Spin Relaxation Boltzmann Distribution

The process of spin relaxation using the wavefunction has already been discussed in Sect. 5.4.2.1. In this section the algorithm is extended to allow the spins to relax 

The first step in the algorithm is to calculate the relaxation probability for each nuclear spin configuration k as 

8 Correlation Between Spin Entanglement and the Spin Relaxation Time 

If the probability of C7(0,1], then the coefficient for the state j with a nuclear spin configuration k collapses as with the exponential term mod- 

Although analytical formulations are available to calculate the intensity of fluorescence, they lack the ability to predict the effect of electron exchange in spurs and their contribution to the spin relaxation times. Currently the only viable method to take these into account is to use numerical simulations such as IRT or Monte Carlo 

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There is one other type of relaxation process that must be mentioned at this point. After irradiation ceases and B, disappears, not only do the populations of the m = + and m = states revert to the Boltzmann distribution, but also the individual nuclear magnetic moments begin to lose their phase coherence and return to a random arrangement around the z axis (Figure 2.1a). This latter process, called spin-spin (or transverse) relaxation, causes decay of MJ>y at a rate controlled by the spin-spin relaxation time T2. Normally, T2 is much shorter than T. A little thought should convince you that if T2 < Th then spin-spin (dephasing) relaxation takes place much faster than spin-lattice (Boltzmann distribution) relaxation. 

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. 

Methods of disturbing the Boltzmann distribution of nuclear spin states were known long before the phenomenon of CIDNP was recognized. All of these involve multiple resonance techniques (e.g. INDOR, the Nuclear Overhauser Effect) and all depend on spin-lattice relaxation processes for the development of polarization. The effect is referred to as dynamic nuclear polarization (DNP) (for a review, see Hausser and Stehlik, 1968). The observed changes in the intensity of lines in the n.m.r. spectrum are small, however, reflecting the small changes induced in the Boltzmann distribution. 

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. 

In metalloproteins, the paramagnet is an inseparable part of the native biomacromolecule, and so anisotropy in the metal EPR is not averaged away in aqueous solution at ambient temperatures. This opens the way to study metalloprotein EPR under conditions that would seem to approach those of the physiology of the cell more closely than when using frozen aqueous solutions. Still the number of papers describing metalloprotein bioEPR studies in the frozen state by far outnumbers studies in the liquid state. Several additional theoretical and practical problems are related to the latter (1) increased spin-lattice relaxation rate, (2) (bio)chemical reactivity, (3) unfavorable Boltzmann distributions, (4) limited tumbling rates, and (5) undefined g-strain. 

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... 

Snider is best known for his paper reporting what is now referred to as the Waldmann-Snider equation.34 (L. Waldmann independently derived the same result via an alternative method.) The novelty of this equation is that it takes into account the consequences of the superposition of quantum wavefunctions. For example, while the usual Boltzmann equation describes the collisionally induced decay of the rotational state probability distribution of a spin system to equilibrium, the modifications allow the effects of magnetic field precession to be simultaneously taken into account. Snider has used this equation to explain a variety of effects including the Senftleben-Beenakker effect (i.e., is, the magnetic and electric field dependence of gas transport coefficients), gas phase NMR relaxation, and gas phase muon spin relaxation.35... 

We described the nuclear Overhauser effect (NOE) among protons in Section 3.16 we now discuss the het-eronuclear NOE, which results from broadband proton decoupling in 13C NMR spectra (see Figure 4.1b). The net effect of NOE on 13C spectra is the enhancement of peaks whose carbon atoms have attached protons. This enhancement is due to the reversal of spin populations from the predicted Boltzmann distribution. The total amount of enhancement depends on the theoretical maximum and the mode of relaxation. The maximum possible enhancement is equal to one-half the ratio of the nuclei s magnetogyric ratios (y s) while the... 

Where an atom has a multiplet ground state, reaction may populate these sublevels with a non-Boltzmann distribution. This is difficult to observe since for light atoms the spin-orbit splitting is small and relaxation is rapid, and also because optical transitions between the components of the multiplet are strongly forbidden. Absorption measurements are possible but have scarcely been applied at all to this particular problem. 

There is another reason why the magnitude of 7) is important. Suppose we have a Boltzmann distribution of nuclei precessing in a magnetic field, and we irradiate the collection with photons of precisely the correct frequency (and energy) to cause transitions (spin flips) between the lower (m = + ) level and the upper (m = -i) states. Because there is initially such a small difference between the populations of the two states, it will not be long before the populations are equalized through the absorption of the photons This, of course, means the spin system has become saturated and no further net absorption is possible. However, if we turn off the source of rf radiation, the system can relax back to the Boltzmann distribution (at a rate controlled by Ti) and absorption can... 

After B, is turned off, nuclei can change their nuclear spin orientations through two types of relaxation processes. Spin-lattice (longitudinal) relaxation (governed by relaxation time 7, ) involves the return of the nuclei to a Boltzmann distribution. Spin-spin (transverse) relaxation (governed by relaxation time 72 or 7 ) involves the dephasing of the bundled nuclear spins. Normally 7 < T2 < 7,. 

The superscript" is used in Scheme 12.2 and throughout this chapter to represent spin polarization, a term applied to situations for which a paramagnehc species possesses a population of spin states that is different from the Boltzmann distribution at the temperature of the experiment. Polarization disappears during the radical paramagnetic relaxation time, usually in the microsecond timescale. Here and below, we will... 

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