Question

The wavelength of the `Na` yellow doublet `(.^(2)P rarr .^(2)S)` are equal to `589.59` and `589.00nm`. Find:
(a) the ratio of the intervals between neighbouring sublevels of the Zeeman splitting of the term `.^(2)P_(3//2)` and `.^(2)P_(1//2)` in a weak magnetic field,
(b) the magnetic field induction `B` at which the interval between neighbouring sublevels of the Zeeman splitting of the term `.^(2)P_(3//2)` is `eta=50` times smaller than the natural splitting of the `.^(2)P`.

Answer 1

(a) For the `.^(2)P_(3//2)` term
`g=1+((3)/(2)xx(5)/(2)+(1)/(2)xx(3)/(2)-1xx2)/(2xx(3)/(2)xx(5)/(2))= 1+(10)/(30)=(4)/(3)`
and the enrgy of the `.^(2)P_(3//2)` sublevels will be
`E(M_(Ƶ))=E_(0)-(4)/(3)mu_(B)BM_(Ƶ)`
where `M_(Ƶ)= +- (3)/(2),+-(1)/(2)`. Thus, between neighbouring sublevels.
`deltaE(.^(2)P_(3//2))=(4)/(3)mu_(B)B`
For the `.^(2)_(1//2)` term
`g=1+((1)/(2)xx(3)/(2)+(1)/(2)xx(3)/(2)-1xx2)/(2xx(1)/(2)xx(3)/(2))`
`=1+(6-8)/(6)=1-(1)/(3)=(2)/(3)`
and the seperation between the two sublevels into which the `.^(2)P_(1//2)` term will split is
`deltaE(.^(2)P_(1//2))=(2)/(3)mu_(B)B`
The ratio of hte two splitting is `2:1`
(b) The interval between neighbouring Zeeman sublevels of the `.^(2)P_(3//2)` term is `(4)/(3)mu_(B)B`. The energy seperation between `D_(1)` and `D_(2)` lines is `(2piħc)/(lambda^(2)) Delta lambda` (this is the natural seperation of the `.^(2)P` them)
Thus `(4)/(3)mu_(B)=(2 piħcDelta lambda)/(lambda^(2)eta)`
or `B=(3 piħcDelta lambda)/(2mu_(B)lambda^(2) eta)`
Substitution gives
`B= 5.46kG`