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 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 .
The fine structure energy corrections can be obtained by using perturbation theory. To do this one adds 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 Darwinian term. The full Hamiltonian is given by
where is the Hamiltonian from the Coulomb interaction. 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 there is a factor of 2, 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 .
Remark: On the (n,l,s)=(n,0,1/2) and (n,l,s)=(n,1,-1/2) energy level, which the fine structure said their level are the same. If we take the g-factor to be 2.0031904622, then, the calculated energy level will be different by using 2 as g-factor. Only using 2 as the g-factor, we can match the energy level in the 1st order approximation of the relativistic correction. When using the higher order approximation for the relativistic term, the 2.0031904622 g-factor may agree with each other. However, if we use the g-factor as 2.0031904622, the result does not agree with the formula, which included every effect.
There is one last term in the non-relativistic expansion of the Dirac equation. It is referred to as the Darwin, and is given by:
The Darwin term affects only the s-orbit. 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-orbit the same energy as the 2p-orbit 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 motivated 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.
The total effect can also be obtained by using the Dirac equation. In this case, it is not necessary to treat the electron as non-relativistic or make other approximations. The exact energies are given by
This expression expands to first order to give the energy corrections derived from perturbation theory, but it contains all higher order terms that were left out in the other calculations. 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.