Spin-lattice relaxation,

The evaluation of relaxation rates in the previous sections was based on the assumption that the energy spacing in the two-level system under consideration is large 

Consider now the case where the energy spacing 21 is very small. Such cases are encountered in the study of relaxation between spin levels of atomic ions embedded in crystal environments, so called spin-lattice relaxation. The spin level degeneracy is lifted by the local crystal field and relaxation between the split levels, caused by coupling to crystal acoustical phonons, can be monitored. The relaxation as obtained from (12.47) and (12.48) is very slow because the density of phonon modes at the small frequency (U21 is small (recall that 

Only terms with a that satisfy a a — u p = C( 2i survive the time integration. We get 

Further analysis is possible only if more information on Fig (co) is available. The theory of spin lattice relaxation leads to Fig(co) co. At low temperature the integral in (12.75) is dominated by the low-frequency regime where we can use g(co) co (see Section 4.2.4). We then have 

In fact, when u 2 0, other relaxation mechanisms should be considered. It is 

Experimentally, one typically observes the rate of lattice-induced energy absorption or loss 

For simplicity, we consider this as an absorption rate to be observed subject to the condition that C is very small, and hence C/=l. (This implies some very efficient spin-polarization mechanism at work. A more general case, although algebraically more complex, is not intrinsically different.) Under these conditions equation (19) may readily be integrated to give 

A change of variable from t to — t cannot affect the value of the first definite integral, which therefore becomes 

Furthermore, (27) will become altogether independent of t once t exceeds the value beyond which/(2 ) no longer correlates significantly with f t). Thus the integral limits in (27) can be altered to oo, so that (27) simply becomes the expression for the Fourier transform of G(r) evaluated at the frequency co ,. Equation (26) is normally described as /(cu /), the spectral density of f t) at (o , i.e.. 

The different contributions of Li and Li to the spin-lattice relaxation of neighbouring nuclei also provide, in suitable cases, a means for the determination of Li-C and Li-H distances in aggregates of organolithium compounds through the use of the isotopic substitution method [42], as has 

If the correlation time Xc is known, the distance rc-x can be determined according to 

Xc is obtained from the dipolar relaxation rate of another nucleus in the same molecule measured under identical conditions and its respective C-H distance. 

Finally, we mention that Li T measurements have been used to study the rate of interconversion between tight and loose ion pairs of lithium fluorenide [44]. The quadrupolar relaxation contribution was derived from Li and Li T data, and a change of the Li quadrupole splitting constant, QSC, obtained via Li quadrupolar and dipolar relaxation rates (see Section 4) with ion pair structure, was indicated. 

In early years of NMR, extensive studies of molecular dynamics were carried out using relaxation time measurements for spin 1/2 nuclei (mainly for 1H, 13C and 31P). However, difficulties associated with assignment of dipolar mechanisms and proper analysis of many-body dipole-dipole interactions for spin 1/2 nuclei have restricted their widespread application. Relaxation behaviour in the case of nuclei with spin greater than 1/2 on the other hand is mainly determined by the quadrupolar interaction and since the quadrupolar interaction is effectively a single nucleus property, few structural assumptions are required to analyse the relaxation behaviour. 

Theoretical expressions for spin-lattice relaxation of 2H nuclei (determined by locally axially symmetric quadrupolar interactions modulated by molecular motions) can be derived for specific dynamic processes, allowing the correct dynamic model to be established by comparison of theoretical and experimental results [34,35]. In addition, T, anisotropy effects, which can be revealed using a modified inversion recovery experiment, can also be informative with regard to establishing the dynamic model [34,35]. 

As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined by the spin Hamiltonian, (6.1). At finite temperamres, the crystal field is modulated because of thermal oscillations of the ligands. This results in spin-lattice relaxation, i.e. transitions between the electronic eigenstates induced by interactions between the ionic spin and the phonons [10, 11, 31, 32]. The spin-lattice relaxation frequency increases with increasing temperature because of the temperature dependence of the population of the phonon states. For high-spin Fe , the coupling between the spin and the lattice is weak because of the spherical symmetry of the ground state. This 

Several types of spin-lattice relaxation processes have been described in the literature [31]. Here a brief overview of some of the most important ones is given. The simplest spin-lattice process is the direct process in which a spin transition is accompanied by the creation or annihilation of a single phonon such that the electronic spin transition energy, A, is exchanged by the phonon energy, hcoq. Using the Debye model for the phonon spectrum, one finds for k T A that 

If the Zeeman splitting is large compared to the crystal field splitting, this leads to cx B T. Usually, the direct process is important only compared to other spin-lattice processes at low temperatures, because only low-energy phonons with hojq = A contribute to the direct process. 

At higher temperatures, the two-phonon (Raman) processes may be predominant. In such a process, a phonon with energy hcOq is annihilated and a phonon with energy HcOr is created. The energy difference TicOq — ha r is taken up in a transition of the electronic spin. In the Debye approximation for the phonon spectrum, this gives rise to a relaxation rate given by 

If optical phonons are responsible for the Raman processes, the Einstein model for the phonon spectrum is more appropriate. In this case, one finds 

At resonance, the rf field B, causes a spin transfer from the lower to the upper level. The equilibrium distribution of the spins in the static field B0 is disturbed. Following any disruption, the nuclear spins relax to be in equilibrium with their surroundings (the lattice ), which, of course, includes B0. This relaxation is assumed to be a first-order rate process with a rate constant l/ T, characterizing each kind of nucleus. Tx is called the spin-lattice relaxation time. It covers a range of about 10-4 to 104 s. 

For liquids with rapid intermolecular motion, 7j is a measure of the life-time of a nucleus in a particular spin state. According to the Heisenberg uncertainty relation, 

It is important now to consider the manner in which the Boltzmann distribution is established. Again we shall for simplicity treat only the case I = Y2. If initially 

The origin of this process may be seen in the following Let n = (na — np) be the difference in population let 0 = (na + np) let W+ be the probability for a nucleus to undergo a transition from the lower to the upper level as a result of an interaction with the environment, and let ILL be the analogous probability for the downward transition. Unlike the radiative transition probabilities, W+ and W are not equal in fact, at equilibrium, where the number of upward and downward transitions are equal, 

This rate equation describes a first-order decay process, characterized by a rate constant 2W, which may be defined as Rt or 1/Tj.The quantity n0fiB0/kT is the value of n at equilibrium or eq. With this notation, Eq. 2.28 becomes 

Thus R, or 1/T, serves as a measure of the rate with which the spin system comes into equilibrium with its environment. 

The magnitude of T, is highly dependent on the type of nucleus and on factors such as the physical state of the sample and the temperature. For liquids Tx is usually between 10 2 and 100, but in some cases may be in the microsecond range. In solids Tx may be much longer—sometimes days. The mechanisms of spin— lattice relaxation and some chemical applications will be taken up in Chapter 8. 

Because the energy separation 2pTfo is very small, the approximation e ] + X is valid, and we have 

consider the case in which the spin system and the lattice are in equilibrium at temperature T in field Hq. This means that the number of transitions upward must equal the number downward. If Wu is the probability of an upward transition per unit time and is the probability of a downward transition, then = 

Therefore, the probability of a downward transition in a magnetie field is slightly greater than the probability of an upward transition. 

Now let energy absorption from an rf field take plaee. The excess number of nuclei in the lower energy state is denoted n, son = N+. — N. The rate of change of n is 

In the earlier treatment we reached the conclusion that resonance absorption occurs at the Larmor precessional frequency, a conclusion implying that the absorption line has infinitesimal width. Actually NMR absorption bands have finite widths for several reasons, one of which is spin-lattice relaxation. According to the Heisenberg uncertainty principle, which can be stated 

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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 ... 

Figure Bl.13.4. The inversion-recovery detennination of the carbon-13 spin-lattice relaxation rates in melezitose. (Reproduced by pemiission of Elsevier from Kowalewski J and Maler L 1997 Methods for Structure Elucidation by High-Resolution N R ed Gy Batta, K E Kover and Cs Szantay (Amsterdam Elsevier) pp 325-47.)...

Canet D, Levy G C and Peat I R 1975 Time saving in C spin-lattice relaxation measurements by inversion-recovery J. Magn. Reson. 18 199-204... 

Attard J J, Doran S J, Flerrod N J, Carpenter T A and Flail L D 1992 Quantitative NMR spin-lattice-relaxation imaging of brine in sandstone reservoir cores J. Magn. Reson. 96 514-25... 

The characteristic time of the tliree-pulse echo decay as a fimction of the waiting time T is much longer than the phase memory time T- (which governs the decay of a two-pulse echo as a function of x), since tlie phase infomiation is stored along the z-axis where it can only decay via spin-lattice relaxation processes or via spin diffusion. 

If the strength of the saturating RF, A B2, and the spin-lattice relaxation time, Jj, are known, then can be measured, again free of magnetic field inliomogeneities. 

The tenn slow in this case means that the exchange rate is much smaller than the frequency differences in the spectrum, so the lines in the spectrum are not significantly broadened. Flowever, the exchange rate is still comparable with the spin-lattice relaxation times in the system. Exchange, which has many mathematical similarities to dipolar relaxation, can be observed in a NOESY-type experiment (sometimes called EXSY). The rates are measured from a series of EXSY spectra, or by perfonning modified spin-lattice relaxation experiments, such as those pioneered by Floflfman and Eorsen [20]. 

If the two sites exchange with rate k during the relaxation, tiien a spin can relax either tlirough nonnal spin-lattice relaxation processes, or by exchanging witli the other site, equation (B2.4.45) becomes (B2.4.46). 

In a second attempt to obtain more insight into the binding location of the dienophile and now also the diene, we have made use of the influence of paramagnetic ions on the spin-lattice relaxation rates of species in their proximity. Qose to these ions the spin-lattice relaxation rate is dramatically enhanced. This effect is highly distance-dependent as is expressed by Equation 5.7, describing the spin-lattice... 

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... 

Figure 5.8. Paramagnetic ion-induced spin-lattice relaxation rates (rp) of the protons of 5.1c and 5.1 f in CTAB solution and of CTAB in the presence of 5.1c or 5.1 f, normalised to rpfor the surfactant -CH-j. The solutions contained 50 mM of CTAB, 8 mM of 5.1c or 5.1f and 0 or 0.4 mM of [Cu (EDTA) f ...

S. spimchromogenes Spin-lattice relaxation Spinnerets Spinnerettes Spinning... 

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. 

The spin-lattice relaxation time, T/, is the time constant for spin-lattice relaxation which is specific for every nuclear spin. In FT NMR spectroscopy the spin-lattice relaxation must keep pace with the exciting pulses. If the sequence of pulses is too rapid, e.g. faster than BT/max of the slowest C atom of a moleeule in carbon-13 resonance, a decrease in signal intensity is observed for the slow C atom due to the spin-lattice relaxation getting out of step. For this reason, quaternary C atoms can be recognised in carbon-13 NMR spectra by their weak signals. 

The technique for measurement which is most easily interpreted is the inversion-recovery method, in which the distribution of the nuclear spins among the energy levels is inverted by means of a suitable 180° radiofrequency pulse A negative signal is observed at first, which becomes increasingly positive with time (and hence also with increasing spin-lattice relaxation) and which... 

Thus, in the series of Ti measurements of 2-octanol (42, Fig. 2.27) for the methyl group at the hydrophobic end of the molecule, the signal intensity passes through zero at Tq = 3.8 s. From this, using equation 10, a spin-lattice relaxation time of Ti = 5.5 s can be calculated. A complete relaxation of this methyl C atom requires about five times longer (more than 30 s) than is shown in the last experiment of the series (Fig. 2.27) Tj itself is the time constant for an exponential increase, in other words, after T/ the difference between the observed signal intensity and its final value is still 1/e of the final amplitude. 

Figure 2.27. Sequence of measurements to determine the C spin-lattice relaxation times of 2-octanol (42) [(CD3)2C0, 75% v/v, 25 °C, 20 MHz, inversion-recovery sequence, stacked plot]. The times at which the signals pass through zero, xo, have been used to calculate, by equation 10, the T values shown above for the nuclei of 2-octanol...

The main contribution to the spin-lattice relaxation of C nuclei which are connected to hydrogen is provided by the dipole-dipole interaction (DD mechanism, dipolar relaxation). For such C nuclei a nuclear Overhauser enhancement of almost 2 will be observed during H broadband decoupling according to ... 

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