Term symbol

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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 (also known as LS coupling) scheme.

A term symbol has the following form:

$^{2S+1}\!L_{J} .\;$

Here:

The symbol 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):

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

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

The subscript J in the term symbol 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: $\scriptstyle 1s\,\,\, ^2S_{\frac{1}{2}}$ . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 0. Spin-orbital angular momentum: J = 1/2. Parity: even.

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

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

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