The level of an atom with a nucleus of spin (such as ) has electronic angular…

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Deriving the Spectrum, Ch Hyperfine structure and isotope shifts

Problem

The

^3P_2

level of an atom with a nucleus of spin

I = 5/2

(such as

^{17}\mathrm{O}

) has electronic angular momentum

J = 2

and magnetic hyperfine constant

A_{\rm hfs} = 10.0\,\mathrm{MHz}

. ((a)) List the possible values of the total angular momentum

F

and check that the sum of their degeneracies

(2F+1)

equals

(2I+1)(2J+1)

. ((b)) Using

E_F = \dfrac{A_{\rm hfs}}{2}[F(F+1) - I(I+1) - J(J+1)]

, find the energy (relative to the unsplit level) of the highest (

F = 9/2

) and lowest (

F = 1/2

) hyperfine levels, in MHz. ((c)) Check the interval rule by computing

E_{9/2} - E_{7/2}

directly and comparing with

A_{\rm hfs}\times F

for

F = 9/2

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