More Reflection Than Incidence?
In 1929, Swedish physicist Oskar Klein applied the Dirac equation—the relativistic equation for electrons—to a simple problem: electron scattering from a step potential. What he found shocked the physics community: for sufficiently strong barriers, the reflection coefficient exceeded 100%.
More electrons seemed to be reflected than were sent in! This appeared to violate the most fundamental principle in quantum mechanics: conservation of probability. How could such an absurdity arise from the mathematics that correctly described electrons in atoms?
The Klein Zone: V₀ > 2mc²
The magic threshold is twice the electron rest mass energy: V₀ > 2mc² ≈ 1.022 MeV. Below this, scattering behaves strangely but doesn't violate conservation laws. Above it, we enter the "Klein zone" where the paradox manifests.
In nonrelativistic quantum mechanics, increasing barrier height always increases reflection and decreases transmission, approaching 100% reflection for infinite barriers. But in the Klein zone, the opposite happens: transmission approaches 100% as the barrier approaches infinity!
Klein zone: R → 0, T → 1 as V₀ → ∞
(Barrier becomes transparent!)
Resolution: The Dirac Sea and Pair Creation
The resolution came from Dirac himself and involves one of the most profound ideas in physics: the Dirac sea. Dirac proposed that the vacuum isn't empty but filled with an infinite sea of negative-energy electrons. The Pauli exclusion principle prevents normal electrons from falling into these states.
When V₀ > 2mc², something remarkable happens: the potential lifts negative-energy states inside the barrier above the vacuum level outside. These states can now be "seen" by the incident electron—and electrons from the Dirac sea can tunnel out.
The Physical Picture
The "extra" reflected current isn't really reflection at all—it's positron creation. When an electron approaches the supercritical barrier:
1. The strong potential creates an electron-positron pair at the barrier
2. The new electron travels backward (appearing as reflection)
3. The positron travels forward into the barrier (appearing as transmission)
4. The original electron is absorbed in the pair creation process
Total charge and probability are conserved—but only if we count both particles and antiparticles. The Klein paradox isn't a paradox at all; it's a dramatic demonstration that strong potentials can create matter from nothing.
Klein Tunneling in Graphene
While ordinary Klein paradox requires extreme energies (over 1 MeV), something remarkable was discovered in 2006: electrons in graphene behave as if they were massless Dirac fermions. They exhibit Klein tunneling at everyday energies.
In graphene's honeycomb lattice, electrons near the Fermi level obey a 2D Dirac equation with zero mass. This means any potential barrier—even a small one—can exhibit 100% transmission at normal incidence. Electrons in graphene pass through barriers like they don't exist.
This has profound implications for graphene electronics: you cannot create conventional transistors by putting potential barriers in the way. The electrons just tunnel through. New approaches using magnetic barriers or bilayer graphene are needed to control current flow.
The Mathematics
The Dirac equation for a step potential V(x) = V₀Θ(x) gives reflection and transmission coefficients:
where p = √(E² - m²c⁴)/c (outside)
p' = √((E - V₀)² - m²c⁴)/c (inside)
When V₀ > E + mc², the quantity (E - V₀)² - m²c⁴ becomes positive again, giving a real momentum p' inside the barrier. But this momentum corresponds to negative-energy states—holes in the Dirac sea—which are positrons moving in the opposite direction.
Supercritical Atoms: Z > 137
The Klein paradox also predicts that sufficiently heavy atomic nuclei should spontaneously create positrons. If Z > 137 (actually about 173 with realistic calculations), the electric field near the nucleus is strong enough to destabilize the vacuum.
Such superheavy elements don't exist naturally, but collisions between uranium nuclei (Z = 92) can briefly create a combined nuclear charge exceeding the critical value. Experiments have searched for spontaneous positron emission, though conclusive detection remains elusive.