The Dipolar Interaction

The dipole interaction arises from the coupling between two magnetic dipoles. Classically the energy of two interacting dipoles p,] and p2, a distance r apart, is given by 

The quantum mechanical Hamiltonian can be derived directly by substitution of Hi - y Li and = 72 /2 which leads to 

This can be usefully rewritten in terms of polar coordinates as 

This leads to the important alphabet expression for the dipolar Hamiltonian 

Only terms A and B commute with E and this is the Hamiltonian the secular part of Hd which is independent of time. 

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As circular domains grow in size or number, the dipolar interactions between them increase until they form a hexagonal array of spacing... 

It is occasionally desirable to retain a small proportion of molecular orientation, in order to quantitate the dipolar interactions present, whilst minimizing their contribution to the linewidth. Partial orientation may be achieved by using a nematic solvent. In large, magnetically anisotropic molecules it may occur naturally at the highest magnetic fields. 

Fig. 11. Scheme of the two possible colhieEir arrangements in the case of a [2Fe-2S] center interacting magneticEilly with a mononucleEir center M. The iron-to-iron distance was tsiken equEil to 2.7 A and the center-to-center distance was assumed to be equal to 12 A. The open Eirrows indicate the location of the equivalent magnetic moment that can be used to describe the dipolar interactions between M Emd both the ferric and ferrous sites of the [2Fe-2S] + center. 

This example shows that dipolar interactions can produce unexpected effects in systems containing polynuclear clusters, so that their complete quantitative description requires a model in which the dipolar interactions between all the paramagnetic sites of the system are explicitly taken into account. Local spin models of this kind can provide a description of the relative arrangement of the interacting centers at atomic resolution and have been worked out for systems containing [2Fe-2S] and [4Fe-4S] clusters (112, 192). In the latter case, an additional complication arises due to the delocalized character of the [Fe(III), Fe(II)] mixed-valence pair, so that the magnetic moments carried by the two sites A and B of Fig. 8B must be written... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

Nuclear spins can be considered as dipoles that interact with each other via dipolar couplings. While this interaction leads to strongly broadened lines in soUd-state NMR spectroscopy, it is averaged out in isotropic solution due to the fast tumbUng of the solute molecules. In Uquid-state NMR spectroscopy, the dipolar interaction can only be observed indirectly by relaxation processes, where they represent the main source of longitudinal and transverse relaxation. 

Fig. 9.5 Illustration of the dipolar interaction. (A and B) The magnetic field induced by spin I adds up to the static magnetic field Bo and leads to a shift of the resonance frequency of the close-by spin S. Since spins parallel and...

The relaxation enhancement displayed for canthaxanthin, 7 -apo-7 -(4-carboxyphenyl)-P-carotene and BI was analyzed to provide interspin distances. The dipolar interactions and the distances can be determined according to procedures described elsewhere (Budker et al. 1995, Rakowsky et al. 1995, Eaton and Eaton 2000, Rao et al. 2000) and based on simulations of the paramagnetic metal ion contribution, WAA. 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

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... 

The coupling term, traditionally denoted by cr B (which has however nothing to do with the screening coefficient of Section 2.2), is the so-called cross-relaxation rate and is a relaxation parameters which depends exclusively on the dipolar interaction between nuclei A and B, contrary to auto-relaxation rates which are compounds of several contributions. For instance, if A is a carbon-13, the auto-relaxation rate can always be written as... 

Conversely, the cross-correlation rates depend solely on the csa mechanism and on the dipolar interaction which is of prime importance here. It arises in fact from correlation functions of the form b (t)b(0), where b (t) refers to the csa mechanism whereas b(t) refers to the dipolar interaction. One has... 

The fact that dynamic 13C polarization is only possible through the indirect way via tire 1H spins suggests the mechanism of polarization transfer. Since the polarization transfer between the electrons and nuclei are driven by the dipolar interactions between them, and the fraction of the guest triplet molecules was small, it would be natural to assume that the polarization of the electron spins in the photo-excited triplet state is given to those H spins which happen to be close to the electron spins, and then the 1H polarization would be transported away over the whole volume of the sample by spin diffusion among the 1H spins. 

Organic Triplet State Molecules and the Dipolar Interaction... 

The last term (which would be zero if D came from the dipolar interaction and thus had zero trace) raises all levels equally and so has no effect on spectroscopy and can be dropped. Thus, again, only two parameters, D and E, are required to completely specify the fine structure interaction. 

C. Elsasser, M. Brecht and R. Bittl, Pulsed electron-electron double resonance on multinuclear metal clusters Assignment of spin projection factors bsed on the dipolar interaction, J. Am. Chem. Soc., 2002, 124, 12606. 

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