Nuclear spin dipolar interaction

We show here the equivalence of the two forms of the nuclear spin dipolar interaction, equations (8.9) and (8.10). The most familiar representation of this interaction is equation (8.9), 

We now expand the scalar products in this expression using the cartesian coordinate system shown in figure 8.52. We obtain 

D is the dipolar coupling (cartesian) tensor, which although not necessarily diagonal in the axis system shown in figure 8.52, can be made so by choosing the z axis to lie along the internuclear vector R 

Now we examine the irreducible spherical tensorial form (8.10) and expand the scalar product in the molecule-fixed coordinate system  

Using the expansion (8.11) we treat each nuclear spin term in (8.437) in turn for 

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Nuclear spin nuclear spin dipolar interaction. 

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

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... 

The dipolar hyperfine interaction is a through-space interaction of the electron and nuclear spin magnetic moments. As such, it is similar to the nuclear spin-nuclear spin dipolar interaction discussed earlier in connection with the H2 molecule in its ground electronic state. We shall meet the dipolar hyperfine interaction in many examples described later, so at the risk of seeming somewhat pedantic and repetitive, we here... 

The magnetic hyperfine interaction is represented by the sum of two terms, representing the Fermi contact interaction, and 3Qiip representing the electron spin-nuclear spin dipolar interaction. They are written as follows ... 

The simplest molecular constant to understand is the nuclear spin dipolar interaction constant, to, which is found to be, within experimental error, that calculated from the classical interaction of two magnetic moments, i.e. gFgHfrwO /47t oc2)(7 3> =o. On the other hand, calculation of scalar electron-coupled spin-spin interaction constants is notoriously difficult, requiring a molecular electronic wave function of the highest quality. The best available calculation for HF quoted by Muenter and Klemperer is one due to O Reilly [96]. 

The third term in equation (8.510) describes only the electron-nuclear spin dipolar interaction, with the first-rank tensor T1 (.S. C2) being constructed so that... 

Finally the electron spin-nuclear spin dipolar interaction, which is more complicated, was given, initially, by equations (8.232) and (8.233) ... 

Note that Jefferts used d for y and / for c/. The first two terms contain contributions from both the Fermi contact interaction and the axial component of the electron spin-nuclear spin dipolar interaction, z being along the direction of the internuclear axis. 

This might appear to be a satisfactory conclusion, so far as the analysis of the observed spectrum is concerned. However, Carrington and Gammie [111] have reexamined the analysis and concluded that there does not appear to be any obvious reason why the nuclear spin-nuclear spin dipolar interaction should be neglected, since it is likely to be similar in magnitude to the nuclear spin-rotation interaction. This interaction was discussed for 112 in chapter 8, where it was represented, in spherical tensor form, by the term... 

Fourth, the nuclear spin dipolar interaction for terms diagonal in J and 7 (8.10) ... 

If the proton spins could be driven to flip at a rate that is rapid compared to the static C- H dipolar interaction, which occurs naturally in mobile polymer solutions, then the resonance lines observed in solid-state NMR spectra would likewise no longer be broadened by these hetero-nuclear, spin-dipolar interactions. The NMR spectrum of PBT shown in Fig. 20.12(b) was recorded by applying an rf field B at the resonance frequency of protons, with a field strength of 50 kHz, in a direction perpendicular to the applied field Bo (analogous to the broadband H scalar-/... 

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