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In [[atomic spectroscopy]], a '''term symbol''' gives the total spin-, orbital-, and spin-orbital [[angular momentum (quantum)|angular momentum]] of the atom. The term symbol has the following form:
In [[atomic spectroscopy]], a '''term symbol''' gives the total spin-, orbital-, and spin-orbital [[angular momentum (quantum)|angular momentum]] of an [[atom]] in a certain quantum  state (often the ground state). The simultaneous eigenfunctions of '''L'''<sup>2</sup> and '''S'''<sup>2</sup> labeled by a term symbol are obtained in the [[Russell-Saunders coupling]] (also known as ''LS'' coupling) scheme.
:<math>
 
   ^{2S+1}X_{J}, \,
A term symbol has the following form:
::<math>
   ^{2S+1}\!L_{J} .\;
</math>
 
Here:
*The symbol ''S'' is the total spin angular momentum of the state and 2''S''+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):
::<math>
S(0), \; P(1),\; D(2),\; F(3),\; G(4),\; H(5),\; I(6),\; K(7), \dots,
</math>  
</math>  
where ''S'' is the total spin angular momentum and 2''S''+1 is the spin multiplicity. The symbol ''X'' represents the total orbital angular momentum. For historical reasons it is coded by a letter as follows (between brackets the ''L'' quantum number designated by the letter):
: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''' &equiv; '''L''' + '''S'''. The value ''J'' satisfies the [[Angular momentum coupling#Triangular conditions|triangular conditions]]:
::<math>
J = |L-S|,\, |L-S|+1, \, \ldots, L+S,
</math>.
 
 
A term symbol is often preceded by the [[Atomic electron configuration|electronic configuration]] that leads to the ''L''-''S'' coupled functions, thus, for example,
:<math>
:<math>
S(0), \, P(1),\, D(2),\, F(3),\, G(4),\, H(5),\, I(6),\, K(7), \dots,
(ns)^k \, (n'p)^{k'}\, (n''d)^{k''}\,\,\, ^{2S+1}L . 
</math>
</math>
and further up the alphabet (excluding ''P'' and ''S''). The value ''J'' is the quantum number of the spin-orbital angular momentum: '''J''' &equiv; '''L''' + '''S'''. The value ''J'' satisfies the [[Angular momentum coupling#Triangular conditions|triangular conditions]]:
The (2''S''+1)(2''L''+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 2''J''+1 components.
:<math>
 
J = |L-S|,\, |L-S|+1, \, \ldots, L+S,
</math>.
Sometimes the [[parity]] of the state is added, as in  
Sometimes the [[parity]] of the state is added, as in  
:<math>
:<math>
   ^{2S+1}X_{J}^o, \,
   ^{2S+1}L_{J}^o, \,
</math>  
</math>  
which indicates that the state has odd parity. This is the case if the sum of the one-electron
which indicates that the state has odd parity. This is the case when the sum of the one-electron
orbital angular momenta is odd.  
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 2''S''+1. The eigenstates labeled by a term symbol are obtained in the [[Russell-Saunders coupling]] scheme.
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 2''S''+1.  
{{editintro}}
{{editintro}}
==Examples==
==Examples==
A few ground state atoms are listed.
A few ground state atoms are listed.
* [[Hydrogen]] atom: <math> ^2S_{\frac{1}{2}}</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 0. Spin-orbital angular momentum: ''J'' = 1/2. Electronic configuration: 1''s''. Parity: even.
* [[Hydrogen]] atom: <math>\scriptstyle 1s\,\,\, ^2S_{\frac{1}{2}}</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 0. Spin-orbital angular momentum: ''J'' = 1/2. Parity: even.
 
* [[Carbon]] atom: <math>\scriptstyle (1s)^2\,(2s)^2\, (2p)^2\,\,\, ^3P_{0}\,</math>. Spin angular momentum: ''S'' = 1. Orbital angular momentum: ''L'' = 1. Spin-orbital angular momentum: ''J'' = 0. Parity even.


* [[Carbon]] atom: <math> ^3P_{0}\,</math>. Spin angular momentum: ''S'' = 1. Orbital angular momentum: ''L'' = 1. Spin-orbital angular momentum: ''J'' = 0. Electronic configuration: [He]2''s''<sup>2</sup>2''p''<sup>2</sup>. Parity even.
* [[Aluminium]] atom: <math>\scriptstyle (1s)^2\,(2s)^2\,(2p)^6\,(3s)^2\,3p\,\,\, ^2P_{\frac{1}{2}}^o\,</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 1. Spin-orbital angular momentum: ''J'' = 1/2. Parity odd.


* [[Aluminium]] atom: <math> ^2P_{\frac{1}{2}}^o\,</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 1. Spin-orbital angular momentum: ''J'' = 1/2. Electronic configuration: [Ne]3''s''<sup>2</sup>3''p''<sup>1</sup>. Parity odd.
* [[Scandium]] atom: <math>\scriptstyle (1s)^2\,(2s)^2\,(2p)^6\,(3s)^2\, (3p)^6\, 3d\, (4s)^2 \,\,\, ^2D_{\frac{3}{2}}\,</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 2. Spin-orbital angular momentum: ''J'' = 3/2. Parity even.


* [[Scandium]] atom: <math> ^2D_{\frac{3}{2}}\,</math>. Spin angular momentum: ''S'' = 1/2. Orbital angular momentum: ''L'' = 2. Spin-orbital angular momentum: ''J'' = 3/2. Electronic configuration: [Ne]3''s''<sup>2</sup>3''p''<sup>6</sup>3''d''<sup>1</sup>4''s''<sup>2</sup>. Parity even.
==External links==
* [http://physics.nist.gov/Pubs/AtSpec/node09.html NIST Atomic Sectroscopy]
* [http://physics.nist.gov/PhysRefData/IonEnergy/tblNew.html A list of term symbols for ground state atoms]


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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:

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):
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: JL + S. The value J satisfies the triangular conditions:
.


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

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

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.

[edit intro]

Examples

A few ground state atoms are listed.

  • Hydrogen atom: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 0. Spin-orbital angular momentum: J = 1/2. Parity: even.
  • Carbon atom: . Spin angular momentum: S = 1. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 0. Parity even.
  • Aluminium atom: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 1. Spin-orbital angular momentum: J = 1/2. Parity odd.
  • Scandium atom: . Spin angular momentum: S = 1/2. Orbital angular momentum: L = 2. Spin-orbital angular momentum: J = 3/2. Parity even.

External links