Multi-electron atomic structure and LS coupling

Intuition Beginner

An atom with many electrons looks hopeless at first: every electron pushes on every other electron, so no single one moves in a simple way. The trick is to pretend, as a first pass, that each electron feels only an average pull. The nucleus attracts it, and the cloud of all the other electrons partly cancels that attraction. The leftover is a smooth inward pull that depends only on how far out the electron sits. Each electron then settles into its own orbit-shape, just like the single electron in hydrogen, and we get the familiar shell picture $1s$, $2s$, $2p$, and so on.

Listing which orbits are filled is the electron configuration, written like $1s^{2}2s^{2}2p^2$ for carbon. But the configuration is not the whole story. Within a partly filled shell the electrons can arrange their spins and their orbital motions in several ways, and these arrangements do not all cost the same energy. Some let the electrons stay farther apart, which lowers the repulsion. Nature picks the cheapest arrangement for the ground state.

Why does this matter? The pattern of cheap and expensive arrangements is exactly what the periodic table encodes. The same average-pull idea, plus a few simple rules about which arrangement wins, predicts why carbon, nitrogen, and oxygen behave the way they do, and it explains the fine splitting of the colors atoms emit.

Visual Beginner

Picture the electrons of an atom as stacked into shells around the nucleus, like seats in a theater arranged in rings. The inner rings fill first. When you reach a partly filled ring, the electrons still have choices: which way each one spins, and which seat it takes. The left side of the picture shows the electrons with spins all pointing the same way, forced by the rules to sit in different seats and so kept apart. The right side shows two electrons crowded with opposite spins, sitting closer and repelling more.

The picture captures the core lesson. The arrangement that spreads the electrons out, with spins aligned, is the cheaper one. Counting how the spins and the orbital motions add up gives each arrangement a short label, and the cheapest label is the ground state of the atom.

Worked example Beginner

Take carbon, configuration $1s^{2}2s^{2}2p^2$. The filled inner parts contribute nothing to the choices; only the two $p$ electrons matter. A $p$ shell has three orbital slots, and each slot can hold one spin-up and one spin-down electron, so six places in all. We place two electrons and ask for the cheapest arrangement.

Step 1. Spread the two electrons over different orbital slots rather than doubling up. Two electrons in the same slot must have opposite spins and sit close, raising the repulsion. Separate slots let them avoid each other.

Step 2. Align the two spins. With both spins up, the electrons are forced by the sign rule from the helium unit to keep apart, lowering the repulsion further.

Step 3. Add up the spins. Two aligned spins give a total spin of $1$ (a "triplet," counting $2\times 1+1=3$ spin orientations).

Step 4. Add up the orbital motions. With the two electrons in two of the three $p$-slots, the cheapest total orbital value works out to $1$ (a "P" state).

Putting the labels together, carbon's ground arrangement is written ${}^{3}P$. What this tells us: by only spreading electrons out and aligning spins, we predicted carbon's ground state without solving any hard equation. The same two moves fix the ground state of every atom.

Check your understanding Beginner

Formal definition Intermediate+

In the central-field approximation, the $N$-electron Hamiltonian is split as $$ \hat H = \sum_{i=1}^N \Big[ -\tfrac12 \nabla_i^2 + U(r_i) \Big] ;+; \Big[ \sum_{i<j} \frac{1}{r_{ij}} - \sum_i \big(U(r_i) + \tfrac{Z}{r_i}\big) \Big], $$ in atomic units, where $U(r)$ is a spherically symmetric one-electron potential chosen to absorb as much of the electron-electron repulsion as possible. The first bracket ${\hat{H}}_0$ has product eigenstates built from single-particle spin-orbitals ${\psi }_{n\ell m_{\ell }{m}_s}$; the second bracket is the residual electrostatic interaction ${\hat{H}}_1$, treated as a perturbation. An electron configuration records the occupation numbers of the $(n,\ell )$ shells, for example $1s^{2}2s^{2}2p^2$. Because $U$ is central, the single-particle orbital angular momentum ${\hat{\mathbf{l}}}_i$ and spin ${\hat{\mathbf{s}}}_i$ are good for each electron at zeroth order, and the configuration is degenerate over the magnetic and spin sublevels of the open shells.

The residual interaction ${\hat{H}}_1$ is a scalar under simultaneous rotation of all electron coordinates and under simultaneous rotation of all spins (the symbol $\sum$ denotes the sum over electrons, ${\nabla }_i$ the gradient in electron $i$'s coordinate, and $r_{ij}=|{\mathbf{r}}_i-{\mathbf{r}}_j|$). It therefore commutes with the total orbital and total spin operators $$ \hat{\mathbf L} = \sum_{i=1}^N \hat{\mathbf l}i, \qquad \hat{\mathbf S} = \sum{i=1}^N \hat{\mathbf s}_i, $$ and with ${\hat{\mathbf{L}}}^2$, ${\hat{\mathbf{S}}}^2$. In the Russell-Saunders (LS) coupling scheme one diagonalises ${\hat{H}}_1$ within a configuration by labelling states with the eigenvalues $L(L+1)$ and $S(S+1)$. A term is the set of states of fixed $(L,S)$; it is written as the term symbol $$ {}^{2S+1}L_J, $$ with $L=0,1,2,3,\dots$ spelled $S,P,D,F,\dots$, the left superscript $2S+1$ the spin multiplicity, and the subscript $J$ the eigenvalue of the total angular momentum $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$, fixed only once the smaller spin-orbit interaction is added. For a given term $J$ ranges over $|L-S|,\dots ,L+S$.

The allowed terms of an open shell are constrained by the Pauli principle: the full $N$-electron state is an antisymmetric combination of spin-orbitals, so not every $(L,S)$ permitted by naive angular-momentum addition survives. The legal terms are the joint $L$-$S$ content of the antisymmetric subspace ${\Lambda }^k$ of the $2(2\ell +1)$-dimensional open shell.

Counterexamples to common slips

• LS coupling is not the statement that each electron has a definite $J$. It is the opposite: the individual ${\hat{\mathbf{l}}}_i$ first couple to a total $\hat{\mathbf{L}}$ and the ${\hat{\mathbf{s}}}_i$ to a total $\hat{\mathbf{S}}$, and only then $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ couple to $\hat{\mathbf{J}}$. Coupling each ${\hat{\mathbf{l}}}_i$ to its own ${\hat{\mathbf{s}}}_i$ first is the opposite (jj) scheme, valid for heavy atoms.
• The Pauli principle is not optional bookkeeping. For two equivalent $p$ electrons naive addition of two $\ell =1$ orbital momenta and two spins gives nine $(L,S)$ combinations; antisymmetry cuts these to the three terms ${}^{1}S$, ${}^{1}D$, ${}^{3}P$ ($1+5+9=15$ states, matching $\left(\genfrac{}{}{0ex}{}{6}{2}\right)$). The forbidden combinations are exactly those that would require two electrons in the same spin-orbital.
• The multiplicity $2S+1$ counts spin orientations only when $S\le L$; otherwise the number of $J$ values is $2L+1$, not $2S+1$. The name "triplet, doublet" still refers to $2S+1$, but the level count of a term is $\min (2S+1,2L+1)$.

Key derivation Intermediate+

Theorem (allowed terms of the $p^2$ configuration and Hund's ground term). Two equivalent $p$ electrons (${\ell }_1={\ell }_2=1$) admit exactly the Russell-Saunders terms ${}^{1}S$, ${}^{1}D$, ${}^{3}P$, accounting for all $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ antisymmetric two-electron states. Among them the term of lowest energy is ${}^{3}P$.

Proof. Each $p$ electron occupies one of six spin-orbitals labelled by $(m_{\ell },m_s)$ with $m_{\ell }\in \{-1,0,1\}$ and $m_s\in \{+\frac{1}{2},-\frac{1}{2}\}$. Two electrons in equivalent orbitals form an antisymmetric pair, so the available states are the $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ unordered pairs of distinct spin-orbitals. Classify them by $M_L=m_{\ell ,1}+m_{\ell ,2}$ and $M_S=m_{s,1}+m_{s,2}$. Tabulating the multiplicities of $(M_L,M_S)$ over the fifteen Pauli-allowed pairs gives the array $$ \begin{array}{c|ccc} M_L \backslash M_S & -1 & 0 & +1 \\hline +2 & 0 & 1 & 0 \ +1 & 1 & 2 & 1 \ 0 & 1 & 3 & 1 \ -1 & 1 & 2 & 1 \ -2 & 0 & 1 & 0 \end{array} $$ Peel off the terms by their largest $(M_L,M_S)$ corners. The entry at $(M_L,M_S)=(+1,+1)$ forces a multiplet with $L=1,S=1$: this is ${}^{3}P$, occupying nine states $M_L\in \{-1,0,1\}$, $M_S\in \{-1,0,1\}$. Remove them. The largest remaining entry is at $(+2,0)$, forcing $L=2,S=0$: this is ${}^{1}D$, five states. Remove them. One state remains at $(0,0)$, giving $L=0,S=0$: this is ${}^{1}S$. The total is $9+5+1=15$, exhausting the configuration with exactly the three terms ${}^{1}S,{}^{1}D,{}^{3}P$.

For the ordering, the residual repulsion ${\hat{H}}_1$ raises the energy of states whose spatial wavefunction is more symmetric (electrons closer) and lowers those whose spatial wavefunction is more antisymmetric (electrons farther apart). Antisymmetry of the full state ties the spatial symmetry to $S$: the maximal-$S$ term ${}^{3}P$ has the most antisymmetric spatial part, so the electrons avoid each other most effectively and the exchange contribution lowers its energy below ${}^{1}D$ and ${}^{1}S$. This is Hund's first rule applied to $p^2$; the explicit exchange computation appears in the next tier. $\square$

Bridge. This term-counting builds toward the entire periodic-table structure, and the bridge is the recognition that the antisymmetry constraint from the two-electron exchange interaction 12.09.02 is exactly what trims the naive $(L,S)$ list down to the physical terms. The foundational reason a configuration splits into terms at all is that the residual repulsion is a rotational scalar in $\hat{\mathbf{L}}$ and $\hat{\mathbf{S}}$ separately, so $L$ and $S$ label its eigenspaces; this is the same Clebsch-Gordan decomposition 12.05.03 used to add two angular momenta, now applied to the open shell. Putting these together, the term symbol ${}^{2S+1}{L}_J$ packages three quantum numbers whose ordering rules — Hund's rules — are the content that appears again in the Master tier as an exchange-energy theorem and as the spin-orbit interval rule.

Exercises Intermediate+

Advanced results Master

Theorem (exchange origin of Hund's first two rules). Within a configuration ${\ell }^k$, treat the residual Coulomb repulsion ${\hat{H}}_1={\sum }_{i<j}{r}_{ij}^{-1}$ in first-order perturbation theory on the degenerate term states. The term of maximal total spin $S$ lies lowest, and among terms of equal $S$ the term of maximal $L$ lies lowest, because the diagonal energy of a term is a sum of direct Coulomb integrals $J_{ab}$ minus exchange integrals $K_{ab}$ over occupied orbital pairs, and the number of subtracted exchange integrals grows with the spatial antisymmetry forced by maximal $S$ and, secondarily, by maximal $L$.

For a determinantal state built from spin-orbitals $\{a\}$, the Slater-Condon rules give the first-order energy $$ E = \sum_a \varepsilon_a + \sum_{a<b}\big( J_{ab} - \delta_{m_s(a),,m_s(b)},K_{ab}\big), \qquad J_{ab} = \langle ab| r_{12}^{-1}|ab\rangle,\quad K_{ab} = \langle ab| r_{12}^{-1}|ba\rangle, $$ where the exchange integral $K_{ab}$ is subtracted only between electrons of parallel spin. The direct sum $\sum J_{ab}$ depends weakly on how the spins and orbital projections are arranged; the exchange sum $\sum K_{ab}$ is the lever. Aligning more spins (raising $S$) creates more parallel-spin pairs, hence subtracts more positive exchange integrals, hence lowers the energy. This is Hund's first rule, and it is the same exchange integral $K$ that split ortho- and para-helium 12.09.02, here summed over the open shell. Among states of maximal $S$ the residual freedom is the orbital arrangement, and maximising $\hat{\mathbf{L}}$ piles the electrons into co-rotating orbits whose mutual avoidance again maximises the magnitude of the relevant exchange terms; this is Hund's second rule. Both rules are statements about the same positive integral $K_{ab}$ that has no classical analogue.

Spin-orbit fine structure and the interval rule. Adding the one-body spin-orbit operator ${\hat{H}}_{\mathrm{SO}}={\sum }_i\xi (r_i){\hat{\mathbf{l}}}_i\cdot {\hat{\mathbf{s}}}_i$ and projecting onto a fixed ${}^{2S+1}L$ term reduces, by the Wigner-Eckart theorem, to a single term-dependent constant: ${\hat{H}}_{\mathrm{SO}}{|}_{\text{term}}=A\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$ with $A$ proportional to the radial expectation $⟨\xi (r)⟩$. The level energies are $$ E(J) = E_0 + \tfrac{A}{2}\big[J(J+1) - L(L+1) - S(S+1)\big], $$ giving Landé's interval rule $E(J)-E(J-1)=AJ$ ^{[Landé 1923]}. The sign of $A$ is positive for shells less than half full (regular multiplet, smallest $J$ lowest, Hund's third rule) and negative for shells more than half full (inverted multiplet, largest $J$ lowest), because past half filling the open shell is better described by holes, which carry the opposite spin-orbit sign. Carbon's ${}^{3}P$ is regular ($A>0$, ground ${}^3{P}_0$) and oxygen's is inverted ($A<0$, ground ${}^3{P}_2$).

The jj-coupling crossover. LS coupling assumes the residual electrostatic splitting (scale $\sim K_{ab}$, growing slowly with $Z$) dominates the spin-orbit splitting (scale $A\sim Z^4/n^3$, growing fast with nuclear charge). For light atoms $K\gg A$ and $L,S$ are good quantum numbers. For heavy atoms $A\gg K$ and the alternative jj coupling holds: each electron's ${\hat{\mathbf{l}}}_i$ couples first to its own ${\hat{\mathbf{s}}}_i$ into a single-electron ${\hat{\mathbf{j}}}_i$, and the ${\hat{\mathbf{j}}}_i$ then couple to total $\hat{\mathbf{J}}$. The two schemes give the same number of levels with the same $J$ values for a configuration — only the intermediate labels $(L,S)$ versus $(j_1,j_2)$ differ — and real atoms interpolate between them. Lead ($6p^2$) is a textbook intermediate case where neither limit is exact. The total $\hat{\mathbf{J}}$ remains a rigorous constant of the motion throughout, because the full Hamiltonian is rotationally invariant; $L$ and $S$ are only approximately conserved.

Synthesis. The structure of every many-electron atom is the foundational reason the periodic table has the shape it does, and the central insight is that a configuration is degenerate at the central-field level and split by two competing scalars: the residual exchange integral $K_{ab}$, which orders terms by Hund's rules, and the spin-orbit coupling $A\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$, which splits each term into the Landé interval ladder. This is exactly the two-electron exchange mechanism 12.09.02 summed over an open shell, packaged with the Clebsch-Gordan addition of angular momenta 12.05.03 that supplies the term labels. Putting these together, the term symbol ${}^{2S+1}{L}_J$, Hund's three rules, the interval rule, and the LS-to-jj crossover are four faces of one perturbative competition, and the ratio $A/K$ — small for light atoms, large for heavy ones — is the dial that selects which coupling scheme and which ground term the atom realises. The bridge from this unit to molecular structure is that the same central-field orbitals and exchange bookkeeping, carried to two nuclei, become the molecular-orbital and valence-bond pictures of the chemical bond.

Full proof set Master

Proposition (microstate count of an open shell ${\ell }^k$). The number of Pauli-allowed states of $k$ equivalent electrons in a shell of orbital angular momentum $\ell$ is $\left(\genfrac{}{}{0ex}{}{2(2\ell +1)}{k}\right)$, and this number equals the sum of $(2S+1)(2L+1)$ over the allowed Russell-Saunders terms.

Proof. A shell of orbital angular momentum $\ell$ has $2\ell +1$ spatial orbitals, each accommodating two spin states, for $2(2\ell +1)$ spin-orbitals. The Pauli principle requires the $N$-electron state to be a single antisymmetric combination, so the legal states are the $k$-element subsets of the $2(2\ell +1)$ spin-orbitals, numbering $\left(\genfrac{}{}{0ex}{}{2(2\ell +1)}{k}\right)$. Each such subset has definite $M_L=\sum m_{\ell }$ and $M_S=\sum m_s$, and the multiset of $(M_L,M_S)$ values is exactly the weight diagram of the antisymmetric tensor power ${\Lambda }^k$ under the diagonal $\mathrm{SO}(3{)}_L\times \mathrm{SU}(2{)}_S$ action. Decomposing that weight diagram into irreducible $(L,S)$ multiplets by the corner-peeling algorithm partitions the $\left(\genfrac{}{}{0ex}{}{2(2\ell +1)}{k}\right)$ states into terms, each contributing $(2S+1)(2L+1)$ states. The sum of these contributions equals the total subset count, since the decomposition is a partition of the same state space. $\square$

Proposition (Landé interval rule and the centre-of-gravity sum). On a fixed ${}^{2S+1}L$ term with spin-orbit operator $A\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$, the fine-structure energies satisfy $E(J)-E(J-1)=AJ$, and the multiplicity-weighted mean shift vanishes: ${\sum }_{J=|L-S|}^{L+S}(2J+1)[E(J)-E_0]=0$.

Proof. From ${\hat{\mathbf{J}}}^2=(\hat{\mathbf{L}}+\hat{\mathbf{S}}{)}^2={\hat{\mathbf{L}}}^2+{\hat{\mathbf{S}}}^2+2\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$, the operator $\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$ has eigenvalue $\frac{1}{2}[J(J+1)-L(L+1)-S(S+1)]$ on the level $J$, so $E(J)=E_0+\frac{A}{2}[J(J+1)-L(L+1)-S(S+1)]$. The difference of consecutive levels is $E(J)-E(J-1)=\frac{A}{2}[J(J+1)-(J-1)J]=AJ$. For the centre of gravity, the weighted sum of the shifts is $\frac{A}{2}{\sum }_J(2J+1)[J(J+1)-L(L+1)-S(S+1)]$; since $\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$ is traceless over the $(2L+1)(2S+1)$-dimensional term space and the trace equals ${\sum }_J(2J+1)⟨\hat{\mathbf{L}}\cdot \hat{\mathbf{S}}{⟩}_J$, the weighted sum vanishes. The unperturbed energy $E_0$ is therefore the multiplicity-weighted mean of the split levels. $\square$

Proposition (LS and jj schemes give identical $J$-content). For a fixed configuration the set of total-$J$ values, counted with multiplicity, is the same whether obtained by coupling $\hat{\mathbf{L}}={\sum }_i{\hat{\mathbf{l}}}_i$ then $\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$, or by coupling each ${\hat{\mathbf{j}}}_i={\hat{\mathbf{l}}}_i+{\hat{\mathbf{s}}}_i$ then $\hat{\mathbf{J}}={\sum }_i{\hat{\mathbf{j}}}_i$.

Proof. Both schemes decompose the same antisymmetric configuration space, a fixed representation of the rotation group $\mathrm{SU}(2{)}_J$ acting diagonally on total angular momentum. The two coupling orders are two associativity bracketings of the same iterated tensor product of single-electron representations, related by the recoupling (Racah) transformation, which is a unitary change of basis within each fixed-$J$ isotypic component 12.05.03. A unitary basis change preserves the dimension of each ${\hat{\mathbf{J}}}^2$ eigenspace, so the multiplicity of every $J$ value is identical in the two schemes. Only the intermediate quantum numbers — $(L,S)$ versus $(j_1,j_2,\dots )$ — differ, and these label different orthonormal bases of the same $J$-multiplets. $\square$

Connections Master

• Exchange interaction and the helium atom 12.09.02. The exchange integral $K_{ab}$ that orders the Russell-Saunders terms by Hund's first rule is the many-electron generalisation of the single $K$ that split ortho- and para-helium. Where helium had one exchange integral between the $1s$ and $2s$ orbitals, an open shell ${\ell }^k$ has a sum of $K_{ab}$ over all parallel-spin orbital pairs, and the Slater-Condon energy expression $E={\sum }_a{\epsilon }_a+{\sum }_{a<b}(J_{ab}-{\delta }_{\parallel }{K}_{ab})$ is the bookkeeping that turns the two-electron result into the multiplet structure of every atom. The sign of $K_{ab}$, positive as in helium, is what makes maximal spin the ground configuration.

• Addition of angular momenta and Clebsch-Gordan coefficients 12.05.03. The entire term-symbol apparatus is repeated Clebsch-Gordan addition: the single-electron ${\hat{\mathbf{l}}}_i$ couple to $\hat{\mathbf{L}}$ and the ${\hat{\mathbf{s}}}_i$ to $\hat{\mathbf{S}}$ by the $\mathrm{SU}(2)$ series, then $\hat{\mathbf{L}}\otimes \hat{\mathbf{S}}$ couples to $\hat{\mathbf{J}}$. The Pauli antisymmetry restricts the naive tensor product to its alternating part, and the equality of the LS and jj level counts is the Racah recoupling identity — the $6j$-symbol change of basis — applied to the open shell.

• Time-independent perturbation theory 12.07.01. The whole hierarchy is a nested degenerate perturbation calculation: the central field gives the degenerate configuration, the residual repulsion (first perturbation) splits it into LS terms, and the spin-orbit coupling (second, smaller perturbation) splits each term into fine-structure levels. The ordering of the two perturbations — electrostatic before magnetic for light atoms, the reverse for heavy atoms — is exactly the LS-versus-jj distinction, decided by which perturbation is larger.

• Fermionic Fock space and the Pauli principle 12.13.02. The restriction of the allowed terms to the antisymmetric subspace is the configuration-space face of the canonical anticommutation relations. The open-shell states are occupation patterns of fermionic creation operators on a fixed set of central-field spin-orbitals, and the corner-peeling enumeration of terms is the decomposition of the $k$-particle antisymmetric Fock sector ${\Lambda }^k$ into joint $\mathrm{SO}(3{)}_L\times \mathrm{SU}(2{)}_S$ irreducibles.

Historical & philosophical context Master

Henry Norris Russell and Frederick Saunders introduced the LS coupling scheme in 1925 ^{[Russell-Saunders 1925]}, analysing the spectra of the alkaline earths and recognising that the observed multiplets are organised by a total orbital and a total spin angular momentum rather than by the individual electrons. Their term notation ${}^{2S+1}{L}_J$ became the standard label for atomic energy levels. In the same period Friedrich Hund extracted, from the empirical spectra of the iron-group elements, the three rules that predict which term of an open shell lies lowest ^{[Hund 1925]}; the rules were stated as regularities before any derivation existed.

Alfred Landé had already, in 1923, found the interval rule and the $g$-factor governing the Zeeman splitting of multiplets ^{[Landé 1923]}, working within the old quantum theory before electron spin was understood. The modern justification came after Heisenberg's exchange mechanism: John Slater's 1929 determinantal theory ^{[Slater 1929]} reduced the multiplet energies to direct and exchange integrals over central-field orbitals, making Hund's first two rules consequences of the positive exchange integral rather than empirical inputs. Landau and Lifshitz give the unified treatment of the periodic system, the LS scheme, and the exchange derivation of Hund's rules in Chapter X of Quantum Mechanics ^{[Landau-Lifshitz 1977]}, and Condon and Shortley's 1935 monograph remains the standard reference for the full term-enumeration and coupling-scheme machinery.

Bibliography Master

@article{RussellSaunders1925,
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  title   = {New Regularities in the Spectra of the Alkaline Earths},
  journal = {The Astrophysical Journal},
  volume  = {61},
  year    = {1925},
  pages   = {38--69}
}

@article{Hund1925,
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}

@article{Lande1923,
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  year    = {1923},
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}

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}

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}

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}

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@book{BransdenJoachain2003,
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