Two-pulse echo

As the spins precess in the equatorial plane, they also undergo random relaxation processes that disturb their movement and prevent them from coming together fiilly realigned. The longer the time i between the pulses the more spins lose coherence and consequently the weaker the echo. The decay rate of the two-pulse echo amplitude is described by the phase memory time, which is the time span during which a spin can remember its position in the dephased pattern after the first MW pulse. Tyy is related to the homogeneous linewidth of the individual spin packets and is usually only a few microseconds, even at low temperatures. 

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. 

In electron spin echo relaxation studies, the two-pulse echo amplitude, as a fiinction of tire pulse separation time T, gives a measure of the phase memory relaxation time from which can be extracted if Jj-effects are taken into consideration. Problems may arise from spectral diflfrision due to incomplete excitation of the EPR spectrum. In this case some of the transverse magnetization may leak into adjacent parts of the spectrum that have not been excited by the MW pulses. Spectral diflfrision effects can be suppressed by using the Carr-Purcell-Meiboom-Gill pulse sequence, which is also well known in NMR. The experiment involves using a sequence of n-pulses separated by 2r and can be denoted as [7i/2-(x-7i-T-echo) J. A series of echoes separated by lx is generated and the decay in their amplitudes is characterized by Ty. ... 

Resolvable modulation is detected on a three-pulse echo decay spectrum of predeuterated 3-carotene radical (Gao et al. 2005) as a function of delay time, T. The resulting modulation is known as ESEEM. Resolvable modulation will not be detected for nondeuterated P-carotene radical since the proton frequency is six times larger. The modulation signal intensity is proportional to the square root of phase sensitive detection and interfering two-pulse echoes and suppressed by phase-cycling technique (Gao et al. 2005). Analysis of the ESEEM spectrum yields the distance from the radical to the D nucleus, a the deuterium coupling constant, and the number of equivalent interacting nuclei (D). The details related to the analysis of the ESEEM spectrum are presented in Gao et al. 2005. 

Acquisition of wide static or MAS spectra of the CT is typically made using a two-pulse echo sequence nil t n (the Hahn echo [55,106]). By setting the strength of the rf-field at mrf = Aco/(I+ 1/2), where Aco is the CT linewidth, these sequences selectively irradiate the CT, while minimizing the contributions from the STs [107, 108]. In such case, the opjj frequencies are given by (29) ... 

A particularly interesting case is when the inhomogeneous width is comparable to the root-mean-square of the fluctuation amplitude (A2n < 2KkBT). In this case the interpretation of the simple two-pulse echo measurement is ambiguous [39, 40]. Again, in the classical limit, in this region,... 

For the dynamical distribution it will in general be necessary to consider both the auto and cross time correlation functions of the 0-1 and the 1-2 frequencies (117). For example, if the fluctuations, <5A(t), in the anhar-monicity are statistically independent of the fluctuations in the fundamental frequency, the oscillating term (1 — elAt3) in Equation (18) would be damped. In a Bloch model the fluctuations in anharmonicity translate into different dephasing rates for the 0-1 and 1-2 transitions that were discussed previously for two pulse echoes of harmonic oscillators. Thus we see that even if A vanishes, the third-order response can be finite (94). 

Investigating molecular dynamics using NMR, in contrast to DS and LS, involves the application of several conceptually quite different techniques. For example, in spin-lattice relaxation studies one is concerned with familiar time correlation functions that are probed as spectral density point by point (Section II.D.2). In the case of line-shape analysis, usually a two-pulse echo sequence is applied, and the... 

Equation (4) shows that modulations of the two-pulse echo amplitude occur at the fundamental hyperfine frequencies and at their sum and difference combination frequencies. The amplitude of the modulations is given by the product of the transition probabilities for the two different transitions associated with branching , Mp wp, while the nonmodulated portion of the echo envelope depends only on the product of the transition probabilities for the nonbranching spins, m " or vf. Substituting the expressions for u and w given by equations (2) and (3) into equation (4) yields... 

From the above discussion, it is clear that observation of ESEEM requires that the microwave pulses affect branching of the EPR transitions. This places a quantum mechanical constraint on the ESEEM experiment, in that each energy level must be involved in at least two different microwave transitions, and an experimental constraint that requires the microwave pulse bandwidth to cover the spread in frequencies needed to fully excite the branching . The experimentally observed ESEEM function is a product of the quantum mechanically derived modulation function and a decay function that describes the loss of magnetization due to spin relaxation. These decay functions are typically modeled with exponential forms exp(-T/To) where n = 1,2 or 0.5. Fora 90° - t - 180° or two-pulse echo experiment, Tq = a time that is typically on the order of 1 qs, as evidenced by the data shown for the Cu(II) center in Figure 1. This... 

A signal which has first vanished with time and then reappears some time later is called an echo. In spectroscopy, the echo is formally associated with a reversal of time, so that the reappearing signal can be understood in terms of time running backwards for a sufficiently isolated ensemble of molecules or spins (Fig. 2.2.9) [Bliil]. For uncoupled spins in simple liquids an NMR echo of the FID is generated by a 180° flip of the phase of all magnetization components. Since the discovery of the original two-pulse echo [Hahl], many other echoes (cf. Section 3.4) have been discovered in spectroscopy based... 

Figure 20. Two-pulse echo decay envelope of Cu(II) complexes of (a) bleomycin, (b) diethylenetriamine and imidazole, and (c) diethylenetriamine and pyrimidine. In traces a and b, one observes a modulation pattern arising from the interaction of Cu(ll) with the remote N of ligated imidazole. The respective magnetic fields and spectrometer frequencies are the following a, 3080 G, 9247 MHz b, 3195 G, 9251 MHz c, 2970 G, 9225 MHz. Lines indicating the periods were added by the authors. From [268], with permission.

Yu. N. Moskvich, N. A. Sergeev, and G. I. Dotsenko, "Two-pulse echo in solids containing isolated three-spin systems," Phys. Stat. Solidi (a) 30, 409-418 (1975). 

Fig. 32. Two-pulse echo decay of triphenylmethyl in triphenylamine at 1.8 K in zero magnetic field. At is the time separation between the exciting pulses.

We show in Fig. 37 a two-pulse echo spectrum, which clearly exhibits intense modulation at the < 1/2 — 3/2> ground state splitting of 8.47 MHz." ... 

Fig. 37. Two-pulse echo intensity vs excitation pulse delay in the transition of...

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