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.