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

(2.22) 
where is the transition probability of the ray transition between two nuclear sublevels, and
is the angular dependence of the radiation probability at an angle to the quantisation axis.
The angular independent term is given by the square of the appropriate ClebschGordan coefficient

(2.23) 
where
is the vector sum
and is the vector sum
.[5]
is the multipolarity of the radiation, being dipolar and being quadrupolar. As the multipolarity of the radiation increases the transition probability decreases.
In Fe the
transition is primarily dipolar and values for this transition are given in Table 2.1.
Table 2.1:
Relative probabilities for a dipole
transition. and are the angular independent and dependent terms arbitrarily normalised. Relative intensities for
and
are shown with arbitrary normalisation.
     


       

  1  3 
 3  6 

 0 
 2 
 4  0 

 
 1 
 1  2 

  0  0  0  0  0 

  0  0  0  0  0 

 
 1 
 1  2 

 0 
 2 
 4  0 

   3 
 3  6 
       
       

In a magnetic spectrum the intensities of the outer, middle and inner lines are in a ratio derived from the product
. Using the values from Table 2.1 gives

(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
with angle. In polycrystalline samples there is no angular dependence and thus the intensity depends only on , giving a sextet of 3:2:1:1:2:3.
Nonmagnetic spectra with quadrupole splitting have several degenerate transitions and the intensity of the two lines are in the ratio

(2.25) 
Dr John Bland, 15/03/2003