Alpha decay and quantum tunnelling

What just happened?

The alpha particle sits in the nucleus as in a well surrounded by a tall Coulomb barrier (charge repulsion). Its energy $E_\alpha \approx 4\text{–}9\,\text{MeV}$ is a few times smaller than the barrier top (tens of MeV). Classically — imprisoned forever. Quantum-mechanically the wavefunction does not vanish abruptly at the wall: it decays exponentially inside the barrier and on the other side has a nonzero value. The particle can “appear” there — this is tunnelling.

Gamow factor and the Geiger–Nuttall law:

The transmission probability decreases exponentially with the barrier “thickness” (Gamow factor $G$). Taking the logarithm gives a linear dependence between $\log_{10} T_{1/2}$ and $Z/\sqrt{E_\alpha}$ — exactly what Geiger and Nuttall noticed in the data back in 1911. Gamow’s model (1928) was the first success of quantum mechanics in the nucleus.

Invariant — exponential sensitivity:

Because $T_{1/2}$ depends on energy through an exponential, a small change in $E_\alpha$ (by ~1 MeV) changes the lifetime by a dozen-odd orders of magnitude. Across the whole 4–9 MeV range that is the difference between microseconds and a time longer than the age of the Universe — and all points lie on the same line (right panel).

Where will you meet this?

Isotopic dating, the radioactive decay series of uranium and thorium, the age of the Earth, and also — the same tunnelling physics — thermonuclear fusion in stars and the tunnel diode in electronics.

Why you need this for the exam:

This is an academic extension of the topic of radioactivity (decay types, the decay law $N = N_0\,2^{-t/T_{1/2}}$). At school you learn that nuclei decay with a given $T_{1/2}$; here you see why these times are so dramatically different — because quantum tunnelling governs them.

Remember: alpha decay is tunnelling through the Coulomb barrier. The lifetime is exponentially sensitive to energy — $\log_{10}T_{1/2} \propto Z/\sqrt{E_\alpha}$ (the Geiger–Nuttall law).