclear Zeeman terms, which is produced by optical pumping (Kastler [31]), by the Overhauser effect [32], or by the effet solide or solid-state effect [33]. In all these cases, the primary effect is a disturbance of the Boltzmann distribution in the electronic-spin system. In the Overhauser effect and the effet solide, this disturbance is caused for example by saturation of an ESR transition. Owing to the hyperfine coupling, a nuclear polarisation then results from coupled nuclear-electronic spin relaxation processes, whereby the polarisation of the electronic spins is transferred to the nuclear spins. 

Radiofrequency pulses are also utilized to measure relaxation times. Three relaxation times have been measured in TPEs, and each is sensitive to different phenomena. Ti, the spin-lattice relaxation time in the laboratory frame, is the relaxation from the nonequilibrium population distribution created by the pulse to the equilibrium Boltzmann distribution. Ti is sensitive to molecular motions that rate in the range of 10 -10 Hz. T2, the spin-spin relaxation time, is the relaxation caused by the establishment of equilibrium between nuclear spins within the system. Spin-spin relaxation measurements also probe motions with rates in the range of 10M0 Hz however, low frequency motions (lOMtPHz) also affect T2. Generally,T2 is one to three orders of magnitude smaller than Ti in solid polymers. Tip, the spin-lattice relaxation time in the rotating frame, probes motions with rates on the order of lO -KfHz. Cross polarization is usually used in Tip measurements. 

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