Elements of Modern Physics
142
Since they all have a similar outer shell, their chemical properties also are
similar. The three series of elements are 21 ≤ Z ≤ 30 for n = 3, 39 ≤ Z ≤ 48 for
n = 4, and 71 ≤ Z ≤ 80 for n = 5. The partially filled inner 3d shell allows some
of these elements to have large magnetic moments. In particular, Fe, Co and Ni
are found to be ferromagnetic materials (see Sec. 8.6). (v) The rare-earths are
a group of 14 elements corresponding to a progressive filling of the 4f subshell,
though the 6s subshell is already complete. These elements have 57 ≤ Z ≤ 70.
In summary, the electronic structure of atoms provides an insight into their
properties.
5.4 ATOMIC SPECTRA
So far only the general features of the electronic structure of an atom have
been considered. These follow from the model in which only the interaction of
the electrons with the nucleus (which produces the single-electron levels with
l-degeneracy) and a term which represents an average interaction between the
electrons (which breaks the l-degeneracy) are included. In this model, the energy
of the atom is the sum of the energies of the electrons. Since the model ignores
the finer details of the interaction, the various energy levels are highly degenerate.
For example, a level with one electron in the 4p shell and another in the 3d shell
will have a degeneracy of 6 × 10 corresponding to a degeneracy of 2(2l + 1) for
each electron. This degeneracy is partially removed if the interaction between
the electrons and their spin-orbit interaction, i.e., the terms H 1 and H 2 in
Eq. (5.14) are included.
In this section, the effect of including the finer details of the mutual interaction
between the electrons (the term H 1 ) and the spin-orbit interaction (the term H 2 )
will be considered. This will allow the characterization of different energy levels
for a given shell configuration, in particular, the ground state. The purpose will
be essentially to enumerate the various energy levels, and to some extent predict
the ordering of the energy levels (the determination of the energies will require
a much more elaborate calculation). These results will be useful not only for
predicting the multiplicity of the spectral lines, but also for describing the behaviour
of the atoms under different physical conditions, e.g. in the presence of an
external magnetic field.
Once the mutual interaction and spin-orbit interaction terms are included,
the total orbital angular momentum is no longer conserved, nor is the total spin
angular momentum. However, since the interaction is a scalar, the total angular
momentum J = L + S is still conserved (i.e. [L, H] ≠ 0, [S, H] ≠ 0, but
[J, H] = 0), and the energy eigenstates can be designated by their total angular
momentum. We will now use l, s and j to designate the individual electron
angular momenta and L, S and J to designate the sums of the angular momenta.
The problem of determining the multiplicity of energy levels corresponding to a
142
Since they all have a similar outer shell, their chemical properties also are
similar. The three series of elements are 21 ≤ Z ≤ 30 for n = 3, 39 ≤ Z ≤ 48 for
n = 4, and 71 ≤ Z ≤ 80 for n = 5. The partially filled inner 3d shell allows some
of these elements to have large magnetic moments. In particular, Fe, Co and Ni
are found to be ferromagnetic materials (see Sec. 8.6). (v) The rare-earths are
a group of 14 elements corresponding to a progressive filling of the 4f subshell,
though the 6s subshell is already complete. These elements have 57 ≤ Z ≤ 70.
In summary, the electronic structure of atoms provides an insight into their
properties.
5.4 ATOMIC SPECTRA
So far only the general features of the electronic structure of an atom have
been considered. These follow from the model in which only the interaction of
the electrons with the nucleus (which produces the single-electron levels with
l-degeneracy) and a term which represents an average interaction between the
electrons (which breaks the l-degeneracy) are included. In this model, the energy
of the atom is the sum of the energies of the electrons. Since the model ignores
the finer details of the interaction, the various energy levels are highly degenerate.
For example, a level with one electron in the 4p shell and another in the 3d shell
will have a degeneracy of 6 × 10 corresponding to a degeneracy of 2(2l + 1) for
each electron. This degeneracy is partially removed if the interaction between
the electrons and their spin-orbit interaction, i.e., the terms H 1 and H 2 in
Eq. (5.14) are included.
In this section, the effect of including the finer details of the mutual interaction
between the electrons (the term H 1 ) and the spin-orbit interaction (the term H 2 )
will be considered. This will allow the characterization of different energy levels
for a given shell configuration, in particular, the ground state. The purpose will
be essentially to enumerate the various energy levels, and to some extent predict
the ordering of the energy levels (the determination of the energies will require
a much more elaborate calculation). These results will be useful not only for
predicting the multiplicity of the spectral lines, but also for describing the behaviour
of the atoms under different physical conditions, e.g. in the presence of an
external magnetic field.
Once the mutual interaction and spin-orbit interaction terms are included,
the total orbital angular momentum is no longer conserved, nor is the total spin
angular momentum. However, since the interaction is a scalar, the total angular
momentum J = L + S is still conserved (i.e. [L, H] ≠ 0, [S, H] ≠ 0, but
[J, H] = 0), and the energy eigenstates can be designated by their total angular
momentum. We will now use l, s and j to designate the individual electron
angular momenta and L, S and J to designate the sums of the angular momenta.
The problem of determining the multiplicity of energy levels corresponding to a