fine structure: Definition from Answers.com

Interference fringes, showing fine structure 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 $\left(Z\alpha\right)^{2}$ , where Z is the atomic number and α is the fine-structure constant.

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

H = H 0 + H k i n e t i c + H s o + H D a r w i n i a n .

1 Kinetic energy relativistic correction
2 Spin-orbit coupling
3 Darwin term
4 See also
5 References
6 External links

Kinetic energy relativistic correction

Classically, the kinetic energy term of the Hamiltonian is:

$T=\frac{p^{2}}{2m}.$

However, when considering special relativity, we must use a relativistic form of the kinetic energy,

$T=\sqrt{p^{2}c^{2}+m^{2}c^{4}}-mc^{2}$

where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Expanding this we find

$T=\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+\dots$

Then, the first order correction to the Hamiltonian is

$H_{kinetic}=-\frac{p^{4}}{8m^{3}c^{2}}$

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 $ψ 0$ is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

$H^{0}\vert\psi^{0}\rangle=E_{n}\vert\psi^{0}\rangle$

$\left(\frac{p^{2}}{2m}+V\right)\vert\psi^{0}\rangle=E_{n}\vert\psi^{0}\rangle$

$p^{2}\vert\psi^{0}\rangle=2m(E_{n}-V)\vert\psi^{0}\rangle$

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 $a 0$ 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

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

Spin-orbit coupling

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

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

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

Darwin term

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

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.

References

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

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fine structure

Contents

Kinetic energy relativistic correction

Spin-orbit coupling

Darwin term

See also

References

External links