130_notes.dvi

a state of definitej. We have assumed that the effect of the field is small compared to the fine
structure corrections. We can write the full energy in a weak magnetic field.

Enjmjℓs=−

1

2

α^2 mc^2

(

1

n^2

+

α^2

n^3

[

1

j+^12

−

3

4 n

])

+gLμBBmj

Thus, in a weak field, thethe degeneracy is completely broken for the statesψnjmjℓs. All
the states can be detected spectroscopically.

In the strong field limit we could use states of definitemℓandmsand calculate the effects of the fine
structure,H 1 +H 2 , as a perturbation. In an intermediate strength field, on the order of 500 Gauss,
the combination of the Hydrogen fine structure Hamiltonian and theterm to the B field must be
diagonalized on the set of states with the same principal quantum numbern.

1.31 Hyperfine Structure

Theinteraction between the spin of the nucleus and the angular momentum of the
electroncauses a further (hyperfine) splitting (See section 24) of atomic states. It is called hyperfine
because it is also orderα^2 like the fine structure corrections, but it is smaller by a factor of about
me
mpbecause of the mass dependence of the spin magnetic moment for particles.

The magnetic moment of the nucleus is

~μN=

ZegN

2 MNc

I~

whereI~is thenuclear spinvector. Because the nucleus, the proton, and the neutron haveinternal
structure, the nuclear gyromagnetic ratio is not just 2. For the proton, it isgp≈ 5 .56.

We computed the hyperfine contribution to the Hamiltonian forℓ= 0 states.

Hhf=

〈e

mc

S~·B~

〉

=

4

3

(Zα)^4

(

m

MN

)

(mc^2 )gN

1

n^3

S~·I~

̄h^2

Now, just as in the case of theL~·S~, spin-orbit interaction, we will define the total angular momentum

F~=S~+~I.

It is in the states of definitefandmfthat the hyperfine perturbation will be diagonal. In essence,
we are doing degenerate state perturbation theory. We could diagonalize the 4 by 4 matrix for the
perturbation to solve the problem or we can use what we know to pickthe right states to start with.
Again like the spin orbit interaction, thetotal angular momentum stateswill be the right states
because we can write the perturbation in terms of quantum numbers of those states.

~S·I~=^1

2

(

F^2 −S^2 −I^2

)

=

1

2

̄h^2

(

f(f+ 1)−

3

4

−

3

4

)

∆E=

2

3

(Zα)^4

(

m

MN

)

(mc^2 )gN

1

n^3

(

f(f+ 1)−

3

2

)

.