Combined Magnetic and Quadrupole Interactions

When dealing with quadrupole or magnetic splitting separately with chemical isomer shifts the recorded spectrum has uniform shifts of resonance lines with no change in their relative separations. However, both the quadrupolar and magnetic interactions depend upon angle and so when they are both present the interpretation of the spectrum can be complex.

The situation can be simplified a great deal if two assumptions are made

1. the electric field gradient is axially symmetric with its principal axis, $V_{zz}$, at an angle $\theta$ to the magnetic axis
2. the strength of the quadrupole interaction is much less than the magnetic interaction, ie $e^{2}qQ \ll \mu{}H$.

The solution to the Hamiltonian can then be solved by treating the quadrupole interaction as a perturbation so that the resultant energy levels are given by

$$E = -q\mu_{N}Hm_{I} + (-1)^{\vert m_{I}\vert+\frac{1}{2}}\frac{e^{2}qQ}{4} \left( \frac{3\cos^{2}\theta - 1}{2} \right)$$ (2.21)

giving a spectrum as in Figure 2.6.[5]

Figure 2.6: The effect of a first-order quadrupole perturbation on a magnetic hyperfine spectrum for a $\nicefrac {3}{2} \rightarrow \nicefrac {1}{2}$ transition. Lines 2,3,4,5 are shifted relative to lines 1,6.

For most $^{57}$Fe spectra the result is a shift in the relative position of lines 1,6 with lines 2,3,4,5. For a positive quadrupole splitting lines 1,6 are shifted positively relative to lines 2,3,4,5 and vice versa. The line separations are equal when there is no quadrupole effect or when $\cos\theta = \nicefrac{1}{\sqrt{3}}$.