Transition Probabilities and Line Strengths

So far we have assumed that the energy levels E. and E are not degenerate and therefore have the statistical weight factor g = 1. In the case of a degenerate level Ej the total transition probability pB j of a transition E - E is the sum 

The Einstein coefficient for a transition E. Ej between two degenerate levels E., E is therefore 

The double sum is called the line strength With this abbreviation, the relations between the Einstein strength f. can be summarized 

Each atomic level Ej with the total angular momentum J = /j(j + l)h has gj = 2j + 1 sublevels (Zeeman levels) which belong to the different projections of the angular momentum J onto the selected z direction and which are degenerate without an external magnetic field. For a transition E. - Ej, the summation in (2.141) extends in this case from n = -j ton = + j and m = - j. to + j. although many of the R.. in the double sum will vanish because of symmetry selection rules, which only allow transitions with n=morn=m l (see Fig.2.31). 

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The previously discussed radiative transition probabilities and line strengths are connected to the peak-stimulated emission cross sections of the rare-earth ion in a glass by... 

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