Alpha Decay

Step 1: The Potential Model

Model the alpha-nucleus system with a nuclear potential inside $r < R$ and a Coulomb potential outside:

where $B = 2(Z-2)k_e$ with $k_e = e^2/(4\pi\epsilon_0) = 1.44$ MeV$\cdot$fm, and $R$is the nuclear radius at which the alpha particle separates. The Coulomb barrier height at the nuclear surface is:

For $^{238}$U ($Z = 92$), $R \approx 9.3$ fm, giving $V_B \approx 28$ MeV, while $Q_\alpha = 4.27$ MeV. The alpha particle must tunnel through ~24 MeV of barrier.

Step 2: WKB Tunneling Integral

The WKB tunneling probability for a particle of energy $E = Q_\alpha$ through a barrier $V(r)$ is:

where $b = B/E$ is the classical outer turning point ($V(b) = E$) and$\mu = m_\alpha m_D/(m_\alpha + m_D)$ is the reduced mass. Substituting $V(r) = B/r$:

Step 3: Evaluating the Integral

Substituting $r = b\sin^2\theta$, so $dr = 2b\sin\theta\cos\theta\,d\theta$:

where $\sin^2\theta_0 = R/b = \rho$. Evaluating:

Therefore the Gamow factor is:

Step 4: Limiting Cases

For thin barriers ($\rho = R/b \ll 1$, i.e., $E \ll V_B$):

The first term is the Sommerfeld parameter $\pi\eta$ where$\eta = Z_\alpha Z_D e^2/(4\pi\epsilon_0 \hbar v)$:

This shows that $\log T \propto -1/\sqrt{E}$ at low energies, which is the essence of the Geiger-Nuttall relation.

From Gamow Factor to Geiger-Nuttall

In the thin-barrier limit, the Gamow exponent is dominated by the Sommerfeld parameter:

Writing $E = Q_\alpha \cdot (A-4)/A \approx Q_\alpha$ and using $v = \sqrt{2Q_\alpha/\mu}$:

where:

For a given element ($Z_D$ fixed), the log of the half-life is linear in$1/\sqrt{Q_\alpha}$. This explains the extraordinary sensitivity: a factor of 2 change in $Q_\alpha$ corresponds to ~20 orders of magnitude change in half-life for heavy nuclei.

Preformation Factor

The simple Gamow model assumes the alpha particle exists as a preformed cluster inside the nucleus. The decay rate is more accurately written as:

where $P_0$ is the preformation probability. For even-even nuclei,$P_0 \approx 0.01\text{-}0.1$. The preformation factor depends on nuclear structure and can be estimated from the overlap of the parent wave function with the alpha-daughter configuration:

The preformation probability is largest for nuclei just above doubly-magic daughters (e.g., alpha emitters above $^{208}$Pb) and accounts for the favored/unfavored alpha decay classification.

Hindrance Factors

The hindrance factor measures the reduction in decay rate relative to an unhindered transition:

Typical values: HF $\approx 1$ for favored (ground-state to ground-state) decays of even-even nuclei; HF $\approx 4\text{-}10$ for decays involving single-particle state changes; HF $\approx 100\text{-}1000$ for transitions requiring rearrangement of multiple nucleon orbits.

Fine Structure

For a parent decaying to the $i$-th excited state of the daughter at excitation energy $E_i^*$:

The branching ratio to state $i$ relative to the ground state is:

Since the tunneling probability is extremely sensitive to energy, even small differences in $Q_i$ lead to large changes in branching ratios. The ground-state transition typically dominates unless angular momentum selection rules suppress it.

Angular Momentum Selection Rules

For alpha decay from parent state $J_P^\pi$ to daughter state $J_D^\pi$, the orbital angular momentum $\ell$ carried by the alpha particle must satisfy:

Non-zero $\ell$ adds a centrifugal barrier $\hbar^2 \ell(\ell+1)/(2\mu r^2)$ to the Coulomb barrier, further suppressing the tunneling probability. Each unit of angular momentum reduces the decay rate by roughly an order of magnitude.

Observed Cluster Emissions

Clusters observed in radioactive decay include: