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# 7.17: How to recognize LS-coupling

$$LS$$-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.