Illustrated Physics
physics, drawn
Concept

Exponential decay — where it appears

A quantity whose rate of decrease is proportional to itself, so that equal intervals remove equal fractions and no finish line exists. It follows from a constant probability per unit time and nothing else, which is why the same curve describes nuclei, capacitors and a cooling cup.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

One equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.

The three ways of coming to rest

An oscillator with friction in it can swing and fade, arrive and stop, or crawl back so slowly it looks stuck. One equation and one number decide which. The number is not what most people would guess, and neither is the value that returns fastest.

mechanics · Pendulum The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term.

The reflection that happens where the glass is not

Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed.

optics · Total internal reflection Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

quantum · Decay Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore.

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

quantum · Decay

Named alongside it

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

DecayHalf-lifeResonanceAbundanceActivityBandwidthBoundary conditionsCharacteristic equationConservation lawsCritical angleCritical dampingDamping

All concepts