Landé's interval rule

The Landé'sche interval rule (according to the German physicist Alfred Landé ) allows in the nuclear physics to estimate the energy difference between two adjacent fine structure - or hyperfine - energy levels . It says that:

in the case of the fine structure, the energy difference of the level with quantum numbers and proportionally is ${\ displaystyle J}$ ${\ displaystyle J-1}$ ${\ displaystyle J}$
in the case of the hyperfine structure the energy difference of the levels with quantum numbers and is proportional to . ${\ displaystyle F}$ ${\ displaystyle F-1}$ ${\ displaystyle F}$

In the hyperfine splitting (right) of the hydrogen atom it can be seen that the energy difference between the levels and is twice greater than that between and .

{\ displaystyle F = 2}

{\ displaystyle F-1 = 1}

{\ displaystyle F = 1}

{\ displaystyle F-1 = 0}

Fine structure

In a fine- multiplet the energy of a level with is the principal quantum number , orbital angular momentum , electron spin , and entire angular momentum envelope given by the formula: ${\ displaystyle n}$ ${\ displaystyle L}$ ${\ displaystyle S}$ ${\ displaystyle J}$

{\ displaystyle \ Delta E_ {n, L, J} = {\ frac {a} {2}} \ left (J (J + 1) -L (L + 1) -S (S + 1) \ right) }

Here is the atom-specific LS coupling constant. ${\ displaystyle a}$

From this it follows for the energy difference of two levels with total angular momentum and : ${\ displaystyle J}$ ${\ displaystyle J-1}$

{\ displaystyle \ Rightarrow \ Delta E _ {\ mathrm {FS}} = \ Delta E_ {n, L, J} - \ Delta E_ {n, L, J-1} = a \ cdot J}

Hyperfine structure

The same applies to the hyperfine structure, except that instead of the envelope angular momentum , the total angular momentum is considered, which includes the nuclear spin : ${\ displaystyle J}$ ${\ displaystyle F}$ ${\ displaystyle I}$

{\ displaystyle \ Delta E _ {\ mathrm {HFS}} = \ Delta E_ {n, J, I, F} - \ Delta E_ {n, J, I, F-1} = A \ cdot F}

Here, as hyperfine structure - coupling constant or intermittent factor also referred. ${\ displaystyle A}$

validity

The interval rule is usually fulfilled to a good approximation for light atoms. It generally loses its validity as soon as the coupling of the angular momentum involved can no longer be treated as a minor disturbance. This is due to the fact that the above formula for the position of the energy levels presupposes LS coupling, which is no longer given with heavy atoms.

Even with light atoms, such as the triplet state of the lightest multi-electron atom helium , the interval rule can be violated due to the spin-spin interaction of the electrons .

The interaction of the electric field of the shell electrons with a non-vanishing quadrupole moment of the atomic nucleus can also cause deviations from the interval rule.

Individual evidence

↑ Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 223 f .
↑ Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . 1st edition. Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 352 .
^ Theo Mayer-Kuckuk: Atomic Physics . 5th edition. BG Teubner, Stuttgart 1997, ISBN 978-3-519-43042-1 , p. 196 .
^ Gerhard Herzberg: Atomic Spectra and Atomic Structure . 2nd Edition. Dower Publications, New York 1944, pp. 178 f .
↑ Wolfgang Demtröder: Experimentalphysik 3 - Nuclear, Particle and Astrophysics . 4th edition. Springer, Berlin / Heidelberg 2010, ISBN 978-3-642-03910-2 , pp. 215 .
↑ Hermann Haken, Hans Christoph Wolf: Atomic and Quantum Physics - Introduction to the experimental and theoretical basics . 7th edition. Springer, Berlin 2001, ISBN 3-540-67453-5 , pp. 379 .

[1] Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 223 f .

[2] Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . 1st edition. Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 352 .

[3] Theo Mayer-Kuckuk: Atomic Physics . 5th edition. BG Teubner, Stuttgart 1997, ISBN 978-3-519-43042-1 , p. 196 .

[4] Gerhard Herzberg: Atomic Spectra and Atomic Structure . 2nd Edition. Dower Publications, New York 1944, pp. 178 f .

[5] Wolfgang Demtröder: Experimentalphysik 3 - Nuclear, Particle and Astrophysics . 4th edition. Springer, Berlin / Heidelberg 2010, ISBN 978-3-642-03910-2 , pp. 215 .

[6] Hermann Haken, Hans Christoph Wolf: Atomic and Quantum Physics - Introduction to the experimental and theoretical basics . 7th edition. Springer, Berlin 2001, ISBN 3-540-67453-5 , pp. 379 .