Atomic Models Lab

Six live simulations that trace how the picture of the atom was built: Thomson's plum pudding and the α-particle test that overturned it, Bohr's quantised orbits, Sommerfeld's ellipses and relativity, space quantisation seen by Stern and Gerlach, the vector atom model, and the normal and anomalous Zeeman effect.

Prepared by Mr. Manoj Devkota
Kamala Science Campus, Dhungrebas, Sindhuli

Thomson's plum-pudding model and the α-particle test

J. J. THOMSON 1904 · GEIGER, MARSDEN & RUTHERFORD 1909–1913

The atom is a ball of uniformly spread positive charge with electrons embedded in it, like plums in a pudding. Drag an electron (or press “Pluck”) and let go: inside the ball the pull back is a spring force, so it vibrates and radiates light of one frequency.

What to show

Number of electrons N 1

Radius of the positive sphere R 1.00 Å

Drag an electron on the canvas and release it.

This atom

In simple words

Basic principle

An atom is neutral. Thomson pictured its positive charge spread evenly through a small ball, with the tiny electrons sitting inside it like plums in a pudding (or seeds in a watermelon).

How it works

Inside the ball, the positive charge pulls a displaced electron back towards the centre like a spring. So electrons settle in rings and, when disturbed, vibrate and give out light of a fixed frequency.

Conclusion

The model explained neutral atoms and the electron, but not the hydrogen line series. When α particles were fired at gold foil, some bounced straight back. That is impossible in a pudding, so Rutherford replaced it with a tiny, heavy nucleus.

How the model works

A sphere of radius \(R\approx10^{-10}\) m carries charge \(+Ze\) spread uniformly; \(Z\) electrons sit inside, so the atom is neutral.
By Gauss's law the field inside grows linearly: \( E(r)=\dfrac{Ze\,r}{4\pi\varepsilon_0R^3} \) for \(r\le R\), and \(\propto 1/r^2\) outside.
An electron displaced by \(r\) feels \( F=-\dfrac{Ze^2}{4\pi\varepsilon_0R^3}\,r \): simple harmonic motion.
Electrons repel each other, so several of them settle in rings (Thomson's 1904 calculation).
A vibrating electron radiates at its vibration frequency:

\[ \omega_0=\sqrt{\frac{e^2}{4\pi\varepsilon_0\,m_eR^3}},\qquad R=1\ \text{Å}\ \Rightarrow\ \lambda=\frac{2\pi c}{\omega_0}\approx118\ \text{nm} \]

α test: Rutherford's formula \( N(\theta)\propto \dfrac{1}{\sin^4(\theta/2)} \); closest approach \( d=\dfrac{2Ze^2}{4\pi\varepsilon_0 K} \).

Try this

With N = 1, pluck the electron. It vibrates at one frequency, so the model predicts one spectral line, not a series. Move R: \(\lambda\propto R^{3/2}\). No single R gives both Lyman α (122 nm) and Hα (656 nm).
Raise N to 5, 6, 8: the electrons form rings. The normal-mode frequencies that appear do not follow the Rydberg formula.
Switch to ②. With the Thomson model no α particle is turned through more than a fraction of a degree. With the nuclear model a few bounce back, and the counts follow \(1/\sin^4(\theta/2)\), matching Geiger and Marsden's 1913 data.
In ②, raise the α energy: the closest approach \(d\) shrinks. For aluminium at 9–10 MeV, \(d\) becomes comparable to the nuclear radius and the pure Coulomb law stops being exact.
Switch to ③: Rutherford's atom, treated classically, collapses in about 16 ps while radiating a continuous spectrum. This is the puzzle Bohr solved.

Successes and failures

Success: first model with electrons (discovered 1897) as parts of the atom; it explains neutrality, gives the right atomic size and the idea of an elastically bound electron, which Lorentz used to explain the normal Zeeman effect.
Failure 1: it gives a few frequencies set by R, not the series \(1/\lambda=R_H(1/n_f^2-1/n_i^2)\).
Failure 2: the largest deflection of an α particle by one atom is about 0.01°. Many small deflections in a foil add to about 1°, so the chance of 90° is about \(10^{-3500}\). Geiger and Marsden saw about 1 in 8000 α particles reflected from a platinum foil (1909).
Rutherford's atom (1911) fixes the scattering but is classically unstable (③), which led directly to Bohr's postulates.

Bohr model of the hydrogen atom

NIELS BOHR · 1913

Click an energy level (right side of the canvas) or a level button to make the electron jump. Upward jumps absorb a photon (a dark line in the absorption strip); downward jumps emit one (a bright line in the emission strip). Choose n = ∞ to ionise the atom, then click a level to recapture the electron.

Atom

Hydrogen-like ion

Jump to level

Or click a level on the energy diagram.

Add a whole series to the spectrum

Show de Broglie standing wave Radii to scale (r ∝ n²)

Current orbit

Last photon

—: make a jump

In simple words

Basic principle

The electron can only move in certain fixed orbits, like steps on a ladder. In these allowed orbits it does not lose energy, so the atom is stable.

How it works

When the electron jumps down a step it gives out one photon whose energy equals the step height, \(h\nu=E_i-E_f\). To jump up it must absorb a photon of exactly that energy. The steps get closer together higher up, so the lines of a series crowd together.

Conclusion

Bohr's model gives the hydrogen spectrum, the 13.6 eV ionisation energy and the size of the atom from basic constants. It fails for atoms with more than one electron and cannot explain the fine details of the lines.

How the model works

The electron moves in a circle; the Coulomb force supplies the centripetal force: \( \frac{mv^2}{r}=\frac{Ze^2}{4\pi\varepsilon_0 r^2} \).
Only orbits with quantised angular momentum are allowed: \( mvr=n\hbar,\ n=1,2,3,\dots \)
In an allowed orbit the electron does not radiate (stationary state).
Light is emitted or absorbed only in a jump: \( h\nu=E_{n_i}-E_{n_f} \).

\[ r_n=\frac{n^2a_0}{Z},\quad v_n=\frac{Z\alpha c}{n},\quad E_n=-\frac{13.6\,Z^2}{n^2}\ \text{eV} \]

\[ \frac{1}{\lambda}=RZ^2\Big(\frac{1}{n_f^2}-\frac{1}{n_i^2}\Big) \]

Try this

Jump 3 → 2, 4 → 2, 5 → 2, 6 → 2. You have built the visible Balmer series (656, 486, 434, 410 nm). Why do the lines crowd together towards 365 nm?
Switch to He⁺. Every wavelength is divided by \(Z^2=4\): the 4 → 3 line of He⁺ lands in the visible near 469 nm.
Turn on the de Broglie wave: the orbit holds exactly \(n\) wavelengths, \(2\pi r_n=n\lambda\). This is the quantum condition in wave language.
Turn on auto cascade and let the spectrum fill; compare the Lyman (UV), Balmer (visible) and Paschen (IR) groups.
Tick “radii to scale”: the n = 6 orbit is 36 times larger than n = 1.
Add the Lyman series: the lines crowd towards the series limit 91.2 nm. Then press n = ∞ from n = 1: the photon needed is exactly 13.6 eV, the ionisation energy.
Choose deuterium and make 3 → 2: Hα moves from 656.47 to 656.29 nm. This 0.18 nm isotope shift, from the reduced mass, is how Urey discovered deuterium in 1932.
Watch the read-out “ν(n→n−1)/f_orbit”: it approaches 1 as n grows. For large n, quantum jumps reproduce classical radiation (Bohr's correspondence principle).

What the model cannot do

It works only for one-electron atoms and ions.
It cannot give line intensities, or explain the fine structure and the Zeeman splitting of lines.
Ground-state angular momentum is \(\hbar\) in Bohr's model but zero in quantum mechanics (the 1s state has \(l=0\)).
Definite orbits violate the uncertainty principle; the modern picture uses probability clouds.
The wavelengths shown use the reduced-mass Rydberg constant \(R_M=R_\infty/(1+m_e/M)\) and are vacuum values (Hα = 656.5 nm in vacuum, 656.3 nm in air).
The Franck–Hertz experiment (1914) confirmed discrete energy levels directly, by electron collisions instead of light.

Sommerfeld's elliptical orbits

ARNOLD SOMMERFELD · 1916

For each principal quantum number \(n\) there are \(n\) orbits, \(k=1,\dots,n\), with the nucleus at one focus. Without relativity they all have the same energy. Switch on relativity: the fast-moving electron near the nucleus gains mass, the ellipse precesses into a rosette and the levels split (fine structure).

Atom

Principal quantum number n 3

Azimuthal quantum number k 1

Show all k orbits for this n Relativistic mass variation (precession)

Exaggerate precession × 2000

Lower-right panel

Selected orbit

In simple words

Basic principle

Like planets, an electron can move in ellipses, not only circles. Sommerfeld allowed elliptical orbits and quantised both the angle and the in-and-out motion, giving a second quantum number \(k\).

How it works

For each \(n\) there are \(n\) orbits of different shape but equal energy. In the thin ellipses the electron swings close to the nucleus and moves very fast, so its mass increases (relativity). The orbit then slowly turns (precesses) and its energy shifts a little.

Conclusion

The tiny shifts split each Bohr level into close sub-levels, which is the fine structure of spectral lines. The size came out right, but the model has no electron spin, so it cannot explain the anomalous Zeeman effect.

How the model works

Two coordinates \( (r,\phi) \) need two quantum conditions (Wilson–Sommerfeld): \( \oint p_\phi\,d\phi=kh \) and \( \oint p_r\,dr=n_rh \).
The principal number is \( n=n_r+k \), with \( k=1,\dots,n \). \(k=0\) is excluded: that “orbit” would pass through the nucleus.
The ellipse has \( a=\frac{n^2a_0}{Z} \) and \( \frac{b}{a}=\frac{k}{n} \); \(k=n\) is the Bohr circle.
Non-relativistically \( E_n=-13.6Z^2/n^2 \) eV for every \(k\): the orbits are degenerate.
Relativity makes the orbit precess and lifts the degeneracy:

\[ E_{n,k}=-\frac{13.6\,Z^2}{n^2}\Big[1+\frac{\alpha^2Z^2}{n^2}\Big(\frac{n}{k}-\frac34\Big)\Big] \]

Precession per revolution \( \approx \pi(\alpha Z/k)^2 \); \( \alpha\approx1/137 \).

Try this

Set n = 3 and step k from 1 to 3. The semi-major axis stays fixed while the orbit becomes rounder (\(b/a=k/n\)).
Watch the speed: the electron rushes past the nucleus and dawdles far away (Kepler's second law). Low-\(k\) orbits come closest, so they feel relativity most and split most.
Compare the fine-structure diagram: the \(n=2\) splitting is 45.3 μeV = 10.9 GHz = 0.365 cm⁻¹, exactly the value later given by Dirac.
Set the exaggeration to 0 and confirm the ellipse closes on itself (no relativity, no precession).
Read the perihelion speed for n = 3, k = 1: about 1.4 % of c, against only 0.04 % at aphelion. That difference is what relativity acts on.
Choose He⁺: every splitting grows by \(Z^4=16\). Paschen's 1916 measurement of the He⁺ 468.6 nm line confirmed this.
Switch the panel to “all k pairs”: without \(\Delta k=\pm1\), six components would appear instead of three.
Switch to “space quantisation”: the orbit's angular momentum \(k\hbar\) may only take orientations with \(k\cos\theta=m\). This prediction was tested by Stern and Gerlach (tab 4).

Where it falls short

The agreement with Dirac's fine-structure formula is partly a coincidence: Sommerfeld had no electron spin. In modern terms \(k=j+\tfrac12\), not \(l\).
The selection rule \( \Delta k=\pm1 \) had to be added by hand.
It still fails for many-electron atoms and cannot explain the anomalous Zeeman effect.
Modern correspondence: \( l=k-1 \) labels the orbital (s, p, d, f).

Stern–Gerlach experiment

OTTO STERN & WALTHER GERLACH · 1922

A beam of silver atoms passes through a strongly inhomogeneous magnetic field. A force \(F_z=\mu_z\,\partial B/\partial z\) deflects each atom by an amount set by its magnetic-moment component. Classical physics predicts a continuous smear; nature gives separate spots.

Experiment

Atoms in the beam

Field gradient ∂B/∂z 1000 T/m

Oven temperature 1323 K

Numbers for this beam

In simple words

Basic principle

Each silver atom is a tiny magnet. A magnet in a field that is stronger at one end than the other is pushed sideways, and the push depends on which way the little magnet points.

How it works

A thin beam of silver atoms from a hot oven passes between specially shaped magnet poles and lands on a glass plate. If the atomic magnets could point any way, the beam would spread into one smear. Instead it splits into two separate traces.

Conclusion

Atomic magnets can point only in a few fixed directions (space quantisation). Two traces need an angular momentum of ½, which is the electron's spin: the result later became the classic proof of spin.

How the experiment works

Silver evaporates in an oven at about 1000 °C; slits 0.03 mm wide select a narrow beam (speed ≈ 540 m/s).
The beam passes for 3.5 cm between a knife-edge pole and a grooved pole: the field is about 0.1 T and its gradient about 10 T/cm.
In a uniform field a magnetic dipole only feels a torque; in a gradient it feels a net force \( F_z=\mu_z\,\partial B/\partial z \).
For silver \( \mu_z=-g\,m_J\,\mu_B=\mp\mu_B \), so the beam splits in two, about 0.2 mm apart on the glass plate.

\[ \Delta z=\frac{\mu_z\,(\partial B/\partial z)\,L^2}{2mv^2},\qquad \text{number of beams}=2J+1 \]

Try this

Compare “Silver” with “Classical magnets”. The deposit's two lips versus a filled lens is the whole discovery.
Turn the gradient to zero: the beam is undeflected even though a field is present.
Raise the oven temperature: faster atoms spend less time in the magnet, so the spots move inward and the deflection spreads less.
Try J = 0, J = 1 and J = 3/2: the plate shows 1, 3 and 4 spots (\(2J+1\)).
Sequential mode: set θ = 90°. Atoms selected as “up” along z come out 50 : 50 along the tilted axis and then 50 : 50 again along z. Measuring one component destroys knowledge of the other.
In sequential mode, compare the counts with \( P=\cos^2(\theta/2) \).
Compare silver with the “1922 expectation” and with hydrogen: all three give two beams of the same size. Because \(mv^2\propto kT\) in the beam, the deflection does not depend on the mass of the atom.

What it tells us (and a historical twist)

Direct proof of space quantisation: the moment's component along a field takes only discrete values.
Two beams (an even number) cannot come from orbital angular momentum, which always gives \(2l+1\), an odd number. The moment of silver comes from the electron's spin, \(s=\tfrac12\).
In 1922 the result was read as confirming Bohr–Sommerfeld space quantisation with \(l=1\) and \(m=\pm1\) (Bohr excluded \(m=0\)). Spin was proposed only in 1925 (Uhlenbeck and Goudsmit).
Phipps and Taylor (1927) repeated the experiment with hydrogen atoms in the ground state (\(l=0\), so no orbital moment at all) and again found two beams: the moment is the electron's spin, with \(g_s\approx2\).
The real deposit has a “lip” shape because the gradient is strongest at the centre of the beam, near the knife edge.

Vector atom model

LANDÉ, UHLENBECK & GOUDSMIT · 1921–1926

Orbital angular momentum \( \vec L \) and spin \( \vec S \) add to \( \vec J \). The two precess quickly about \( \vec J \), while \( \vec J \) precesses slowly about the field \( \vec B \) with a quantised projection \( m_J\hbar \). Drag on the 3-D view to rotate it.

Orbital quantum number l

Spin quantum number s

Total j

Projection on B (m_J)

Magnetic field B 1.0 T

Lower-right panel

Show magnetic moment μ Animate precession

This state

In simple words

Basic principle

The electron has two kinds of spin-like motion: going round the nucleus (orbital angular momentum \(\vec L\)) and spinning on itself (spin \(\vec S\)). Both behave like arrows (vectors) that add up to a total \(\vec J\).

How it works

The two arrows can only add in a few allowed ways, giving levels such as ²P₁/₂ and ²P₃/₂. In a magnetic field the total arrow \(\vec J\) can only take a few tilts, each with its own small energy shift \(g_J m_J\mu_BB\).

Conclusion

The vector model explains fine-structure doublets, term symbols and the strange (anomalous) Zeeman patterns through the Landé g-factor. It is a picture of the quantum-mechanical results, and it works best for weak fields and light atoms.

How the model works

Each angular momentum has magnitude \( \sqrt{x(x+1)}\,\hbar \): \( |\vec L|=\sqrt{l(l+1)}\hbar \), \( |\vec S|=\sqrt{s(s+1)}\hbar \).
In LS coupling \( \vec J=\vec L+\vec S \), with \( j=|l-s|,\dots,l+s \). The term is written \( ^{2s+1}L_j \).
In a field, \( \vec J \) can only point so that \( J_z=m_J\hbar \), \( m_J=-j,\dots,+j \) (space quantisation).
Because \( g_s\approx2 \), \( \vec\mu=-\frac{\mu_B}{\hbar}(\vec L+2\vec S) \) is not parallel to \( \vec J \). Only its component along \( \vec J \) survives the fast precession, giving the Landé factor:

\[ g_J=1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)},\qquad \Delta E=g_J\,m_J\,\mu_B B \]

Try this

Set l = 1, s = ½ and compare j = ½ (g = 2/3) with j = 3/2 (g = 4/3). The dashed μ vector leans away from J differently.
Set s = 0 (singlet): g = 1 for every level, so every level splits with the same spacing \( \mu_B B \). This is the root of the normal Zeeman effect (tab 6).
Choose the panel “Spin–orbit splitting”. For l = 1, s = ½ the two levels are separated by \(\tfrac32A\) (the sodium D₁/D₂ doublet). For s = 1 the three ³P levels obey Landé's interval rule: gaps in the ratio 1 : 2.
Choose “Two valence electrons” and slide from LS to jj. The levels regroup, but the number of levels with each J never changes. The sp configuration of C, Si, Ge, Sn and Pb moves along this diagram.
Set B = 0: J stops precessing about a fixed axis and the \(m_J\) levels merge (degenerate).
Check \(\vec L\cdot\vec S\): the angle between \( \vec L \) and \( \vec S \) is fixed by \(l, s, j\) through the cosine rule.

Limits and extensions

The picture of definite vectors is semiclassical; only \( |\vec J| \) and \( J_z \) are sharp, the other components are not.
The model assumes the weak-field limit (Zeeman energy ≪ fine-structure splitting). In strong fields L and S decouple: the Paschen–Back effect.
LS (Russell–Saunders) coupling suits light atoms, where the electrostatic interaction between electrons dominates; in heavy atoms spin–orbit coupling of each electron dominates and \(\vec j_1+\vec j_2=\vec J\) (jj coupling).
The rule \(\sqrt{j(j+1)}\) (not \(j\)) came from experiment and was later derived by quantum mechanics.

The Zeeman effect: normal and anomalous

ZEEMAN 1896 · LORENTZ 1897 · LANDÉ 1921 · PASCHEN & BACK 1912

Put a light source in a magnetic field and each spectral line splits into several close, polarised components. Choose a line and raise B: watch every energy level split into 2J + 1 sub-levels and the allowed jumps (Δm = 0, ±1) build the pattern seen in the spectrometer.

What to show

Spectral line

Observe the light

Polariser

Magnetic field B 1.00 T

This line

In simple words

Basic principle

An atom behaves like a tiny magnet because its electrons orbit and spin. In a magnetic field, a magnet's energy depends on which way it points, and an atom may point only in a few allowed ways. So every energy level splits into several close levels, and every spectral line splits into several close lines.

How it works

A level with total angular momentum J splits into 2J + 1 sub-levels, spaced by \(g_J\mu_BB\). Light is emitted only in jumps with Δm = 0 (π light, polarised along B) or Δm = ±1 (σ light, polarised across B, or circular when seen along B). If both levels have the same spacing, the lines bunch into three: the normal effect. If the spacings differ (spin present), many lines appear: the anomalous effect.

Conclusion

The splitting proved that light comes from electrons inside atoms and gave their e/m. The anomalous patterns could only be explained with electron spin and the Landé g-factor. In very strong fields the pattern simplifies again (Paschen–Back effect). Astronomers use the effect to measure magnetic fields on the Sun.

Normal Zeeman effect (spin S = 0)

For singlet levels \(g_J=1\), so every level splits with the same spacing: \( \Delta E=m_J\,\mu_BB \).
Jumps obey \(\Delta m_J=0,\pm1\), so the photon energy changes by only \(0\) or \(\pm\mu_BB\): three lines however many sub-levels there are.
Lorentz's classical picture gives the same answer: the field speeds up one circular motion of the bound electron and slows the other by the Larmor frequency.

\[ \Delta\nu=\frac{eB}{4\pi m_e}=\frac{\mu_BB}{h}=14.0\ \text{GHz per tesla},\qquad \Delta\lambda=\frac{\lambda^2}{c}\,\Delta\nu \]

Cd 643.8 nm at 1 T: Δλ ≈ 0.019 nm. A Fabry–Pérot or Lummer–Gehrcke plate is needed to see it.

Anomalous Zeeman effect (spin S ≠ 0)

Because the spin moment counts double (\(g_s\approx2\)), the magnetic moment is not parallel to \(\vec J\); its average along \(\vec J\) gives the Landé factor \(g_J\).
Upper and lower levels have different \(g_J\), so the lines fall at many places:

\[ h\nu=h\nu_0+\big(g_u m_u-g_l m_l\big)\mu_BB,\qquad g_J=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)} \]

Na D₁: \(g=\tfrac23\to2\): 4 lines at \(\pm\tfrac23,\pm\tfrac43\) (in units of \(\mu_BB\)).
Na D₂: \(g=\tfrac43\to2\): 6 lines at \(\pm\tfrac13,\pm1,\pm\tfrac53\).
Relative strengths follow the Wigner 3j coefficients; the simulation uses them.

Try this

Cadmium 643.8 nm has 9 allowed jumps, but they fall on only 3 positions: count them on the level diagram.
Switch to longitudinal viewing: the middle (π) line disappears and the outer lines become circularly polarised in opposite senses.
Use the polariser in transverse view: “∥ B” leaves only π lines, “⊥ B” only σ lines. This is how the components are identified in practice.
Compare Na D₁ and D₂. Then build your own: ¹P₁ → ¹S₀ or ¹D₂ → ¹P₁ (S = 0) always gives the normal triplet, while ³P₁ → ³S₁ (S = 1) is anomalous.
Mode ②: watch why the frequency changes: the magnetic (Lorentz) force adds to the pull of the atom for one sense of rotation and subtracts for the other.
Mode ③ with lithium: raise B past about 1 T. The 10 lines regroup into a normal-looking triplet: the Paschen–Back effect.

Conclusion and importance

Zeeman (1896) and Lorentz showed that spectral lines come from oscillating charged particles. From the size of the splitting they obtained \(e/m\approx10^{11}\) C/kg and a negative charge, in 1897, the same year Thomson measured \(e/m\) of cathode rays. Nobel Prize 1902.
The splitting into \(2J+1\) components is direct evidence of space quantisation, as in the Stern–Gerlach experiment.
The anomalous effect, unexplained for 25 years, led to electron spin (1925) with \(g_s\approx2\).
Selection rules: \(\Delta m_J=0\) (π, linear ∥ B), \(\pm1\) (σ, linear ⊥ B seen across the field, circular seen along it); \(\Delta J=0,\pm1\) but not \(0\to0\).
Strong fields (\(\mu_BB\) ≫ fine-structure splitting) decouple \(\vec L\) and \(\vec S\): the Paschen–Back effect. Hale (1908) measured sunspot fields of about 0.3 T this way.

Teaching guide

FOR B.SC. AND M.SC. CLASSES

How to use these simulations in a lecture, in a laboratory class, on a website and from PowerPoint, with learning outcomes, a suggested lesson sequence, discussion questions and the common misconceptions to watch for.

The story the six tabs tell

Model	New idea	Explains	Fails on	Quantum numbers
Thomson (1904)	Electrons embedded in a uniform positive sphere; elastically bound electron	Neutral atom, atomic size, electron as a constituent	Line series; large-angle α scattering (Geiger–Marsden 1909–13)	none
Rutherford (1911)	Tiny massive nucleus, \(N\propto1/\sin^4(\theta/2)\)	α scattering, nuclear size < 30 fm	Stability (collapse in ≈ 16 ps), sharp lines	none
Bohr (1913)	Stationary orbits, \(mvr=n\hbar\), photon in a jump	Hydrogen series, Rydberg constant, ionisation energy 13.6 eV	Fine structure, intensities, many-electron atoms	\(n\)
Sommerfeld (1916)	Elliptic orbits; relativistic mass variation	Fine structure of hydrogen (correct magnitude)	Spin, anomalous Zeeman effect	\(n,\ k\)
Stern–Gerlach (1922)	Space quantisation observed directly	Discrete orientations of atomic moments	(an experiment) its two beams needed spin	\(m_J\)
Vector atom (1921–26)	\(\vec J=\vec L+\vec S\), Landé \(g\)	Anomalous Zeeman effect, term symbols, fine structure	Strong fields (Paschen–Back), heavy atoms (jj)	\(n,l,s,j,m_J\)
Zeeman effect (1896–1912)	Levels split into \(2J+1\) sub-levels in B; \(\Delta m=0,\pm1\)	Normal triplet (classical and quantum), anomalous patterns (\(g_J\)), polarisation, Paschen–Back	(an effect) the anomalous form needed spin to explain	\(J, m_J, g_J\)

Learning outcomes

After the six sessions a student should be able to:

describe Thomson's model, derive the oscillation frequency of a bound electron, and explain why α scattering ruled it out;
derive \(r_n\), \(v_n\), \(E_n\) and the Rydberg formula, and identify the series;
state the Wilson–Sommerfeld rules, give \(b/a=k/n\) and explain the degeneracy and its removal by relativity;
explain why a gradient (not a uniform field) is needed, estimate the Stern–Gerlach deflection, and relate the number of beams to \(2J+1\);
build \(\vec J\) from \(\vec L\) and \(\vec S\), compute \(g_J\), apply the interval rule, and contrast LS and jj coupling;
explain the normal Zeeman effect classically (Larmor) and quantum mechanically, predict anomalous patterns with intensities and polarisation, and describe the Paschen–Back limit.

Suggested sequence (6 × 50 min)

Thomson → Rutherford. Plum pudding and SHM → one-line prediction → α test: predict, then reveal the counts → classical collapse.
Bohr. 10 min postulates → 15 min derivation → 15 min live: build the Balmer series, then He⁺ → 10 min de Broglie and limitations.
Sommerfeld. Two quantum conditions → ellipse family for n = 3 → degeneracy → switch on relativity; read the n = 2 splitting.
Stern–Gerlach. Torque vs force → predict the classical smear → reveal two lips → number of beams → sequential magnets (M.Sc.).
Vector model. Magnitudes \(\sqrt{x(x+1)}\) → coupling → Landé \(g\) derivation → spin–orbit interval rule → LS vs jj.
Zeeman effect. Apparatus → Lorentz's classical triplet and \(e/m\) → quantum levels → Na D₁/D₂ → polarisation → Paschen–Back.

Discussion and assessment questions

For Thomson's hydrogen atom with R = 1 Å, find the emitted wavelength. Why can it not explain the Balmer series?
Find the distance of closest approach of a 7.7 MeV α particle to a gold nucleus. (Answer: 29.6 fm.)
Why is the Balmer series the only one seen by eye? What limits the series at 364.7 nm?
Show that \(v_1=\alpha c\). What does this imply about relativistic corrections in heavy atoms?
Why does the k = 1 orbit split most? Relate it to the closest approach \(a(1-e)\).
Estimate the Stern–Gerlach splitting for v = 540 m/s, ∂B/∂z = 10³ T/m, L = 3.5 cm. (Answer: about 0.22 mm.)
Why would orbital angular momentum alone never give two beams?
Derive \(g_J\) from the projection of \(\vec\mu\) on \(\vec J\). Find the D₁ and D₂ line positions in units of \(\mu_BB\).
Show classically that \(\Delta\nu=eB/4\pi m\). Find \(\Delta\lambda\) for the Cd 643.8 nm line at 1 T. (Answer: 0.019 nm.)
Why is the central component missing when a Zeeman pattern is viewed along the field?

Misconceptions to address

“The electron really moves in a circle.” The orbit is a model; quantum mechanics gives probability distributions.
“A uniform magnetic field splits the beam.” It only causes precession; splitting needs a gradient.
“Stern and Gerlach discovered spin.” They observed space quantisation; spin was recognised three years later.
“\(|\vec J|=j\hbar\).” The magnitude is \(\sqrt{j(j+1)}\hbar\); the maximum projection is \(j\hbar\).
“\(\vec\mu\) is parallel to \(\vec J\).” Not when both L and S contribute, because \(g_s\neq g_l\).
“Most α particles bounce back from gold.” About 1 in 10⁴ – 10⁵ is turned through more than 90°; nearly all pass straight through.
“Normal means common.” The anomalous effect is far more common; “normal” only means the classical theory could explain it.

Using the simulations

Projector: press “Full screen” on any canvas. The canvases redraw at any size.
Direct links: add #thomson, #bohr, #sommerfeld, #sg, #vector, #zeeman or #guide to the page address to open that tab.
PowerPoint: the companion deck plays recorded versions of each simulation; the “Open live simulation” button on each video slide opens the matching tab (keep this file next to the deck).
Website: the page is one HTML file with the equation renderer built in. Upload it as is; it needs internet only for the web fonts (system fonts are used otherwise).

Embedding in another page:

<iframe src="atomic_models.html#sg" width="100%" height="820"
        style="border:0" allowfullscreen></iframe>

What is exaggerated, and what is exact

Exact: all energies, radii, wavelengths, fine-structure values, Landé factors, Zeeman patterns and intensities (3j coefficients), the Paschen–Back levels (exact 2 × 2 diagonalisation), the Rutherford trajectories and the Stern–Gerlach numbers in the read-outs.
Schematic: the LS ↔ jj correlation diagram (level order and spacing are illustrative; the J-counting is exact) and the Thomson multiple-scattering width (scaled from Geiger's ≈ 1° for gold).
Time-compressed: the classical collapse (15.6 ps shown in 9 s) and the α-counting rates.
Slowed down: orbital motion (real period of the n = 1 orbit is 1.5 × 10⁻¹⁶ s) and atom flight times.
Narrowed: the velocity spread of the silver beam in the Stern–Gerlach view (±15 %), so the two lips stay distinct; the real thermal spread blurs them more.
Exaggerated: Sommerfeld precession (the true advance for k = 1 is 0.0096° per turn) and the Stern–Gerlach deflection on the side view.
Simplified: in the auto cascade the electron chooses lower levels at random; real branching ratios differ.

References

J. J. Thomson, “On the structure of the atom”, Phil. Mag. 7, 237 (1904); E. Rutherford, Phil. Mag. 21, 669 (1911).
H. Geiger and E. Marsden, Proc. R. Soc. A 82, 495 (1909); “The laws of deflexion of α particles through large angles”, Phil. Mag. 25, 604 (1913).
N. Bohr, “On the constitution of atoms and molecules”, Phil. Mag. 26, 1 (1913).
P. Zeeman, Phil. Mag. 43, 226 (1897); F. Paschen and E. Back, Ann. Phys. 39, 897 (1912); T. E. Phipps and J. B. Taylor, Phys. Rev. 29, 309 (1927).
A. Sommerfeld, “Zur Quantentheorie der Spektrallinien”, Ann. Phys. 51, 1 (1916).
W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld”, Z. Phys. 9, 349 (1922).
B. Friedrich and D. Herschbach, “Stern and Gerlach: how a bad cigar helped reorient atomic physics”, Physics Today 56(12), 53 (2003).
H. Schmidt-Böcking et al., “The Stern–Gerlach experiment re-examined by an experimenter”, Europhysics News 50(3), 15 (2019).
H. E. White, Introduction to Atomic Spectra (McGraw-Hill); A. Beiser, Concepts of Modern Physics; C. J. Foot, Atomic Physics (Oxford).