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7.17: How to recognize LS-coupling

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LS-coupling is in practice a good approximation in light atoms, but there are appreciable departures from LS-coupling in the heavier atoms. Generally the several lines in a multiplet in LS-coupling are fairly close together in wavelength for LS-coupling, but, as departures from LS-coupling become more pronounced, the lines in a multiplet may become more widely separated and may appear in quite different parts of the spectrum.

In LS-coupling, multiplets always connect terms with the same value of S. Thus, while ^3 \text{D} − \ ^3 \text{P} would be "allowed" for LS-coupling, ^3 \text{D} − \ ^1 \text{P} would not. \Delta S = 0 is a necessary condition for LS-coupling, but is not a sufficient condition. Thus while a multiplet with \Delta S \neq 0 certainly indicates departure from LS-coupling, \Delta S = 0 by no means guarantees that you have LS-coupling. In spectroscopy, the term "forbidden" generally refers to transitions that are forbidden to electric dipole radiation. Transitions that are forbidden merely to LS-coupling are usually referred to as "semi-forbidden", or as "intersystem" or "intercombination" transitions. We shall have more on selection rules in section 7.24.

The energies, or term values, of the levels (each defined by LSJ) within a term are given, for LS-coupling, by a simple formula:

T = \frac{1}{2} a [ J(J+1) - L(L+1) - S(S+1)]. \label{7.17.1} \tag{7.17.1}

Here a is the spin-orbit coupling coefficient, whose value depends on the electron configuration. What is the separation in term values between two adjacent levels, say between level J and J −1? Evidently (if you apply equation \ref{7.17.1}) it is just aJ. Hence Landé's Interval Rule, which is a good test for LS-coupling: The separation between two adjacent levels within a term is proportional to the larger of the two J-values involved. For example, in the KL3s (^2 S) 3 p^3 P^{\text{o}} term of \text{Mg} \ _\text{I} (the first excited term above the ground term), the separation between the J = 2 and J = 1 levels is 4.07 \ \text{mm}^{-1}, while the separation between J = 1 and J = 0 is 2.01 \ \text{mm}^{-1}. Landé's rule is approximately satisfied, showing that the term conforms closely, but not exactly, to LS-coupling. It is true that for doublet terms (and all the terms in \text{Na} \ _\text{I} and \text{K} \ _\text{I} for example, are doublets) this is not of much help, since there is only one interval. There are, however, other indications. For example, the value of the spin-orbit coupling coefficient can be calculated from LS-theory, though I do not do that here. Further, the relative intensities of the several lines within a multiplet (or indeed of multiplets within a polyad) can be predicted from LS-theory and compared with what is actually observed. We discuss intensities in a later chapter.

The spin-orbit coupling coefficient a can be positive or negative. If it is positive, the level within a term with the largest J lies highest; such a term is called a normal term, though terms with negative a are in fact just as common as "normal" terms. If a is negative, the level with largest J lies lowest, and the term is called an inverted term. Within a shell (such as the L-shell) all the s electrons may be referred to as a subshell, and all the p electrons are another subshell. The subshell of s electrons can hold at most two electrons; the subshell of p electrons can hold at most six electrons. If the outermost subshell (i.e. the electrons responsible for the optical spectrum) is less than half full, a is positive and the terms are normal. If it is more than half full, a is negative and the terms are inverted. If the subshell is exactly half full, a is small, the term is compact and may be either normal or inverted. For example in \text{Al} \ _\text{I}, the term 3 p^2 \ ^4\text{P} (which has three levels - write down their J-values) is normal. There are only two p electrons out of six allowed in that subshell, so the subshell is less than half full. The term 2s 2 p^4 \ ^4\text{P} of \text{O} \ _\text{II} has four p electrons, so the subshell is more than half full, and the term is inverted. The term 2s^2 2 p^3 \ ^2\text{P}^{\text{o}} of the same atom has a subshell that is exactly half full. The term happens to be normal, but the two levels are separated by only 0.15 \ \text{mm}^{-1}, which is relatively quite tiny.


This page titled 7.17: How to recognize LS-coupling is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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