The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum numbern. 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 the 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.
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. The two magnetic vectors, and 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.
the expectation value for the Hamiltonian is:
Thus the order of magnitude for the spin-orbital coupling is .
When weak external magnetic fields are applied, the spin-orbit coupling contributes to the Zeeman effect.
There is one last term in the non-relativistic expansion of the Dirac equation. It is referred to as the Darwin term, as it was first derived by Charles Galton Darwin, and is given by:
The Darwin term affects only the s orbitals. This is because the wave function of an electron with vanishes at the origin, hence the delta function has no effect. For example, it gives the 2s orbital the same energy as the 2p orbital by raising the 2s state by 6977145109124427590♠9.057×10−5 eV.
The Darwin term changes the effective potential at the nucleus. It can be interpreted as a smearing out of the electrostatic interaction between the electron and nucleus due to zitterbewegung, or rapid quantum oscillations, of the electron. This can be demonstrated by a short calculation.
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.
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.
The total effect can also be obtained by using the Dirac equation. In this case, the electron is treated as non-relativistic. The exact energies are given by
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.