Hamiltonian dipolar

The dipolar-coupling Hamiltonian (TTy) describes the through-space coupling between two nuclear spins l and Ij. The dipolar coupling has an rk- dependence, and is key to the determination of internuclear distances in both solid-state and solution-state NMR. The high-field truncated form of the dipolar Hamiltonian is given by... 

Rotational-echo double resonance (REDOR), originally introduced by Gullion and Schaefer [102], is a method to recouple heteronuclear spin pairs. The sequence relies on a train of rotor-synchronized n pulses applied to the I spin to interrupt the spatial averaging of the heteronuclear dipolar coupling under MAS to give a nonvanishing dipolar Hamiltonian over a full rotor cycle (Fig. 11.8). Typically, REDOR data are collected by col-... 

The longitudinal cross-relaxation rate (see Eq. (13)) originates solely from the terms in the dipolar Hamiltonian involving both spins, namely those terms corresponding to zero-quantum and double-quantum transitions so that... 

The angles a and P define the orientation of the sample relative to the Bo-field cor denotes the rotation frequency. For the REDOR reference experiment, the rotor-synchronised spin-echo experiment for the S nuclei (cf. Figure lA), the dipolar Hamiltonian integrated over one rotor period Tr averages to zero... 

Since in the case of an isolated pair of spin nuclei the dipolar dephasing, and hence the REDOR evolution curve, is exclusively governed by the dipolar Hamiltonian, the data analysis proves to be straightforward employing a universal REDOR curve, in which the normalised difference intensity AS/Sq is plotted as a function of the dimensionless product NTRxd.2 ... 

As previously observed, the dominant terms in the Hamiltonian which describe a spin system in the solid state are the dipolar and quadrupolar terms. In the case of nuclei with 1 = 1/2 (such as H, 13C, 19F and 29Si) the quadrupolar interaction is zero. The dipolar Hamiltonian HD (for a homonuclear spin system) has the general form ... 

The broad line spectra of nuclei with spin I = 1/2 in the solid state are mainly a consequence of the dominant contribution of the dipolar Hamiltonian HD (Eq. (4)), which gives rise to a local field B)oc. Its magnitude varies as a function of the angle 0.. between the intemuclear vector r.j and the applied magnetic field B0. Depending on the nature of spin system, two general types of interactions can be distinguished ... 

The main purpose of the sequences is to obtain an averaged Hamiltonian H in which the dipolar term is very small compared with the chemical shift term. The zeroth-order of the average dipolar Hamiltonian term is given by the following equation ... 

It is largely accepted that the dominant mechanism of nuclear spin relaxation in condensed polymers is due to dipolar interactions between the spins. The truncated homonuclear dipolar Hamiltonian has the form [15] ... 

The sum must be made over all spin pairs in the proton-rich solid. In the absence of large-amplitude molecular motion this Hamiltonian describes a line shape of width of up to 100 kHz. In the presence of molecular motion the angular part of Equation 13.1 becomes time-dependent, and the partial averaging of this term results in reduced linewidths. In polymers the geometry of main-chain motion is limited by the structure of the polymer chain, and is inherently anisotropic. As a general rule, as the measurement temperature is increased the motion tends to become more isotropic in nature as the free volume increases, and the extent of averaging of the dipolar Hamiltonian increases. This... 

Using time-dependent perturbation theory and taking full account of the symmetry and commutation relations for the high-order dipolar Hamiltonians, Hohwy et al.61 69 gave a systematic analysis of homonuclear decoupling under sample rotation and proposed a novel approach to the design of multiple-pulse experiments. Based on the theoretical analysis, they proposed a pulse sequence that can average dipolar interaction up to the fifth order. One example of these pulse sequences is shown at the top of Fig. 3. This sequence is sufficiently powerful that it is possible to obtain precise measurement of proton chemical shift anisotropies, as shown in Fig. 3. 

The relationships between the various forms of the dipolar Hamiltonian are explained in appendix 8.2. As we see from (1.59), the dipolar interaction has various components in the molecule-fixed axis system but the most important one, and often the only one to be determined from experiment, is Tq(C). This leads us to define a constant to, the axial dipolar hyperfine component, given in SI units by,... 

We now show that the nuclear spin dipolar interaction has matrix elements of exactly the same form. We take the dipolar Hamiltonian to have the form given previously in equation (8.10) and find that its matrix elements are given by... 

This form of the electron spin-spin dipolar Hamiltonian is discussed in appendix 8.3. The diagonal (q = 0) component of (8.192) may be written... 

Note that this is similar to the Hamiltonian describing the dipolar interaction of two nuclear magnetic moments, presented in equation (8.9), but is opposite in sign. Using the results derived in the first part of appendix 8.1, we see that the dipolar Hamiltonian (8.227) may be written as a cartesian tensorial operator ... 

Third, we deal with the dipolar coupling of the nuclear spins, which is evaluated below note the form of the dipolar Hamiltonian (equation (8.282)), which is the one appropriate for the particular angular momentum coupling scheme used. 

We first make use of the results derived for the dipolar coupling of two proton spins in appendix 8.1. If we replace I by I, I2 by S and note the overall change in sign, we see that the conventional dipolar Hamiltonian,... 

Use the dipolar Hamiltonian in Eq. 7.8 to compute the energy levels and spectra for the A part of an AX spin system. Ignore chemical shifts and indirect coupling. Use wave functions and other pertinent results from Chapter 6. 

Use the dipolar Hamiltonian for a homonuclear spin system, Eq. 7.6, together with symmetrized wave functions from Chapter 6, to compute the energy levels and spectrum for a single H20 molecule in a solid. 

One can see from Eq. (3) that the van Vleck dipolar Hamiltonian is the product of a spatial part and a spin part. In liquids, the rapid isotropic molecular tumbling motion, which occurs at frequencies well above the dipolar linewidth, averages the spatial part (1—3 cos2 6tj) to zero, thus nulling the dipolar broadening. In solids, the spins are constrained to vibrate and rotate about their mean positions, resulting in an effective dipolar Hamiltonian which is generally non-zero, and consequently in... 

This leads to the important alphabet expression for the dipolar Hamiltonian... 

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