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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.