Landé Interval Rule
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In
atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, the Landé interval rule Landé, A. Termstruktur und Zeemaneffekt der Multipletts. Z. Physik 15, 189–205 (1923). https://doi.org/10.1007/BF01330473 states that, due to weak angular
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
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).


Background

The rule assumes the Russell–Saunders coupling and that interactions between spin
magnetic Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...
moments can be ignored. The latter is an incorrect assumption for light
atoms Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
. As a result of this, the rule is optimally followed by atoms with medium
atomic numbers The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
.E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1959, p 193. The rule was first stated in 1923 by
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-
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physicist
Alfred Landé Alfred Landé (13 December 1888 – 30 October 1976) was a German-American physicist known for his contributions to quantum theory. He is responsible for the Landé g-factor and an explanation of the Zeeman effect. Life and achievements Alf ...
.


Derivation

As an example, 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 In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole int ...
. The interactions between electrons couple their orbital and spin angular momentums. Let's denote the spin and orbital angular momentum as \mathbf and \mathbf for each electrons. Thus, the total orbital angular momentum is \mathbf = \mathbf_1 + \mathbf_2 and total spin momentum is \mathbf = \mathbf_1 + \mathbf_2. Then the coupling in the LS-scheme gives rise to a Hamiltonian: H_=\beta_ \mathbf_ \cdot \mathbf_+\beta_ \mathbf_ \cdot \mathbf_ where \beta_1 and \beta_2 encode the strength of the coupling. The Hamiltonian H_ acts as a perturbation to the state \vert L m_L S m_S \rangle . The coupling would cause the total orbital \mathbf and spin \mathbf angular momentums to change directions, but the total angular momentum \mathbf = \mathbf + \mathbf would remain constant. Its z-component J_z would also remain constant, since there is no external torque acting on the system. Therefore, we shall change the state to \vert L m_L J m_J \rangle , which is a linear combination of various \vert L m_L S m_S \rangle . 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 \mathbf as a vector. \mathbf_1 and \mathbf_2 precesses around the total orbital angular momentum \mathbf . Consequently, the component perpendicular to \mathbf averages to zero over time, and thus only the component along \mathbf needs to be considered. That is, \mathbf_ \rightarrow\left \mathbf, ^\right\mathbf . We replace , \mathbf, ^ by L(L+1) and \left(\overline\right) by the expectation value \left\langle\mathbf_ \cdot \mathbf\right\rangle. Applying this change to all the terms in the Hamiltonian, we can rewrite it as \begin H_ &=\beta_ \frac \mathbf \cdot \frac \mathbf+\beta_ \frac \mathbf \cdot \frac \mathbf \\ &=\beta_ \mathbf \cdot \mathbf \end The energy shift is then E_=\beta_\langle\mathbf \cdot \mathbf\rangle. Now we can apply the substitution \mathbf \cdot \mathbf=(\mathbf \cdot \mathbf-\mathbf \cdot \mathbf-\mathbf \cdot \mathbf) / 2 to write the energy as E_=\frac\. Consequently, the energy interval between adjacent J sub-levels is: \Delta E_=E_-E_=\beta_ J This is the Landé interval rule. As an example, consider a ^3P term, which has 3 sub-levels ^ \mathrm_,^ \mathrm_,^ \mathrm_ . The separation between J = 2 and J = 1 is 2 \beta , twice as the separation between J = 1 and J = 0 is \beta . As for the spin-spin interaction responsible for the hyperfine structure, because the Hamiltonian of the hyperfine interaction can be written as H_ = A_ \mathbf \cdot \mathbf where \mathbf is the nuclear spin and \mathbf is the total angular momentum, we also have an interval rule: E_-E_= A_ F where F is the total angular momentum \mathbf = \mathbf + \mathbf . The derivation is essentially the same, but with nuclear spin \mathbf , angular momentum \mathbf and total angular momentum \mathbf .


Limitations

The interval rule holds when the coupling is weak. In the LS-coupling scheme, a weak coupling means the energy of spin-orbit coupling E_ is smaller than residual electrostatic interaction: E_ \ll E_. Here the residual electrostatic interaction refers to the term including electron-electron interaction after we employ the
central field approximation In atomic physics, the central field approximation for many-electron atoms takes the combined electric fields of the nucleus and all the electrons acting on any of the electrons to be radial and to be the same for all the electrons in the atom. ...
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 A \gg \Delta E_. For example, in helium, the spin-spin interactions and spin-other-orbit interaction have an energy comparable to that of the spin-orbit interaction.


References

{{DEFAULTSORT:Lande interval rule Atomic physics