120 Hyperfine structure and isotope shift
Exercises
(6.1)The magnetic field in fine and hyperfine structure
Calculate the magnetic flux densityBat the cen-
tre of a hydrogen atom for the 1s^2 S 1 / 2 level and
also for 2s^2 S 1 / 2.
Calculate the magnitude of the orbital magnetic
field experienced by a 2p-electron in hydrogen
(eqn 2.47).
(6.2)Hyperfine structure of lithium
The figure shows the energy levels of lithium in-
volved in the 2s–2p transitions for the two isotopes(^6) Li and (^7) Li. (The figure is not to scale.)
Explain in simple terms why the hyperfine split-
ting is of orderme/Mp smaller than the fine-
structure splitting in the 2p configuration of
lithium.
Explain, using the vector model or otherwise, why
the hyperfine interaction splits a givenJlevel into
2 J+ 1 hyperfine levels ifJIand 2I+ 1 levels
ifIJ. Hence, deduce the nuclear spin of^6 Li
and give the values of the quantum numbersL,J
andFfor all its hyperfine levels. Verify that the
(^6) Li (^7) Li
interval rule is obeyed in this case.
Determine from the data given the nuclear spin of
(^7) Li and give the values ofL,JandFfor each of
the hyperfine levels on the figure. Calculate the
hyperfine splitting of the interval marked X.
(For the hyperfine levels a to d the parameterAnlj
is positive for both isotopes.)
(6.3)Hyperfine structure of light elements
Use the approximate formula in eqn 6.17 to esti-
mate the hyperfine structure in the ground states
of atomic hydrogen and lithium. Comment on the
difference between your estimates and the actual
values given for hydrogen in Section 6.1.1 and for
Li in Exercise 6.2.
(6.4)Ratio of hyperfine splittings
The spin and magnetic moment of the proton are
(1/ 2 , 2. 79 μN), of the deuteron (1, 0. 857 μN)andof
(^3) He (1/ 2 ,− 2. 13 μN). Calculate the ratio of the
ground-state hyperfine splittings of (a) atomic hy-
drogen and deuterium and (b) atomic hydrogen
and the hydrogen-like ion^3 He+.