Calculate the magnetic moment of an atom (in Bohr magnetons)
(a) in `.^(1)F` state,
(b) in `.^(2)D_(3//2)` state,
(c ) in the state in which `S=1, L=2`, and Lande factor `g=4//3`.

Explanation

(a) For the `(1)F` states `S=0,L=3=J`
`g=1+(3xx4-3xx4)/(2xx3xx4)=1`
Hence `mu=sqrt(3xx4) mu_(B)=2sqrt(3)mu_(B)`
(b) For the `.^(2)D_(3//2)` state `S=(1)/(2),L=2,J=(3)/(2)`
`g=1+((15)/(4)+(3)/(4)-6)/(2xx(15)/(4))=1+(18-24)/(30)=(4)/(5)`
Hence `mu=(4)/(5)sqrt(15//4)mu_(B)=(2)/(5)sqrt(15)mu_(B)=2sqrt((3)/(5))mu_(B)`
(c ) We have
`(4)/(3)=1+(J(J+1)+2-6)/(2J(J+1))`
or `(4)/(3)J(J+1)=J(J+1)-4`
or `J(J+1)=12implies J=3`
Hence `mu=(4)/(3)sqrt(12)mu_(B)=(8)/(sqrt(3))mu_(B)`.