Topic E.1 — Structure of the Atom — opens the IB Physics quantum & nuclear block, and it is one of the most reliably examined nuclear topics across Paper 1, Paper 2 and Paper 3. The SL portion (Geiger–Marsden, nuclear notation, atomic spectra) returns every year as conceptual MCQs, while the HL extension (nuclear radius and density, deviations from Rutherford, the Bohr model) carries the heavier calculation marks.
This cheatsheet condenses every formula, definition, trick and trap for Topic E.1 SL + HL onto one page you can revise from before any mock or final paper. Scroll to the bottom for the printable PDF, the full notes pack, and the gated tutorial library used by Photon Academy students in Singapore.
§1 — Geiger-Marsden-Rutherford Experiment E.1 SL+HL
Setup and observations
Setup: $\alpha$ source $\rightarrow$ lead collimator $\rightarrow$ thin gold foil $\rightarrow$ ZnS scintillation screen; all in vacuum.
| Observation | Conclusion |
|---|
| Most $\alpha$ pass straight through | Atom is mostly empty space |
| Small fraction deflected $> 90^\circ$ | Positive charge & mass concentrated in tiny nucleus |
| Very few ($\sim$ 1 in 8000) back-scatter | Nucleus is extremely small and dense |
TrapThe Geiger-Marsden experiment does NOT provide evidence for discrete energy levels. That requires spectral line evidence. Only the empty-space and concentrated positive-charge claims can be deduced. (IB May 2023 P1 Q22 — answer A.)
NoteScale: atomic radius $\sim 10^{-10}$ m; nuclear radius $\sim 10^{-15}$ m; ratio $\sim 10^5$. "If the nucleus were a football, the atom would be 30 km across."
§2 — Nuclear Notation & Structure E.1 SL+HL
Nuclear notation:$^{A}_{Z}\text{X}$ with $A =$ nucleon number, $Z =$ proton number, $N = A - Z$.
Particle summary
| Particle | Charge | Mass | Location |
|---|
| Proton | $+e$ | $\approx 1$ u | In nucleus |
| Neutron | $0$ | $\approx 1$ u | In nucleus |
| Electron | $-e$ | $\approx m_p / 1836$ | Orbits nucleus |
Isotopes: same $Z$, different $A$ (same element, different neutron count). Atomic mass unit: $1\,\text{u} = 1.661 \times 10^{-27}$ kg $= \tfrac{1}{12}$ mass of a $^{12}_{6}$C atom.
TrickIn nuclear reactions: conserve $A$ (top numbers) AND $Z$ (bottom numbers) separately. The $\gamma$ photon carries no $A$ or $Z$.
§3 — Atomic Spectra & Photon Transitions E.1 SL+HL
Photon energy:$E = hf = \dfrac{hc}{\lambda}, \quad h = 6.626 \times 10^{-34}\,\text{J s}$
Transition energy:$E_{\text{photon}} = E_{\text{upper}} - E_{\text{lower}}$ ($E_{\text{upper}} > E_{\text{lower}}$)
Emission vs absorption spectra
- Emission: hot gas emits; bright discrete lines on a dark background.
- Absorption: white light through a cool gas; dark lines at the same $\lambda$ as the emission lines.
- Why discrete? Electrons occupy only specific energy levels; transitions produce only specific $\Delta E$ values $\Rightarrow$ only specific $\lambda$.
- Chemical fingerprinting: each element has a unique set of energy levels, so spectra identify elements in stars, nebulae and distant galaxies.
NoteAbsorption spectrum of the Sun (Fraunhofer lines) lets us identify the composition of its atmosphere. Helium was discovered in the Sun before it was found on Earth.
TrapDo NOT say electrons absorb "any" energy and release "some" later. They absorb photons of exactly the right energy to bridge an energy gap. Any other photon energy passes straight through.
§4 — Nuclear Radius & Density E.1 HL only
Nuclear radius:$R = R_0 A^{1/3}$, $R_0 = 1.2 \times 10^{-15}\,\text{m} = 1.2$ fm
Nuclear density:$\rho = \dfrac{m_u}{\tfrac{4}{3}\pi R_0^3} \approx 2.3 \times 10^{17}\,\text{kg m}^{-3}$ (constant for ALL nuclei)
TrickNuclear density is CONSTANT because $V = \tfrac{4}{3}\pi R_0^3 A$ and mass $= A m_u$, so $\rho = m/V = m_u/(\tfrac{4}{3}\pi R_0^3)$ — the $A$ cancels. Exam question types: find the radius of nucleus B given nucleus A using $R \propto A^{1/3}$, or confirm density is the same for different nuclei.
TrapDo NOT say density increases with $A$. Nuclear density is the SAME for all nuclei (approximately) — about $10^{14}$ times the density of ordinary matter.
Quick ratio method
To compare radii of two nuclei $X$ and $Y$:
$$\frac{R_Y}{R_X} = \left(\frac{A_Y}{A_X}\right)^{1/3}$$
Example: $A_X = 2$ (deuterium), $A_Y = 16$ (oxygen-16): $R_Y/R_X = (16/2)^{1/3} = 8^{1/3} = 2$. So $R_Y = 2 R_X$. Densities identical.
§5 — Deviations from Rutherford Scattering E.1 HL only
- Low-energy $\alpha$ particles: pure Coulomb (electric) repulsion only. Rutherford formula works perfectly.
- High-energy $\alpha$ particles: alpha enters nuclear range $\Rightarrow$ strong nuclear force acts $\Rightarrow$ fewer large-angle deflections than Rutherford predicts.
What deviations tell us:
- Evidence for the strong nuclear force. ✓
- Allow estimation of the nuclear radius (energy at onset of deviation gives $d \approx R$). ✓
- Deviations occur at low energies — FALSE (only at HIGH energy). ✗
Closest approach:$d = \dfrac{k\, q_\alpha\, q_Z}{E_k} = \dfrac{k(2e)(Ze)}{E_k}$
TrickAt closest approach ALL kinetic energy converts to electrostatic PE: $E_k = k q_1 q_2 / d$. Just rearrange for $d$. Only the electric force matters in the IB derivation.
Trap$d$ is measured from the centre of the nucleus. If $d \gg R_{\text{nucleus}}$ Rutherford applies; if $d \approx R_{\text{nucleus}}$ expect deviations.
§6 — The Bohr Model of Hydrogen E.1 HL only
Energy levels:$E_n = \dfrac{-13.6}{n^2}\,\text{eV}$, $n = 1$ (ground), $2, 3, \ldots$
Angular momentum:$mvr = \dfrac{nh}{2\pi} = n\hbar$
Orbital radius:$r_n = n^2 a_0$, $a_0 = 0.529$ Å $= 5.29 \times 10^{-11}$ m
Orbital speed:$v_n \propto 1/n$ $\Rightarrow$ higher $n$ $\Rightarrow$ slower speed
Hydrogen energy levels (memorise)
| $n$ | $E_n$ | Name |
|---|
| 1 | $-13.6$ eV | Ground state |
| 2 | $-3.40$ eV | 1st excited |
| 3 | $-1.51$ eV | 2nd excited |
| 4 | $-0.85$ eV | 3rd excited |
| $\infty$ | 0 eV | Ionised |
Series: Lyman (down to $n = 1$, UV); Balmer (down to $n = 2$, visible); Paschen (down to $n = 3$, IR). Ionisation energy from level $n$: $E_{\text{ion}} = |E_n| = 13.6/n^2$ eV.
TrickAngular momentum ratio: $L_n = n\hbar$, so $L_2/L_1 = 2$. Speed ratio: $v_n \propto 1/n$, so $v_2/v_1 = 1/2$ (use $mvr = nh/2\pi$ with $r_2 = 4 r_1$). Radius ratio: $r_n \propto n^2$, so $r_2/r_1 = 4$.
TrapThe Bohr model ONLY works for hydrogen (one-electron atoms). Do not apply $-13.6/n^2$ to helium or heavier atoms. The model also contradicts classical electrodynamics but gives correct spectral predictions for H.
Worked Example — Closest Approach & Bohr Transition
Question (HL Paper 2 style — 6 marks)
(a) An $\alpha$ particle with kinetic energy $E_k = 5.00$ MeV is fired head-on at a gold nucleus ($Z = 79$). Calculate the distance of closest approach $d$. [3]
(b) An electron in a hydrogen atom drops from $n = 4$ to $n = 2$. Calculate the wavelength of the emitted photon and state the spectral series. [3]
Solution
- Convert $E_k$: $E_k = 5.00 \times 10^6 \times 1.60 \times 10^{-19} = 8.00 \times 10^{-13}$ J. (M1)
- Apply $E_k = k(2e)(Ze)/d \Rightarrow d = k(2e)(Ze)/E_k$. (M1)
- $d = \dfrac{(8.99 \times 10^9)(2)(79)(1.60 \times 10^{-19})^2}{8.00 \times 10^{-13}} \approx 4.55 \times 10^{-14}$ m. (A1) (Note: $d \gg R_{\text{Au}} \approx 7.0 \times 10^{-15}$ m, so Rutherford scattering still applies.)
- $\Delta E = E_4 - E_2 = -0.85 - (-3.40) = 2.55$ eV $= 4.08 \times 10^{-19}$ J. (M1)
- $\lambda = hc/\Delta E = (6.63 \times 10^{-34})(3.00 \times 10^8)/(4.08 \times 10^{-19}) \approx 4.87 \times 10^{-7}$ m $\approx 487$ nm. (A1)
- This is in the visible range with the lower level $n = 2$, so the line belongs to the Balmer series. (A1)
Examiner's note: Two classic mistakes — (i) forgetting that $E_n$ is negative when computing $\Delta E$, and (ii) using the alpha kinetic energy in MeV directly without converting to joules. Always convert to SI before plugging into $hc/\lambda$.
Common Student Questions
What does the Geiger-Marsden experiment actually prove?
Only two things: (i) the atom is mostly empty space, and (ii) positive charge and almost all the mass are concentrated in a tiny, dense nucleus. It does not prove discrete energy levels — that requires spectral line evidence. This is the IB May 2023 Paper 1 Q22 trap (correct answer: A). If a multiple choice option mentions "energy levels" alongside Geiger-Marsden, eliminate it.
Is nuclear density the same for all nuclei?
Yes. Because $R = R_0 A^{1/3}$, the nuclear volume is proportional to $A$, and the mass is also proportional to $A$, so density $\rho = m/V$ cancels the $A$ and stays constant at about $2.3 \times 10^{17}$ kg m$^{-3}$ for every nucleus. Saying density rises with $A$ is a guaranteed mark loss in any HL E.1 question.
When does Rutherford scattering break down?
At high alpha energies — when the alpha particle gets close enough that the strong nuclear force starts to act. You then see fewer large-angle deflections than the pure Coulomb (Rutherford) prediction. This deviation gives evidence for the strong nuclear force AND lets you estimate the nuclear radius (the closest-approach distance at the energy of onset is roughly $R$).
Can I use $E_n = -13.6/n^2$ for helium or heavier atoms?
No. The Bohr model only works for hydrogen and other one-electron systems (e.g. He$^+$, Li$^{2+}$). Applying $-13.6/n^2$ to neutral helium or heavier atoms is wrong. The IB only ever asks Bohr-level calculations on hydrogen — if a question gives you helium, expect to be using energy-level diagrams from the data booklet, not the $-13.6/n^2$ formula.
What is the difference between emission and absorption spectra?
Emission: a hot gas emits bright discrete lines on a dark background. Absorption: white light passes through a cool gas, leaving dark lines on a continuous spectrum — at the same wavelengths as the emission lines for the same element. Both arise because electrons can only occupy discrete energy levels, so only specific photon energies are absorbed or emitted. The same element therefore "fingerprints" both ways.
What's NOT in this cheatsheet
This page gives you the formulas and the traps. The full Photon Academy E.1 Structure of the Atom library (only available to enrolled students or via the resource library subscription) adds:
- 40-page Notes PDF — every concept worked through in full, with derivations and intuition.
- Tutorial booklet — 30+ IB-style questions sequenced from foundation to AHL difficulty.
- Tutorial Solutions — full mark-scheme-style worked solutions with M1/A1/R1 annotations.
- Practice Solutions — extra past-paper-style problems with detailed walk-throughs.
- Cheatsheet PDF — print-ready, brand-formatted, the same one our students take into mock exams.
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