what is the INNER QUANTUM number ?
You will get much better answers if you could point us to the source where you came across this term.
I found this while studying the Dirac Equation for the hydrogenic atoms from the book Relativistic quantum mechanics for atoms and molecules by grant page no 148
http://www.merriam-webster.com/dictionary/inner quantum number ?
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Elsewhere, I see it defined as, "...number J which gives an atom's total angular momentum, excluding the nuclear spin..."
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