Nuclear Structure & Radiation

Rutherford scattering, the nuclear model, properties of alpha, beta and gamma radiation, nuclear instability, decay equations, and nuclear radius.

Spec Points Covered
  • Describe Rutherford scattering and explain what each observation reveals about atomic structure.
  • State the properties of alpha, beta and gamma radiation including ionising ability, penetrating powerThe rate of energy transfer. Measured in watts (W). and deflection in fields.
  • Write balanced nuclear decay equations for alpha, beta-minus, beta-plus emission and electron capture.
  • Use the N-Z graph to predict which decay mode an unstable nucleus will undergo.
  • Describe how electron diffraction and closest approach methods are used to estimate nuclear radius.
  • Apply $R = R_{0} A^{1/3}$ to calculate nuclear radius and show that nuclear densityMass per unit volume of a material. Measured in kg m⁻³. is constant.
Notes
01 Rutherford fired alpha particles at gold foil to probe atomic structure Rutherford scattering 3.8.1.1 → 02 The nuclear model replaced the plum pudding model through experimental evidence 3.8.1.1 → 03 Alpha particles are helium nuclei: highly ionising, weakly penetrating Alpha particle 3.8.1.2 → 04 Beta-minus is an electron emitted when a neutron converts to a proton Beta-minus particle 3.8.1.2 → 05 Beta-plus is a positron emitted when a proton converts to a neutron Beta-plus particle 3.8.1.2 → 06 Gamma radiation is an electromagnetic wave: weakly ionising, highly penetrating Gamma radiation 3.8.1.2 → 07 Comparing alpha, beta and gamma: a summary table 3.8.1.2 → 08 Background radiation is always present and must be subtracted from measurements Background radiation 3.8.1.2 → 09 Radiation safety: minimise time, maximise distance, use shielding 3.8.1.2 → 10 Applications of radiation: smoke detectors use alpha, thickness gauges use beta 3.8.1.2 → 11 The N-Z graph shows which nuclei are stable and predicts their decay mode Nuclear stability graph 3.8.1.4 → 12 Alpha decay reduces A by 4 and Z by 2 ${}^{A}_{Z}X \rightarrow {}^{A-4}_{Z-2}Y + {}^{4}_{2}\alpha$ 3.8.1.3 → 13 Beta-minus decay keeps A constant and increases Z by 1 ${}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + {}^{0}_{-1}\beta + \bar{\nu}_e$ 3.8.1.3 → 14 Beta-plus decay keeps A constant and decreases Z by 1 ${}^{A}_{Z}X \rightarrow {}^{A}_{Z-1}Y + {}^{0}_{+1}\beta + \nu_e$ 3.8.1.3 → 15 Electron capture converts a proton to a neutron using an orbital electron Electron capture 3.8.1.3 → 16 Gamma emission releases energy without changing the nucleus composition 3.8.1.4 → 17 Closest approach gives an upper limit for nuclear radius $E_k = \frac{Qq}{4\pi\varepsilon_0 r}$ 3.8.1.5 → 18 Electron diffraction gives a more accurate measurement of nuclear radius $\sin \theta = 1.22 \frac{\lambda}{2R}$ 3.8.1.5 → 19 Nuclear radius depends on mass number: R = R₀\(A^{1/3}\) $R = R_0 A^{1/3}$ 3.8.1.5 → 20 Nuclear density is constant and independent of the size of the nucleus $\rho = \frac{3u}{4\pi R_0^3}$ 3.8.1.5 → 21 Required practical: verifying the inverse square law for gamma radiation 3.8.1.2 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Nuclear radius equation
$$R = R_0 A^{1/3}$$
  • Where:
    • $R$ = nuclear radius (m)
    • $R₀$ = constant ≈ 1.05 fm
    • $A$ = nucleon (mass) number
Verified by plotting R vs \(A^{1/3}\) (gradient = R₀) or ln R vs ln A (gradient = 1/3, intercept = ln R₀).
Inverse square law for gamma radiation
$$I = \frac{k}{x^2}$$
  • Where:
    • $I$ = intensity (W m⁻²)
    • $k$ = constant of proportionality
    • $x$ = distance from source (m)
Applies to gamma only. Can substitute count rate C for intensity I.
de Broglie wavelength
$$\lambda = \frac{h}{mv}$$
  • Where:
    • $λ$ = de Broglie wavelength (m)
    • $h$ = Planck's constant (J s)
    • $m$ = mass of electron (kg)
    • $v$ = speed of electron (m s⁻¹)
As speed increases, wavelength decreases. Used to determine if electrons will diffract around nuclei.
Closest approach (kinetic energy = potential energy)
$$E_k = \frac{Qq}{4\pi\varepsilon_0 r}$$
  • Where:
    • $Eₖ$ = kinetic energy of alpha particle (J)
    • $Q$ = charge of alpha particle = 2e (C)
    • $q$ = charge of target nucleus = Ze (C)
    • $ε₀$ = permittivity of free space (F m⁻¹)
    • $r$ = distance of closest approach (m)
Derived by equating Eₖ = ½mv² to Coulomb potential energy. Gives an upper limit for nuclear radius.
Electron diffraction first minimum
$$\sin \theta = 1.22 \frac{\lambda}{2R}$$
  • Where:
    • $θ$ = angle of first minimum (°)
    • $λ$ = de Broglie wavelength (m)
    • $R$ = nuclear radius (m)
The 1.22 factor accounts for circular aperture diffraction.
Nuclear density
$$\rho = \frac{3u}{4\pi R_0^3}$$
  • Where:
    • $ρ$ = nuclear density (kg m⁻³)
    • $u$ = atomic mass unit = 1.661 × 10⁻²⁷ kg
    • $R₀$ = constant ≈ 1.05 × 10⁻¹⁵ m
Derived from ρ = m/V = Au / (4/3)πR₀³A. The A cancels, proving density is constant at ~3.4 × 10¹⁷ kg m⁻³.
Q Retrieval Practice All 15 Questions →
Q1. What were the three key observations from Rutherford's alpha scattering experiment?
  • Most alpha particles passed straight through (atom is mostly empty space).
  • Some were deflected through small angles (positive chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). concentrated at the centre).
  • Very few bounced back at >90° (nucleus is extremely small, dense, and contains most of the mass).
Q2. State the properties of an alpha particle.
  • Helium nucleus: 2 protons + 2 neutrons.
  • Mass = 4 u, chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). = +2e.
  • Highly ionising (~10,000 ion pairs per cm), weakly penetrating (3-7 cm range in air, stopped by paper).
Q3. State the properties of a beta-minus particle.
  • High-energyThe capacity to do work. Measured in joules (J). electron emitted from the nucleus.
  • ChargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). = -e.
  • Moderately ionising (~100 ion pairs per cm), moderately penetrating (20 cm - 3 m in air, stopped by ~3 mm aluminium).
Q4. What is the difference between beta-plus emission and electron capture?
  • Both convert a proton to a neutron.
  • Beta-plus: a positron and neutrino are emitted.
  • Electron capture: an orbital electron is absorbed by the nucleus, emitting a neutrino and often a gamma ray.
  • Both decrease Z by 1 and increase N by 1.
Q5. Why does the inverse square law apply to gamma radiation but not alpha or beta?
  • Gamma is not easily absorbed, so it spreads out uniformly as a sphere.
  • Alpha and beta are absorbed quickly by matter before they can spread out.