On the Biot–Savart law of electromagnetism applied to the atomic circulation current

The law of Biot and Savart, derived by observation of the magnetic field produced by electric current flowing in a macroscopic conductor, is shown to yield the correct expression for the nuclear hyperfine interaction, when applied to the electron circulation current derived for hydrogenic atoms from the theories of Pauli and Dirac, as further developed independently by Darwin, Gordon and Hartree. The law thus applies also on the microscopic scale to the magnetic field generated by the electron probability flux of the various atomic eigenstates, and may clearly be generalised formally for application to many-electron atoms, as outlined by Hartree. The present, fully relativistic, treatment is advantageous in avoiding the usual artificial splitting of the hyperfine interaction into a magnetic dipole–dipole contribution and a separate contribution arising from electron orbital motion. The formula obtained forms the basis for the inclusion of higher order (quantum electrodynamic and nuclear recoil) corrections. Interest in this subject is enhanced by observations of ‘strongly forbidden’ (magnetic dipole) atomic transitions arising from the nuclear magnetic interaction, notably the 21 cm line of hydrogen, which has played an important role in astrophysics, in both measurements of galactic rotation and studies of the early Universe.