Illustrated Physics
physics, drawn
Concept

Barrier — where it appears

A region a system has not the energy to cross classically, whose height and width decide how rarely it is crossed. What sits in the exponent is an integral over its shape rather than its height alone, which is why the width matters far more than the peak does.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 10, 20, 30 times kT are factors of 10^-4.3, 10^-8.7, 10^-13.0. At 300 K, kT is 25.9 meV, so a barrier of 0.35 eV is 13.5 kT and a factor of 1.3e-6. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The exponential that decides everything

Maximising the number of ways a reservoir can arrange what is left after taking E out of it gives one factor, e to the minus E over kT. Its exponent is a ratio, which is why a barrier of a third of an electronvolt — nothing at all by chemical standards — is the difference between instantly and never.

thermodynamics · Entropy Twenty-four decades of lifetime from a factor of two in energy. The half-lives of 7 alpha emitters against the reciprocal square root of the alpha's energy — the Geiger–Nuttall coordinates — with the measured values as points and a one-line tunnelling model as the open ones. The energies span a factor of 2.2 and the half-lives span 24 decades, which is what an exponent does. The model has no fitted parameter in it and reproduces every lifetime to within 0.5 decades — bad arithmetic by any ordinary standard, and a hundred-thousandth of the range it is predicting.

A wall that a factor of two makes impassable

Polonium-212 lives three tenths of a microsecond. Thorium-232 lives fourteen billion years. The alpha particles they emit differ in energy by a factor of two, and the lifetimes differ by twenty-four decades — because the quantity that decides is not the energy but an exponent built from it, and an exponent is where small differences go to become enormous.

quantum · Tunnelling A current that grows thirty decades with the field. The current density of electrons tunnelling out of a cold metal against the field at its surface, on a logarithmic scale, from the Fowler–Nordheim law, for work functions of 2.7 eV (lanthanum hexaboride), 4.5 eV (tungsten) and 5.6 eV (a surface barely willing to let electrons go). For tungsten the current is 0.009 A/m² at 2 GV/m, 5·10⁵ at 4 GV/m and 6·10⁹ at 8 GV/m — rising by more than twenty orders of magnitude between 1 and 4 GV/m. The work function sits in the exponent as a three-halves power, so lowering it from 4.5 to 2.7 eV raises the current at 3 GV/m by a factor of 2·10⁵. A current density of a million amperes per square metre, an ordinary working value for a field emitter, needs about 4.2 GV/m on tungsten.

The electrons a field pulls from cold metal

A metal holds its electrons behind a step a few electronvolts high, and to get them out the usual way is to heat the metal white-hot until some climb over it. Put a strong enough field outside instead and the step tilts into a triangle a couple of nanometres wide, and electrons leave a cold metal by tunnelling through it. The current grows thirty decades as the field rises tenfold, a sharp needle at a thousand volts reaches the field that a flat plate would need a million volts for, and the electrons that come out all have nearly the same energy — which is why the brightest, sharpest electron microscopes are fed from a single etched tungsten point.

quantum · Tunnelling

Named alongside it

The objects these essays reach for when they reach for this one.

TunnellingActivation energyAlpha decayArrhenius lawThe Boltzmann factorCoulomb barrierCross-sectionDecayElectric fieldElectron microscopeEntropyExponential

All concepts