Exponential recovery

This represents an exponential recovery of strain which is a reversal of the predicted creep. 

Fig. 1.—Diagrammatic Representation of the Recovery of Magnetization along the z-Axis (Mj), from Its Initial Value (-M ) to +Mo, Following Its Inversion by a 180° Pulse. The exponential recovery curve shown in [A] depicts the return of magnetization that would be found in a typical inversion-recovery experiment. The curve in [B] would be obtained from a three-pulse sequence, and is a plot of which decreases from an initial value of...

When Ti is long enough it may be obtained by observation of the exponential recovery of z direction magnetization after removal of a large Hi field or the loss of magnetization after application of the rf field. 

The most important relaxation processes in NMR involve interactions with other nuclear spins that are in the state of random thermal motion. This is called spin-lattice relaxation and results in a simple exponential recovery process after the spins are disturbed in an NMR experiment. The exponential recovery is characterised by a time constant Tj that can be measured for different types of nuclei. For organic liquids and samples in solution, Tj is typically of the order of several seconds. In the presence of paramagnetic impurities or in very viscous solvents, relaxation of the spins can be very efficient and NMR spectra obtained become broad. 

Fig. 3.18. Exponential recovery (A) of Mz(t) of a nuclear spin / dipole coupled to a paramagnetic metal ion. When I is also coupled to another nuclear spin J, the latter also coupled to the metal ion, non-exponentiality occurs. If J relaxes slower than /, curves B and C are obtained for a selective and a non-selective experiment respectively. If J relaxes slower than /, curves D (selective) and E (non-selective) are obtained. If J relaxes at the same rate as /, a selective experiment gives an intermediate behavior between curves B and D (not shown), while a non-selective experiment gives pure exponential recovery (A). It is apparent that in all cases non-selective experiments perform better than selective experiments, as they are less sensitive to the non-exponentiality introduced by I-J coupling. Conditions R m = 10 s l R M = 20 s l (B,C), 5 s l (D,E) and 10 s l (A). The I-J cross-relaxation rate ou (Chapter 7) is —20 s"1.

The linear drop and exponential recovery shape of these transformations also appear in power-compensated DSC traces, but for different reasons. The temperature measuring device (RTD) measures its own temperature, which is influenced by all substances in the chamber, the housing, the sample crucible, as well as the melting sample. The device adds power to the sample side as needed to compensate for the cooling effect on the chamber due to sample melting. This energy requirement increases lineaxly since the setpoint sample temperature increases linearly. When melting is over, the need for extra heat flow to the sample chamber side drops exponentially as the chamber temperature quickly catches up to the setpoint. 

Rb and 1H SLR rate as a function of temperature is a very important parameter which shows the suppression of phase transition and reveals the frustration in the mixed system. Temperature dependence of Ti in any ordered system can be described by the well known Bloembergen-Purcell-Pound (BPP) type expression. However, disordered systems show deviations from BPP behaviour, showing a broad distribution of relaxation times. The magnetization recovery shows a stretched exponential recovery of magnetization following M(t)=Mo(1 — 2 exp (— r/Ti) ) where a is the stretched exponent. 

Sobol et al.8 have measured proton SLR time (7 ) in Rb1 x(NH4)xH2As04 systems in the range 100-4.2 K. Magnetization recovery was found to be non-exponential in the entire range of temperature. The MR data fit to a stretched exponential recovery and the exponent a was found to be temperature dependent implying broadening of the distribution of microscopic correlation times p(r ) with decreasing temperature. 

Fig. 16. Single-exponential recovery of the photoinduced bleaching of a single crystal of all-rmns-iS-carotene recorded at 77 K. The crystal was excited at 575 nm and the recovery was probed at 1300 nm.

M/tan0 for various 6 values to yield a straight line with a slope exp(-At/Tj) for magnetization obeying exponential recovery. In this case, the pulse interval At is fixed and the adjustable parameter is the nutation angle which is most conveniently changed by the rf pulse width. This may make the method awkward for most existing apparatus. 

Eq. 9 leads to spin diffusion with the flip-flop rate limiting the observed T. Below 10 s, the relaxation for the A spin is limited by the relaxation sink, which corresponds to the bonded methylene carbon. The effect of the siqall deviation In the coefficients from 1.0 and 0.0 near t s 10 s is non-exponential recovery, with the negative coefficient leading to a curve which is concave downward. For the M spin, the decay Jumps from theX curve for correlation times <10 " s to the Xg curve for correlation times >10 s. 

As shown in Figure Ib-d, the electron spin relaxation kinetics observed at 22 K for the tyrosine radical D" " in PSII are markedly non-exponential. Consistent with the results of ESE studies of Hoff and coworkers (5,9) and of Britt etal (10), the saturation-recovery transients observed for D" " in PSII membrane samples poised in the Sj or the S2 state (Figure Ic and d) are better fit by a rate law including two evenly weighted exponential recovery rate constants rather than by a law with a single exponential rate constant. Evelo et al. (5) account for this behavior in... 

Two methods of time-domain ESR will be briefly discussed. The oldest method is that of saturation recovery, which is a direct method to determine spin-lattice relaxation times. The idea is to perturb the steady-state population of spins with a partially saturating pulse of microwaves and then to observe with a very weak microwave field the recovery of the perturbed spin population to equilibrium. In the absence of complications, the recovery process is exponential and can be related to the time constant for spin-lattice relaxation. Exponential recoveries are generally observed in liquids and in some cases in solids. If the spin-lattice relaxation time is not much longer than the spin-spin relaxation time, which is atypical in paramagnetic systems, the interpretation of saturation recovery data becomes more complex. 

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