Nuclear radius (A-level only) — AQA A-Level Physics

The nuclear radius is a fundamental property of atomic nuclei, describing the effective size of the nucleus. It is determined experimentally through scattering experiments, most notably using high-energy electrons or alpha particles. The radius is found to follow the empirical relationship R = r0 * A^(1/3), where r0 is approximately 1.2 fm and A is the nucleon number. This topic is crucial for understanding nuclear structure, density, and the limits of the strong nuclear force.

In the AQA A-Level Physics specification, this topic falls under 'Nuclear Physics' and is essential for explaining phenomena such as nuclear stability, alpha decay, and the liquid-drop model. It also connects to other areas like particle physics and quantum mechanics, particularly the wave-particle duality of electrons used in scattering. Mastery of this topic enables students to perform calculations, interpret experimental data, and explain why nuclear density remains constant across all elements.

• →Nuclear radius R is proportional to the cube root of the mass number A: R = r0 * A^(1/3), with r0 ≈ 1.2 fm.
• →Nuclear density is constant for all nuclei because volume is proportional to A, leading to density independent of A.
• →Electron scattering is used to measure nuclear radius because electrons interact electromagnetically and have a de Broglie wavelength small enough to resolve the nucleus.
• →The first minimum in the electron diffraction pattern is related to the nuclear radius by sin(theta) = 0.61 * lambda / R.
• →Alpha particle scattering (Rutherford scattering) provided early evidence for a small, dense nucleus but is less precise for radius measurement due to the strong force and finite size of alpha particles.

• States that for a head-on alpha collision all initial kinetic energy becomes electric potential energy at the distance of closest approach.
• Uses Coulomb potential energy U = (1/4πε₀)(q₁q₂/d) with q₁ = 2e for the alpha and q₂ = Ze for the nucleus.
• Rearranges to d = 2Ze²/(4πε₀Ek) and explains that d is an upper estimate because the alpha never actually touches the nuclear surface.
• Explains that electron diffraction determines radius from the angular position of the first minimum in the scattered electron intensity pattern.
• Uses the de Broglie relation λ = h/p, with p from accelerating voltage via eV = ½mv², to find the electron wavelength.
• Applies sinθ ≈ 1.22λ/(2R) or equivalent geometry to calculate R from the measured first-minimum angle.
• Compares methods: alpha scattering gives only an estimate limited by the point-charge and strong-force approximations, whereas electron diffraction gives a more precise value because electrons interact with the nuclear charge distribution.
• States that nuclear radii are of the order of 10⁻¹⁵ m to 10⁻¹⁴ m, or 1 fm to 10 fm.
• Recalls the relationship R = R₀A^(1/3) with R₀ approximately 1.2 fm.
• Uses the relationship to calculate a typical radius for a given nucleon number A.
• Explains that radius increases with the cube root of nucleon number, so volume is proportional to A.
• Deduces that nuclear density is approximately constant across different nuclei because mass and volume both scale with A.
• Compares nuclear size with atomic size, noting that nuclear radii are about 10⁻⁵ to 10⁻⁴ times atomic radii.
• State that at closest approach the alpha particle's kinetic energy is fully converted into electrostatic potential energy.
• Use the Coulomb potential energy expression E = kQq/r, with k = 1/(4πε₀), and equate it to ½mv².
• Substitute Q = Ze for the nucleus and q = 2e for the alpha particle, then rearrange to make r the subject.
• Recognise that the calculated distance is an upper limit for the nuclear radius because the alpha particle does not penetrate the nucleus.
• Carry out the arithmetic with consistent SI units, converting MeV to joules where necessary.
• State the relationship R = r₀A^(1/3) and identify r₀ as a constant.
• Cube the radius relationship to show R³ ∝ A.
• Substitute R³ into the sphere volume formula V = (4/3)πR³ to obtain V ∝ A.
• Explain that nuclear mass is approximately proportional to A, so density ρ = m/V is approximately constant.
• Use the constant-density conclusion to compare different nuclei or to estimate a nuclear radius from a known value.
• States the nuclear radius relationship R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m and identifies A as the nucleon number.
• Calculates the nuclear volume using V = (4/3)πR³, substituting the expression for R to obtain V = (4/3)πr₀³A.
• Uses nuclear mass m ≈ A × 1.66 × 10⁻²⁷ kg, recognising that the mass of a nucleus is approximately A atomic mass units.
• Divides mass by volume and cancels A to show ρ = (1.66 × 10⁻²⁷)/((4/3)πr₀³), giving a value of about 2.3 × 10¹⁷ kg m⁻³.
• Explains that because A cancels, nuclear density is approximately the same for all nuclei, independent of nucleon number.
• Describes the shape of the graph: a large central maximum at zero angle, followed by minima and smaller secondary maxima at larger angles.
• Identifies that the first minimum occurs at an angle related to the nuclear diameter by sin θ ≈ λ/D, where λ is the electron de Broglie wavelength.
• Explains that higher electron energy gives a shorter de Broglie wavelength and therefore a narrower diffraction pattern with the first minimum at a smaller angle.
• Uses the position of the first minimum to estimate the nuclear diameter or radius, giving values of the order of 10⁻¹⁵ m.
• Recognises that the diffraction pattern is evidence for the wave nature of electrons and for the small size of the nucleus.

• 💡Write the energy-balance equation first, then substitute values, so the method is clear even if arithmetic slips.
• 💡When using electron diffraction, calculate the electron momentum from the accelerating voltage before finding λ.
• 💡State explicitly why the alpha-scattering result is an estimate, as this is often worth a marking point.
• 💡Quote R₀ ≈ 1.2 fm when using R = R₀A^(1/3), as the value may be expected in the answer.
• 💡Check whether the question asks for a value in metres or femtometres and convert consistently.
• 💡Use the cube-root relationship to justify constant nuclear density when asked to explain it.
• 💡Write the energy balance equation first, then substitute values, so the method is clear even if arithmetic slips.
• 💡Check that every charge is expressed as a multiple of e and converted to coulombs before calculating.
• 💡State explicitly that the result is an estimate of the upper limit of the nuclear radius, not an exact measurement.
• 💡Show the intermediate step R³ ∝ A before substituting into the volume formula, so the proportional reasoning is visible.
• 💡Use the phrase 'approximately proportional' for mass and A, since binding energy causes a small difference.
• 💡When comparing two nuclei, write the ratio of densities and cancel the constants to demonstrate constancy.
• 💡Write the full algebraic expression for density before substituting numbers, so the cancellation of A is visible and can earn method credit.
• 💡Keep powers of ten separate from the numerical calculation to reduce arithmetic errors when cubing r₀.
• 💡State the final answer with the unit kg m⁻³ and an appropriate number of significant figures, typically two or three.
• 💡Sketch the graph with intensity on the vertical axis and angle on the horizontal axis, labelling the central maximum and first minimum clearly.
• 💡Quote the relationship sin θ ≈ λ/D when explaining how the graph is used to estimate nuclear size.
• 💡Link changes in electron energy to changes in wavelength and then to changes in the angular width of the pattern.
• 💡Always write the formula R = r0 * A^(1/3) and define r0 as the proportionality constant with value ~1.2 fm. When calculating, show all steps clearly, including cubing or cube rooting as needed.
• 💡In explanations, explicitly state that electrons are used because they are point-like and interact electromagnetically, and that high energies are needed to produce a de Broglie wavelength smaller than the nucleus.
• 💡When interpreting diffraction patterns, remember that the angle of the first minimum is used, and the formula sin(theta) = 0.61 * lambda / R is an approximation. Do not confuse the angle with the scattering angle in Rutherford scattering.

• Using the alpha particle charge as e instead of 2e; correct by remembering an alpha is a helium nucleus with charge 2e.
• Forgetting to convert kinetic energy from MeV or keV into joules before substituting; correct by multiplying by 1.60 × 10⁻¹³ J per MeV.
• Treating the closest-approach distance as the exact nuclear radius; correct by stating it is an upper limit because the alpha is repelled before reaching the surface.
• Writing R ∝ A instead of R ∝ A^(1/3); correct by remembering volume is proportional to A, so radius scales as the cube root.
• Using R₀ = 1.2 × 10⁻¹⁰ m; correct by using R₀ ≈ 1.2 × 10⁻¹⁵ m.
• Confusing nuclear radius with atomic radius; correct by noting atomic radii are around 10⁻¹⁰ m, roughly 10⁵ times larger.
• Using the force equation F = kQq/r² instead of the potential energy equation when equating to kinetic energy; correct by using E = kQq/r for the energy balance.
• Forgetting that the alpha particle carries a charge of 2e rather than e; correct by substituting q = 2e.
• Leaving kinetic energy in MeV while r is required in metres; correct by converting MeV to joules using 1 MeV = 1.6 × 10⁻¹³ J.
• Treating R as directly proportional to A rather than to A^(1/3); correct by cubing both sides to relate volume to A.
• Assuming nuclear density increases with nucleon number; correct by showing that V ∝ A and m ∝ A, so density is constant.
• Using the electron radius or atomic radius in place of the nuclear radius; correct by using r₀ ≈ 1.2 × 10⁻¹⁵ m for nuclear calculations.
• Using the atomic radius (about 10⁻¹⁰ m) instead of the nuclear radius (about 10⁻¹⁵ m); the correction is to use R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m.
• Forgetting to cube the radius when calculating volume; the correction is to apply V = (4/3)πR³ fully, including cubing the numerical value of R.
• Treating the mass of the nucleus as A grams rather than A atomic mass units in kilograms; the correction is to convert using 1 u ≈ 1.66 × 10⁻²⁷ kg.
• Confusing the angle of the first minimum with the angle of the first maximum; the correction is to identify the first minimum as the first point of zero intensity after the central peak.
• Assuming the pattern broadens as electron energy increases; the correction is that higher energy means shorter wavelength and a narrower pattern.
• Using the wavelength of visible light instead of the electron de Broglie wavelength; the correction is to calculate λ = h/p from the accelerated electron momentum.
• Students often think nuclear radius is directly proportional to mass number (R ∝ A) rather than R ∝ A^(1/3). Correction: The volume is proportional to A, so radius is proportional to A^(1/3).
• Students may believe that electrons are scattered by the strong nuclear force. Correction: Electrons are leptons and do not experience the strong force; they scatter via the electromagnetic force.
• Students sometimes confuse nuclear radius with atomic radius, which is about 10^5 times larger. Correction: Nuclear radius is of the order 10^-15 m, while atomic radius is about 10^-10 m.

1. 1Start by reviewing the Rutherford alpha-particle scattering experiment and its limitations, then learn the electron scattering method and why it is superior for measuring nuclear radius.
2. 2Memorise the formula R = r0 * A^(1/3) and practice calculations for different nuclei, ensuring you can convert between fm and m and handle cube roots.
3. 3Understand the derivation that nuclear density is constant: density = mass/volume = (A * m_proton) / ((4/3) * pi * R^3) = (A * m_proton) / ((4/3) * pi * r0^3 * A) = constant.
4. 4Work through past paper questions on electron scattering, including those that require using the diffraction formula to estimate R from given data.
5. 5Create a summary sheet comparing alpha scattering and electron scattering, highlighting the advantages of electrons and the meaning of the first minimum in the diffraction pattern.

• 📋Calculation questions: Given A and r0, calculate the nuclear radius or density. Advice: Write down the formula, substitute values, and give units.
• 📋Explanation questions: Explain why electron scattering is used instead of alpha scattering to determine nuclear radius. Advice: Mention electromagnetic interaction, point-like nature, and de Broglie wavelength.
• 📋Data analysis questions: Interpret a graph of intensity against angle for electron scattering and identify the first minimum to calculate R. Advice: Use the formula sin(theta) = 0.61 * lambda / R and ensure consistent units.
• 📋Derivation questions: Show that nuclear density is independent of mass number. Advice: Start from density = mass/volume, substitute R = r0 * A^(1/3) and mass = A * u, and simplify.