Questions: Alpha Decay and Tunneling Through the Coulomb Barrier

Question 1

Nucleus A emits alpha particles with 5 MeV kinetic energy and has a half-life of 10,000 years. Nucleus B has a similar structure but emits alphas at 6 MeV. What should you expect for nucleus B's half-life?

• A. Slightly shorter — perhaps around 8,000 to 9,000 years, since 6 MeV is only modestly higher
• B. Dramatically shorter — possibly many orders of magnitude shorter — due to the exponential sensitivity of the Gamow factor
• C. Longer — higher-energy alphas face a wider effective barrier because of larger Coulomb repulsion
• D. Essentially the same — a 20% energy increase has a small effect on a quantum-mechanical process

Answer: (B)

This is the key insight of the Gamow factor: the tunneling probability is exponentially sensitive to the alpha energy. The Gamow exponent G = ∫κ(r) dr decreases as alpha energy increases (the classically forbidden region narrows), but because G enters as exp(−2G), even a small change in G produces an enormous change in decay rate. The Geiger-Nuttall law captures this empirically: alpha energies span only a factor of ~2 (4–9 MeV) while half-lives span 25 orders of magnitude (microseconds to billions of years). A 1 MeV increase in alpha energy can easily reduce the half-life by a factor of 10⁶ or more.

Question 2

Why can't classical mechanics account for alpha decay in heavy nuclei?

• A. Classical mechanics doesn't include the strong nuclear force, so it cannot model nuclear binding
• B. The alpha particle's kinetic energy (4–9 MeV) is less than the Coulomb barrier height (~25–30 MeV), so classically the alpha is permanently trapped inside the nucleus
• C. Classical mechanics predicts too fast a decay rate, since it allows the alpha to bounce off the barrier indefinitely
• D. Classical mechanics cannot handle Coulomb interactions at the femtometer scale

Answer: (B)

The Coulomb barrier between the alpha particle and the daughter nucleus peaks at roughly 25–30 MeV for heavy nuclei, yet the alpha's kinetic energy (set by Q-value) is only 4–9 MeV. Classically, a particle cannot cross an energy barrier higher than its kinetic energy — it would bounce back every time. Yet alpha decay does occur, in some nuclei with extraordinary speed. The only explanation is quantum tunneling: the alpha's wavefunction has nonzero amplitude in the classically forbidden barrier region and emerges on the other side with finite probability. This is not a correction to classical mechanics; it is a fundamentally quantum phenomenon.

Question 3

The Geiger-Nuttall law — which shows that alpha-decay half-lives span 25 orders of magnitude while emitted alpha energies vary by only a factor of two — is a direct consequence of the exponential sensitivity of tunneling probability to the Gamow factor.

• T. True
• F. False

Answer: True

Yes. The Gamow factor G enters the tunneling probability as exp(−2G), so small changes in G (driven by small changes in alpha energy) produce enormous changes in the decay rate. The Gamow-Sommerfeld factor for the Coulomb barrier decreases steeply with increasing alpha energy, amplifying a factor-of-2 energy range into a 10²⁵ range of half-lives. The Geiger-Nuttall law is not an independent empirical coincidence — it is a quantitative prediction of the WKB tunneling calculation applied to the Coulomb barrier.

Question 4

Uranium-238 and polonium-212 have vastly different half-lives (4.5 billion years vs. 300 nanoseconds) because they decay by fundamentally different nuclear mechanisms.

• T. True
• F. False

Answer: False

Both nuclei decay by exactly the same mechanism: quantum tunneling of an alpha particle through the Coulomb barrier. The 25-order-of-magnitude difference in half-life arises entirely from the Gamow factor. Polonium-212 emits alphas at ~8.8 MeV; uranium-238 at ~4.3 MeV. This ~2× energy difference makes the Gamow exponent much smaller for Po-212, giving a tunneling probability that is roughly 10²⁵ times larger. Same mechanism, same physics — only the Gamow factor differs, and exponential amplification does the rest. This is one of the most striking demonstrations of exponential sensitivity in all of physics.

Question 5

Why does a small increase in the kinetic energy of the emitted alpha particle produce such a dramatic decrease in the nucleus's half-life?

Model answer: The tunneling probability is T ≈ exp(−2G), where G is the Gamow factor — the integral of the local inverse decay length through the classically forbidden barrier region. A higher alpha energy means the particle's energy is closer to the barrier height, so the classically forbidden region (where E < V(r)) is narrower and shallower. This reduces G, but because G enters as an exponent, even a modest reduction in G produces an exponentially large increase in T. The decay rate λ = ν × T (attempt frequency times tunneling probability), so an exponentially larger T means an exponentially larger decay rate — and exponentially shorter half-life. The factor-of-2 range in alpha energies translates into a 10²⁵ range in half-lives through this exponential mechanism.

This exponential sensitivity is the defining feature of quantum tunneling in nuclear physics. It means that half-life is extraordinarily sensitive to the nuclear structure details that determine the Q-value (the mass-energy difference available for the decay). Nuclei that are 'almost stable' against alpha decay can be pushed to instability by surprisingly small changes in the alpha energy, and this sensitivity makes the Geiger-Nuttall law both a striking empirical regularity and a precise quantitative prediction of WKB theory.