L-S coupling is a quantum mechanical scheme in multi-electron atoms where individual orbital angular momenta (l<sub>i</sub>) couple together to form a resultant total orbital angular momentum (L), and individual spin angular momenta (s<sub>i</sub>) couple to form a resultant total spin angular momentum (S). These resultant vectors then interact via spin-orbit coupling to yield the total angular momentum (J).
Basic understanding of quantum numbers n, l, m<sub>l</sub>, m<sub>s</sub>, vector atom model representation, and total angular momentum quantum number J in multi-electron systems.
Detailed derivation of vector addition for L-S coupling, determination of term symbols (<sup>2S+1</sup>L<sub>J</sub>), Hund's rules for ground state determination, and application to multi-electron atomic spectra.
Conditions: Assuming vectors L and S precess rapidly about their resultant J due to spin-orbit interaction.
Result: J(J+1) = L(L+1) + S(S+1) + 2(L\cdot S)
Conditions: First-order perturbation theory applied to spin-orbit interaction energy Eso.
Result: \Delta $E_{so} = \frac{1}{2}\zeta_{LS}[J(J+1) - L(L+1) - S(S+1)]$
| Formula / law | Equation | Variables and units | Conditions |
|---|---|---|---|
| Total Orbital Angular Momentum Magnitude | $$|L| = \sqrt{L(L+1)}\hbar$$ | L = total orbital quantum number J s | Valid in quantum mechanics |
| Total Spin Angular Momentum Magnitude | $$|S| = \sqrt{S(S+1)}\hbar$$ | S = total spin quantum number J s | Valid in quantum mechanics |
| Total Angular Momentum Magnitude | $$|J| = \sqrt{J(J+1)}\hbar$$ | J = total angular momentum quantum number J s | L-S coupling regime |
| Range of Total Angular Momentum Quantum Number J | $$J = |L - S|, |L - S| + 1, \dots, L + S$$ | L, S = total orbital and spin quantum numbers Dimensionless | Vector triangle rule |
| Spin Multiplicity | $$Multiplicity = 2S + 1$$ | S = total spin quantum number Dimensionless | Number of possible orientations of S relative to L |
| Spin-Orbit Interaction Energy Shift | $$E_{so} = A_{LS} \frac{J(J+1) - L(L+1) - S(S+1)}{2}$$ | $A_{LS} = coupling constant$, J, L, S = quantum numbers Joules (J) or eV | Russell-Saunders coupling |
| Landé g-factor for L-S Coupling | $$g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$$ | g = Landé splitting factor, J, S, L = quantum numbers Dimensionless | Weak magnetic field (Zeeman effect) |
| Number of Valence Electrons Spin Sum | $$S = \sum s_i$$ | $s_i = spin quantum number of individual electrons$ Dimensionless | Vector addition of spins |
Draw: Vectors L and S precessing rapidly around their resultant vector J, while J itself precesses slowly around an external axis (or in space).
Interpret: Illustrates conservation of angular momentum components and the hierarchical coupling strength where L and S couple first before interacting with J.
Exam use: Frequently asked in conceptual multiple-choice questions regarding vector orientations and precession cones.
Draw: A beam of silver atoms passing through an inhomogeneous magnetic field splitting into two distinct traces on a detector plate.
Interpret: Proves space quantization of angular momentum and existence of electron spin ($s = 1/2$, giving 2s+1 = 2 components).
Exam use: Directly tested as foundational background for understanding vector atom models and angular momentum coupling.
Draw: Energy levels of a term symbol splitting into multiple J-levels under spin-orbit interaction (e.g., 2P splitting into 2P1/2 and 2P3/2).
Interpret: Demonstrates the Landé interval rule where the separation between adjacent J levels is proportional to the larger J value.
Exam use: Used in numerical and theoretical questions regarding fine structure splitting in atomic spectra.
Answer: J = 1, 2, 3
Answer: 3P0
Answer: $g = 4/3 (or 1.33)$
Common error: L-S coupling means individual electron spins couple directly to individual orbital angular momenta of other electrons.
Correct: Individual orbital momenta couple to form L, and individual spins couple to form S; L and S then couple together.
Russell-Saunders coupling assumes intra-atomic electrostatic forces group all li together and all si together before spin-orbit interaction takes place.
Common error: The term symbol 2S+1LJ has J as the principal quantum number.
Correct: J represents the total angular momentum quantum number of the atom.
J is the vector sum of L and S, completely distinct from the principal quantum number n.
| Feature | Shells Less Than Half-Filled | Shells More Than Half-Filled |
|---|---|---|
| Lowest Energy J Value | $J = |L - S|$ | $J = L + S$ |
| Physical Reason | Antiparallel coupling minimizes energy for sparse states | Parallel coupling minimizes energy for dense states |
Choose an option to reveal the explanation.
Explanation: L-S coupling is valid primarily for light and medium-weight atoms where electrostatic Coulomb repulsion between electrons is much stronger than the magnetic spin-orbit interaction.
Explanation: Individual orbital angular momenta ($l_i$) vector-add to form a total orbital angular momentum vector $L$.
Explanation: The magnitude of total angular momentum is given by $|J| = \sqrt{J(J+1)}\hbar$, where $J$ is the total angular momentum quantum number.
Explanation: The range of $J$ is from $|L - S|$ to $L + S$ in integer steps. Here, $|2 - 1| = 1$ up to $2 + 1 = 3$, giving $J = 1, 2, 3$.
Explanation: The term $^{2S+1}L_J$ uses $2S+1$ to represent the spin multiplicity of the atomic state.
Explanation: Letters $S, P, D, F, G\dots$ correspond to total orbital quantum numbers $L = 0, 1, 2, 3, 4\dots$ respectively. Thus, $L = 2$ corresponds to $D$.
Explanation: Hund's Rule 1 states that for a given electron configuration, the term with the maximum multiplicity $(2S+1)$ lies lowest in energy due to exchange energy stabilization.
Explanation: For shells less than half-filled, the level with the minimum $J$ value ($J = |L - S|$) has the lowest energy; for shells more than half-filled, the level with maximum $J$ value ($J = L + S$) has the lowest energy.
Explanation: The Landé splitting factor is given by $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$.
Explanation: Spin multiplicity is given by $2S + 1$. For $S = \frac{3}{2}$, multiplicity $= 2(\frac{3}{2}) + 1 = 4$ (quartet state).
Explanation: It is incorrect to think individual spins couple directly to individual orbital momenta of other electrons; rather, all $l_i$ couple to form $L$ and all $s_i$ couple to form $S$ first.
Explanation: For one electron, $L = l = 1$ (P state) and $S = s = 1/2$, giving multiplicity $2S+1 = 2$. $J$ can be $|1 - 1/2| = 1/2$ and $1 + 1/2 = 3/2$. Thus, $^2P_{1/2}$ and $^2P_{3/2}$.
Explanation: The resultant orbital vector $L$ and resultant spin vector $S$ interact via spin-orbit interaction to form total angular momentum $J$, causing fine structure splitting.
Explanation: Hund's Rule 2 states that for a given multiplicity, the term with the largest value of total orbital angular momentum $L$ lies lowest in energy.
Explanation: Two electron spins vector-add to give total spin $S = |s_1 - s_2|$ to $s_1 + s_2$, which is $0$ (singlet) and $1$ (triplet).
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Explanation: For $p^2$, possible terms are $^3P$, $^1D$, $^1S$. By Hund's Rule 1, max multiplicity is $3$ ($^3P$). For $p^2$ (less than half-filled), Rule 3 gives the lowest $J$ value $J = |L - S| = |1 - 1| = 0$. Thus, $^3P_0$.
Explanation: The maximum value of $J$ is given by $L + S = 2 + \frac{3}{2} = \frac{7}{2}$.
Explanation: For a single $s$ electron ($l = 0$), $L = 0$ (which corresponds to symbol $S$), $S = 1/2$ (multiplicity $2S+1 = 2$), and $J = 0 + 1/2 = 1/2$. Thus, $^2S_{1/2}$.
Explanation: Using $g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$, substituting $L=0, S=1/2, J=1/2$: $g = 1 + \frac{(3/4) + (3/4) - 0}{2(3/4)} = 1 + \frac{3/2}{3/2} = 2$.
Explanation: For shells more than half-filled, the spin-orbit interaction causes the level with the maximum $J$ value ($J = L + S$) to lie lowest in energy.
Explanation: Three unpaired electrons give total spin $S = 3 \times (1/2) = 3/2$. The spin multiplicity is $2S + 1 = 2(3/2) + 1 = 4$ (quartet).
Attempt these questions under a ten-minute limit using a separate timer.
Explanation: L-S coupling is valid primarily for light and medium-weight atoms where electrostatic Coulomb repulsion dominates over spin-orbit interactions.
Explanation: For two $p$ electrons ($l_1 = 1, l_2 = 1$), the possible values of $L$ are $2, 1, 0$. The maximum value of $L$ is $2$ (corresponding to a $D$ term).
Explanation: The spin-orbit interaction energy shift is given by $E_{so} = A_{LS} \frac{J(J+1) - L(L+1) - S(S+1)}{2}$.
Explanation: Multiplicity $2S+1 = 4 \Rightarrow S = 3/2$. Letter $D \Rightarrow L = 2$. Subscript gives $J = 7/2$. (Note: $L+S = 2 + 3/2 = 7/2$, which matches).
Explanation: For heavy atoms, individual spin-orbit interactions ($l_i$ with $s_i$ for each electron) become stronger than inter-electronic electrostatic repulsion, leading to $j-j$ coupling.
Explanation: For $p^2$, number of ways to put 2 electrons in 6 states is $\binom{6}{2} = \frac{6 \times 5}{2} = 15$ microstates.
Explanation: For $d^1$, $l = 2 (D), s = 1/2 (multiplicity = 2)$. Shell is less than half-filled ($1 < 5$), so $J = |L - S| = |2 - 1/2| = 3/2$. Thus, $^2D_{3/2}$.
Explanation: The total angular momentum quantum number $J$ remains a good quantum number in both coupling schemes because total angular momentum is always conserved in a rotationally invariant isolated atom.
Explanation: Multiplicity is $2(1)+1 = 3$ (triplet). $L = 1$ corresponds to $P$. $J$ ranges from $|1-1|=0$ to $1+1=2$, giving $J = 0, 1, 2$. Thus, $^3P_0, ^3P_1, ^3P_2$.
Explanation: In L-S coupling, the vectors $L$ and $S$ vector-add to form $J$ and precess rapidly about the direction of the total angular momentum vector $J$ due to the internal spin-orbit torque.