Relaxation, spin

Relaxation rates of nuclear spins can also be related to aspects of molecular structure and behaviour in favourable circumstances, in particular internal molecular motions. It is true to say, however, that the relationship between relaxation rates and structural features is not as well defined as those of the chemical shift and spin-spin coupling constants and is not used on a routine basis. The problem of reliable interpretation of relaxation data arises largely from the numerous extraneous effects that influence experimental results, meaning empirical correlations for using such data are not generally available, and this aspect of NMR will not be further pursued. 

The Bloch theory of NMR assumes that the recovery of the - -z-magnetisation, M, follows exponential behaviour, described by  

There are a number of different experiments devised for the determination of the longitudinal relaxation times of nuclear spins, and only the most commonly applied method, inversion recovery, will be considered here. The full procedure is described first, followed by the quick-and-dirty approach that is handy for experimental set-up. 

the procedure is to run an experiment with r = 0 and adjust (phase) the spectrum to be negative absorption. After having waited 5Ti, repeat the experiment with an incremented r using the same phase adjustments, until the signal passes through the null condition (Fig. 2.23), thus defining Tnuu, which may be different for each resonance in the spectrum. Errors may be introduced from inaccurate 180° pulses, from off-resonance effects (see Section 3.2) and from waiting for insufficient periods between acquisitions, so the fact that these values are estimates cannot be overemphasised. 

One great problem with these methods is the need to know something about the Ti s in the sample even before these measurements. Between each new 

Because careful analysis of the decay process reveals details of molecular structure and of interactions between molecules, we need to understand how a spin system returns to equilibrium after the application of a radiofrequency pulse. 

As resonant absorption continues, the population of the upper state rises to match that of the lower state. From eqn 13.13, we can expect the intensity of the absorption signal to decrease with time as the populations of the spin states equalize. This decrease due to the progressive equalization of populations is called saturation. 

The fact that saturation is often not observed must mean that there are non-radiative processes by which p nuclear spins can become a spins again and hence help to maintain the population difference between the two sites. The nonradi-ative return to an equilibrium distribution of populations in a system (eqn 13.9) is an aspect of the process called relaxation. If we were to imagine forming a system of spins in which all the nuclei were in their p state, then the system returns exponentially to the equilibrium distribution (a small excess of a spins over p spins) with a time constant called the spin-lattice relaxation time, T, (Fig. 13.25). 

What causes each type of relaxation In each case the spins are responding to local magnetic fields that act to twist them into different orientations. However, there is a crucial difference between the two processes. 

The best kind of local magnetic field for inducing a transition from p to a (as in spin-lattice relaxation) is one that fluctuates at a frequency close to the resonance frequency. Such a field can arise from the tumbling motion of the molecule in 

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S spin remains in tliennal equilibrium on die time scale of the /-spin relaxation. This situation occurs in paramagnetic systems, where S is an electron spin. The spin-lattice relaxation rate for the / spin is then given by ... 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

Table Bl.13.2 Interactions giving rise to nuclear spin relaxation.

B1.13.3.1 SPIN-LATTICE AND SPIN-SPIN RELAXATION RATES... 

The spin-spin relaxation time, T, defined in the Bloch equations, is simply related to the width Av 2 Lorentzian line at the half-height T. Thus, it is in principle possible to detennine by measuring... 

Woessner D E 1962 Nuclear spin relaxation in ellipsoids undergoing rotational Brownian motion J. Chem. Rhys. 37 647-54... 

Woessner D E 1962 Spin relaxation processes in a two-proton system undergoing anisotropic reorientation J. Chem. Rhys. 36 1-4... 

Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

Maler L, Widmalm G and Kowalewski J 1996 Dynamical behavior of carbohydrates as studied by carbon-13 and proton nuclear spin relaxation J. Phys. Chem. 100 17 103-10... 

Dayie K T, Wagner G and Lefeevre J F 1996 Theory and practice of nuclear spin relaxation in proteins Anna. Rev. Phys. Chem. 47 243-82... 

Kowalewski J 1990 Nuclear spin relaxation in diamagnetic fluids Part 1 Annu. Rep. NMR Spectrosc. 22 308-414 1991 Part 2 Annu. Rep. NMR Spectrosc. 23 289-374... 

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

The following diseussion of the time dependenee of the EPR response in a TREPR experiment is based on the assumption that the transient paramagnetie speeies is long lived with respeet to the spin relaxation parameters. 

M continually decreases under the influence of spin-spin relaxation which destroys the initial phase coherence of the spin motion within they z-plane. In solid-state TREPR, where large inliomogeneous EPR linewidths due to anisotropic magnetic interactions persist, the long-time behaviour of the spectrometer output, S(t), is given by... 

We make one important note here regarding nomenclature. Early explanations of CIDNP invoked an Overhauser-type mechanism, implying a dynamic process similar to spin relaxation hence the word dynamic m the CIDNP acronym. This is now known to be incorrect, but the acronym has prevailed in its infant fomi. 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

Schell S A J, Mehran F, Eaton G R, Eaton S S, Viehbeck A, O Toole T R and Brown C A 1992 Electron spin relaxation times of< , in solution Chem. Phys. Lett. 195 225-32... 

Here Ti is the spin-lattice relaxation time due to the paramagnetic ion d is the ion-nucleus distance Z) is a constant related to the magnetic moments, i is the Larmor frequency of the observed nucleus and sis the Larmor frequency of the paramagnetic elechon and s its spin relaxation time. Paramagnetic relaxation techniques have been employed in investigations of the hydrocarbon chain... 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

FID Free induction decay, decay of the induction (transverse magnetisation) back to equilibrium (transverse magnetisation zero) due to spin-spin relaxation, following excitation of a nuclear spin by a radio frequency pulse, in a way which is free from the influence of the radiofrequency field this signal (time-domain) is Fourier-transformed to the FT NMR spectrum (frequency domain)... 

ESR can detect unpaired electrons. Therefore, the measurement has been often used for the studies of radicals. It is also useful to study metallic or semiconducting materials since unpaired electrons play an important role in electric conduction. The information from ESR measurements is the spin susceptibility, the spin relaxation time and other electronic states of a sample. It has been well known that the spin susceptibility of the conduction electrons in metallic or semimetallic samples does not depend on temperature (so called Pauli susceptibility), while that of the localised electrons is dependent on temperature as described by Curie law. 

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