IB DP Physics: Atomic structure, scattering, spectra and Bohr model (DP Physics) – Revision Notes

Condensed recall for IB DP Physics sub-topic E.1 (Structure of the atom), from the IB Physics guide, first assessment 2025. The core ideas are SL and HL; parts marked HL only are the additional higher level. For full explanations and worked examples use the study guide, and for exam-style practice the practice questions. Not sure where to start? Try the free 10-minute diagnostics. Also see the course hub and the printable checklist.

Constants used below: c = 3.00 × 10⁸ m s⁻¹, h = 6.63 × 10⁻³⁴ J s, e = 1.60 × 10⁻¹⁹ C, k = 8.99 × 10⁹ N m² C⁻², u = 1.66 × 10⁻²⁷ kg and m(electron) = 9.11 × 10⁻³¹ kg.

Geiger-Marsden-Rutherford experiment (SL and HL)

Observation	Meaning
Nearly all alpha particles go straight through	The atom is largely empty space
A few deflect through large angles	A concentrated positive charge repels them
Very few bounce almost straight back	The charge sits in a tiny, massive region: the nucleus

Set-up: a narrow alpha beam, very thin gold foil, evacuated chamber, movable detector. Conclusion: a small, dense, positive nucleus with electrons around it. At SL, a qualitative account is enough.

Nuclear notation (SL and HL)

Symbol	Meaning
A	Nucleon number (protons + neutrons)
Z	Proton number
X	Chemical symbol
N = A − Z	Neutron number

A neutral atom has Z electrons. You do not need to recall chemical symbols.

Spectra, levels and photons (SL and HL)

Spectrum	Appearance	Cause
Continuous	All wavelengths in a band	Hot, dense solid
Emission	Bright lines on dark	Excited gas: electrons fall to lower levels
Absorption	Dark lines on continuous	Cool gas takes out photons that match its gaps

• Lines at fixed wavelengths are evidence that atomic energy levels are discrete.
• Emission: one photon per transition, with energy equal to the gap between the two levels.
• Absorption: a photon is taken in only if its energy equals a gap up from the current level.
• Photon energy: E = hf. With c = fλ, λ = hc / ΔE.
• Widen the gap and the frequency rises while the wavelength shrinks.
• Each element has its own set of lines, so spectra identify the chemical composition, for example of a star’s atmosphere.

Describing and explaining spectra

• Say what you see: bright lines (emission) or dark lines (absorption), at specific wavelengths.
• Say what it means: the atom has only certain allowed energies, so only certain photon energies can be emitted or absorbed.
• Say how it identifies a gas: each element has a unique set of level gaps, so a unique pattern of lines. Matching a star’s dark lines to laboratory lines shows which elements are in its outer gas.
• Count lines: from a level diagram, count each distinct pair of levels. With three levels there are three possible downward transitions; with four, six.

Worked reminder: levels at −8.0 eV, −3.0 eV and −1.0 eV. The shortest-wavelength line comes from the largest gap, −1.0 to −8.0 eV, which is 7.0 eV. A 4.0 eV photon is not absorbed from the ground state, because no gap equals 4.0 eV (the gaps are 5.0 eV, 7.0 eV and 2.0 eV).

Method for a transition calculation

1. Read the two energy levels and subtract: ΔE = E_upper − E_lower.
2. Convert eV to J by multiplying by 1.60 × 10⁻¹⁹.
3. Find f = ΔE / h.
4. Find λ = c / f if wanted.
5. State whether the photon is emitted (falling) or absorbed (rising).

Nuclear radius and density (HL only)

Result	Formula
Radius	R = R₀A^(1/3)
Volume	Proportional to A
Density	ρ = 3u / (4πR₀³), the same for every nucleus

Worked reminder: with R₀ = 1.2 fm, A = 125 gives R = 1.2 × 5 = 6.0 fm. Doubling A multiplies R by 2^(1/3) = 1.26, not 2. Constant density means closely packed nucleons. Typical value: about 2.3 × 10¹⁷ kg m⁻³ when R₀ = 1.2 fm.

Closest approach and high energies (HL only)

• Head-on, point charges, nucleus fixed, only the electric force: all the initial kinetic energy is stored as electric potential energy at the closest point.
• d = 2kZe² / E_k for an alpha particle (charge 2e) and a nucleus of proton number Z.
• Higher energy gives a smaller d.
• When d becomes comparable to the nuclear radius, the strong force acts, and measured scattering deviates from the prediction that uses only electric repulsion.

Worked reminder: a 6.0 MeV alpha particle fired at a nucleus with Z = 20 reaches d = 2 × 20 × 1.44 / 6.0 = 9.6 fm. If the nucleus has A = 40, then R = 1.2 × 40^(1/3) = 4.1 fm, so d is more than twice R and the electric force alone describes the scattering.

Useful shortcut: ke² = 1.44 MeV fm. Then d (fm) = 2 × Z × 1.44 / E_k (in MeV) for an alpha particle.

Method in steps:

1. Write the energy balance: E_k = kq₁q₂/d.
2. Convert the energy to joules (MeV × 1.60 × 10⁻¹³).
3. Rearrange for d and substitute the charges 2e and Ze.
4. Compare d with R = R₀A^(1/3) to decide whether the electric force alone applies.

Bohr model of hydrogen (HL only)

Result	Formula
Energy levels	E = −13.6 / n² eV
Angular momentum	mvr = nh / 2π
Ground state	n = 1, E = −13.6 eV
Ionised	E = 0

• Only orbits with whole-number n exist, so only certain energies exist.
• Photon energy in a transition: ΔE = 13.6 (1/n_lower² − 1/n_upper²) eV.
• Worked reminder: n = 3 to n = 2 gives 13.6 × (1/4 − 1/9) = 1.89 eV, which is a photon of wavelength about 658 nm.

Must-know distinctions

• Emission versus absorption spectrum: bright lines on dark versus dark lines on bright, at the same wavelengths for one gas.
• Photon energy versus level energy: the photon carries the difference, not either level.
• Z versus A: Z fixes the element, A counts all nucleons.
• Closest approach versus nuclear radius: d is where the alpha stops; it equals R only at the threshold for deviations.
• SL versus HL: the experiment and spectra are SL and HL; R = R₀A^(1/3), closest approach and Bohr are HL only.

Quick self-test

1. A nuclide has A = 127 and Z = 53. How many neutrons does it have?
2. What does the very small number of alpha particles scattered almost straight back tell you?
3. Why does the absorption spectrum of a gas have dark lines at the same wavelengths as that gas’s emission lines?
4. Find the frequency and wavelength of a 2.5 eV photon.
5. (HL only) Find the radius of a nucleus with A = 343 when R₀ = 1.2 fm.
6. (HL only) A nucleus has radius 7.2 fm. Find A.
7. (HL only) Find the distance of closest approach of a 4.0 MeV alpha particle fired head-on at a nucleus with Z = 50.
8. (HL only) Find the photon wavelength for the hydrogen transition n = 6 to n = 2.
9. (HL only) Which Bohr level of hydrogen has energy −0.85 eV?
10. (HL only) Find the angular momentum of the electron in the n = 3 orbit.

Answers

1. N = 127 − 53 = 74 neutrons.
2. The positive charge is concentrated in a very small, massive region; a spread-out charge could not reverse the particle.
3. The gas absorbs photons whose energies match its own level gaps, which are the same gaps that give its emission lines.
4. E = 2.5 × 1.60 × 10⁻¹⁹ = 4.0 × 10⁻¹⁹ J. f = 4.0 × 10⁻¹⁹ / 6.63 × 10⁻³⁴ = 6.03 × 10¹⁴ Hz. λ = 3.00 × 10⁸ / 6.03 × 10¹⁴ = 4.97 × 10⁻⁷ m.
5. R = 1.2 × 343^(1/3) = 1.2 × 7 = 8.4 fm.
6. A = (7.2 / 1.2)³ = 6³ = 216.
7. d = 2 × 50 × 1.44 / 4.0 = 36 fm (3.6 × 10⁻¹⁴ m).
8. ΔE = 13.6 (1/4 − 1/36) = 3.02 eV = 4.84 × 10⁻¹⁹ J. λ = hc / ΔE = 6.63 × 10⁻³⁴ × 3.00 × 10⁸ / 4.84 × 10⁻¹⁹ = 4.11 × 10⁻⁷ m (411 nm).
9. n² = 13.6 / 0.85 = 16, so n = 4.
10. L = nh / 2π = 3 × 6.63 × 10⁻³⁴ / 2π = 3.17 × 10⁻³⁴ J s.

Where marks are usually lost

• Using a level energy instead of the energy difference for the photon.
• Leaving energies in eV when substituting into E = hf.
• Describing the experiment without saying what each observation shows about the atom.
• Writing that electrons were found to orbit a nucleus, when the evidence is about the nucleus.
• Saying the absorption lines are “missing colours” without linking them to level gaps.
• Treating the nuclear radius formula as linear in A.
• Forgetting the charge 2e of the alpha particle in the closest-approach calculation.
• Giving the Bohr energy with a positive sign.
• Mixing up n in the energy formula with the number of photons or lines.

Official syllabus

IB Physics guide, first assessment 2025, published by the International Baccalaureate Organization: sub-topic E.1 Structure of the atom.