Terms and term symbol in atoms. Term symbol, Hund rules LS coupling hierarchy jj coupling hierarchy Hyperfine coupling

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1 Terms and term symbol in atoms Term symbol, Hund rules LS coupling hierarchy jj coupling hierarchy Hyperfine coupling

2 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.

3 Term symbol and Hund rules 2S+1 L J 2S+1----multiplicity L angular momentum J total angular momentum In Hunds rules for obtaining the ground state of an atom: 1. the state with the largest total angular momentum L 2. the largest total spin momentum S 3. 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

4 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:

5 Example: C (1s) 2 (2s) 2 (2p) 2. Only the partially filled 2p subshell needs to be considered. In this case l 1 = 1, l 2 = 1 and s 1 = s 2 = 1/2. Possible terms: Taking the (2J + 1) multiplicity of each term (which corresponds to all possible values of M J ), 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 M L, M S and M J values resulting from all 15 possible occupations of the six 2p spin-orbitals with two electrons compatible with the Pauli principle.

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7 The maximum value of M L is 2 and occurs in combination with M S = 0. This implies a 1 D term with five M J components corresponding to (M L M S ) = (2 0), (1 0), (0 0), (-1 0) and (-2 0). Eliminating these states, the one with the highest M L value has M L = 1 and comes in combination with a maximal M S value of 1. We can conclude that the term is 3 P (consisting of 3 P 0, 3 P 1 and 3 P 2 ). There are 9 components corresponding to (M L M S ) = (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 1 S 0 state. The terms corresponding to the (2p) 2 configuration allowed by the Pauli principle are therefore 1 D 2, 3 P 2, 3 P 1, 3 P 0 and 1 S 0. 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 structure of the (2p) 2 conguration in LS coupling.

8 The jj coupling hierarchy In heavy atoms relativistic effects become large and the spin-orbit coupling between l i and s i dominates: In jj coupling, one obtains the possible terms by first adding vectorially the orbital angular momentum l i and the electron spin s i of each electron to form a resultant electronic angular momentum j i for each electron. The total electronic angular momentum J results from the vectorial addition of all j i.

9 For a two electron system, one gets:

10 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.

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12 Examples: electron configuration and term symbol

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

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15 The He atom: spectral complexity

16 Helium - remarks Helium exists as parahelium 1 S 0 and orthohelium 3 S 1. 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 metals or 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

17 Atoms and ions in astrophysics

18 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:

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