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Any nuclear S>0 spin is a magnetic moment, hence it couples with other spins by means of the bare magnetic dipolar interaction. The coupling can be expressed as the classical energy of a magnetic moment m in the magnetic induction field B_d, i.e. -m·B_d. The source of the induction is another magnetic moment, m₁, and the dipolar field is written classically as

{$(1) \quad\quad {\mathbf B}_d({\mathbf r})= -\frac {\mu_0}{4\pi}. \frac{3{\hat r}({\mathbf m}_1\cdot\hat{r}) -{\mathbf m }_1}{r^3}$}

where r is the distance between the two moments.

The sources for this field can either be other nuclear spins or any unpaired electron spins of the compound. In the second case the field is called hyperfine. In a liquid all these interactions are very effectively mediated by fast translations and rotations and may be neglected other than by their average effect.

In the crystalline state they have to be taken into account carefully. A molecular solid may represent an intermediate case: mesophases exist, in between the liquid and the solid phase, where molecules are translationally static but they undergo rotational dynamics.

Let us consider the crystal case. The dipolar field has the well known dependence on the relative orientation of m₁ and r. The field lines are shown in the figure.

Nuclei. One nucleus will experience the field of all other nuclear spins, that may form a dense or a diluted lattice (depending on the natural abundance of the species). When nuclei are the only source present, they are slowly precessing around similar fields, hence they are quasi static and with random orientations. In this case the statistical ensemble of the nuclei is subject to a nearly Gaussian distribution of local field.

Unpaired electrons. They are present in metals and in insulating paramagnets, as well as in magnetically ordered materials. On the time-scale of NMR the electron dynamics is very much faster than the nucler spin dynamics and the effective field B_d can be computed as the spatial average of eq. (1) over electron wave functions. A few cases may be distinguished. The first is non magnetic metals in an external field, where, besides the spatial average, the fast electron spin fluctuations must also be considered. We deal with that later (Knight shift). Non magnetic insulators are trivial: they have no unpaired electrons. Finally, in magnetic insulators the unpaired electron spin may be on the same site of the probe nucleus, or on a neighbour site. We shall start from these two last cases.

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