Download Atomic Term Symbols and Energy Splitting

Chemistry 362
Fall 2015
Dr. Jean M. Standard
November 4, 2015
Atomic Term Symbols and Energy Splitting
1. Atomic Term Symbols and the Sodium D-Line
The sodium D-line is responsible for the familiar orange glow of many street lights. The origin of the glow is
emission of photons in the visible region of the electromagnetic spectrum from excited sodium atoms. The excited
atoms emit light and return to their ground electronic states.
The sodium D-line gets its name because there are really two closely-spaced emissions possible, or a "doublet", as
shown in Figure 1. These transitions occur at wavelengths of 5890 and 5896 Å.
Na 3p1
λ=5896 Å
λ=5890 Å
E
Na 3s1
Figure 1. Sodium atom atomic emissions that produce the so-called D-line.
The doublet observed in the sodium D-line transition involves the outer electron in the sodium atom which
undergoes a transition from an excited 1s22s22p63p1 configuration to the ground state 1s22s22p63s1 configuration.
To see why this electronic transition corresponds to a doublet, the atomic term symbols for the different electronic
configurations must be determined. In both cases, only the outer open shell need be considered.
Sodium Atom Ground State 3s1 Term Symbol
Since the ground state of sodium only has one outer electron, the total orbital angular momentum quantum number L
and total spin angular momentum quantum number S are identical to the orbital and spin angular momentum
quantum numbers of the outer electron. Thus,
S = s1 =
1
2
and L = ℓ1 = 0.
The multiplicity 2S+1 is therefore 2 (a doublet) and the state corresponds to a 2 S state. Then, all that is needed is to
determine the total angular momentum
J.
€
€ quantum number
2
The total angular momentum quantum number J ranges from L − S €to L + S . For the S state, L=0 and S =
leads to J = 12 .
2
1 €
S1/ 2 .
Therefore, the only possible term symbol for the sodium
state is €
€ 3s ground
€
€
€
1
2
2
Sodium Atom Excited State 3p1 Term Symbol
The 3p1 excited state of sodium only has one outer electron, so the total orbital angular momentum quantum number
L and total spin angular momentum quantum number S are identical to the orbital and spin angular momentum
quantum numbers of the outer electron. Thus,
S = s1 =
1
2
and L = ℓ1 = 1.
2
The multiplicity 2S+1 is therefore 2 (a doublet) and the state corresponds to a P state. Then, all that is needed is to
determine the total angular momentum
J.
€
€ quantum number
2
The total angular momentum quantum number J ranges from L − S €to L + S . For the P state, L=1 and S =
leads to two possible values of J, J =
1
2
and J = 32 .
2
€ 2
1
Therefore, there are two possible term symbols for the
€ sodium€3p excited state: P1/ 2 and P3/€2 . Spin-orbit
coupling leads to energy€splitting between
these two terms.
€
€
€
Energy Level Diagram
The term symbols determined for the ground and excited states of sodium can be used to label the transitions
responsible for the sodium D-line emission, as shown in Figure 2.
Na 2P3/2
Na 2P1/2
λ=5896 Å
λ=5890 Å
E
Na 2S1/2
Figure 2. Atomic term symbols for transitions involved in the sodium D-line.
1
2
3
3. Another Example of Energy Splitting of Atomic Terms
Consider an example of an atomic electron configuration 1s12p1. There are 12 ways of choosing the individual
quantum numbers for the two electrons in this configuration. In the absence of electron-electron repulsions, all these
states are degenerate.
1
3
The possible term symbols for the 1s12p1 configuration are P and P (not including the J value). Hund's first rule
3
states that terms with higher multiplicity will be lower in energy. Thus, including electron-electron repulsion, P
1
will be lower in energy than P.
1
1
3
For the P term, the only possible value of J is 1; thus, the only term symbol for this state is P1. For the P term, the
3
3
3
1
possible values of J are 0, 1, and 2; this leads to term symbols P0, P1, and P2. The total degeneracy of the P and
3
P terms is 3, 1, 3, and 5, respectively, for a total of 12 (in agreement with the 12 sets of individual quantum
numbers discussed above).
3
3
3
The P0, P1, and P2 states are split in energy by a very small amount. This splitting is due to the coupling of spin
angular momentum (S) with total orbital angular momentum (L). This spin-orbit coupling splits levels within the
same term (that is, the same values of L and S) that have different values of J.
Finally, if the atom is placed in a magnetic field, the levels with the same values of L, S, and J, but with different
values of MJ are split. All of the energy splittings for the 1s12p1 electron configuration are summarized in Figure3.
no elec-elec
repulsion
elec-elec
repulsion
spin-orbit
coupling
magnetic
field
Figure 3. Energy splitting of atomic terms in the 1s12p1 configuration.