Nuclear Energy Levels and Relaxation Times

We have already calculated the energy gap between these two spin states [Eq. (2.4)], and this must equal the energy of the absorbed photon [Eq. (1.3)]. Combining these with Eq. (2.6) gives us 

as we expected from Chapter 1, for resonance to occur, the radiation frequency must exactly match the precessional frequency. 

But there is a fly in the ointment. Quantum mechanics tells us that, for net absorption of radiation to occur, there must be more particles in the lower energy state than in the higher one. If the two populations happen to be equal, Einstein predicted theoretically that transition from the upper (m = state to the lower (m = +f) state (a process called stimulated emis- 

Is there any reason to expect that there will be an excess of nuclei in the lower spin state The answer is a qualified yes. For any system of energy levels at thermal equilibrium, there will always be more particles in the lower state(s) than in the upper state(s). However, there will always be some particles in the upper state(s). What we really need is an equation relating the energy gap (AE) between the states to the relative populations of (numbers of particles in) each of those states. This time, quantum mechanics comes to our rescue in the form of the Boltzmann distribution  

Our NMR theory is almost complete, but there is one more thing to consider before we set about designing a spectrometer. We indicated previously that at equilibrium in the absence of an external magnetic field, all nuclear spin states are degenerate and, therefore, of equal probability and population. Then, when immersed in a magnetic field, the spin states establish a new (Boltzmann) equilibrium distribution with a slight excess of nuclei in the lower energy state. 

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Transitions between Zeeman energy levels, and hence nuclear magnetic relaxation, are caused by fluctuations (time variations) in the local interactions at the nucleus that can cause transitions. The transition rates between these energy levels that cause nuclear spin relaxation depend on two factors ... 

Firstly, unless there is adequate relaxation during the resonance experiment, the (21 + 1) energy levels tend to become equally populated and the signal disappears saturation occurs. Secondly, line widths are inversely proportional to the average times nuclei spend in the upper energy levels, and so depend on relaxation rates. Finally the effect of nuclear or electron spins upon the energy levels of neighbouring nuclear spins is influenced by their respective relaxation rates. 

The temperature-dependent shift of the resonance line is strongly connected with the nuclear magnetic relaxation times and the linewidth. The fluctuating magnetic field at the nucleus usually has components perpendicular to the external magnetic field. The Fourier spectrum of these field components contains terms with resonance frequencies, which induce transitions between nuclear energy levels. One may foresee the temperature... 

Relaxation effects in Mossbauer spectroscopy are of a different nature from those in NMR. The term relaxation effects or relaxation spectra in nuclear gamma resonance spectroscopy refers to averaging effects that occur in the hyperfine spectrum when the hyperfine interactions fluctuate at a rate more rapid than the nuclear frequency characteristic of the hyperfine interaction itself. This situation is a consequence of the rapid relaxation of the host ion among its energy levels, and the relaxation time for such effects is characteristic of the ion and not of the nuclear spins. The relaxation processes involved also affect electron spin resonance spectra, and their discussion is best considered in that context (see sections 3.3. and 3.4.). In the following subsections the principal interactions which contribute to the nuclear spin relaxation times in NMR experiments are briefly considered, and the connections between these and the parameters characterizing the steady-state spectrum are outlined. 

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... 

Symmetries of local electrical environments of quadrupolar nuclei (/ 1) profoundly influence relaxation times and resonance line shapes of such nuclei (9, 116). Consider a nucleus for which I = % (Br79, Bn). In the absence of quadrupolar perturbation, the nuclear spin levels are evenly spaced, as shown in I below, and the three possible nuclear resonance transitions have equal energies (Am = 1). If, however, eqQ 0... 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

Energy levels with Overhauser effect (a) Relaxation due to a time-dependent isotropic contact electron-spin-nuclear-spin hyperfine interaction a(t)l S which has a zero time-average, but allows processes X and Y and enhances nuclear spin transitions when the electron populations are made equal by saturation, (b) Relaxation is due to all dipole-dipole interactions, which allow processes X, V, and PNi nuclear spin transitions are forced into emission by the Overhauser effect. In (a) the relative Boltzmann populations before saturation are shown. 

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

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