Spectrum Line Intensities

The hyperfine interactions thus far have given the relative energies of the various transitions taking place but have not given information on the relative intensities of these transitions in the recorded spectrum. The intensities arise from the coupling of two angular momentum states, which can be expressed as the product of both an angular dependent term and an angular independent term by

$$A(L,\theta) = C^{2}(J)\Theta(J,\theta)$$ (2.22)

where $C^{2}(J)$ is the transition probability of the $\gamma$-ray transition between two nuclear sub-levels, and $\Theta(J,\theta)$ is the angular dependence of the radiation probability at an angle $\theta$ to the quantisation axis.

The angular independent term is given by the square of the appropriate Clebsch-Gordan coefficient

$$C^{2}(J) = \langle I_{1}\boldsymbol{J} - m_{1}m \arrowvert I_{2}m_{2} \rangle^{2}$$ (2.23)

where $\boldsymbol{J}$ is the vector sum $\boldsymbol{J} = \boldsymbol{I}_{1} + \boldsymbol{I}_{2}$ and $m$ is the vector sum $m = m_{1} - m_{2}$.[5] $\boldsymbol{J}$ is the multipolarity of the radiation, $J=1$ being dipolar and $J=2$ being quadrupolar. As the multipolarity of the radiation increases the transition probability decreases.

In $^{57}$Fe the $14.41\ensuremath{\unskip\,\mathrm{keV}}$ transition is primarily dipolar and values for this transition are given in Table 2.1.

Table 2.1: Relative probabilities for a dipole $\nicefrac {3}{2} \rightarrow \nicefrac {1}{2}$ transition. $C^{2}$ and $\Theta$ are the angular independent and dependent terms arbitrarily normalised. Relative intensities for $\theta =90^{\circ }$ and $\theta =0^{\circ }$ are shown with arbitrary normalisation.

$$m_{2} -m_{1} m C C^{2} \Theta \theta =90^{\circ } \theta =0^{\circ }$$

$$+\nicefrac{3}{2} +\nicefrac{1}{2} +1 1+\cos^{2}\theta$$ 1 3 3 6

$$+\nicefrac{1}{2} +\nicefrac{1}{2} \sqrt{\nicefrac{2}{3}} 2\sin^{2}\theta$$ 0 2 4 0

$$-\nicefrac{1}{2} +\nicefrac{1}{2} -1 \sqrt{\nicefrac{1}{3}} 1+\cos^{2}\theta$$ 1 1 2

$$-\nicefrac{3}{2} +\nicefrac{1}{2} -2$$ 0 0 0 0 0

$$+\nicefrac{3}{2} -\nicefrac{1}{2} +2$$ 0 0 0 0 0

$$+\nicefrac{1}{2} -\nicefrac{1}{2} +1 \sqrt{\nicefrac{1}{3}} 1+\cos^{2}\theta$$ 1 1 2

$$-\nicefrac{1}{2} -\nicefrac{1}{2} \sqrt{\nicefrac{2}{3}} 2\sin^{2}\theta$$ 0 2 4 0

$$-\nicefrac{3}{2} -\nicefrac{1}{2} -1 1 1+\cos^{2}\theta$$ 3 3 6

In a magnetic spectrum the intensities of the outer, middle and inner lines are in a ratio derived from the product $C^{2}(J)\Theta(J,\theta)$. Using the values from Table 2.1 gives

$$3\left(1+\cos^{2}\theta\right) : 4\sin^{2}\theta : 1+\cos^{2}\theta$$ (2.24)

from which it can be seen that the outer and inner lines are always in the ratio of 3:1 whilst the middle line varies between $0 \rightarrow 4$ with angle. In polycrystalline samples there is no angular dependence and thus the intensity depends only on $C^{2}(J)$, giving a sextet of 3:2:1:1:2:3.

Non-magnetic spectra with quadrupole splitting have several degenerate transitions and the intensity of the two lines are in the ratio

$$3\left(1+\cos^{2}\theta\right) : 2+3\sin^{2}\theta$$ (2.25)