Chain Evolution

One obvious way to examine the output data is by decay chain.  The mix of fission fragments changes over time, and the exponential view of this evolution makes nice charts.  Several principles of decay become apparent.

A Simple Case

The decay along the main chain of atomic weight 67 is well behaved.  The chart below follows the entire run.  English time units at the bottom lend perspective.  The relevant isotope names are on the top, but it is their Rung number that charts best.  Here, the most popular initial product sits on Rung 2.  It gets a small contribution from two hotter isotopes, 67-Co (red) and 67-Fe (dark blue).  The Rung 2 isotope has a half-life of 21 seconds, so after a minute, its daughter dominates the mix.  The Rung 1 isotope decays in a few days to the Sink.  By the end of the first year of reactor operation 99.9% of the population of isotopes of atomic weight 67 is 67-Zn.

Normalized yield for Decay Chain 67

Note the Y-axis!

These charts show the relative contribution of each isotope over time.  The cumulative yield is normalized by dividing it by the exponential (2^Timestep).  The total number of normalized atoms at each TimeStep is 2.00.  This is different from the chart of 135-Xe population in the sidebar.  There, the y-axis is total yield, measured in atoms.  After a few days, the creation and decay (and capture!) of 135-Xe atoms come to equilibrium at a constant yield of about 1600 atoms.  The flat lines on the right of that sidebar chart indicate a constant population.

In the chart above, the line after a year is flat, but 67-Zn has not come to equilibrium.  A flat line here means the isotope represents a constant proportion of the entire waste stream.  The total waste stream is growing in mass (a term we need to use carefully) at a rate of two atoms per second.  Therefore, the population of 67-Zn keeps growing. As the 67 chain is near the tail of the distribution, the 67-Zn population (relative or absolute) is tiny – 3.5 parts per billion.  That’s seven atoms out of the 2 billion fission fragments in the waste stream at TimeStep 30.

Chain 171 is another very symmetric chart.  The near equal spacing of the transfers from one rung to the next is striking.  (Most other chains are not so neat.)

Normalized yield for Decay Chain 171

Most of the nuclei “born” at atomic weight 171 are on Rung 3.  The decays from 3 to 2, 2 to 1 and 1 to 0 are also shown here with red arrows.  Rung 1 has a half-life long enough that the Sink takes years to dominate the mix.  Even after 34 years, there is still some 171-Tm.

A Significant Neutron Poison

The two chains above have Sinks that do nothing but collect mass from higher rungs.  The last parts of the curves of the Sinks are rising or flat.  The population curve for a strong neutron poison’s curve will fall.  149-Sm is the second-most important poison.  As the Sink for the main chain at atomic weight 149, it takes everything with it as it goes.

Chain 149 yield flows to Chain 150, due to 149-Sm neutron capture

A minor point is that the population stops dropping at the very end, and even rises a bit.  147-Pm and 148-Pm both capture neutrons readily, but 148-Pm most often decays before it plumps to 149.  148-Nd, the Main Sink for chain 148, has a small capture cross section, but the last time frame is long enough for it to contribute a bit to decay chain 149.

A Plumper Recipient

The main Sink for chain 114 inherits more from chain 113 than from its own parents.  This comes from the plumping of 113-Cd, which has the fifth highest capture barns.  Calling it a Major poison is a bit inflated, though.  Chain 113 is in the Sparse Valley, so 113-Cd doesn’t capture all that many neutrons.

Chain 114 population swells due to 113-Cd neutron capture

Short-lived Daughter

In general, the lower the rung, the longer the half-life.  There are a few dozen exceptions.  When the parent lives a lot longer than the daughter, it is noticeable on these charts as a small bump to the right of the parent.

Chain 97: A daughter outlives its parent

For Chain 144 the bump can only be seen in the numerical data.

Chain 144: A very, very short-lived daughter

Chain 135 shows the small bump of 135-Xe to the right of its parent, 135-I.  The half-life of 135-Xe is longer, but the HLE is shorter!  The presence of 135-I complicates the 135-Xe problems, as it keeps feeding the 135-Xe population.

Chain 135: Parent outlives daughter – if capture is included!

Long-Lived Fission Products (LLFP)

Chain 135 (above) does not end at 135-Ce (Rung 1).  There are seven fission products with half-lives in the “million-year” range.  These are the main long-term radioactive hazard after failed fuel.  Fortunately, 90% of the radioactivity is due to one isotope. 99-Tc, and this has a decent capture cross-section.  Keep it in the reactor!

Chain 126 has another LLFP, 126-Sn.  This is, at Rung 2, the most popular of the initial fission fragments at atomic weight 126.  Therefore, one can assume a non-zero initial yield for “nearby” isotopes.  In this case, that includes isomers at rungs 1.2 and 1.1.  These are lower on the energy ladder than 126-Sn, so avoid getting hung up there.  Instead, they reach stability fairly quickly, supplying the slight rise in 126-Te at Rung 0 at the right.

LLFP delays the evolution to the real Sink

Quick to stability

Most of Chain 126 settles to stability very quickly.  The flatness of the Rung 2 curve starts early.  Only a few chains become stable as quickly.  Chain 100 is the fastest, with the longest half-life of an unstable isotope being 7.1 seconds.  100-Mo flattens out faster than any other Sink.

Stable within minutes

Group Analysis

This analysis groups fission fragment populations with an odd kluge of aggregation methods.  The first group consists of the nine half-life bins, Time_Bins 0-8.  The numbers for these exclude isomers and shadow-side nuclides, which are listed separately.  The shadowed numbers include isomers, so the given isomer values are for the main decay chains only.

Contribution by Group

GroupIsotopesYield (atoms)Contribution (%)
Yield & Contribution per TimeBin + Shadowed (Rung>9) & Isomers (Level>0)
Yields for Timestep 20 (12 days), 235-U/Slow with capture


There are a total of 2,097,152 (2Meg) atoms in the waste stream at Snapshot 20. About 45% 0f those are stable by then, and 16% are primordial.


The contribution of shadow-side nuclides increases with neutron capture.  Some stable isotope can capture a neutron and move to the right on the NuDat chart.  This can place them on an Isolated Island.  From there, a B- decay will move them to the second sink.



Standard Physics Terms

Nuclide:  An isotope or isomer.  Nuclide is the primary key that links the data across the various tables in the IFFY database.

Isotope:  A specific combination of protons and neutrons in the nucleus of an atom.  The sum of the number of protons and neutrons is the isotope’s atomic weight.  Isotopes of the same element have the same number of protons, but differ in the number of neutrons.  IFFY designates an isotope by its atomic weight, followed by a dash, followed by one or two letters symbolizing the element.  (14-C won’t be found here.  It’s outside the range of fission fragments.)

Isomer:  A nucleus in an excited energy state.  Designated by -1 or -2 after the isotope name.  Think of them as football-shaped, which will soon relax to the ground state.

Internal Transform (IT):  The decay mode of an isomer to the ground state of the isotope.

Primordial:  An isotope (or the longest-lived isomer, 180-Ta-1) with a half-life long enough to survive since the formation of Earth.  Primordial isotopes are considered stable here.

Fission:  The process of splitting an atom.  Unless otherwise stated, the fuel atom involved will be 235-U.

Fission fragment/product:  One of the two nuclei created in a fission event.  A few neutrons are also freed, which keep the chain reaction going.

Fissile Fuel: An atom that will fission when it absorbs a Slow neutron, such as 235-U.  (A fast neutron will split just about anything.)

Fertile fuel (or fuelstock): An atom that will become fissile after it captures one neutron.  232-Th and 238-U are the primary sources of atoms that will someday become fuel.

Neutron Poison:  An isotope that likes to absorb the free neutrons needed to keep the chain reaction going.  The premier poison is 135-Xe.

Barn:  A unit of area about the size of a 235-U nucleus in cross-section.  Apparently, a neutron doesn’t have to hit the nucleus head-on to cause fission.  The cross-section for neutron absorption for 135-Xe is over 2.6 million barns, so it will reach into the next county to snatch a neutron.

Radioactive Decay Modes

There are a number of ways an unstable nuclide can change to a different nuclide.  The Internal Transform mode is mentioned above.

Beta Minus (B-):  This is easily the most popular decay mode for fission fragments.  Technically, it uses the weak nuclear force to switch a quark inside a neutron from down to up.  More practically, it turns a neutron into a proton.  This is the old alchemists’ dream, changing one element into another.  This transmutation moves the nucleus one cell to the right on the periodic table.

An isomer can skip over its ground state cousin by decaying via the B- mode.  This happens for 85-Kr-1, which reduces the population of 85-Kr by ~78%.  85-Kr is one of the bothersome waste products with a half-life in the range of ten to thirty years.  (This is practically the only reason we need the extra complexity that comes with isomers!)

B-m: This differs from B- because the result is an isomer.  85-Br becomes 85-Kr-1, so the nucleus does not drop all the way to the next rung (see Ladder analogy below).  The “m” means meta-stable, which describes the isomer’s energy state.

Beta Plus (B+) and Electron Capture (EC):  These are symmetric with B-, and relevant to the shadowed decay chains.  The nucleus becomes the next lighter element.  Some nuclides decay by EC, some B+, and some both.  In either case a proton changes to a neutron.

EC-m/B+m: This is the shadowed side’s equivalent to B-m.  This is the only way to get from a Shadowed chain to a main chain!  Due to this, the shadowed side is processed first, so the evolution of fission fragments can be done in one pass.

B-N: This is the most common form of neutron (N) emission.  136-Sb, 100-Rb, and 98-Rb are so unstable they eject two neutrons in B-2N decay.

B-mn: A neutron is emitted and the result is an isomer.

IFFY Analogies

Stable valley:  A “proper” combination of neutrons and protons is stable.  Fission usually results in nuclei that have too many neutrons.  These are located on the neutron-heavy side of the stable valley, energetically “uphill” of the stable isotopes.  They roll downhill, radiating energy via radioactive decay as they spontaneously seek to become stable.  The NuDat website has a wonderful interactive graphic of the valley.

Decay chain:  The evolutionary path of an unstable nucleus generally stays at the same atomic weight.  For example, the lightest IFFY decay chain starts at 65-Fe and transmutes through 65-Co and 65-Ni, and stops at stable 65-Cu.

A highly unstable nucleus can eject a neutron, which pushes the nucleus down to the next decay chain.  In fact, the lightest decay chain in the data is for atomic weight 66.  11% of the time 66‑Mn emits a neutron, which changes it to atomic weight 65.  (We had to add this chain by hand.)  The heaviest decay chains in the source data are of atomic weight 172.

Shadowed chains:  The main decay chains cover neutron-heavy nuclei plus the stable isotope at the end of the chain.  The shadowed chains involve the rare neutron-poor isotopes.  The analogy is that of a mountain and the rain shadow it causes, which is exactly opposite of the valley analogy!  The terminating stable isotope does not have to be the same as the isotope at the end of the main chain at that atomic weight, because there are sometimes more than one stable isotope at a given atomic weight.

Sink:   Unstable nuclei “drain” to a stable terminator.  Again, this is energetically downhill, which is the only possible direction for a spontaneous physical process such as radioactive decay.  There are three sinks for atomic weight 138 and 96, and two for many others, especially when the atomic weight is an even number..

Unstable islands:  Nature likes even numbers, apparently.  If there are two sinks at a given atomic weight, they usually have an even number of protons.  The “odd” isotope between them might decay either way.  These are the islands in the stream of the stable valley.

Decay Ladder:  This analogy emphasizes the discrete, as opposed to continuous, nature of radioactive decay, and its downward direction with respect to energy.  The sink at the bottom of each major decay chain is assigned a Rung value of zero.  Isotopes with too many neutrons are given higher rung values, always integers.  The data for one chain has nine rungs.

Shadowed chains and unstable islands are given larger rung values.  When decay is processed, these come first.

Isomers are given an extra 0.1 or 0.2 in their Rung value (and the analogy suffers a bit).

Family relationships: The decaying fission fragment is called the parent.  It decays to a daughter, which becomes a parent when/if it decays.  Isomers are called cousins.

New IFFY Terms

Rung:  In a simple case, the rung is equal to the number of destabilizing neutrons above the stable number in the Sink.  Rungs provides the (descending) order in which the decay chains can be processed.  A more complex treatment adds 15 or 20 for shadow-side nuclides, or 100 for unstable islands, in order to process all decaying nuclides in one pass.

Time_Bin:  A method of aggregating nuclides by how long they live.  Most half-life references are xx sec, xx min, xx years.  We keep the unit, but not the xx.

Half-Life Equivalent (HLE):  A measure of how long a nucleus lives when subjected to a steady neutron flux.  The neutrons can cause fission or plumping.  The neutrons come from outside the nucleus affected, whereas the changes due to natural radioactivity are due to internal instability.

HLE_Bin:  Similar to Time_Bin but only covers neutron absorbing isotopes, including fuels and failed fuels.  Time frames are slightly different, using weeks, months and decades but not seconds or minutes.

(Absorption) Class:  A roman numeral I, X, C, or M denoting the range of neutron absorption cross-section.  Z is for zero, but actually means less than one. I = 1…9,  X=10…99, C=100…999, M means >= 1000 barns.

Plump:  A nucleus will get heavier by one atomic number when it absorbs a neutron, and this often triggers transmutation to another element.  Some stable isotopes will plump up to another stable isotope of the same element, which is still referred to as transmutation.  We don’t like to use the verb transmute in this case, so we found a one syllable substitute.

Failed Fuel:  A fissile fuel atom that fails to fission will plump.  It may also transmute.  Either way, it will no longer be fuel.  It may become fertile fuel or sludge.

Sludge:  An heavy atom that requires more than one neutron to evolve into a fuel atom.  236-U needs three more before becoming fissile 239-Pu.  This is wasteful of neutrons, which are really expensive.  Many sludge isotopes have a low cross-section for capturing neutrons, so they sit in the reactor for a long time.

Evolution Charts

After 12 Days

The evolution of the waste stream becomes apparent when we analyze the relative contribution of groups of isotopes over time.  The Time_Bin characteristic aggregates the 1322 nuclides into nine groups.  Here’s an Excel chart that shows the composition of the waste getting proportionally more stable as the reactor gets older.  After a MegaSec, the hot isotope population (HL < 1 day) is no longer growing.  Their proportional contribution to the whole (normalized yield) is dropping.  Their population has leveled out,  but their daughters continue to grow.  Stable isotopes will only grow in population, unless they capture neutrons.

The Mass is 2Meg atoms at MSec and 2Gig atoms at GSec.  “Mass” is approximate.

Yikes! A bit of sensitivity testing— Average AtWt varies a lot!

  • Day: 122
  • Year: 129
  • Millenia: 111
  • Primordial: 123
  • Stable: 118

Guess I should change the Y-axis title to Population, or rather relative population.

The Full Span

The chart above focuses on the last ten (eleven) time steps.  A chart of the entire sim run is more colorful.

Evolution to Stability

Each chart is tailored to color the shorter-lived isotopes in hot colors, fading to black as the nuclei become stable.

These charts are still in development.  The y-axis isn’t quite right yet.  Normalized Yield runs from 0 to 2.0 atoms, not 0-100%.  The x-axis of the upper chart is more descriptive but,

  1. Not all of the TimeSteps have names, and
  2. They wouldn’t fit if they did.

Rung Analysis

IFFY introduces a new characteristic of isotopes, the Rung.  To mix three metaphors, the end of a decay chain is the bottom of the energy ladder, and the sink into which unstable nuclei drain.  The sink for the major decay chains is given a Rung value of Zero.  This is a stable isotope, which means it has the “proper” proportion of protons and neutrons.  Isotopes with too many neutrons have Rung values from one to nine, one for each excess neutron.

The table below shows how rapidly excess neutrons make the nucleus unstable.

Isotopes @ Rung per Time_Bin

Rungs 7, 8, 9 have only Zero bin isotopes
No shadowed isotopes or Isomers


  • All Rung 0 isotopes are stable or as good as stable for this analysis.
  • At Rung 3, there are no isotopes with half-lives longer than a day.
  • At Rung 4, there are no isotopes with half-lives longer than an hour.
  • At Rung 5, there are no isotopes with half-lives over a minute.
  • At Rung 6, there is just one isotope with a half-life over a second.

Gifts from Mother-of-Ra

Mother-of-Ra was the supernova that exploded a bit less than five billion years ago.  Our solar system was just a diffuse cloud of gas and dust before Mother-of-Ra pushed it all sideways.  This push compressed the gas so that it started to collapse of its own weight.  We call the result Sol, but the Egyptians were first.

There might have been a few atoms heavier than iron in that cloud from previous supernovae, but most of the heavy stuff we have now was gathered then.  The cloud’s population of Carbon and Oxygen was higher, since these elements can be puffed out into space by smaller stars in their red giant phase.  Red giants can make traces of the heavier elements through the slow s-process.  Supernovae use the rapid r-process to create most of the heavy elements.

The r-process started when Mother-of-Ra’s nuclear fire went out.  When the supernova collapsed, the outer layers rushed inward, picking up a lot of kinetic energy.  Some atoms may have approached 1/3 the speed of light in their fall!

If nature finds too much energy in one place it can relieve the stress by converting it to matter.  This is E = mc2, but the opposite of an A-bomb.  The gravitational energy converted to a flood of new neutrons that piled up on the atoms inside Mother-of-Ra.

  • Too many neutrons make a nucleus unstable.  It can relieve the stress by converting one of the neutrons to a proton.  The atom becomes the next element up on the periodic table.  This is the secret behind transmutation of the elements!

The flood of neutrons kept piling on.  Some nuclei obtained so many excess neutrons they transmuted several times per second.  This created the Tin, Gold, and Lead we use today.  The entire element-building process was over in about fifteen seconds!  Mother-of-Ra was generous in her gifts for those few seconds.

Thorium and Uranium are the heaviest elements that have survived since the supernova.  They can be considered nature’s energy storage device.  The energy they store is gravity!  What could be cleaner?

Relevant absorption

The IFFY yields do not yet take into account the absorption of neutrons by fission products (FP).  The Barns_FP table has data for 241 isotopes (no isomers), some of which are not fission fragments but might be useful someday.  There are 165 isotopes in the FullSnaps table that might be relevant neutron absorbers.  We can ignore many of these for three reasons:

  • 65 change from a Stable/Primordial isotope directly to another stable/primordial isotope of the same element,
  • 39 have a yield less than 2, which is 1 ppm at Snap20 (for 235-U/Slow),
  • 32 have a cross-section less than 1.0, which take at least 1200 years to transmute

This leaves 29 isotopes that might be relevant with respect to neutron absorptions that significantly change the composition of the waste.  These are listed in the Absorbers table with a Relevant value of 1.  In my copious free time…


A form of neutron absorption has now been incorporated into the model.

Primary keys: What’s that IZA stuff?

Table of Radioactive Isotopes (TORI_2) Issues

The primary key for the TORI_2 database is an ugly kluge called IZA, which may stand for “Isotope designator based on Z and A numbers”.  That itself requires further explanation.  Physicists, and possibly chemists, use Z to designate an element by its atomic number.  (IFFY does the same, but calls it Protons, not Z.)  Physicists use A for atomic weight.  IZA multiplies the Z by 10,000 and adds the A, creating an integer that identifies each isotope uniquely.  Isomers modify this by adding 300 for each energy level.  There are 277 atomic weights, so these will be unique as well.  Some isomers are four energy levels above the ground state, which means +1200.  This is why Z is multiplied by 10,000.  Multiplying by only 1000 might lead to duplicate IZA identifiers.

It is very common to use integers for the primary key.  A key must be unique, and integers do that.  It is also easy to add keys by adding one to the last key that was created.  Database engines have an auto-number function to do that.  When a new customer comes along, the database will associate her name with a new ID in the Customer table.  Her records (purchases, payments, deliveries, etc.) will be spread across several tables.  For ease of maintenance, you want to use her ID rather than her name in all these different places.  If she gets married and her name changes, it only needs to be changed in the Customer table.

Since they can change, names are not often used as a primary key.  But the names of isotopes/elements are very unlikely to change.  (Exception: Super-heavy elements have place-holder Latin names until they are confirmed to exist.  These are way beyond the range needed for IFFY.)  So, we decided on a simpler and far more readable key – the name of the nuclide.

Just for grins, though, we have morphed IZA to EZA.  In this scheme, the Energy (level) of the isomer is multiplied by a million.  The proton number, multiplied by 1000, is added to that and then atomic weight.  When sorted, the isomers will be lumped at the top or bottom of the list.