Echo intensity

The mathematical description of the echo intensity as a fiinction of T2 and for a repeated spin-echo measurement has been calculated on the basis that the signal before one measurement cycle is exactly that at the end of the previous cycle. Under steady state conditions of repeated cycles, this must therefore equal the signal at the end of the measurement cycle itself For a spin-echo pulse sequence such as that depicted in Figure B 1.14.1 the echo magnetization is given by [17]... 

The electron-spm echo envelope modulation (ESEEM) phenomenon [37, 38] is of primary interest in pulsed EPR of solids, where anisotropic hyperfme and nuclear quadnipole interactions persist. The effect can be observed as modulations of the echo intensity in two-pulse and three-pulse experiments in which x or J is varied. In liquids the modulations are averaged to zero by rapid molecular tumbling. The physical origin of ESEEM can be understood in tenns of the four-level spin energy diagram for the S = I = model system... 

The main advantage of tlie tln-ee-pulse ESEEM experiment as compared to the two-pulse approach lies m the slow decay of the stimulated echo intensity detemiined by T, which is usually much longer than the phase memory time Ty that limits the observation of the two-pulse ESE. 

An alternative 2D ESEEM experiment based on the four-pulse sequence depicted in figure B 1.15.12(B) has been proposed by Meliring and coworkers [40]. In the hyperfme sublevel correlation (HYSCORE) experiment, the decay of the echo intensity as a fimction of is governed by -relaxation, whereas the echo decay along the t2-axis is... 

Muns ENDOR mvolves observation of the stimulated echo intensity as a fimction of the frequency of an RE Ti-pulse applied between tlie second and third MW pulse. In contrast to the Davies ENDOR experiment, the Mims-ENDOR sequence does not require selective MW pulses. For a detailed description of the polarization transfer in a Mims-type experiment the reader is referred to the literature [43]. Just as with three-pulse ESEEM, blind spots can occur in ENDOR spectra measured using Muns method. To avoid the possibility of missing lines it is therefore essential to repeat the experiment with different values of the pulse spacing Detection of the echo intensity as a fimction of the RE frequency and x yields a real two-dimensional experiment. An FT of the x-domain will yield cross-peaks in the 2D-FT-ENDOR spectrum which correlate different ENDOR transitions belonging to the same nucleus. One advantage of Mims ENDOR over Davies ENDOR is its larger echo intensity because more spins due to the nonselective excitation are involved in the fomiation of the echo. 

Figure 6 NMR proton spin-echo intensity vs. time.

The most important advantage of ESE-ENDOR lies in the fact that it does not critically depend on spin relaxation rates like the conventional cw-ENDOR technique, i.e. ESE-ENDOR might be more sensitive than the latter one. In ESE-ENDOR changes in the echo intensities up to 100% have been reported109). 

Three analytical expressions for the spin-echo intensity as a function of the gradient in a pulsed field gradient NMR experiment for spins diffusing in a sphere with reflecting walls are reinvestigated. It is found that none of the published formulas are completely correct. By numerical comparisons the correct formula is found. 

Since the Murday-Cotts paper in 1968 much progress has been made toward the theoretical description of the spin-echo intensity E g, A) for spins diffusing in well-defined geometries. Tanner and Stejskal derived already in 1968 the exact expression of ii(g. A) for spins diffusing in a rectangular box. The derivation of an exact expression for E g, A) for diffusion in a sphere with reflecting walls is not a trivial mathematical problem and it took between 1992 and 1994 when three expressions were published. All three expressions are only valid in the short-gradient-pulse approximation (see below). 

Callaghan points out the analogy between PFGE NMR experiments in the short-gradient-pulse limit and diffraction experiments. Under the short gradient pulse condition the echo intensity obtained from a PFG NMR experiment can be written as ... 

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

Figure 2 shows a multiple-echo, intensity-weighted cyclohexane MRM image of a 300 pm cross-sectional slice of a 4 mm diameter coke bean. The density-weighted H images are collected with an unprocessed image resolution of 24 pm pixel"1. Since the coke is packed in wet gypsum, the area outside the coke shows very little cyclohexane intensity and appears black. Several feature arise from this image which complement the SEM images in Figure 1. Firstly, a dark band at the perimeter of the coke corresponds to the exterior shell depicted in deal in Figure lc. This dark feature reflects the absence of cyclohexane and thus indicates the...

By chosing short values of x, the effect of diffusion may be eliminated, because the diffusion is effective in reducing the echo intensity only during the 2x period. However, there is a modified version of spin-echo pulse sequences which provides the measurement of the spin diffusion coefficients (D) by NMR, as it will be shown later. 

The effect of curing on the diffusion of polymer and the curing agent is studied for the system of hydroxyl-terminated polybutadiene (R-45-M)/isophorone disso-cyanate (IPDI). Both components contribute to the echo intensity and the plot of In P(x)/I(0)] vs (G5)2 (A — 5/3) consists of two exponentials (Eq. (22)) the fast component (the steep intial slope) is attributed to the IPDI, and the long component to the R-45-M. The dependence of both diffusion constants on the curing time is shown in Fig. 19. The accuracy for Dfast data is less pronounced than for the polymer D(Mn), because only the first few data points are relevant for its determination. Furthermore, the low tail of the R-45-M molecular weight distribution nearly coin-... 

Sweep the magnetic field until an echo appears in the Specjet. Optimize the echo intensity by varying the mw power and by adjusting the phase of the detection channels. Decrease the shot repetition time until the echo intensity decreases. Sweep the magnetic field until the echo is maximal. This magnetic field is kept throughout the PELDOR experiment. Readjust the sample position in the cavity to maximize the echo intensity. 

Create a Hahn-echo sequence with 16/32 ns in one channel and optimize the echo intensity. Do the same in a second channel and adjust its phase 180° to the first cannel. Both channels are used for the two-step phase cycle. Running the experiment with the two-step phase cycle eliminates receiver offsets and allows reading off the real echo signal intensity, which is important for accurately determining Vx in spin counting experiments (Section 5.1). 

---