Alpha Decay Explained via Gamow and the WKB Method

Alpha decay, the emission of a helium nucleus (\( \alpha \)-particle) from a heavy nucleus, was initially mysterious because classically the alpha particle’s energy is insufficient to overcome the Coulomb barrier. Gamow (1928) used quantum mechanics and the concept of tunneling to explain how alpha particles escape.

1. Nuclear Potential and Physical Setup

Consider an alpha particle inside a nucleus of charge \( Z \) and mass number \( A \), leaving a daughter nucleus with charge \( Z_d = Z-2 \). The radial potential is:

  • Inside the nucleus: attractive nuclear potential \( V(r) \approx -V_0 \).
  • Outside the nucleus: Coulomb potential \( V(r) = \dfrac{Z_d Z_\alpha e^2}{4\pi \varepsilon_0 r} \), where \( Z_\alpha = 2 \).

The alpha particle energy \( E \) is much less than the barrier, so classical escape is impossible.

2. Schrödinger Equation and WKB Approximation

For radial motion (\( l = 0 \)), the Schrödinger equation is:

\[
-\frac{\hbar^2}{2m} \frac{d^2 u}{dr^2} + V(r) u = E u, \quad u(r) = r \psi(r)
\]
\[
k(r) = \frac{\sqrt{2m |E-V(r)|}}{\hbar}, \quad
\text{allowed: } E>V \Rightarrow k = \frac{\sqrt{2m(E-V)}}{\hbar}, \quad
\text{forbidden: } E \]
\[
\text{Turning points: } r_1 \approx R, \quad r_2 = \frac{Z_d Z_\alpha e^2}{4\pi \varepsilon_0 E}
\]

WKB gives the tunneling probability:

\[
T \approx \exp\left[-2 \int_{r_1}^{r_2} \kappa(r) \, dr \right] = \exp[-2G], \quad
G \approx \pi \eta – k R
\]
\[
\eta = \frac{Z_d Z_\alpha e^2}{\hbar v}, \quad
k = \frac{\sqrt{2 m E}}{\hbar}, \quad
R = r_0 A^{1/3}, \quad r_0 = 1.2 \text{ fm}
\]
\[
v = \sqrt{\frac{2 E}{m_\alpha}}, \quad
\lambda = P_\alpha \nu T, \quad
\nu \approx \frac{v}{2 R}, \quad
T_{1/2} = \frac{\ln 2}{\lambda}
\]

3. Geiger–Nuttall Law

The WKB result yields the empirical relation:

\[
\log_{10} T_{1/2} \approx a \frac{Z_d}{\sqrt{E_\alpha}} + b
\]

It shows the strong exponential sensitivity of half-life to alpha particle energy.

4. Numeric Examples

Constants used:

  • \( \hbar c = 197.327~\mathrm{MeV \cdot fm} \)
  • \( e^2/(4\pi \varepsilon_0) = 1.440~\mathrm{MeV \cdot fm} \)
  • \( m_\alpha c^2 = 3727.379~\mathrm{MeV} \)
  • Assumed \( r_0 = 1.2~\mathrm{fm} \)
Derived quantities and half-lives for two isotopes
Isotope Z A Eα (MeV) Zd R (fm) v/c η k (fm⁻¹) G T = e^{-2G} ν (Hz) λ (Pα=1) (s⁻¹) T₁/₂ (Pα=1) (s) Pα required to match experiment
^{238}U 92 238 4.267 90 7.437 0.04785 27.451 0.9038 79.519 8.518×10⁻⁷⁰ 9.645×10²⁰ 8.215×10⁻⁴⁹ 8.437×10⁴⁷ ≈ 5.98×10³⁰
^{212}Po 84 212 8.954 82 7.155 0.06931 17.266 1.3093 44.874 1.055×10⁻³⁹ 1.452×10²¹ 1.531×10⁻¹⁸ 4.526×10¹⁷ ≈ 1.51×10²⁴

Notes and Interpretation

  • The simple WKB/Gamow model with P_α = 1 greatly overestimates half-lives compared with experiment.
  • Experimental half-lives:
    • ^{238}U: 4.468×10⁹ yr ≈ 1.41×10¹⁷ s
    • ^{212}Po: 2.99×10⁻⁷ s
  • The preformation probability \(P_α\) required to match experiments is extremely large in this naive calculation.
  • Gamow’s major contribution was explaining the functional form of the Geiger–Nuttall law: \( \log_{10} T_{1/2} \propto Z_d / \sqrt{E_\alpha} \).