11.2.2: Spin-Orbit Coupling
- Page ID
- 352416
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The magnetic fields created by \(S\) and \(L\) are not isolated from one another; they interact through spin-orbit coupling (aka Russell-Saunders coupling). We will consider only this simple form of coupling in this text. Its application is limited to the elements with \(z<40\), including the first row of the transition elements. In the case of heavier elements, we must also consider \(jj\) coupling; however, we will not discuss the latter here.
In the Russell-Saunders spin-orbit coupling scheme, the interaction between \(S\) and \(L\) is expressed by an additional quantum number, the total angular momentum quantum number (\(J\)). The possible values of \(J\) are values between \(L+S\) and \(|L-S|\).
\[J = L+S, \; L+S-1, \; L+S-2, \; ..., \; |L-S|\]
A value of \(J\) must be positive or zero for a multielectron system. J values can fall into series \(\frac{1}{2}, \frac{3}{2}, \frac{5}{2},...\) or \(0, 1, 2,...\). The quantum number \(J\) is added to the term symbol as a subscript to the right of the letter describing the term. A full term symbol is as follows:
\[^{(2S+1)}L_J\]The result of spin-orbit coupling is that a term for the free ion is split into states of different energies. For example, a \(^3P\) state of a carbon atom with a \(p^2\) electron configuration would be split into three different energy states (according to the three possible J values 0, 1, and 2): \(^3P_0, ^3P_1, ^3P_2\).
The relative energies of the states can be predicted from Hund's Third Rule.
Hund's third Rule:
- For subshells that are less than half-filled, the lowest energy state has the lowest \(J\) value.
- For subshells that are exactly half-filled, there is only one J value, thus it is the lowest energy.
- For subshells that are more than half-filled, the lowest energy state has the largest \(J\) value.
Thus, in this case where the \(p\) subshell is less than half full, the lowest energy state from the \(^3P\) free ion term would be that with \(J=0\), \(^3P_0\), followed by \(J=1\) and \(J=2\). The splitting and relative energies are depicted in Figure \(\PageIndex{1}\). Spin-orbit coupling and the splitting of the free ion terms have important implications for electronic spectra because they affect the energies of electronic transitions.
