Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation.

It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887,[1][2] laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.

For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines.

The scale of the fine structure splitting relative to the gross structure energies is on the order of (Zα)2, where Z is the atomic number and α is the fine-structure constant, a dimensionless number equal to approximately 1/137.

The fine structure energy corrections can be obtained by using perturbation theory.

To perform this calculation one must add three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin–orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron.

These corrections can also be obtained from the non-relativistic limit of the Dirac equation, since Dirac's theory naturally incorporates relativity and spin interactions.

This section discusses the analytical solutions for the hydrogen atom as the problem is analytically solvable and is the base model for energy level calculations in more complex atoms.

The gross structure assumes the kinetic energy term of the Hamiltonian takes the same form as in classical mechanics, which for a single electron means

However, when considering a more accurate theory of nature via special relativity, we must use a relativistic form of the kinetic energy,

Expanding the square root for large values of

Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.

Therefore, the first order relativistic correction for the hydrogen atom is

On final calculation, the order of magnitude for the relativistic correction to the ground state is

The spin–orbit correction can be understood by shifting from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it.

In this case the orbiting nucleus functions as an effective current loop, which in turn will generate a magnetic field.

However, the electron itself has a magnetic moment due to its intrinsic angular momentum.

couple together so that there is a certain energy cost depending on their relative orientation.

This gives rise to the energy correction of the form

Notice that an important factor of 2 has to be added to the calculation, called the Thomas precession, which comes from the relativistic calculation that changes back to the electron's frame from the nucleus frame.

When weak external magnetic fields are applied, the spin–orbit coupling contributes to the Zeeman effect.

[4] Quantum fluctuations allow for the creation of virtual electron-positron pairs with a lifetime estimated by the uncertainty principle

Another mechanism that affects only the s-state is the Lamb shift, a further, smaller correction that arises in quantum electrodynamics that should not be confused with the Darwin term.

The Darwin term gives the s-state and p-state the same energy, but the Lamb shift makes the s-state higher in energy than the p-state.

The total effect, obtained by summing the three components up, is given by the following expression:[5]

is the total angular momentum quantum number (

It is worth noting that this expression was first obtained by Sommerfeld based on the old Bohr theory; i.e., before the modern quantum mechanics was formulated.

This expression, which contains all higher order terms that were left out in the other calculations, expands to first order to give the energy corrections derived from perturbation theory.

However, this equation does not contain the hyperfine structure corrections, which are due to interactions with the nuclear spin.

Other corrections from quantum field theory such as the Lamb shift and the anomalous magnetic dipole moment of the electron are not included.

Interference fringes , showing fine structure (splitting) of a cooled deuterium source, viewed through a Fabry–Pérot interferometer .
Energy diagram (to scale) of the hydrogen atom for n =2 corrected by the fine structure and magnetic field. First column shows the non-relativistic case (only kinetic energy and Coulomb potential), the relativistic correction to the kinetic energy is added in the second column, the third column includes all of the fine structure, and the fourth adds the Zeeman effect (magnetic field dependence).
Relativistic corrections (Dirac) to the energy levels of a hydrogen atom from Bohr's model. The fine structure correction predicts that the Lyman-alpha line (emitted in a transition from n = 2 to n = 1 ) must split into a doublet.