Fine structure (physics)

In physics, fine structure describes the composition of a spectral line from several distinguishable lines or an energy level from several distinguishable energy values. These can not yet be distinguished with low spectral resolution or in a theoretical approximation, which explains the relatively late discovery of the fine structure towards the end of the 19th century.

description

The term fine structure originated in optical spectroscopy towards the end of the 19th century because many lines in the line spectra of atoms turned out to be composed with increasing measurement accuracy ( fine structure splitting ).

The fine structure of a spectral line is explained by the fact that its position in the spectrum corresponds to the energy difference between two energy levels, and that many levels, which appear uniform when viewed approximately, show up as a multiplet if the measurement or modeling is more accurate , i.e. H. as a group of several levels with closely adjacent energies.

The term is mainly used in relation to the energy levels of individual electrons in atoms and solids , but also in an analogous manner in other areas. In the line spectra of the atoms, the fine structure split in the optical range is about 1/1000 to 1 / 100,000 of the typical wavelength or transition energy. In the X-ray range , it increases to about 1/10 in the internal electrons of the atoms of heavy elements.

Even more precise measurements on atoms show further, but even smaller splits:

The hyperfine structure discovered in 1924 is caused by the interaction between the electrons and the nuclear moments ,
the Lamb shift from quantum electrodynamic vacuum polarization discovered in 1947 .

Physical causes

The fine structure is explained by the fact that the energy levels, which are correctly described according to Bohr's atomic model or non-relativistic quantum mechanics without spin when measured in low resolution , are shifted and partially split when calculated according to the relativistically correct quantum mechanical Dirac equation . This can largely be traced back to the kinematics, which were changed according to the theory of relativity , and to the spin-orbit coupling caused by the electron spin and the associated anomalous magnetic moment .

In order to make these contributions individually visible, the Dirac equation is approximated by a series expansion and thus correction terms for the non-relativistic Hamilton operator are obtained . Apart from the constant rest energy of the electron, the Hamilton operator then reads in first order: ${\ displaystyle H_ {0}}$ ${\ displaystyle m _ {\ mathrm {e}} c ^ {2}}$

{\ displaystyle H = H_ {0} + W _ {\ mathrm {M}} + W _ {\ mathrm {SB}} + W _ {\ mathrm {D}} + \ ldots.}

The correction terms are in detail:

${\ displaystyle W _ {\ mathrm {M}} = - {\ frac {{\ vec {p}} ^ {\; 4}} {8m _ {\ mathrm {e}} ^ {3} c ^ {2}} }}$ - the relativistic correction of the kinetic energy
${\ displaystyle W _ {\ mathrm {SB}} = {\ frac {1} {2m _ {\ mathrm {e}} ^ {2} c ^ {2}}} \, {\ vec {s}} \ cdot { \ vec {\ ell}} \ {\ frac {1} {r}} {\ frac {\ mathrm {d} V} {\ mathrm {d} r}}}$ - the spin-orbit coupling
${\ displaystyle W _ {\ mathrm {D}} = {\ frac {\ hbar ^ {2}} {8m _ {\ mathrm {e}} ^ {2} c ^ {2}}} \ Delta V}$ - the Darwin term as correction of the potential energy with the Laplace operator ${\ displaystyle \ Delta}$

The energy shift , which is called the fine structure, is then corresponding ${\ displaystyle \ Delta E}$

{\ displaystyle \ Delta E = E _ {\ mathrm {M}} + E _ {\ mathrm {SB}} + E _ {\ mathrm {D}}.}

Hydrogen atom

Fine structure splitting as one of the corrections to the energy levels of the hydrogen atom

In the case of the hydrogen atom, relativistic effects, spin-orbit interaction and Darwin's term can be combined into a formula for the correction of the energy levels:

{\ displaystyle \ Delta E _ {\ mathrm {FS}} = E _ {\ mathrm {n}} \ left [{\ frac {Z ^ {2} \ alpha ^ {2}} {n}} \ left ({\ frac {1} {j + {\ frac {1} {2}}}} - {\ frac {3} {4n}} \ right) \ right]}

With

the energy of the levels in the hydrogen atom without fine structure ${\ displaystyle E _ {\ mathrm {n}} = - E_ {R} {\ frac {Z ^ {2}} {n ^ {2}}}}$ ${\ displaystyle E _ {\ mathrm {n}} = - E_ {R} {\ frac {Z ^ {2}} {n ^ {2}}}}$
- the Rydberg energy ${\ displaystyle E_ {R} \ approx 13 {,} 6 \, {\ text {eV}}}$
- the atomic number ${\ displaystyle Z}$
- the principal quantum number ${\ displaystyle n}$
the fine structure constant ${\ displaystyle \ alpha}$
the total angular momentum . ${\ displaystyle j}$

This formula causes for every possible one and a lowering of the energy. It agrees with Sommerfeld's fine structure formula , which had been developed years before the discovery of spin and the Dirac formula as part of the semi-classical conception of the Bohr-Sommerfeld atomic model . ${\ displaystyle n}$ ${\ displaystyle j}$

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^ H. Friedrich: Theoretical Atomic Physics, Third Edition, p. 88 f.
↑ Wolfgang Demtröder : Experimentalphysik 3 . 5th edition. Springer, ISBN 3-540-21473-9 , pp. 158-161 .
↑ A. Sommerfeld: On the fine structure of hydrogen lines. History and current state of theory; Natural Sciences, July 1940, Volume 28, Issue 27, pp 417-423; https://link.springer.com/article/10.1007/BF01490583

[1] H. Friedrich: Theoretical Atomic Physics, Third Edition, p. 88 f.

[2] Wolfgang Demtröder : Experimentalphysik 3 . 5th edition. Springer, ISBN 3-540-21473-9 , pp. 158-161 .

[3] A. Sommerfeld: On the fine structure of hydrogen lines. History and current state of theory; Natural Sciences, July 1940, Volume 28, Issue 27, pp 417-423; https://link.springer.com/article/10.1007/BF01490583

Fine structure (physics)

contents

description

Physical causes

Hydrogen atom

See also

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