Terms and term symbol in atoms

Term symbol, Hund rulesLS coupling hierarchyjj coupling hierarchyHyperfine coupling

For all atoms extensive lists of term values are tabulated (Ch. Moore, NBS atomic data tables, see literature list).

Each angular momentum leads to a magnetic moment. Electrostatic and magnetic interactions between these moments couple the angular momenta. There are two limiting cases of angular momentum coupling hierarchy used to label the terms of atoms:

the LS coupling hierarchy, which is important for the lighter atoms,

the jj coupling hierarchy, which is important for the heaviest atoms.

Term symbol and Hund rules

2S+1LJ

2S+1----multiplicityL---------angular momentumJ----------total angular momentum

In Hunds rules for obtaining the ground state of an atom:

1. the state with the largest total angular momentum L2. the largest total spin momentum S3. relates to the the vectorial connection between L and S

- if an electron shell is less than half filled J=L-S- if an electron shell is less than half filled J=L+S-if an electron shell is half filled J=L+S=S, L=0

LS coupling hierarchy

In LS coupling, one obtains the possible terms by first adding vectorially the orbital angular momenta li of the electrons to form a resultant total orbital angular momentum L. Then the total electron spin S is determined by vectorial addition of the spins si of all electrons. Finally the total angular momentum J is determined by adding vectorially S and L. For a two electron system one gets:

Example:C (1s)2(2s)2(2p)2. Only the partially filled 2p subshell needs to be considered. In this case l1 = 1, l2 = 1 and s1 = s2 = 1/2.

Possible terms:

Taking the (2J + 1) multiplicity of each term (which corresponds to all possible values of MJ ), we obtain a total of 36 states corresponding to the 36 states predicted without consideration of the Pauli principle (previous lecture).As discussed above, there are only 15 allowed states for the configuration (1s)2(2s)2(2p)2. We now determine which of all terms listed above are allowed by the Pauli principle. This is done by first finding the ML , MS and MJ values resulting from all 15 possible occupations of the six 2p spin-orbitals with two electrons compatible with the Pauli principle.

The maximum value of ML is 2 and occurs in combination with MS = 0. This implies a 1D term with five MJ components corresponding to (ML MS ) = (2 0), (1 0), (0 0), (-1 0) and (-2 0). Eliminating these states, the one with the highest ML value has ML = 1 and comes in combination with a maximal MS value of 1. We can conclude that the term is 3P (consisting of 3P0 , 3P1 and 3P2 ). There are 9 components corresponding to (ML MS ) = (1 1),(1 0), (1 -1), (0 1), (0 0), (0 -1), (-1 1), (-1 0) and (-1 -1). Eliminating these states from the table, only one component remains, (0 0), which corresponds to a 1S0 state. The terms corresponding to the (2p)2 configuration allowed by the Pauli principle are therefore 1D2 , 3P2 , 3P1 , 3P0 and 1S0 . The following diagram shows that the terms have different energies and that the (2J +1) term components of each term (which are degenerate in zero field) can be observed in a magnetic field.

Schematic energy level structureof the (2p)2 conguration in LS coupling.

The jj coupling hierarchy

In heavy atoms relativistic effects become large and the spin-orbit coupling between li and si dominates:

In jj coupling, one obtains the possible terms by first adding vectorially the orbital angular momentum li and the electron spin si of each electron to form a resultantelectronic angular momentum ji for each electron. The total electronic angular momentum J results from the vectorial addition of all ji .

For a two electron system, one gets:

One can observe the evolution from LS coupling to jj coupling by looking at the evolution of the energy level structure associated with a given configuration as one moves down a column in the periodic table. The correlation diagram shows the evolution of the energy level structure associated with the excited (np)((n+1)s) configuration in C, Si, Ge, Sn and Pb (data taken from Ch. Moore's Tables, see literature list). The zero point of the energy scale has been placed at the center of gravity of the energy level structure.

Examples: electron configuration and term symbol

Term symbols of elements in the first three periods of the Periodic Table

The He atom: ‘‘spectral complexity‘‘

Helium - remarks

Helium exists as parahelium 1S0 and orthohelium 3S1 .

As there are two electrons in the helium atom, two spin states, singlet and triplet, are possible.

Spectral transitions between the two spin states are forbidden and only take place within one system having the same spin state.

The spectra of all two-electron systems such as helium, alkaline earth metalsor a hydrogen molecule display singlet and triplet series.

In optical spectroscopy and in the photochemistry and biology of larger molecules,the differential roles of the triplet and singlet states are of interest

Atoms and ions in astrophysics

Hyperfine coupling

The coupling of the total angular momentum without nuclear spin J with the nuclear spin I is typically much weaker than the spin-orbit coupling. It can be viewed as an interaction between the magnetic moment of the nuclear spin:

and the total magnetic moment of the electrons: