Flat, then exponential, then a power law
At its defaults it draws 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.
survival-decay is one function in lib/figures/atomic.js —
what happens once one is bound by the other. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
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.
Decay, and the ensemble it is a property of
The options are the ones A nucleus with no clock passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.
Decay, and the ensemble it is a property of
The options are the ones A nucleus with no clock passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
2000 nuclei followed for 5 half-lives. The smooth curve is the exponential; the stepped traces are 2 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 1000, 500, 250, 125, 63 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 63 are left and the scatter is a visible fraction of that.
Decay, and the ensemble it is a property of
The options are the ones A nucleus with no clock passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
800 nuclei followed for 5 half-lives. The smooth curve is the exponential; the stepped traces are 4 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 400, 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.
Decay, and the ensemble it is a property of
The options are the ones A nucleus with no clock passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
40 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 4 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 20, 10, 5, 3 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 3 are left and the scatter is a visible fraction of that.
Everything in the chain decaying at the parent's rate
The options are the ones The chain that runs at its slowest member's rate passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What checks it
physicscheck asserts something about survival-decay that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A nucleus with no clock
A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.
QuantumThe 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.
QuantumThe clock read by counting atoms
Radiocarbon dating reads a clock that ticks slowly: a gram of living carbon holds sixty thousand million carbon-14 atoms, and only thirteen and a half of them decay each minute. Counting those decays to date a milligram of bone would take years. Counting the atoms themselves, one by one in an accelerator, takes an hour, because in any measurement shorter than the clock's mean life almost every atom is still there, undecayed and countable. And the method's famous limit of about fifty thousand years turns out not to be set by the half-life at all. It is set by how much modern carbon a sample has picked up, because a trace of contamination outweighs whatever carbon-14 a very old sample has left.
QuantumThe energy that did not all arrive
A nucleus emitting one particle has no choice about the energy it comes out with — momentum conservation fixes it, and the spectrum is a line. Beta decay gives a continuum instead, from zero to the full available energy, and the shape of that continuum is a count of the ways the energy can be shared. Counting it required a third body nobody had seen, and the way the count approaches its endpoint is still the best place to weigh one.
QuantumThe 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.
QuantumThe half-life that chemistry can change
A nucleus has no clock, the usual account goes, and nothing outside it touches the rate. That is very nearly true and it is not exactly true. Electron capture takes an electron from the nucleus's own position, so its rate is proportional to how many electrons are there — which is chemistry, worth a per cent. And a nucleus stripped of every electron can gain a decay channel it did not have: rhenium-187 goes from forty-two billion years to thirty-three.
QuantumThe lifetime that depends on where the mirror is
A half-life is treated as the most private fact about an atom or a nucleus, and for a nucleus that is nearly true. An excited atom beside a mirror decays faster or slower depending on how many half-wavelengths away the mirror stands, and between two mirrors closer than half a wavelength it does not emit at all. The atom has not changed. A decay is a transition to a state with a photon in it, and the room decides whether that state exists.
QuantumThe spin a photon has to carry away
An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.