Key Equations

Nuclear Structure & Radiation — AQA A-Level Physics

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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).
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.
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.
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.
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.
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.
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