Spin-lattice relaxation transition

We have seen that in a steady field Hq a small excess, no, of nuclei are in the lower energy level. The absorption of rf energy reduces this excess by causing transitions to the upper spin state. This does not result in total depletion of the lower level, however, because this effect is opposed by spin-lattice relaxation. A steady state is reached in which a new steady value, n, of excess nuclei in the lower state is achieved. Evidently n can have a maximum value of o and a minimum value of zero. If n is zero, absorption of rf energy will cease, whereas if n = no, a steady-state absorption is observed. It is obviously desirable that the absorption be time independent or. in other words, that s/no be close to unity. Theory gives an expression for this ratio, which is called Zq, the saturation factor ... 

Fig. 19. Experimental spin alignment decay curves of chain deuterated PS-d3 at temperatures above and below the glass transition for various evolution times t,. Note the different timescales of t2 at the different temperatures. The straight lines indicate the decays of the plateau values on the timescale of the spin-lattice relaxation time T,. Sample characterization Mw = 141000, Mw/Mn = 1.13, atactic...

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

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... 

Ammonium alums undergo phase transitions at Tc 80 K. The phase transitions result in critical lattice fluctuations which are very slow close to Tc. The contribution to the relaxation frequency, shown by the dotted line in Fig. 6.7, was calculated using a model for direct spin-lattice relaxation processes due to interaction between the low-energy critical phonon modes and electronic spins. 

When, however, phonons of appropriate energy are available, transitions between the various electronic states are induced (spin-lattice relaxation). If the relaxation rate is of the same order of magnitude as the magnetic hyperfine frequency, dephasing of the original coherently forward-scattered waves occurs and a breakdown of the quantum-beat pattern is observed in the NFS spectrum. 

In the theory of deuteron spin-lattice relaxation we apply a simple model to describe the relaxation of the magnetizations T and (A+E), for symmetry species of four coupled deuterons in CD4 free rotators. Expressions are derived for their direct relaxation rate via the intra and external quadrupole couplings. The jump motion between the equilibrium positions averages the relaxation rate within the same symmetry species. Spin conversion transitions couple the relaxation of T and (A+E). This mixing is included in the calculations by reapplying the simple model under somewhat different conditions. The results compare favorably with the experimental data for the zeolites HY, NaA and NaMordenite [6] and NaY presented here. Incoherent tunnelling is believed to dominate the relaxation process at lowest temperatures as soon as CD4 molecules become localized. 

Familiar to most chemists is the notion of spin-lattice relaxation [25]. Labeled as T, the spin-lattice relaxation time is defined as the amount of time for the net magnetization (A/J to return to its equilibrium state (M0) after a spin transition is induced by a radiofrequency pulse ... 

Furthermore, the method of orientation selection can only be applied to systems with an electron spin-spin cross relaxation time Tx much larger than the electron spin-lattice relaxation time Tle77. In this case, energy exchange between the spin packets of the polycrystalline EPR spectrum by spin-spin interaction cannot take place. If on the other hand Tx < Tle, the spin packets are coupled by cross relaxation, and a powder-like ENDOR signal will be observed77. Since T 1 is normally the dominant relaxation rate in transition metal complexes, the orientation selection technique could widely be applied in polycrystalline and frozen solution samples of such systems (Sect. 6). 

The reason why one chose to follow the main liquid-crystalline to gel phase transition in DPPC by monitoring the linewidth of the various or natural abundance resonance is evident when we consider the expressions for the spin-lattice relaxation time (Ti) and the spin-spin relaxation time T2). The first one is given by 1/Ti oc [/i(ft>o) + 72(2ft>o)] where Ji coq) is the Fourier transform of the correlation function at the resonance frequency o>o and is a constant related to internuclear separation. The relaxation rate l/Ti thus reflects motions at coq and 2coq. In contrast, the expression for T2 shows that 1/T2 monitors slow motions IjTi oc. B[/o(0) -I- /i(ft>o) + /2(2u>o)], where /o(0) is the Fourier component of the correlation function at zero frequency. Since the linewidth vi/2 (full-width at half-maximum intensity) is proportional to 1 / T2, the changes of linewidth will reflect changes in the mobility of various carbon atoms in the DPPC bilayer. 

Also spin-lattice relaxation times T and spin-spin relaxation times T2 were measured as a function of pressure on different selectively deuterated DPPC (at C2, Cg and Ci3, respectively) by Jonas and co-workers (Fig. 14). The spin-latticed relaxation time T is sensitive to motions with correlation times tc near Uo i e., motions with correlation times in the range from 10 to 10 " s. In comparison with Ti, the spin-spin relaxation time T 2 is more sensitive to motions with correlation times near (e qQlh), i.e., in the intermediate to slow range (10 " to 10 s). The Ti and T2 values obtained showed characteristic changes at various phase transition pressures, thus indicating abrupt changes... 

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