Spectroscopic Notation

Spectroscopic Notation

Neutral atoms consist of a heavy nucleus with charge $Z$ surrounded by $Z$ electrons. Positively charged atomic ions generally have structure similar to the atom with the same number of electrons except for a scale factor; negative ions lack the attractive Coulomb interaction at large electron-core separation and hence have few if any bound levels. Thus the essential feature of an atom is its number of electrons, and their mutual arrangement as expressed in the quantum numbers. An isolated atom has two good angular momentum quantum numbers, $J$ and $M_{J}$. (This is strictly true only for atoms whose nuclei have spin $I=0$. However, $J$ is never significantly affected by coupling to $I$ in ground state atoms.) In zero external field the atomic Hamiltonian possesses rotational invariance which implies that each $J$ level is degenerate with respect to the $2J+1$ states with specific $M_{J}$ (traditional atomic spectroscopists call these states "sublevels"). For each $J$,$M_{J}$ an atom will typically have a large number of discrete energy levels (plus a continuum) which may be labeled by other quantum numbers. If Russell-Saunders coupling ($L-S$ coupling) is a good description of the atom (true for light atoms), then $L$ and $S$, where

$L=\sum _{i=1}^{N}{\mathbf {\ell } }_{i}$

$S=\sum _{i=1}^{N}S_{i}$

are nearly good quantum numbers and may be used to distinguish the levels. In this case the level is designated by a Term symbolized $^{2S+1}L_{J}$ where $2S+1$ and $J$ are written numerically and $L$ is designated with this letter code:

		$L$:	O	1	2	3	4	...
		Letter:	$S$	$P$	$D$	$F$	$G$

The Letters stand for Sharp, Principal, Diffuse, and Fundamental - adjectives applying to the spectral lines of one electron atoms. When describing a hydrogenic atom, the term is preceded by the principle quantum number of the outermost electron. For example, in sodium, the ground state is designated $3^{2}S_{1/2}$, while the D-line excited states are $3^{2}P_{1/2}$ and $3^{2}P_{3/2}$. This discussion of the term symbol has been based on quantum numbers for all electrons of the atom. Alternatively one may have or assume knowledge of the internal structure - the quantum numbers of each electron. These are specified as the {\it configuration}, e.g..

$1s^{2}2s^{2}2p^{2},$

a product of symbols of the form $n\ell ^{k}$ which represents $k$ electrons in the orbital $n$, $\ell$. $n$ is the principal quantum number, which characterizes the radial motion and has the largest influence on the energy. $n$ and $m$ are written numerically, but the $spdf$ ... coding is used for $\ell$. Returning to the example of sodium, the configuration is $1s^{2}2s^{2}2p^{6}3s$, which is often abbrevated to simply $3s$. In classifying levels, the term is generally more important that the configuration because it determines the behavior of an atom when it interacts with $E$ or $B$ fields. Selection rules, for instance, generally deal with $\Delta J,\Delta L,\Delta S$. Furthermore the configuration may not be pure - if two configurations give rise to the same term (and have the same parity) then intra-atomic electrostatic interactions can mix them together. This process, called configuration interaction, results in shifts in the level positions and intensities of special lines involving them as well as in correlations in the motions of the electrons within the atoms.