Magnetism and relativity

The classification of interactions according to their relativistic character is based on the energy of a free particle, which is given by ε² = m²_ec⁴ + p²c²:

..........(1)

The order of magnitude of the velocity of electrons in solids is αc, where c is the velocity of light and α is the fine-structure constant defined above. Expanding in powers of α shows the hierarchy of interactions:

.......(2)

Here m_ec² = 511 keV; the second and third terms, which represent the order of magnitude of electrostatic and magnetostatic energies, are respectively 13.6 eV (the Rydberg, R₀) and 0.18 meV. Magnetic dipolar interactions are therefore of order 2 K.
Note that the Pauli susceptibility can also be written in terms of the fine-structure constant as χP = α²k_Fa₀/π, where k_F is the Fermi wavevector and a0 is the Bohr radius. The force on a charged particle given by the Lorentz does not change in a moving frame, but the relative contributions of the electric and magnetic fields to the force are modified. What looks like an electric field to a stationary electron acquires a magnetic field component in the frame of a moving electron, and vice versa.
The electric field acting on an electron subject to a magnetic field in a frame in which the electron is stationary is

........(3)

Conversely, the magnetic field acting on an electron subject to an electric field in a frame where the electron is stationary is

.......(4)

This field leads to the Larmor precession of the spin of a moving electron in an electric field, which is known as the Rashba effect