Dipolar Coupling Between Two Spins

The Hamiltonian for the dipolar coupling between two magnetic moments associated with the spins I and J is given by an equation similar to Eq. (1.1) 

In general, dipolar relaxation occurs whenever these quantities are modulated by time-dependent phenomena. In solids r and y are fixed for a single I-J pair. In liquids, y varies upon rotation of the molecule bearing the I-J pair, and r may vary because of chemical exchange. 

It is interesting to note that on rapid (with respect to the anisotropy of the interaction energy in frequency units) rotation, the dipolar interaction energy averages to zero. In contrast, the square of this quantity (which determines the extent of the contribution to relaxation) does not average to zero. The average values obtained by integration of the functions results in 

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Within this framework, the dipolar coupling between two spins I and S may be described in terms of the Hamiltonian... 

The most common interactions are dipolar coupling between two spins, chemical shielding and quadrupolar interaction for spins I> 1. The chemical-shielding interaction arises from the motion of electrons around a nucleus induced by the external applied static magnetic field. This motion generates local magnetic fields that modify the total field experienced by the nucleus, and are characteristic of the local chemical environment of the nuclei. All nuclei with a spin /> 1 possess a quadrupolar moment Q that interacts with the EFG Vrs — d2 V/dXfdx at the nuclear site. The EFG tensor Vrs is a symmetric tensor with zero trace Tr( E) = A E = 0 (from Laplace equation). 

To illustrate the effect of MAS, we consider in Fig. 9.7 the specific example of a dipolar coupling between two spins. The four cones represent the range of positions adopted over the course of one rotor period for four different orien-... 

Whenever the system is no longer constituted by single non-interacting spins, the simple Bloch Equation (2) must be completed by additional coupling terms. Let us consider the dipolar interaction between two spins... 

A special case occurs if the solid contains nuclear spin pairs that are sufficiently far apart from each other such that intrapair coupling dominates interpair coupling. In this situation, instead of a featureless broad lineshape, we observe a well-defined powder pattern known as a Pake doublet (Figure 15.15). Since the dipolar coupling between two nuclei... 

Rotational resonance is an effect whereby the dipolar coupling between two nuclei is restored under MAS by matching the isotropic chemical shift difference between the two spins (expressed in Hz) to an integer multiple of the MAS frequency (p ), i.e.. 

DeRose and coworkers have explored conformational changes of TAR RNA upon binding of divalent metal ions (Ca " ) by measuring the dipolar coupling between two attached spin labels 20 using CW EPR (Fig. 2). The U25-U40 distances obtained from Fourier deconvolution methods are 11.9 0.3 A for TAR RNA in the absence of divalent metal cations and 14.2 0.3 A when 50 mM Ca " was added [45]. These results are in accordance with the proposed coaxial stacking of the two TAR helices upon addition of metal ions based on the X-ray crystal structure [80]. 

For heteronuclear dipolar relaxation, the dipole-dipole coupling between two unlike spin- nuclei / and S (e.g., 13C-H pair) separated by an internuclear distance rIS is considered. The Zeeman spin-lattice (7jz) and spin-spin (T2) relaxation times for the / spin are given, respectively, by... 

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... 

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