Term symbol: Difference between revisions

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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 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]] (or ''LS'' coupling) scheme.
 
A term symbol has the following form:
:<math>
:<math>
   ^{2S+1}L_{J}, \,
   ^{2S+1}L_{J}, \, \qquad\qquad\qquad\qquad\qquad\qquad(1)
</math>  
</math>  
where ''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 designated by the letter):
where ''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 designated by the letter):
:<math>
:<math>
S(0), \, P(1),\, D(2),\, F(3),\, G(4),\, H(5),\, I(6),\, K(7), \dots,
S(0), \, P(1),\, D(2),\, F(3),\, G(4),\, H(5),\, I(6),\, K(7), \dots,
</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]]:
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''' &equiv; '''L''' + '''S'''. The value ''J'' satisfies the [[Angular momentum coupling#Triangular conditions|triangular conditions]]:
:<math>
:<math>
J = |L-S|,\, |L-S|+1, \, \ldots, L+S,
J = |L-S|,\, |L-S|+1, \, \ldots, L+S,
</math>.
</math>.
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]] (or ''L''-''S'' coupling) scheme.
 


A term symbol is often preceded by the [[Atomic electron configuration|electronic configuration]] that leads to the ''L''-''S'' coupled functions, thus, for example,  
A term symbol is often preceded by the [[Atomic electron configuration|electronic configuration]] that leads to the ''L''-''S'' coupled functions, thus, for example,  

Revision as of 11:53, 21 February 2008

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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 (or LS coupling) scheme.

A term symbol has the following form:

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):

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: 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.

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