Why is the transition from 3P1 to 3S1 in Helium forbidden?

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SUMMARY

The transition from 3P1 (m=0) to 3S1 (m=0) in Helium is forbidden due to the antisymmetry of the Clebsch-Gordan coefficients, which results in a vanishing matrix element for this transition. In contrast, the transition from 3P1 (m=0) to 3S1 (m=+/-1) is allowed. This discrepancy highlights the limitations of atomic selection rules, which do not fully account for the conditions under which certain transitions can occur. The precise measurement of the fine structure constant was achieved by observing this forbidden transition.

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Chaosmarch
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The transition in the n=2 state of Helium from 3P1 (m=0) to 3S1 (m=0) is forbidden. I know this, because I went to a lecture where a group measured the fine structure constant very precisely using this fact. However, as far as I can see, according to atomic selection rules (available here: http://en.wikipedia.org/wiki/Selection_rule#Summary_table) this should be a perfectly acceptable E1 transition. Can anyone explain why this transition is forbidden?

For reference, the 3P1 (m=0) to 3S1 (m=+/-1) transition is allowed.
 
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The Clebsch Gordan coefficient for 1+1=1 with all m=0 vanishes.
This is because 1+1=1 is antisymmetric.
 
So the selection rules in that table do not account for matrix elements vanishing? I was under the impression the selection rules were derived from the CG coefficients.
 
Wikipedia just left some out. Not unusual for W.
 

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