Decay Chain Simulator

What Is the Decay Chain Simulator?

How to use the radioactive decay chain simulator

1. Start from the state it opens in. The lab boots at rest on 20 d, 2.0 d, 500 MBq and 1.00 T1, so every reading is already on the screen before you touch anything. Reset returns all four sliders to exactly that and stops the sweep if it is running.
2. Set the parent half-life. Parent half-life · T₁ runs from 1 to 100 d in whole days, and it stretches the horizontal axis with it, because the plot always covers eight of them. At 20 d the axis ticks read 0, 80 and 160; at 100 d they read 0, 400 and 800. If you want the single-nuclide version of that idea first, the guide to half-life and the decay constant covers it.
3. Set the daughter half-life. Daughter half-life · T₂ runs from 0.1 to 50.0 d in tenths. These two sliders together decide every shape on the screen; nothing else does. Put the daughter below the parent and it climbs and then follows; put it above and it never catches up.
4. Choose how strong the source is. Parent activity at the start runs from 10 to 1000 MBq in steps of 10 and means the activity at the moment the daughter was last stripped out. It multiplies the two activity cards and the total in the strip, and it leaves the ratio, the ceiling, the peak time and the three percentages exactly where they were.
5. Move the clock. Time, in parent half-lives runs from 0 to 8 in steps of 0.02, and the figure beside it reads in multiples of T₁ rather than days. Time now gives you the same instant in days, so 1.00 T₁ is 20.00 d on the opening chain and 13.00 d on the Ba-140-like one.
6. Read the four cards down the right. Parent activity and Daughter activity carry their own formula lines, A1 = λ1N1 and A2 = λ2N2, in MBq to one decimal place. Activity ratio · A₂/A₁ carries four decimal places at every size it can reach, and Which regime carries one of exactly three phrases.
7. Read the eight cells under the graph. Both decay constants to six decimal places per day, the three atom percentages to two, then Time now, Daughter peaks at and Limiting ratio. That last one prints the word none rather than a number whenever the daughter is the longer-lived of the pair.
8. Read the strip below them. Total activity now is what a detector that cannot separate the two lines would see, and beside it a sentence names the peak. If the dashed marker is drawn the sentence points at it; if the peak sits too close to the vertical axis to mark, the sentence gives the time in words instead and says that no marker was drawn.
9. Press Play the sweep to let the clock run. It advances four tenths of a parent half-life every second, wraps back to the start past the right-hand edge, and becomes Pause the sweep while it is going. Only the clock moves: the half-lives and the source strength stay where you put them.

When the daughter is stable, or its fate is not the thing you are after, this is more lab than you need. The half-life lab draws the single falling curve with nothing feeding it, which is the picture this one is built to contrast with. Come back here the moment the second nuclide starts to matter.

Worked example: change one thing at a time

Every row below is one setting of the four sliders, and every cell is a string the running lab printed there. The first two rows are the state it opens in and the nearest stop to its peak; the next four are the three regimes with a fourth chain that has no ceiling at all; the last four go looking for the corners. Where a cell and the lab ever part company, believe the lab.

The two ratio columns are the pair to read together. One is where the chain stands now and the other is where it is going, and they are never the same question. On the Ba-140-like row the live ratio is 1.1390 while the ceiling is 1.1504; push that same chain on to 2.70 T₁ and the live figure has crept up to 1.1504 as well, which is the ratio arriving rather than the two quantities being one thing.

Row 6 is the one with no ceiling. A parent of 5 d feeding a daughter of 50.0 d never settles into anything, so the Limiting ratio cell prints none instead of a number, and the daughter does not even reach its own peak until 18.46 d — by which time the parent, on a 5 d half-life, has very little left to give.

Row 7 is the case most textbook formulae cannot print. Give the parent and the daughter the same half-life and the usual expression divides by zero; this lab reads 0.6931 instead, because when the two decay constants match the ratio is simply the decay constant times the elapsed time. One parent half-life in, that product is ln 2.

Rows 1, 10 and 11 are the same chain at 10, 500 and 1000 MBq. The ratio, the ceiling, the peak time and the three percentages print identical strings in all three, and that agreement is not a result worth having: the source strength cancels out of every one of those quantities algebraically. What it tells you is narrower and more useful — the shape of a chain is set by the two half-lives, and the source only sets the height of the picture.

Rows 8 and 9 are the two ends of the clock. At 0.00 T₁ the daughter has not been made yet, so its card reads 0.0 MBq and the ratio reads 0.0000. At 8.00 T₁ the pair is down to 2.0 and 2.2 MBq and the live ratio prints 1.1111, the same four characters as the ceiling. If you need those figures for a real pair of half-lives rather than round ones, the decay chain calculator takes them to more decimal places and will run the sum backwards to date the sample.

Formula and symbol reference

The lab works from one relation. The ratio of the two activities is A2/A1 = λ2 t f((λ2 - λ1)t), where f(x) = (1 - e-x)/x and f(0) = 1. Written that way the difference of the two decay constants never sits underneath anything, which is why row 7 of the table above has an answer instead of an error.

Each decay constant is ln 2 divided by that member's half-life, so the two sliders you move and the two constants the panel prints carry the same information twice over. What kind of decay each step is never enters the arithmetic: a beta emitter and an alpha emitter with the same half-life draw the same pair of curves. The account of beta decay covers what is actually leaving the nucleus, which is the part this lab does not draw.

Two of those rows deserve a second look. The Activity ratio card keeps four decimal places from 0.0000 right up to 21.0607, the largest figure these sliders can produce, with no switch to fewer digits at any size along the way. And none in the Limiting ratio cell is a real answer, not a gap: it is the lab declining to name a destination for a ratio that has none.

The physics: why the two curves cross at the daughter's peak

The daughter is being made and destroyed at the same moment. Its population changes at the parent's activity minus its own, so it grows for exactly as long as the parent is the stronger of the two and shrinks once it has overtaken. The turning point is therefore the instant the two activities are equal, which on a graph is the instant the two curves cross.

That is why the marker and the crossing are always at the same place, and why the caption under the graph says so in one sentence rather than two. On the opening state it reads: The dashed marker at 7.38 d is where the daughter activity peaks, and the two curves cross there because the peak is exactly where the two activities are equal.

You cannot quite land on it with the sliders, and that is worth knowing before you go hunting. The clock moves in steps of 0.02 T₁, so on the opening chain the nearest stop to a peak at 7.38 d is 0.36 T₁, which is 7.20 d. There the two cards read 389.6 MBq and 387.1 MBq and the ratio reads 0.9935 — near, and not the same.

After the peak the daughter stops keeping its own time. It is fed by a parent that is itself falling, so the pair lock into a fixed ratio and from then on both decline at the parent's rate. That fixed value is what the Limiting ratio cell has been printing all along, and the live ratio spends the rest of the graph creeping up to it.

How close that fixed value sits to 1 is the only thing separating the three names on the regime card. A parent far longer-lived than its daughter gives a ceiling barely above 1; a parent that still outlives its daughter but not by much gives one noticeably above 1; a parent that goes first gives no ceiling at all. The full account of decay chains works all three through with real nuclides and worked problems.

The hundredfold line between the first two names is a stated convention and nothing more. Nothing happens to the physics as a chain crosses it. The table below holds the daughter at 0.1 d and walks the parent across it one day at a time.

Nine days against a tenth of a day is called transient and eleven days is called secular, while the ceiling moves from 1.0112 to 1.0092 — about two parts in a thousand. The name changed; nothing measurable did. That is exactly why the lab prints the ceiling next to the name instead of leaving you with the word.

And the two activities never do become equal. Take the parent to 100 d against the same 0.1 d daughter, the deepest secular state these sliders reach, and both the Activity ratio and the Limiting ratio read 1.0010. They agree with each other to every digit printed, and neither of them is 1.0000. That last thousandth is not a rounding artefact: it is the margin by which the parent stays ahead, and the daughter is supplied out of it.

Where the decay chain model breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the situation it stands for, or of the way the figures are printed, and each item says what the lab does about it.

The daughter has to start at nothing

Every curve here begins with the daughter absent, which is what the phrase since separation means. If your sample already held some of the daughter when your clock started, the true curve sits above this one by the amount that was there, decaying on its own schedule. There is no control for it, because the model has no term for it.

Exactly three members, and no branches

A parent, one radioactive daughter, one stable end. A nuclide that decays two ways at once splits into two partial paths, and this lab has no way to divide the flow between them. Bismuth-210, which one of the presets is named after, has a tiny alpha branch alongside its main beta route, and nothing on this page models it.

Real series are much longer than three

Uranium-238 runs a long way down to lead-206, and a natural series like that is a queue of chains rather than one. Each neighbouring pair in it behaves much as this lab shows, but the whole series does not, and a three-member tool cannot tell you what the eleventh member is doing. Look the intermediate half-lives up in published nuclear data rather than guessing them.

The sliders cannot reach a published half-life

The parent moves in whole days and the daughter in tenths, so barium-140 at 12.751 d and lanthanum-140 at 1.67858 d simply are not on the grid; 13 d and 1.7 d are the nearest stops, and that is why the preset says Ba-140-like. Bismuth-210 at 5.012 d misses a whole-day stop in the same way, which is why the inversion preset is Bi-210-like and never the nuclide itself. When the figures have to be the published ones, the decay chain calculator takes them exactly, in whichever time unit they were quoted in.

Nothing here is counted, so nothing here is noisy

These curves are smooth because the model is deterministic. A real measurement of the same chain is a count, and a count carries its own scatter that grows relatively worse as the activity falls. The far right of this graph, where the pair are down to a couple of megabecquerels, is exactly where a real detector would be struggling most and this drawing is at its most confident.

The three percentages are rounded one at a time

Parent atoms left, Daughter atoms and Stable atoms each get two decimal places of their own, so the printed trio need not close on 100.00. At 13 d and 1.7 d with the clock on 1.30 T₁ they read 40.61, 6.09 and 53.29 per cent. The lab's own fixed note says they are not presented as a sum, and this page does not present them as one either.

Recomputing one reading from another will usually not come out

Activities are rounded to one decimal place and the ratio to four, each on its own, so dividing the two figures you can see rarely reproduces the ratio card exactly. On the opening state 277.2 divided by 250.0 gives 1.1088 against a printed 1.1089. The lab carries full precision internally and only the display is rounded.

The far right of the graph is where the ratio stops moving

Once the live ratio has crept up to the limiting ratio, later instants print the same four characters as each other. The far right of the opening chain reads 1.1089 at one parent half-life and 1.1111 at eight, and beyond that it has nowhere left to move. A measured ratio can date a sample only while it is still climbing.

Every number here is a setting, not a measurement

The four sliders are things you chose, and nothing in this lab has measured a source, a shipment or a sample. It will happily draw a 1000 MBq parent with a one-day half-life; whether such a thing sits in anybody's laboratory is a separate question the arithmetic has no view on.

Where decay chain readings are actually used

Generators that supply a short-lived daughter

A long-lived parent is held on a column and the daughter that grows in on it is washed off when it is wanted, after which it starts growing back. The shape of that regrowth is the daughter curve on this graph read from the left-hand edge, and the useful question is how long to wait before the next draw. Load Secular, deepest reachable to see the limiting case, where the daughter comes back almost to the parent's own activity.

Radon indoors is a daughter, not a source

Radon-222 has a half-life of 3.8235 d, which is far too short for any of it to be left over from the formation of the rock. It is there because radium-226, at 1600 y, is in the ground making more of it, which is the same parent-feeding-daughter arrangement this lab draws. That is also why airing a room out once does not settle it: the supply in the ground has not been touched, so the gas builds back up along the same climb this lab draws from the left-hand edge of the graph.

Dating a sample from the ratio of two lines

If you can measure the parent and the daughter separately, the ratio between them is a clock, because on a chain that has not yet reached its ceiling each elapsed time gives a different value. That is a different method from counting how much of one nuclide is left, which is what radiocarbon dating does, and it needs no assumption about the starting amount.

A fission-product pair that arrives still growing

A barium-140 and lanthanum-140 pair is a fission product that arrives at a laboratory still building its daughter, so the dose rate from a freshly separated batch goes up before it goes down. Load A Ba-140-like chain and run the sweep to watch that happen. The reactor lab shows where such pairs come from in the first place.

Shielding a source that is really two sources

The Total activity now figure in the strip is what an instrument that cannot separate the two nuclides responds to, and on the opening state it is 527.2 MBq against a parent of 250.0 MBq. Planning around the parent alone understates it by more than a factor of two. The two nuclides also rarely emit at the same energy, so the shielding calculation is genuinely two calculations.

Checking a delivered source against its quoted date

A delivered source is quoted at an activity on a date, and if its daughter was separated at that moment the pair have been growing together ever since. Setting the two half-lives and the elapsed time here gives the shape of what should have happened; the half-life calculator is the right tool instead when the daughter is stable and one curve is the whole story.

Teaching which curve is which

Chain problems go wrong when a student reads the daughter curve as though it were a second parent curve. Drag the daughter half-life from 0.1 d up to 50.0 d and watch the same picture turn from two lines lying on top of one another into a late, low hump that peaks long after the parent has gone. That is hard to see on paper and immediate here.

Where to go next

For the method itself, with the three regimes worked through on real nuclides, eight problems of rising difficulty and the diagrams that go with them, read Radioactive Decay Chains Explained. If you would rather type published half-lives than drag sliders, or you need the sum run backwards to date a sample, the calculator is the first card under Related tools below.

The single-nuclide background this lab assumes is in What Is Half-Life in Physics?, with the half-life lab and the half-life calculator beside it. For what is actually being emitted at each step there is Beta Decay Explained, Gamma Rays: Properties and Uses and the survey in Radioactivity: Alpha, Beta and Gamma.

Further afield, Carbon Dating: The Physics Behind It and the carbon dating lab take the dating idea the other way about, while How a Nuclear Reactor Works and Fission vs Fusion cover where fresh chains come from. The rest of the collection is in the library of physics simulations and on the blog, and the site search will find a topic by name.