Nuclear spin/rotation interaction

The magnetic moments given above will interact with an applied magnetic field, and these interactions are discussed extensively in chapter 8. In some diatomic molecules both nuclei have non-zero spin and an associated magnetic moment. The magnetic interactions which then occur are the nuclear spin-rotation interactions, represented by the operator... 

This would imply a very simple linear Zeeman effect but, as we show in chapter 8, additional terms describing the nuclear spin rotation interaction and the spin-spin interaction make the system much more interesting. The nuclear spin transitions are induced by an oscillating magnetic field applied perpendicular to the static magnetic field, the perturbation being represented, for example, by the term... 

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 calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The nuclear spin-rotation interaction becomes very simple for a diatomic molecule. The principal components of the tensor a for a polyatomic molecule were described in equation (8.163) this expression reveals that for a diatomic system the axial component (c/)zz is zero and, of course, the two perpendicular components are equal. The nuclear spin rotation interaction for a diatomic molecule is therefore described by a single parameter c/. The appropriate term in the effective Hamiltonian, first presented in equation (8.7), is... 

The most important terms in the effective hyperftne Hamiltonian are those which describe the nuclear quadrupole and nuclear spin-rotation interactions ... 

The electron spin rotation interaction is similar to the nuclear spin rotation interaction we met earlier, and may be written as a simple scalar interaction,... 

As we shall see, each of these two terms, one for each nucleus, describes a second-rank scalar interaction between the electric field gradient at each nucleus and the nuclear quadrupole moment. De Santis, Lurio, Miller and Freund [44] included two other terms which involve the nuclear spins. One is the direct dipolar coupling of the 14N nuclear magnetic moments, an interaction which we discussed earlier in connection with the magnetic resonance spectrum of D2 its matrix elements were given in equation (8.33). The other is the nuclear spin-rotation interaction, also discussed in connection with H2 and its deuterium isotopes. It is represented by the term... 

The final term in the effective Hamiltonian describes the nuclear spin-rotation interaction and its matrix elements are relatively straightforward ... 

The predominant isotope of cesium is 133Cs which has a nuclear spin I of 7/2 its quadrupole moment and g-factor will be denoted by Q and gi. The 19F nucleus has spin /2 of 1 /2 (and therefore no quadrupole moment) and a nuclear g-factor denoted g2. The nuclear hyperfine Hamiltonian used by English and Zorn [51] was the sum of five terms representing the 133Cs quadrupole interaction, the 133Cs nuclear spin-rotation interaction, the 19F nuclear spin-rotation interaction, the dipolar (tensorial) interaction between the 133Cs and 19F nuclear spins, and the scalar interaction between the two nuclear spins. Consistent with the conventions in use at the time, this Hamiltonian was written in the following form ... 

The molecular physics underlying the nuclear spin-rotation interaction has been discussed by Flygare [107], In the general case of a polyatomic molecule the spin-rotation interaction is represented by a second-rank tensor in a molecule fixed coordinate system x, y, z, the diagonal component in the x direction may be written as the sum of a nuclear part (k labelling the nucleus under consideration) and an electronic part ... 

The effective Hamiltonian contained terms for both nuclei describing the orbital hy-perfine interaction, the electric quadrupole interaction and the nuclear spin-rotation interaction ... 

The four terms now represent uniquely the Fermi contact, dipolar, electron spin rotation and nuclear spin rotation interactions. We will examine the relationships between the parameters in (11.78) and (11.79) in due course. 

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... 

The rotational levels are extrapolated to high J values according to the method of Khachkuruzov (7) who proposed a simpler form of Wooley s method (8). The rotational levels are weighted in accordance with the nuclear spin-rotation interaction (9) as follows even J weight = 0.625, odd J weight = 0.375. The so-called quasibound rotational levels were not Included in the calculation. No suitable spectroscopic information exists for any of the excited states of BSg. It should be noted that the... 

Absolute Shielding Scales. - Gee and Wasylishen have recommended that the microwave study of AlH be repeated, as the previous one failed to account for the Al nuclear spin-rotation interaction. Their ab initio calculations give a range of values depending on the method used but are all of the order of 300 kHz. The value of the spin-rotation constant for Al in AIF, on the other hand, has been measured, Cj = 8.2 1.3 kHz, which leads to (ct — CTj ) = 320 50 ppm for Al in AIF and... 

NUCLEAR MAGNETIC SHIELDING, NUCLEAR SPIN-ROTATION INTERACTIONS, AND THE MOLECULAR ZEEMAN EFFECT IN THE PRESENCE OF NUCLEAR INTERACTIONS... 

The other is the nuclear spin-rotation interaction, also discussed in connection with H2 and its deuterium isotopes. It is represented by the term... 

---