Atomic Term Symbols and Energy Splitting. λ=5890 Å

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1 Chemistry 362 Spring 2018 Dr. Jean M. Standard April 18, 2018 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 3p 1 λ=5896 Å λ=5890 Å E Na 3s 1 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 1s 2 2s 2 2p 6 3p 1 configuration to the ground state 1s 2 2s 2 2p 6 3s 1 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 3s 1 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 = s 1 = 1 2 and L = l 1 = 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 quantum number J. The total angular momentum quantum number J ranges from L S to L + S. For the 2 S state, L=0 and S = 1 2 leads to J = 1 2. Therefore, the only possible term symbol for the sodium 3s 1 ground state is 2 S 1/ 2.

2 2 Sodium Atom Excited State 3p 1 Term Symbol The 3p 1 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 = s 1 = 1 2 and L = l 1 = 1. The multiplicity 2S+1 is therefore 2 (a doublet) and the state corresponds to a 2 P state. Then, all that is needed is to determine the total angular momentum quantum number J. The total angular momentum quantum number J ranges from L S to L + S. For the 2 P state, L=1 and S = 1 2 leads to two possible values of J, J = 1 2 and J = 3 2. Therefore, there are two possible term symbols for the sodium 3p 1 excited state: 2 P 1/ 2 and 2 P 3/ 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 2 P 3/2 Na 2 P 1/2 λ=5896 Å λ=5890 Å E Na 2 S 1/2 Figure 2. Atomic term symbols for transitions involved in the sodium D-line.

3 3 2. Another Example: Excited 1s 1 2p 1 Configuration Consider an example of an atomic electron configuration 1s 1 2p 1. 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. The possible term symbols for the 1s 1 2p 1 configuration are 1 P and 3 P (not including the J value). Hund's first rule states that terms with higher multiplicity will be lower in energy. Thus, including electron-electron repulsion, 3 P will be lower in energy than 1 P. For the 1 P term, the only possible value of J is 1; thus, the only term symbol for this state is 1 P 1. For the 3 P term, the possible values of J are 0, 1, and 2; this leads to term symbols 3 P 0, 3 P 1, and 3 P 2. The total degeneracy of the 1 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). The 3 P 0, 3 P 1, and 3 P 2 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 M J are split. All of the energy splittings for the 1s 1 2p 1 electron configuration are summarized in Figure 3. Note that in this case, because this configuration corresponds to an excited state, the spin-orbit splitting of the J levels does not follow Hund's Rules. no elec-elec elec-elec spin-orbit magnetic repulsion repulsion coupling field Figure 3. Energy splitting of atomic terms in the 1s 1 2p 1 configuration.

4 4 3. Atomic Terms for Partially Filled Subshells We have seen previously how to determine atomic terms and term symbols for electron configurations with one outermost electron (such as s 1 or p 1 ), as well as for electron configurations with two outermost electrons in different subshells (such as 1s 1 2p 1 or 2p 1 3p 1 ). For systems in which two or more electrons occupy the same partially filled subshell, atomic term symbols may be constructed via vector addition as previously described. The process must be carried out by combining the angular momentum of two electrons at a time, so it is rather tedious. However, some of the term symbols obtained for these electron configurations through the vector addition process must be excluded due to violation of the Pauli Principle. The remaining allowed atomic terms for these electron configurations are presented in the table below. Electron Configuration s 1 Atomic Terms 2 S p 1, p 5 p 2, p 4 2 P 1 S, 1 D, 3 P p 3 2 P, 2 D, 4 S d 1, d 9 d 2, d 8 d 3, d 7 d 4, d 6 2 D 1 S, 1 D, 1 G, 3 P, 3 F 4 F, 4 P, 2 H, 2 G, 2 F, 2 D, 2 P 5 D, 3 H, 3 G, 3 F, 3 D, 3 P, 1 I, 1 G, 1 F, 1 D, 1 S d 5 6 S, 4 G, 4 F, 4 D, 4 P, 2 I, 2 H, 2 G, 2 F, 2 D, 2 P, 2 S An example of an energy level diagram for a p 2 electron configuration, corresponding to atomic carbon, is shown on the next page.

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