--- title: The Periodic Table and Atomic Spectra module: Many-Electron Atoms moduleNumber: 5 lessonNumber: 1 order: 501 summary: > Identical electrons demand antisymmetric wave functions, which is the Pauli exclusion principle: no two electrons share all four quantum numbers. Filling shells in order of increasing energy — shifted by penetration and shielding — builds the periodic table and its recurring ionization pattern. Selection rules govern optical spectra, and an external field splits lines by the Zeeman effect. topics: [Many-Electron Atoms] sources: - book: Tipler & Llewellyn ref: "Ch. 7 — Atomic Physics; §7-6 The Schrödinger Equation for Two or More Particles; §7-7 Ground States of Atoms: The Periodic Table" - book: Tipler & Llewellyn ref: "§7-8 Excited States and Spectra of Atoms, The Zeeman Effect" draft: false --- Hydrogen is a one-electron problem solved exactly. Every heavier atom adds electrons that repel one another, and the Schrödinger equation no longer separates. Two ideas carry the analysis through. Electron-electron repulsion shifts the energies of the hydrogenic states, breaking the degeneracy in $\ell$; and the indistinguishability of electrons imposes a rule — the exclusion principle — that decides how many electrons a shell can hold. Together they generate the periodic table and the optical spectra of the elements. ## Identical particles and exchange symmetry For two particles the Schrödinger equation involves both coordinates. If the particles do not interact, solutions are products of single-particle states, $$ \psi_{nm}(x_1, x_2) = \psi_n(x_1)\,\psi_m(x_2), $$ but this form distinguishes the particles: swapping the labels gives a different function. Identical particles cannot be told apart, so the probability density must be unchanged under exchange:[^tl-76] $$ |\psi(x_1, x_2)|^2 = |\psi(x_2, x_1)|^2. $$ This forces $\psi$ to be either **symmetric** or **antisymmetric** under the swap. Neither the bare product nor a single term qualifies; the acceptable combinations are $$ \psi_S = C\left[\psi_n(x_1)\psi_m(x_2) + \psi_n(x_2)\psi_m(x_1)\right] \quad\text{(symmetric)}, $$ $$ \psi_A = C\left[\psi_n(x_1)\psi_m(x_2) - \psi_n(x_2)\psi_m(x_1)\right] \quad\text{(antisymmetric)}. $$ > **Theorem (Pauli exclusion principle).** Electrons (and protons, neutrons, and > other spin-$\tfrac12$ particles) have _antisymmetric_ total wave functions. If > two electrons occupied the same single-particle state, $n = m$, the > antisymmetric combination $\psi_A$ vanishes identically. Therefore no two > electrons may share the same set of quantum numbers $n, \ell, m_\ell, m_s$. Particles with symmetric wave functions — $\alpha$ particles, deuterons, photons, mesons — do not obey the exclusion principle; they may pile into one state. The antisymmetry of electrons is what gives matter its shell structure. $$ % caption: The exclusion principle in orbital boxes: each orbital holds two % electrons of opposite spin; a same-spin pair in one orbital is forbidden. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % allowed box \draw[thick] (0,0) rectangle (1.2,0.7); \draw[very thick, ->] (0.35,0.12) -- (0.35,0.58); \draw[very thick, ->] (0.85,0.58) -- (0.85,0.12); \node[anchor=north] at (0.6,-0.05) {allowed}; % forbidden box \begin{scope}[xshift=3.0cm] \draw[black, thick] (0,0) rectangle (1.2,0.7); \draw[black, very thick, ->] (0.35,0.12) -- (0.35,0.58); \draw[black, very thick, ->] (0.85,0.12) -- (0.85,0.58); \draw[black, thick] (0.05,0.05) -- (1.15,0.65); \node[black, anchor=north] at (0.6,-0.05) {forbidden}; \end{scope} \node[black!70, anchor=west] at (4.8,0.35) {one orbital, two opposite spins}; \end{tikzpicture} $$ ## Filling the shells Neglecting electron-electron interaction, each electron in a heavier atom occupies a hydrogen-like state labeled by $n, \ell, m_\ell, m_s$. The energy rises with both $n$ and $\ell$. The $\ell$ dependence is new: it comes from **penetration and shielding**.[^tl-77] Inner electrons screen the nuclear charge, so an outer electron sees an effective charge $Z_{\text{eff}}$ between $1$ and $Z$. A low-$\ell$ orbital, with its inner probability bump (recall the [2s radial distribution](/atomic-physics/quantum-hydrogen-atom/hydrogen-wave-functions)), penetrates the screening cloud, sees more nuclear charge, and binds more tightly. For a given $n$, energy therefore increases with $\ell$: $3s < 3p < 3d$. The shift is large enough to reorder shells. Beyond argon the $4s$ level drops below $3d$, so potassium and calcium fill $4s$ before the $3d$ subshell begins. $$ % caption: Subshell energy ordering. Penetration lowers low-ℓ subshells; the 4s % level falls below 3d, setting the filling sequence of the periodic table. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % subshell levels at increasing height following the Madelung order \draw[->, black] (-0.3,-0.2) -- (-0.3,6.4) node[above, black!70] {energy}; \draw[thick] (0,0.0) -- (1.0,0.0); \node[anchor=west] at (1.05,0.0) {1s}; \draw[thick] (0,0.9) -- (1.0,0.9); \node[anchor=west] at (1.05,0.9) {2s}; \draw[thick] (0,1.5) -- (1.0,1.5); \node[anchor=west] at (1.05,1.5) {2p}; \draw[thick] (0,2.3) -- (1.0,2.3); \node[anchor=west] at (1.05,2.3) {3s}; \draw[thick] (0,2.9) -- (1.0,2.9); \node[anchor=west] at (1.05,2.9) {3p}; \draw[acc, thick] (0,3.7) -- (1.0,3.7); \node[acc, anchor=west] at (1.05,3.7) {4s}; \draw[acc, thick] (0,4.1) -- (1.0,4.1); \node[acc, anchor=west] at (1.05,4.1) {3d}; \draw[thick] (0,4.7) -- (1.0,4.7); \node[anchor=west] at (1.05,4.7) {4p}; \draw[thick] (0,5.3) -- (1.0,5.3); \node[anchor=west] at (1.05,5.3) {5s}; \draw[thick] (0,5.7) -- (1.0,5.7); \node[anchor=west] at (1.05,5.7) {4d}; % highlight 4s below 3d \draw[black, dashed] (1.5,3.7) -- (3.0,3.7); \draw[black, dashed] (1.5,4.1) -- (3.0,4.1); \node[black!70, anchor=west] at (3.0,3.9) {4s lower than 3d}; \end{tikzpicture} $$ Walking up $Z$ and applying the exclusion principle gives the electron configurations:[^tl-77] - **Helium** ($Z=2$): $1s^2$. Two electrons of opposite spin fill the K shell; the total spin is zero. The high ionization energy makes it inert. - **Lithium** ($Z=3$): $1s^2 2s$. The third electron cannot join the full K shell, so it enters $n=2$, choosing $2s$ over $2p$ because $2s$ penetrates the core. - **Beryllium** ($Z=4$): $1s^2 2s^2$. The exclusion principle pairs the fourth electron with the second $2s$ electron, opposite spin. - **Boron to neon** ($Z=5$ to $10$): the $2p$ subshell fills its six slots ($3$ values of $m_\ell$, $2$ of $m_s$), ending at neon, $1s^2 2s^2 2p^6$, inert. - **Sodium to argon** ($Z=11$ to $18$): the $3s$ then $3p$ subshells fill, ending at argon, $1s^2 2s^2 2p^6 3s^2 3p^6$. The recurrence of a filled outer shell is the periodicity. Elements sort into blocks by which subshell is filling. $$ % caption: Periodic-table blocks by the subshell being filled: two s columns, % six p columns, ten d columns, and the f block set below. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % all blocks share the bottom baseline; heights show which periods they span % s block (2 columns, tallest) \draw (0,0) rectangle (0.9,3.5); \node at (0.45,1.75) {s}; % d block (10 columns, middle, shorter) \draw (2.5,0) rectangle (5.5,2.0); \node at (4.0,1.0) {d}; % p block (6 columns, right) \draw (6.0,0) rectangle (8.7,3.0); \node at (7.35,1.5) {p}; % f block set below \draw (2.5,-1.3) rectangle (6.5,-0.5); \node at (4.5,-0.9) {f}; \end{tikzpicture} $$ The clearest signature of shell closing is the first ionization energy, the energy to remove the outermost electron. It climbs across a period and drops sharply at each alkali metal, where a new shell starts one electron far from a screened core. The peaks fall at the noble gases $Z = 2, 10, 18, 36, 54, 86$. $$ % caption: First ionization energy versus Z peaks at the closed-shell noble % gases and drops at the alkali metals, tracing the periodicity. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$Z$}; \draw[->, black] (0,0) -- (0,3.4) node[above, black!70] {ionization energy}; % sawtooth peaks at He, Ne, Ar, Kr, Xe \draw[acc, very thick] (0.2,1.9) -- (0.5,3.0) % up to He (0.5,3.0) -- (0.9,0.7) % drop to Li (0.9,0.7) -- (2.1,2.7) % rise to Ne (2.1,2.7) -- (2.5,0.85) % drop to Na (2.5,0.85) -- (3.6,2.35) % rise to Ar (3.6,2.35) -- (3.95,1.0) % drop to K (3.95,1.0) -- (5.1,2.05) % rise to Kr (5.1,2.05) -- (5.4,1.05) % drop to Rb (5.4,1.05) -- (6.5,1.85) % rise to Xe (6.5,1.85) -- (6.8,1.0); % drop to Cs \node[black, anchor=south] at (0.5,3.0) {He}; \node[black, anchor=south] at (2.1,2.7) {Ne}; \node[black, anchor=south] at (3.6,2.35) {Ar}; \node[black, anchor=south] at (5.1,2.05) {Kr}; \node[black, anchor=south] at (6.5,1.85) {Xe}; \node[black, anchor=north] at (0.9,0.65) {Li}; \node[black, anchor=north] at (2.5,0.8) {Na}; \end{tikzpicture} $$ ### Worked example: the effective charge in lithium The measured first ionization energy of lithium is $5.39$ eV, and its outer electron has $n = 2$. Modeling it as a hydrogenic electron of effective charge $Z_{\text{eff}}$, $$ E = \frac{Z_{\text{eff}}^2\,(13.6~\text{eV})}{n^2} = 5.39~\text{eV}, \qquad n = 2, $$ gives $$ Z_{\text{eff}}^2 = \frac{4\,(5.39)}{13.6} = 1.585, \qquad Z_{\text{eff}} \approx 1.3. $$ The $2s$ electron sees an effective charge slightly above $1$: the two K-shell electrons screen most of the $+3$ nucleus, but the $2s$ penetration adds a little.[^tl-77] ## Atomic spectra Optical spectra come from transitions of the outermost electron; core transitions land in the ultraviolet and X-ray. The alkali metals — Li, Na, K, Rb, Cs — behave almost hydrogenically, one electron over a closed core, so their spectra resemble hydrogen's. Two rules govern which transitions appear:[^tl-78] $$ \Delta\ell = \pm 1, \qquad \Delta j = 0, \pm 1 \ \ (\text{no } j=0 \to j=0). $$ In sodium the outer $3s$ electron excites to $3p$, about $2.1$ eV up. Spin-orbit coupling splits $3p$ into $^2P_{3/2}$ and $^2P_{1/2}$, separated by only about $0.002$ eV. The two transitions to the ground state produce the famous sodium yellow doublet, the color of sodium street lamps: $$ 3p(^2P_{3/2}) \to 3s(^2S_{1/2}): \ \lambda = 589.0~\text{nm}, $$ $$ 3p(^2P_{1/2}) \to 3s(^2S_{1/2}): \ \lambda = 589.6~\text{nm}. $$ $$ % caption: Sodium optical levels. The 3p level is a spin-orbit doublet; both % components decay to 3s, giving the two yellow D lines near 589 nm. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % ground 3s \draw[black!70, thick] (0,0) -- (2.4,0); \node[black!70, anchor=east] at (0,0) {$3s\ ^2S_{\tfrac{1}{2}}$}; % 3p doublet (spread apart so the term labels do not collide) \draw[acc, thick] (0.4,3.4) -- (2.4,3.4); \node[acc, anchor=west] at (2.45,3.4) {$3p\ ^2P_{\tfrac{3}{2}}$}; \draw[acc, thick] (0.4,2.5) -- (2.4,2.5); \node[acc, anchor=west] at (2.45,2.5) {$3p\ ^2P_{\tfrac{1}{2}}$}; % higher levels sketch \draw[black, thick] (0.4,4.4) -- (2.4,4.4); \node[black, anchor=west] at (2.45,4.4) {$4s$, $3d$ ...}; % transitions (D lines) \draw[acc, ->] (1.0,3.4) -- (1.0,0.05); \draw[acc, ->] (1.7,2.5) -- (1.7,0.05); \node[acc, anchor=north east] at (1.05,0.5) {589.0}; \node[acc, anchor=north west] at (1.7,0.5) {589.6}; \node[black, anchor=north] at (1.4,-0.1) {D doublet (nm)}; \end{tikzpicture} $$ ## The Zeeman effect An external magnetic field $\vec B$ supplies the preferred direction that free space lacks. The total angular momentum $\vec J$ quantizes along $\vec B$, and a level of quantum number $j$ splits into $2j + 1$ sublevels, one for each $m_j$, with energy shift proportional to $m_j$. Split levels split the spectral lines. In the simplest (normal) case a single line becomes three, governed by $\Delta m = 0, \pm 1$.[^tl-78] Zeeman and Lorentz shared the Nobel Prize for the discovery and its explanation. $$ % caption: Zeeman effect. A magnetic field splits the upper level into sublevels; % the transitions allowed by Δm = 0, ±1 turn one line into a triplet. \begin{tikzpicture}[scale=1.0, font=\footnotesize, >=stealth] \definecolor{acc}{HTML}{4A6FA5} % without field \draw[black, thick] (0,3.0) -- (1.4,3.0); \node[black!70, anchor=east] at (0,3.0) {upper}; \draw[black, thick] (0,0) -- (1.4,0); \node[black!70, anchor=east] at (0,0) {lower}; \draw[black, ->] (0.7,2.95) -- (0.7,0.05); \node[black, anchor=north] at (0.7,-0.1) {no B}; % with field: upper splits into 3 \draw[black, thick] (3.0,3.4) -- (4.6,3.4); \draw[black, thick] (3.0,3.0) -- (4.6,3.0); \draw[black, thick] (3.0,2.6) -- (4.6,2.6); \node[black, anchor=west] at (4.75,3.75) {$2j+1$ sublevels}; \draw[black!70, thick] (3.0,0) -- (4.6,0); % three transitions \draw[acc, ->] (3.5,3.4) -- (3.5,0.05); \draw[acc, ->] (3.8,3.0) -- (3.8,0.05); \draw[acc, ->] (4.1,2.6) -- (4.1,0.05); \node[black, anchor=north] at (3.8,-0.1) {with B}; % resulting triplet spectrum \draw[black] (6.2,0.2) -- (6.2,3.4); \foreach \x in {6.5,6.9,7.3}{\draw[acc, very thick] (\x,0.4) -- (\x,1.4);} \node[black, anchor=north] at (6.9,0.3) {triplet}; \end{tikzpicture} $$ The atomic physics of this module is thus a chain of consequences from the Schrödinger equation: three quantum numbers from the three coordinates, a fourth from electron spin, a rule of antisymmetry that stacks electrons into shells, and selection rules that decide which lines an atom emits. The same shell idea, with protons and neutrons in place of electrons, reappears in the [shell model of the nucleus](/nuclear-physics/nuclear-force-deuteron/nuclear-force-shell-overview). [^tl-76]: **Tipler & Llewellyn**, §7-6 — the two-particle Schrödinger equation, the indistinguishability condition $|\psi(x_1,x_2)|^2 = |\psi(x_2,x_1)|^2$, the symmetric and antisymmetric combinations, and the antisymmetry of electron wave functions giving the exclusion principle. [^tl-77]: **Tipler & Llewellyn**, §7-7 — energy ordering by $n$ and $\ell$, penetration and shielding raising energy with $\ell$, the $4s$-below-$3d$ reordering, the shell-by-shell ground-state configurations, and the ionization-energy pattern ($Z_{\text{eff}} \approx 1.3$ for lithium). [^tl-78]: **Tipler & Llewellyn**, §7-8 — alkali spectra, the selection rules $\Delta\ell = \pm 1$ and $\Delta j = 0, \pm 1$, the sodium yellow doublet from the spin-orbit-split $3p$ level, and the Zeeman splitting of levels into $2j+1$ sublevels in an external field.