 Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.
The gross structure of line spectra is the line spectra predicted by nonrelativistic 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 splitting is on the order of (Zα)^{2}, where Z is the atomic number and α is the finestructure constant, a dimensionless number equal to approximately 7.297×10^{−3}.
The fine structure can be separated into three corrective terms: the kinetic energy term, the spinorbit term, and the Darwinian term. The full Hamiltonian is given by
Contents
Kinetic energy relativistic correction
Classically, the kinetic energy term of the Hamiltonian is
However, when considering special relativity, we must use a relativistic form of the kinetic energy,
where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Expanding this in a Taylor series, we find
Then, the first order correction to the Hamiltonian is
Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.
where ψ^{0} is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see
We can use this result to further calculate the relativistic correction:
For the hydrogen atom, , , and where a_{0} is the Bohr Radius, n is the principal quantum number and l is the azimuthal quantum number. Therefore the relativistic correction for the hydrogen atom is
where we have used:
On final calculation, the order of magnitude for the spinorbital coupling for ground state is .
Spinorbit coupling
The spinorbit correction arises when we shift 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, which is come from the relativistic calculation of change back to electron frame from nucleus frame by Llewellyn Thomas. This factor also called the Thomas factor.
since
the expectation value for the Hamiltonian is:
Thus the order of magnitude for the spinorbital 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 gfactor to be 2.0031904622, then, the calculated energy level will be different by using 2 as gfactor. Only using 2 as the gfactor, 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 gfactor may agree with each other. However, if we use the gfactor as 2.0031904622, the result does not agree with the formula, which included every effect.
Darwin term

 ψ(0) = 0 for l > 0
Thus, the Darwin term affects only the sorbit. For example it gives the 2sorbit the same energy as the 2porbit by raising the 2sstate by .
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.
Another mechanism that affects only the sstate is the Lamb shift. The reader should not mix up the Darwin term and the Lamb shift. The Darwin term makes the sstate and pstate the same energy, but the Lamb shift makes the sstate higher in energy than the pstate.
Total effect
The total effect, obtained by summing the three components up, is given by the following expression ^{[1]}:
where j is the total angular momentum (j = 1 / 2 if l = 0 and otherwise). It is worth noting that this expression was first obtained by A. Sommerfeld based on the old Bohr theory, i.e., before the modern quantum mechanics was formulated.
See also
References
 Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.
 Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.
External links
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fine structure — n microscopic structure of a biological entity or one of its parts esp. as studied in preparations for the electron microscope fine structural adj * * * ultrastructure … Medical dictionary
fine structure — /fuyn/, Physics. a group of lines that are observed in the spectra of certain elements, as hydrogen, and that are caused by various couplings of the azimuthal quantum number and the angular momentum quantum number. Cf. hyperfine structure. [1915… … Universalium
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fine structure — smulkioji sandara statusas T sritis fizika atitikmenys: angl. fine structure vok. Feinstruktur, f rus. тонкая структура, f pranc. structure fine, f … Fizikos terminų žodynas
fine structure — smulkioji sandara statusas T sritis chemija apibrėžtis Atomų, molekulių spektro linijų ar smailių skaidos vaizdas. atitikmenys: angl. fine structure rus. тонкая структура … Chemijos terminų aiškinamasis žodynas
fine structure — noun the presence of groups of closely spaced spectrum lines observed in the atomic spectrum of certain elements the fine structure results from slightly different energy levels • Hypernyms: ↑spectrum line • Part Holonyms: ↑atomic spectrum … Useful english dictionary
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fine structure — noun Date: 1935 microscopic structure of a biological entity or one of its parts especially as studied in preparations for the electron microscope • fine structural adjective … New Collegiate Dictionary
fine structure — Смотри тонкая структура … Энциклопедический словарь по металлургии
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