Fine structure

Fine structure
Interference fringes, showing fine structure (splitting) of a cooled deuterium source, viewed through a Fabry-Pérot étalon.

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 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 splitting is on the order of ()2, where Z is the atomic number and α is the fine-structure 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 spin-orbit term, and the Darwinian term. The full Hamiltonian is given by



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.

E_{n}^{(1)}=\langle\psi^{0}\vert H'\vert\psi^{0}\rangle=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{4}\vert\psi^{0}\rangle=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{2}p^{2}\vert\psi^{0}\rangle

where ψ0 is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see


We can use this result to further calculate the relativistic correction:

E_{n}^{(1)}=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{2}p^{2}\vert\psi^{0}\rangle
E_{n}^{(1)}=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert (2m)^{2}(E_{n}-V)^{2}\vert\psi^{0}\rangle
E_{n}^{(1)}=-\frac{1}{2mc^{2}}(E_{n}^{2}-2E_{n}\langle V\rangle +\langle V^{2}\rangle )

For the hydrogen atom, V=\frac{e^{2}}{r}, \langle V\rangle=\frac{-e^{2}}{a_{0}n^{2}}, and \langle V^{2}\rangle=\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}} where a0 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

E_{n}^{(1)}=-\frac{1}{2mc^{2}}\left(E_{n}^{2}+2E_{n}\frac{e^{2}}{a_{0}n^{2}} +\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}}\right)=-\frac{E_{n}^{2}}{2mc^{2}}\left(\frac{4n}{l+1/2}-3\right)

where we have used:

 E_n = - \frac{e^2}{2 a_0 n^2}

On final calculation, the order of magnitude for the spin-orbital coupling for ground state is  -9.056 \times 10^{-4}\ \text{eV}.

Spin-orbit coupling

H_{so}=\frac{1}{2} \left(\frac{Ze^2}{4\pi \epsilon_{0}}\right)\left(\frac{g_s}{2m_{e}^{2}c^{2}}\right)\frac{\vec L\cdot\vec S}{r^{3}}

The spin-orbit 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, \vec B and \vec\mu_s 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

 \Delta E_{SO} = \xi (r)\vec L \cdot \vec S

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.


 \left\langle \frac {1}{r^3} \right\rangle = \frac {1}{n^3 a_0^3} \frac {1} {l (l+\frac{1}{2}) (l + 1)}
 \left\langle \vec L \cdot \vec S \right\rangle = \frac {\hbar^2} {2} ( j(j+1) - l(l+1) - s(s+1) )

the expectation value for the Hamiltonian is:

 \left\langle H_{SO} \right\rangle = \frac{E_n{}^2}{m_e c^2} \left( n \frac{j(j+1)-l(l+1)-\frac{3}{4}}{l \left( l+\frac{1}{2}\right) (l+1) } \right)

Thus the order of magnitude for the spin-orbital coupling is  \frac{Z}{n^3} 10^{-5}\text{ eV}.

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.

Darwin term

 E_{\mathrm{Darwin}}=\frac{\hbar^{2}}{8m_{e}^{2}c^{2}}\,4\pi\left(\frac{Ze^2}{4\pi \epsilon_{0}}\right)\delta^{3}\left(\vec r\right)

 E_{\mathrm{Darwin}}=\frac{\hbar^{2}}{8m_{e}^{2}c^{2}}\,4\pi\left(\frac{Ze^2}{4\pi \epsilon_{0}}\right)| \psi(0)^2|
ψ(0) = 0 for l > 0
 \psi (0) = \frac{1}{\sqrt{4\pi}}\,2 \left( \frac {Z}{n a_0} \right)^\frac {3}{2} \text{ for } l = 0
 E_{\mathrm{Darwin}}=\frac{2n}{m_e c^{2}}\,E_n^2

Thus, the Darwin term affects only the s-orbit. For example it gives the 2s-orbit the same energy as the 2p-orbit by raising the 2s-state by  9.057 \times 10^{-5}\text{ 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.

Another mechanism that affects only the s-state is the Lamb shift. The reader should not mix up the Darwin term and the Lamb shift. The Darwin term makes the s-state and p-state the same energy, but the Lamb shift makes the s-state higher in energy than the p-state.

Total effect

The total effect, obtained by summing the three components up, is given by the following expression [1]:

\Delta E = \frac{m_{e}c^{2}(Z\alpha)^{4}}{2n^3}\left( \frac{1}{j + 1/2} - \frac{3}{4n} \right)\,,

where j is the total angular momentum (j = 1 / 2 if l = 0 and j = l \pm 1/2 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.

New Microsoft Office PowerPoint Presentation.png

See also


  1. ^ Berestetskii, V. B.; E. M. Lifshitz, L. P. Pitaevskii (1982). Quantum electrodynamics. Butterworth-Heinemann. ISBN 9780750633710. 
  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. 
  • Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. 

External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • 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

  • fine structure — smulkioji sandara statusas T sritis Standartizacija ir metrologija apibrėžtis Atomų, molekulių, kristalų spektrų artimų linijų sistema, susidariusi dėl elektronų energijos lygmenų skaidos. atitikmenys: angl. fine structure vok. Feinstruktur, f… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • 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

  • fine structure — fine′ struc′ture [[t]faɪn[/t]] n. 1) phs a group of lines that are observed in the spectra of certain elements, as hydrogen 2) cbl the aggregate of components of the cytoskeleton • Etymology: 1915–20 …   From formal English to slang

  • 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

  • fine structure genetic mapping — fine structure genetic mapping. См. тонкое генетическое картирование. (Источник: «Англо русский толковый словарь генетических терминов». Арефьев В.А., Лисовенко Л.А., Москва: Изд во ВНИРО, 1995 г.) …   Молекулярная биология и генетика. Толковый словарь.

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.