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THEORY

According to classical thinking, the electron moving around the nucleus has a magnetic dipole moment, because it is charged. The interaction of this magnetic dipole moment with the magnetic moment of the nucleus (due to its Spin ) leads to hyperfine splitting.

However, due to the electron's spin, there is also hyperfine splitting for s-shell electrons, which have zero orbital angular momentum. In this case, the magnetic dipole interaction is even stronger, as the electron probability density does not vanish inside the nucleus (r=0).

The amount of correction to the Bohr Energy levels due to hyperfine splitting of the hydrogen atom is on the order of:

: rac{m}{m_p} \alpha^4 m c^2

where
m

mp

α

c


For atoms other than hydrogen, the Nuclear Spin ec{I} and the Total Electron Angular Momentum ec{J} = ec{L} + ec{S} get coupled, giving rise to the total angular momentum ec{F} = ec{J} + ec{I}. The hyperfine splitting is then
:\Delta E_{hfs} = - ec{\mu}_I ec{B}_J = rac{a}{2} F(F+1) - I(I+1) - J(J+1) ,
where
:a = rac{g_I ec{\mu}_N ec{B}_J}{\sqrt{J(J+1)}},
with ec{\mu}_N the magnetic dipole moment of the nucleus.