In atomic physics, the Landé interval rule [1] states that, due to weak angular momentum coupling (either spin-orbit or spin-spin coupling), the energy splitting between successive sub-levels are proportional to the total angular momentum quantum number (J or F) of the sub-level with the larger of their total angular momentum value (J or F). [2][3]

Background

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The rule assumes the Russell–Saunders coupling and that interactions between spin magnetic moments can be ignored. The latter is an incorrect assumption for light atoms. As a result of this, the rule is optimally followed by atoms with medium atomic numbers.[4]

The rule was first stated in 1923 by German-American physicist Alfred Landé.[1]

Derivation

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As an example,[2] consider an atom with two valence electrons and their fine structures in the LS-coupling scheme. We will derive heuristically the interval rule for the LS-coupling scheme and will remark on the similarity that leads to the interval rule for the hyperfine structure.

The interactions between electrons couple their orbital and spin angular momentums. Let's denote the spin and orbital angular momentum as   and   for each electrons. Thus, the total orbital angular momentum is   and total spin momentum is  . Then the coupling in the LS-scheme gives rise to a Hamiltonian:

 

where   and   encode the strength of the coupling. The Hamiltonian   acts as a perturbation to the state   . The coupling would cause the total orbital   and spin   angular momentums to change directions, but the total angular momentum   would remain constant. Its z-component   would also remain constant, since there is no external torque acting on the system. Therefore, we shall change the state to  , which is a linear combination of various  . The exact linear combination, however, is unnecessary to determine the energy shift.

To study this perturbation, we consider the vector model where we treat each   as a vector.   and   precesses around the total orbital angular momentum   . Consequently, the component perpendicular to   averages to zero over time, and thus only the component along   needs to be considered. That is,   . We replace   by   and   by the expectation value  .

Applying this change to all the terms in the Hamiltonian, we can rewrite it as

 

The energy shift is then

 

Now we can apply the substitution   to write the energy as

 

Consequently, the energy interval between adjacent   sub-levels is:

 

This is the Landé interval rule.

As an example, consider a   term, which has 3 sub-levels   . The separation between   and   is   , twice as the separation between   and   is   .

As for the spin-spin interaction responsible for the hyperfine structure, because the Hamiltonian of the hyperfine interaction can be written as

 

where   is the nuclear spin and   is the total angular momentum, we also have an interval rule:

 

where   is the total angular momentum   . The derivation is essentially the same, but with nuclear spin   , angular momentum   and total angular momentum   .

Limitations

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The interval rule holds when the coupling is weak. In the LS-coupling scheme, a weak coupling means the energy of spin-orbit coupling   is smaller than residual electrostatic interaction:  . Here the residual electrostatic interaction refers to the term including electron-electron interaction after we employ the central field approximation to the Hamiltonian of the atom. For the hyperfine structure, the interval rule for two magnetic moments can be disrupted by magnetic quadruple interaction between them, so we want  . [2]

For example, in helium, the spin-spin interactions and spin-other-orbit interaction have an energy comparable to that of the spin-orbit interaction. [2]

References

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  1. ^ a b Landé, A. Termstruktur und Zeemaneffekt der Multipletts. Z. Physik 15, 189–205 (1923). https://doi.org/10.1007/BF01330473
  2. ^ a b c d Foot, Christopher J (2005). Atomic Physics (1st ed.). ISBN 978-0-19-850695-9. Retrieved December 11, 2020.
  3. ^ Morris, Christopher G. (1992). Academic Press dictionary of science and technology. Academic Press. pp. 1201. ISBN 0-12-200400-0.
  4. ^ E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1959, p 193.