NIST: Atomic Spectroscopy - Helium-like Ions; LS Coupling

6. Helium and Helium-like Ions; LS Coupling

The energy structure of the normal 1snl configurations is dominated by the electron-nucleus and electron-electron Coulomb contributions [4]. In helium and in helium-like ions of the lighter elements, the separations of levels having the same n and having l = s, p, or d are mainly determined by direct and exchange electrostatic interactions between the electrons - the spin-orbit, spin-other orbit, and other relativistic contributions are much smaller. This is the condition for LS coupling, in which:

(a) The orbital angular momenta of the electrons are coupled to give a total orbital angular momentum L = Σ_il_i.
(b) The spins of the electrons are coupled to give a total spin S = Σ_i s_i.

The combination of a particular S value with a particular L value comprises a spectroscopic term, the notation for which is ^2S+1L. The quantum number 2S + 1 is the multiplicity of the term. The S and L vectors are coupled to obtain the total angular momentum, J = S + L, for a level of the term; the level is denoted as ^2S+1L_J.

For 1snl configurations, L = l and S = 0 or 1, i.e., the terms are singlets (S = 0) or triplets (S = 1). As examples of the He I structure, the ionization energy (energy required to remove one of the 1s electrons in the 1s² ground configuration) is 24.5874 eV, the 1s2s ³S - ¹S separation is 0.7962 eV, the 1s2p ³P� - ¹P� separation is 0.2539 eV, and the 1s2p ³P�₂ - ³P�₀ fine-structure spread is only 1.32 � 10^-4 eV.

7. Hierarchy of Atomic Structure in LS Coupling

The centrality of LS coupling in the analysis and theoretical interpretation of atomic spectra has led to the acceptance of notations and nomenclature well adapted to discussions of particular structures and spectra [2]. The main elements of the nomenclature are shown in the table below, most of the structural entities having already been defined in the above discussions of simple spectra. The quantum numbers in the table represent a full description for complex configurations, and the accepted names for transitions between the structural elements are also given.

**Atomic structural hierarchy in LS coupling and names for the groups of all transitions between structural entities.**
Structural entity	Quantum numbers ^a	Group of all transitions


Configuration	(n_il_i)^N_i	Transition array
Polyad	(n_il_i)^N_i γS₁ L₁ nl S L, S L′ ...	Supermultiplet
Term	(n_il_i)^N_i γS L	Multiplet
Level	(n_il_i)^N_i γS L J	Line
State	(n_il_i)^N_i γS L J M	Line component

^a The configuration may include several open subshells, as indicated by the i subscripts. The letter γ represents any additional quantum numbers, such as ancestral terms, necessary to specify a particular term.

As an example, the Ca I 3d4p ³D�₂ level belongs to the ³D� term which, in turn, belongs to the 3d4p ³(P� D� F�) triplet triad. The 3d4p configuration also has a ¹(P� D� F�) singlet triad. The 3d4s configuration has only monads, one ¹D and one ³D. The 3d4s ³D₂ - 3d4p ³D�₃ line belongs to the corresponding ³D - ³D� triplet multiplet, and this multiplet belongs to the great Ca I 3d4s ³D - 3d4p ³(P� D� F�) supermultiplet of three triplet multiplets discussed by Russell and Saunders in their classic paper on the alkaline-earth spectra [6]. The 3d4s - 3d4p transition array includes both the singlet and triplet supermultiplets, as well as any (LS-forbidden) intercombination or intersystem lines arising from transitions between levels of the singlet system and those of the triplet system. The order of the two terms in the transitions as written above, with the lower-energy term on the left, is standard in atomic spectroscopy. Examples of notations for complex configurations are given in Notations for Different Coupling Schemes.