How to find Reduced matrix elements for the rare earth ion transitions? (for example in case of sm3+, the laser transition is 4G5/2→6H7/2)
The formulas for getting the reduced matrix elements of tensor operators are given in many places, I attach a link to the thesis of Maximiliano Colarieti-Tosti look for "fulltext" somewhere near the bottom of the page (they are of course given in any textbook on the subject, but this is the only place I know to find it reliably online). Equations 3.21 and 3.22 should do the trick, the reduced matrix element is basically everything independent of M in eq. 3.22 (compare equation 3.11 for the definition of the reduced matrix element from the Wigner-Eckart theorem). This is of course the same answer that Kai gave you, just sum up the 3j (and 6j) symbols.
http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4257
If you are looking at dipole transition, the rank of the tensor is 1.
Use Wigner Eckart theorem, you will get a phase factor and a 3j symbol along with the reduced matrix element.
The reduced matrix element is independent of M values.
For eg. for dipole transition, the reduced matrix element will be
C^1(m,n) = (-1)^(jm + 1/2) (2jm+1)^(1/2) (2jn+1)^1/2 (jm 1 jn)
(1/2 0 -1/2)
The first term is a phase factor
second term is a factor
Third term is the 3j values
Hope this helps
For quadrapole transition the same equation holds except that the rank changes to 2.
Just use tables by Carnall:
https://www.osti.gov/servlets/purl/6417825/
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