 # Theory of spin-orbit coupling and fine structure

In this particular article Theory of Spin-orbit coupling and fine structure, we are going to discuss about basic information about the fine structure. We will also discuss about Schrodinger theory’s failure to explain fine structure along with Sommerfeld’s attempt.

## Fine structure

The fine structure is a splitting of the spectral lines, into several distinct components, which is found in all atomic spectra. The principal series yellow sodium line 3s-3p is not a simple line but is a doublet. Which is consisting of two components 5890 Aº and 5896 Aº. The double separation in wavenumber units is  Δ¯ν =2.5 cm⁻¹ etc. Hence the levels 3p, 4p, 5p etc, must be double, with the above doublet separation. Similar doublet fine structure is seen in the sharp series lines of alkali spectral lines. A compound doublet structure (three lines) is seen in the spectral lines of diffuse and fundamental series. Examples of fine structure (doublets) in Na spectrum (schematic). The unit of wavelength is A⁰(angstron).  Further analysis of the lines of alkali-like spectra shows that the s-levels are single and the p,d, f,…levels are doublets. Many lines of the optical spectra of hydrogen, also, when viewed with the high resolving power spectrometer, are found to be composed of a closely spaced pair of lines. The line structure of atoms is of the order of 10⁻³ eV∼10 cm⁻¹. The separation, in terms of reciprocal wavelength, between adjacent components of a single spectral line is the order of 10⁻⁴ times the separation between adjacent lines.

### Schrodinger(Bohr)Theory’s failure to explain fine structure

By solving the non-relativistic Schrodinger equation,

HΨnlm1=EnΨnlm1

where the non-relativistic Hamiltonian is

H=((p)/2m)+(-Ze/4πεr)

with linear momentum operator p=-ih∇ and energy eigenvalues En. ¹A state, ψnlm1, of the atom, is specified by three quantum numbers, n the principle (total) quantum number, l the orbital angular momentum quantum number and m1 the magnetic quantum number. The above theory reproduces the energy levels predicted by Bohr theory of hydrogen-like atoms. The energy of a stationary state of the H atom and hydrogen-like ions depends solely on n. States of different l but equal n have the same energy. They are degenerate and do not predict splitting of spectral lines. Thus, the fine structure cannot be explained by the Coulomb interaction between the nucleus and the electrons.

For alkali-atoms, there is a single electron outside the core consisting of the nucleus and the closed shell of electrons. In the central-field approximation. It is assumed that each electron moves in a cental-field approximation. It is assumed that each electron moves in a central, or spherically symmetrical, force field, produced by the nucleus and the other electrons. This approximation is good enough to explain the spectrum of alkalies and the dependence of energy levels on two quantum numbers n and l. But, it can not explain the fine structure splitting of the energy levels.

#### Sommerfeld’s attempt

The first attempt to explain fine structure was made by Sommerfield by treating the problem relativistically by incorporating. The relativistic variation of mass with velocity in the central force problem, particularly in the case of the hydrogen atom. For an electron in a hydrogen atom, ν/c∼10⁻² or less. Thus we would expect the relativistic corrections to the total energy to be of the order of ν²/c²∼10⁻⁴. This is just the order of magnitude of the splitting in the energy states of hydrogen that would be needed to explain the fine structure of the hydrogen spectrum.

Sommerfeld showed that incorporation of relativistic effects removes the degeneracy of energy levels and a small difference in energy then occurs between states with different l and equal n. This difference can account for the fine structure of the Balmer lines in H-spectrum. The results of Sommerfeld were in good numerical agreement with the observed fine structure of hydrogen.

But the situation was not as satisfactory for the alkalis. In these atoms, the electron responsible for the optical spectrum would be expected to move in a Bohr-like orbit of the large radius at low velocity. So the relativistic variation of mass would be expected to be small. However, the structure splitting was observed to be very much larger than in hydrogen. Consequently, the doubt arose concerning the validity of Sommerfeld’s explanation of the origin of fine structure.

##### Conclusion

In this article ‘Theory of Spin-orbit coupling and fine structure’, we have discussed various important topics like basic information about fine structure. Theories related to fine structure etc in most easiest way possible.