Elements of Modern Physics
144
exchange of spatial coordinates. There is a tendency for the electrons to avoid
each oher, thus minimizing the mutual repulsion. As a result, the lowest energy
state is generally the state with the largest total spin S. Extending these qualitative
results, it is deduced that as the total spin, decreases, the energy of the states
increases. Similar arguments can be used to determine the ordering of the
L levels for a given S. Of the states with different L values, the state with the
largest L will correspond to electrons moving in the same sense, i.e. they can
avoid coming together. Therefore, the ground state has the largest possible
L value compatible with the largest possible S value. This is known as Hund’s
rule. In addition, the energy of the states for a given S increases as L decreases.
Finally, the degeneracy of the states with different J values but the same L and
S values is removed by the spin-orbit interaction, i.e. H 2 , whose effect can be
represented by (this can be rigorously justified by using the theorem in Sec. 6.2),
H 2 = C LS L . S
(5.25)
where C LS is constant for a given L and S. This term gives a perturbative
contribution to the energy,
∆E J =
1 [ (
1)
(
1)
(
1)]
2
C J J
L L
S S
+ −
+ −
+
(5.26)
where the subscripts of C have dropped. It is found that the constant C is
positive for multiplets formed from a subshell that is half or less than half-filled,
and negative for multiplets formed from a subshell which is more than
half-filled. Therefore, for a subshell which is half-filled or less, the energy within
the multiplet increases as J increases, and for a subshell which is more than
half-filled, the energy within the multiplet decreases as J increases. This is
known as the multiplet rule. In partiular, this means that the ground state of an
atom has the smallest J value subject to Hund’s rule if the subshell is half-filled
or less, and the largest J value if the subshell is more than half-filled. The
separation between the levels with values J + 1 and J (but the same L and S) is
obtained as
E J + 1 – E J = C (J + 1)
(5.27)
This is known as the Lande interval rule which states that the spacing
between consecutive levels of a fine-structure multiplet is proportional to
the larger of the two J values of the levels. These ideas are made explicit by
the following two examples.
Consider an atom with two valence electrons, one in the ns state and the
second in the n′ l state where l ≠ 0. This state has a total degeneracy of
4(2l + 1) corresponding to two states for the s electron and 2(2l + 1) states for
the l electron. The total wave function is a product of the spatial part
u n,n′,L (r 1 , r 2 ) = R n (y 1 ) Y 0
0
(θ 1 , φ 1 ) R n ′ (r 2 ) Y l
m
(θ 2 , φ 2 )
(5.28)
and the spin part
144
exchange of spatial coordinates. There is a tendency for the electrons to avoid
each oher, thus minimizing the mutual repulsion. As a result, the lowest energy
state is generally the state with the largest total spin S. Extending these qualitative
results, it is deduced that as the total spin, decreases, the energy of the states
increases. Similar arguments can be used to determine the ordering of the
L levels for a given S. Of the states with different L values, the state with the
largest L will correspond to electrons moving in the same sense, i.e. they can
avoid coming together. Therefore, the ground state has the largest possible
L value compatible with the largest possible S value. This is known as Hund’s
rule. In addition, the energy of the states for a given S increases as L decreases.
Finally, the degeneracy of the states with different J values but the same L and
S values is removed by the spin-orbit interaction, i.e. H 2 , whose effect can be
represented by (this can be rigorously justified by using the theorem in Sec. 6.2),
H 2 = C LS L . S
(5.25)
where C LS is constant for a given L and S. This term gives a perturbative
contribution to the energy,
∆E J =
1 [ (
1)
(
1)
(
1)]
2
C J J
L L
S S
+ −
+ −
+
(5.26)
where the subscripts of C have dropped. It is found that the constant C is
positive for multiplets formed from a subshell that is half or less than half-filled,
and negative for multiplets formed from a subshell which is more than
half-filled. Therefore, for a subshell which is half-filled or less, the energy within
the multiplet increases as J increases, and for a subshell which is more than
half-filled, the energy within the multiplet decreases as J increases. This is
known as the multiplet rule. In partiular, this means that the ground state of an
atom has the smallest J value subject to Hund’s rule if the subshell is half-filled
or less, and the largest J value if the subshell is more than half-filled. The
separation between the levels with values J + 1 and J (but the same L and S) is
obtained as
E J + 1 – E J = C (J + 1)
(5.27)
This is known as the Lande interval rule which states that the spacing
between consecutive levels of a fine-structure multiplet is proportional to
the larger of the two J values of the levels. These ideas are made explicit by
the following two examples.
Consider an atom with two valence electrons, one in the ns state and the
second in the n′ l state where l ≠ 0. This state has a total degeneracy of
4(2l + 1) corresponding to two states for the s electron and 2(2l + 1) states for
the l electron. The total wave function is a product of the spatial part
u n,n′,L (r 1 , r 2 ) = R n (y 1 ) Y 0
0
(θ 1 , φ 1 ) R n ′ (r 2 ) Y l
m
(θ 2 , φ 2 )
(5.28)
and the spin part