greatitself.
Continue
advanced34 min studyLesson 583 in Physics

Alpha decay and Gamow penetration

Separate energy release, barrier tunneling, cluster formation, and the measured partial decay rate.

Before you begin
On this page

The question for today

What you’ll understand

  • Construct a leading WKB penetration factor with a dimensionless action.
  • Relate preformation and assault frequency to a partial alpha rate.
  • Check recoil energy and distinguish half-life, mean lifetime, and width.

An energetically allowed alpha decay can remain extremely slow because the emerging charged cluster faces a Coulomb barrier. The Gamow picture connects that suppression to a forbidden-region action, but it does not predict every lifetime from Q alone. The alpha cluster must also be present in the parent wavefunction with the appropriate quantum numbers. Recoil, cluster formation, and barrier shape each enter a physically distinct part of the calculation.

The relative energy crosses a forbidden region

In the parent rest frame, the alpha and daughter have equal and opposite momenta. The Q-value is their total kinetic energy, while the alpha alone receives a fraction determined by the daughter mass. Use the relative-motion energy, not an uncorrected measured alpha energy, in a relative-coordinate barrier equation. Outside a chosen nuclear radius, a leading potential is the repulsive Coulomb interaction between charge two and the daughter charge; angular momentum can add a centrifugal barrier.

For an alpha energy below the barrier, the radial wavefunction is evanescent between turning points. WKB estimates the transmitted probability by twice the integral of the imaginary wave number. The integral is dimensionless, and the factor two arises because probability is the squared amplitude. Smooth turning-point matching and a substantial action support the leading exponential; corrections near the barrier top, angular-momentum matching, and the poorly known interior wavefunction can affect the prefactor.

P≃e−2G,G=∫r1r22μ[V(r)−Q]ℏ dr,VC(r)=2Zde24πϵ0rP\simeq e^{-2G},\qquad G=\int_{r_1}^{r_2}\frac{\sqrt{2\mu[V(r)-Q]}}{\hbar}\,dr,\qquad V_C(r)=\frac{2Z_de^2}{4\pi\epsilon_0r}
μ is the alpha–daughter reduced mass, V and Q are relative energies, and r is length. With consistent SI units G and P are dimensionless. The Coulomb form is for the external region; centrifugal terms and nuclear structure need separate treatment. The leading WKB exponential requires a sufficiently opaque barrier.↔ Scroll wide equations horizontally.

Penetrability is not yet a decay rate

A simple factorized model multiplies the penetration probability by an assault frequency and a preformation factor. The frequency estimates repeated encounters with the barrier from the nuclear interior. The preformation factor represents the overlap with an alpha-plus-daughter configuration and depends on nuclear structure; it is not an extra universal constant. This separation is phenomenological, because microscopic definitions can redistribute physics between the cluster amplitude, matching radius, and prefactor.

A nearly exponential isolated decay has a rate in inverse seconds, a mean lifetime in seconds, and a natural energy width equal to Planck’s reduced constant times the rate. The half-life differs from the mean lifetime by the logarithm of two. If several branches compete, partial rates add to the total rate and a branching fraction is a partial rate divided by that total. A measured total half-life therefore does not by itself fix the alpha partial lifetime.

λα≃SανP,λtot=∑iλi,bα=λαλtot,T1/2=ln⁡2λtot,Γi=ℏλi\lambda_\alpha\simeq S_\alpha\nu P,\qquad\lambda_{\rm tot}=\sum_i\lambda_i,\qquad b_\alpha=\frac{\lambda_\alpha}{\lambda_{\rm tot}},\qquad T_{1/2}=\frac{\ln2}{\lambda_{\rm tot}},\qquad\Gamma_i=\hbar\lambda_i
Sα and branching fractions are dimensionless, ν and λ are in s⁻¹, half-life is seconds, and Γi is an energy width. These relations use constant exponential partial decay rates. Preformation times assault frequency times penetrability is a model, not a general exact lifetime theorem.↔ Scroll wide equations horizontally.

Exponential sensitivity has limits as a structural explanation

Increasing Q shortens the classically forbidden region and lowers its action, often changing lifetimes by many orders of magnitude. This mechanism underlies the approximate Geiger–Nuttall systematics across related alpha emitters. It does not mean a universal straight line determines all nuclei: odd nucleons, different daughter excitations, and required orbital angular momentum can hinder an otherwise favorable channel. Comparing partial widths across branches helps distinguish barrier effects from a changing structural overlap.

The square-barrier limit is useful for checking the exponent without claiming it is the nuclear Coulomb field. There the imaginary wave number is constant, so action equals that wave number times barrier width. Doubling width doubles the action, not the decay probability. For realistic analysis, use measured masses and a declared barrier model, propagate Q uncertainties through the exponential, and keep a fitted preformation factor separate from an independently established microscopic prediction.

Work it through

Convert a toy penetration model into a lifetime

An illustrative alpha model has G=10ln⁡10G=10\ln10, preformation Sα=0.1S_\alpha=0.1, and assault frequency ν=1020 s−1\nu=10^{20}\,\mathrm{s}^{-1}. Assume alpha decay is the only branch. Find penetrability, decay rate, half-life, and width using ℏ=6.58×10−16 eV s\hbar=6.58\times10^{-16}\,\mathrm{eV\,s}.

  1. The probability exponent is −2G=−20ln⁡10-2G=-20\ln10, so P=10−20P=10^{-20}. Using e−Ge^{-G} instead would give an amplitude-scale number and the wrong probability.

  2. Multiply all three factors: λα=0.1(1020)(10−20)=0.1 s−1\lambda_\alpha=0.1(10^{20})(10^{-20})=0.1\,\mathrm{s}^{-1}. Since this is the only branch, the total rate is also 0.1 per second and the mean lifetime is 10 s.

  3. The half-life is (ln⁡2)/0.1≈6.93 s(\ln2)/0.1\approx6.93\,\mathrm{s}. The energy width is ℏλ=6.58×10−17 eV\hbar\lambda=6.58\times10^{-17}\,\mathrm{eV}, not 0.1 eV; a frequency and an energy width differ by the factor ℏ.

The result

The toy penetrability is 10⁻²⁰, its rate is 0.1 s⁻¹, half-life is about 6.93 s, and width is 6.58 × 10⁻¹⁷ eV. These parameters illustrate the model rather than identify a real isotope.

The living notebook

Explore the model

Interactive lab

Interactive lab · markov

From repeated survival to an exponential law

Identify state 0 with an undecayed parent and state 1 with a decayed system. Reset and set both controls to p = 0.20 and q = 0. Predict surviving state-0 probabilities 0.80 after one Step and 0.64 after two Steps. This discrete constant-probability survival model illustrates repeated opportunities for decay; its clock is not an alpha lifetime calibration and it contains no Coulomb barrier or preformation calculation.

From repeated survival to an exponential lawExact finite-step probabilities for a two-state chain starting in state 0, not sampled trajectories. Playback advances two transitions per second. When p = q = 0 every distribution is stationary; when p = q = 1 the chain alternates forever. A single zero transition creates an absorbing state. Current readings: P(state 1) at step 0: 0%; Stationary P(state 1): 60%; Chain behavior: Monotone convergence. All parameters are adjustable with the labeled controls below.Two-state Markov distribution after 0 steps: state zero probability 1, state one probability 0.Initial distribution (1, 0) · step 0p = 0.3q = 0.201100%0%1−p1−qP(0)P(1)Monotone convergence
P(state 0)P(state 1)Stationary distribution
P(state 1) at step 0
0%
Stationary P(state 1)
60%
Chain behavior
Monotone convergence
t = 0 s

Exact finite-step probabilities for a two-state chain starting in state 0, not sampled trajectories. Playback advances two transitions per second. When p = q = 0 every distribution is stationary; when p = q = 1 the chain alternates forever. A single zero transition creates an absorbing state. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA repeated survival factor approaches exponential decay, but penetrability and a physical attempt rate must be supplied separately.

Pause & reason

Check your understanding

In leading WKB tunneling, why is the probability factor exp(−2G) rather than exp(−G)?

Your turn

Try it yourself

For a toy rectangular forbidden region, the imaginary wave number is 0.5 fm−10.5\,\mathrm{fm}^{-1}. Compare widths 4 fm and 6 fm, with identical assault and preformation factors. Find both WKB penetrabilities and the ratio of the second half-life to the first.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

For a constant imaginary wave number, G is its product with the barrier width.

Compare with the solution

The actions are G1=0.5(4)=2G_1=0.5(4)=2 and G2=0.5(6)=3G_2=0.5(6)=3. Their leading penetrabilities are e−4e^{-4} and e−6e^{-6}. The second rate is smaller by e−2e^{-2}, so the second half-life is larger by e2≈7.39e^2\approx7.39. The exponent difference controls the ratio, even though the width increases by only a factor 1.5. These are illustrative leading WKB values, not an exact finite-barrier transmission calculation.

Keep the idea

Take it with you

  • Alpha recoil and daughter recoil share the total Q-value.
  • The Gamow action controls a probability exponentially.
  • A partial width needs structural and frequency information beyond penetrability.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.