Dipolar coupling between two spins

APPENDIX

DIPOLAR COUPLING BETWEEN TWO SPINS

II

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)



H=−

µ 0  2γ I γ J [3( I ⋅ r )( J ⋅ r ) − I ⋅ J ] 4π r 3

(II.1)

where r is a unit vector oriented along the line of length r connecting I and J. In the most general case, the I and J vectors are not parallel, and each of them will form different angles with r and with the applied magnetic field, B0. If the Zeeman interaction is the dominant among all interactions involving the spins, the quantization axis of both spins will be along B0 (Fig. II.1). The first spin I creates a dipole µ1 which precesses around the field B0 and thus has a static component along the field and a rotating component in the plane perpendicular to the field. The static component of µ1 produces at the site of the dipole µ2, related to the second spin J, a small static field that depends on the relative position of the spins. For large B0, only the component parallel or antiparallel to B0 significantly changes the net static field. From Eq. (II.1), when θI = θJ = γ, it results that the dipolar interaction energy depends on the interspin distance r, on the eigenstates of the I and J spins, and on the angle γ  between the interspin vector r and the external magnetic field (along which both I and J are quantized)

E ∝ 3 cos 2 γ (t ) − 1.

(II.2)

FIG. II.1  Reciprocal orientation of a nuclear spin I and an electron spin J, in the presence of an external magnetic field

NMR of Paramagnetic Molecules. http://dx.doi.org/10.1016/B978-0-444-63436-8.00014-4 Copyright © 2017 Elsevier B.V. All rights reserved.

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APPENDIX II  Dipolar coupling between two spins

In general, dipolar relaxation occurs whenever these quantities are modulated by time-dependent phenomena. In solids r and γ are fixed for a single I–J pair. In liquids, γ 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 1

1

1 1 (3cos 2 γ − 1)d(cos γ ) = ( x 3 − x ) = 0 ∫ 2 −1 2 −1

1

(II.3)

1

1 1 (3 cos 2 γ − 1)2 d(cos γ ) = ∫ (9 x 4 + 1 − 6 x 2 ) d x 2 −∫1 2 −1

=

19 5 6 3  x + x − x  25 3

1

= −1

4 . 5

(II.4)