Hyperfine structure

Hyperfine structure
1. Nuclear spin effect
The nuclear spin I interacts with the magnetic field at the nucleus due to the orbiting electrons:
…..(1)
Where: ?I is magnetic moment of the nucleus and A is vector potential.
The magnetic moment of the nucleus is proportional to I:
….(2)
The proportionality constant is the nuclear magneton defined as:

…..(3)
Where ?B is Bohr magneton. This definition is used since magnetic moments are typically inversely proportional to the mass of the particle, so this results in gI values of order 1. The truly total angular momentum of the atom becomes F = J + I and it obtain the exact expression
Hence the hyperfine splitting is:

…..(4)

fine: magnetic field at electron from nucleus nuclear charge Ze orbits with speed -v
hyperfine: magnetic field at nucleus from electron electron charge e orbits with speed v
The part of the wave function inside the nucleus provides an essentially uniform magnetization, since to very good approximation for r of order the nuclear size. The field inside a uniformly magnetized sphere is:

…(5)

Notice that by dimensional analysis, is proportional to and for hydrogen-like systems, it find:

…..(6)

2. Nuclear quadrupole term
There is an electrostatic effect which arises from the departure of the shape of the nucleus from spherical symmetry. An isolated nucleus cannot carry a permanent electric dipole moment, so the first correction to the electrostatic potential is proportional to the electric quadrupole moment of the nucleus. One finds:

….(7)

where Q is the nuclear electric quadrupole moment operator, is the electric field produced by the electrons at the nucleus, and I is the nuclear spin.