Term symbol

In atomic spectroscopy, a term symbol gives the total spin-, orbital-, and spin-orbital angular momentum of an atom in a certain quantum state (often the ground state). The simultaneous eigenfunctions of L² and S² labeled by a term symbol are obtained in the Russell-Saunders coupling (or LS coupling) scheme.

A term symbol has the following form:

$^{2S+1}L_{J},\,\qquad \qquad \qquad \qquad \qquad \qquad (1)$

where S is the total spin angular momentum of the state and 2S+1 is the spin multiplicity.

The symbol L represents the total orbital angular momentum of the state. For historical reasons L is coded by a letter as follows (between brackets the L quantum number designated by the letter):

$S(0),\,P(1),\,D(2),\,F(3),\,G(4),\,H(5),\,I(6),\,K(7),\dots ,$

and further up the alphabet (excluding P and S).

The subscript J in the term symbol (1) is the quantum number of the spin-orbital angular momentum: J ≡ L + S. The value J satisfies the triangular conditions:

$J=|L-S|,\,|L-S|+1,\,\ldots ,L+S,$.

A term symbol is often preceded by the electronic configuration that leads to the L-S coupled functions, thus, for example,

$(ns)^{k}\,(n'p)^{k'}\,(n''d)^{k''}\,\,\,^{2S+1}L.$

The (2S+1)(2L+1) different functions referred to by this symbol form a term. When the quantum number J is added (as a subscript) the symbol refers to an energy level, comprising 2J+1 components.

Sometimes the parity of the state is added, as in

$^{2S+1}L_{J}^{o},\,$

which indicates that the state has odd parity. This is the case when the sum of the one-electron orbital angular momentum numbers in the electronic configuration is odd.

For historical reasons, the term symbol is somewhat inconsistent in the sense that the quantum numbers L and J are indicated directly, by a letter and a number, respectively, while the spin S is indicated by its multiplicity 2S+1.

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Examples

A few ground state atoms are listed.

• Hydrogen atom: $\scriptstyle 1s\,\,\,^{2}S_{\frac {1}{2}}$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 0. Spin-orbital angular momentum: J = 1/2. Parity: even.

• Carbon atom: $\scriptstyle (1s)^{2}\,(2s)^{2}\,(2p)^{2}\,\,\,^{3}P_{0}\,$. Spin angular momentum: S = 1. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 0. Parity even.

• Aluminium atom: $\scriptstyle (1s)^{2}\,(2s)^{2}\,(2p)^{6}\,(3s)^{2}\,3p\,\,\,^{2}P_{\frac {1}{2}}^{o}\,$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 1/2. Parity odd.

• Scandium atom: $\scriptstyle (1s)^{2}\,(2s)^{2}\,(2p)^{6}\,(3s)^{2}\,(3p)^{6}\,3d\,(4s)^{2}\,\,\,^{2}D_{\frac {3}{2}}\,$. Spin angular momentum: S = 1/2. Orbital angular momentum: L = 2. Spin-orbital angular momentum: J = 3/2. Parity even.

External links

• NIST Atomic Sectroscopy
• A list of term symbols for ground state atoms