Tag: radioactive decay

A corny teaching analogy

Pop goes the comparison

Keith S. Taber

The order of corn popping is no more random than the roll of a dice.

I was pleased to read about a 'new' teaching analogy in the latest 'Education in Chemistry' (the Royal Society of Chemistry's education magazine) – well, at least it was new to me. It was an analogy that could be demonstrated easily in the school science lab, and, according to Richard Gill (@RGILL_Teach on Twitter), went down really well with his class.

Teaching analogies

Analogies are used in teaching and in science communication to help 'make the unfamiliar familiar', to show someone that something they do not (yet) know about is actually, in some sense at least, a bit like something they are already familiar with. In an analogy, there is a mapping between some aspect(s) of the structure of the target ideas and the structure of the familiar phenomenon or idea being offered as an analogue. Such teaching analogies can be useful to the extent that someone is indeed highly familiar with the 'analogue' (and more so than with the target knowledge being communicated); that there is a helpful mapping across between the analogue and the target; and that comparison is clearly explained (making clear which features of the analogue are relevant, and how).

Read about analogies in science

The analogy is discussed in the July 2022 Edition of Education in Chemistry, and on line.

Richard Gill suggests that 'Nuclear decay is a tough concept' to teach and learn, but after making some popcorn he realised that popping corn offered an analogy for radioactive decay that he could demonstrate in the classroom.

Richard Gill describes how

"I tell the students I'm going to heat up the oil; I'm going to give the kernels some energy, making them unstable and they're going to want to pop. I show them under the visualiser, then I ask, 'which kernel will pop first?' We have a little competition. Why do I do this? It links to nuclear decay being random. We know an unstable atom will decay, but we don't know which atom will decay or when it will decay, just like we don't know which kernel will pop when."

Gill, 2022

In the analogy, the corn (maize) kernels represents atoms or nuclei of an unstable isotope, and the popped corn the decay product, daughter atoms or nuclei. 1

'Parents' before 'decay
(Image by Paulo Marcos pmarcos6821 from Pixabay)
'Daughters' after 'decay'
(Image by Leonardo Alvarado from Pixabay)

Richard Gill homes in on a key feature of radioactive decay which may seem counter-intuitive to learners, but which is actually a pattern found in many different phenomena – exponential decay. The rate of radioactive decay falls (decays, confusingly) over time. Theoretically the [radioactive] decay rate follows a very smooth [exponential] decay curve. Theoretically, because of another key feature of radioactive decay that Gill highlights – its random nature!

It may seem that something which occurs by random will not lead to a regular pattern, but although in radioactivity the behaviour of an individual nucleus (in terms of when it might decay) cannot be predicted, when one deals with vast numbers of them in a macroscopic sample, a clear pattern emerges. Each different type of unstable atom has an associated half-life which tells us when half of a sample will have decayed. These half-lives can vary from fractions of a second to vast numbers of years, but are fixed for a particular nuclide.

Richard Gill notes that he can use the popping corn demonstration as background for teaching about half-life,

I usually follow this lesson with the idea of half-lives. The concept of half-lives now makes sense. Why are there fewer unpopped kernels over time? Because they're popping. Why do radioactive materials become less radioactive over time? Because they're decaying.

Gill, 2022

Perhaps he could even develop his demonstration to model the half-life of decay?

Modelling the popcorn decay curve

The Australian Earth Science Education blog suggests

"Popcorn can be used to model radioactive decay. It is a lot safer than using radioactive isotopes, as well as much tastier"

and offers instructions for a practical activity with a bag of corn and a microwave to collect data to plot a decay curve (see https://ausearthed.blogspot.com/2020/04/radioactive-popcorn.html). Although this seems a good idea, I suspect this specific activity (which involves popping the popping corn in and out of the oven) might be too convoluted for learners just being introduced to the topic, but could be suitable for more advanced learners.

However, The Association of American State Geologists suggests an alternative approach that could be used in a class context where different groups of students put bags of popcorn into the microwave for different lengths of time to allow the plotting of a decay curve by collating class results (https://www.earthsciweek.org/classroom-activities/dating-popcorn).

Another variants is offered by The University of South Florida's' Spreadsheets Across the Curriculum' (SSAC) project. SSAC developed an activity ("Radioactive Decay and Popping Popcorn – Understanding the Rate Law") to simulate the popping of corn using (yes, you guessed) a spreadsheet to model the decay of corn popping, as a way of teaching about radioactive decay!

This is more likely to give a good decay curve, but one cannot help feeling it loses some of the attraction of Richard Gill's approach with the smell, sound and 'jumping' of actual corn being heated! One might also wonder if there is any inherent pedagogic advantage to simulating popping corn as a model for simulating radioactive decay – rather than just using the spreadsheet to directly model radioactive decay?

Feedback cycles

The reason the popping corn seems to show the same kind of decay as radioactivity, is because it can be represented with the same kind of feedback cycle.

This pattern is characteristic of simple systems where

  • a change is brought about by a driver
  • that change diminishes the driver

In radioactive decay, the level of activity is directly proportional to the number of unstable nuclei present (i.e., the number of nuclei that can potentially decay), but the very process of decay reduces this number (and so reduces the rate of decay).

So,

  • when there are many unstable nuclei
  • there will be much decay
  • quickly reducing the number of unstable nuclei
    • so reducing the rate of decay
    • so reducing the rate at which unstable nuclei decay
      • so reducing the rate at which decay is reducing
      • …

and so forth.

Exponential decay is a characteristic of systems with a simple negative feedback cycle
(source: ASCEND project)

Recognising this general pattern was the focus of an 'enrichment' activity designed for upper secondary learners in the Gatsby SEP supported ASCEND project which presented learners with information about the feedback cycle in radioactive decay; and then had them set up and observe some quite different phenomena (Taber, 2011):

  • capacitor discharge
  • levelling of connected uneven water columns
  • hot water cooling

In each case the change driven by some 'driver' reduced the driver itself (so a temperature difference leads to heat transfer which reduces the temperature difference…).

Read about the classroom activity

In Richard Gill's activity the driver is the availability of intact corn kernels being heated such that water vapour is building up inside the kernel – something which is reduced by the consequent popping of those kernels.

A negative feedback cycle

Mapping the analogy

A key feature of an analogy is that it can be understood as a kind of mapping between two conceptual structures. The making popcorn demonstration seems a very simple analogue, but mapping out the analogy might be useful (at least for the teacher) to clarify it. Below I present a representation of a mapping between popping corn and radioactive decay, suggesting which aspects of the analogue (the popping corn) map onto the target scientific concept.

Mapping an analogy between making pop-corn and radioactive decay

In this mapping I have used colour to highlight differences between the two (conceptual) structures. Perhaps the most significant difference is represented by the blue (target concept) versus red (analogue) features.

Most analogies only map to a limited extent

There will be aspects of an analogue that do not map onto anything on the target, and sometimes there will be an important feature of the target which has no analogous feature in the analogue. There is always the possibility that irrelevant features of an analogue will be mapped across by learners.

As one example, the comparison of the atom with a tiny solar system was once an image often used as a teaching analogy, yet it seems learners often have limited understandings of both analogue and target, and may be transferring across inappropriately – such as assuming the electrons are bound to the atom by gravity (Taber, 2013a). Where students have an alternative conception of the analogue (the earth attracts the sun, but not vice versa) they will often assume the same pattern in the target (the nucleus is not attracted to the electrons).

Does this matter? Well, yes and no. A teaching analogy is used to introduce a technical scientific concept by making it seem familiar. This is a starting point to be built upon (so, Richard Gill tells us that he will build upon the diminishing activity of cooking corn in his his popcorn demonstration to introduce the idea of half-life), so it does not matter if students do not fully understand everything immediately. (Indeed, it is naive to assume most learners could acquire a new complex set of ideas all at once: learning is incremental – see Key ideas for constructivist teaching).

Analogies can act as 'scaffolds' to help learners venture out from their existing continents of knowledge towards new territory. Once this 'anchor' in learners' experience is established one can, so to speak, disembark from the scaffolding raft the onto the more solid ground of the shore.

Read about scaffolding learning

However, it is important to be careful to make sure

  • (a) learners appreciate the limitations of models (such an analogies) – that they are thinking and learning tools, and not absolute accounts of the natural word; and that
  • (b) the teacher helps dismantle the 'scaffolding' once it is not needed, so that it is not retained as part of the learners 'scientific' account.
Weak anthropomorphism

An example of that might be Gill's use of anthropomorphism.

…unstable atoms/nuclei need to become stable…

…I'm going to give the kernels some energy, making them unstable and they're going to want to pop…

Anthropomorphism

This type of language is often used to offer narratives that are more readily appreciated by learners (making the unfamiliar familiar, again) but students can come to use such language habitually, and it may come to stand in place of a more scientific account (Taber & Watts, 1996). So, 'weak' anthropomorphism used to help introduce something abstract and counter-intuitive is useful, but 'strong' anthropomorphism that comes to be adopted as a scientific explanation (e.g., nuclei decay because they want to be stable) is best avoided by seeking to move beyond the figurative language as soon as students are ready.

Read about anthropomorphism

The 'negative' analogy

The mapping diagram above may highlight several potential teaching points that may be considered (perhaps not to be introduced immediately, but when the new concepts are later reinforced and developed).

Where does the energy come from?

One key difference between the two systems is that radioactive decay is (we think) completely spontaneous, whereas the corn only pops because we cook it (Gill used a Bunsen burner) and left to its own devices remains as unpopped kernels.

Related to this, the source of energy for popping corn is the applied heat, whereas unstable nuclei are already in a state of high energy and so have an 'internal' source for their activity. This a key difference that will likely be obvious to some, but certainly not all learners in most classes.

When is random, random?

A more subtle point relates to the 'random' nature of the two events. I suggest subtle, because there are many published reports written by researchers in science education which suggests even supposed experts can have a pretty shaky ideas of what counts as random (Taber, 2013b).

Read 'Nothing random about a proper scientific evaluation?'

Read about the randomisation criterion

As far as scientists understand, the decay of one unstable nucleus in a sample of radioactive material (rather than another) is a random process. It is not just that we are not yet able to predict when a particular nucleus will decay – according to current scientific accounts it is not possible to predict in principle. This is an idea that even Einstein found difficult to accept.

That is not true with the corn. Presumably there are subtle differences between kernels – some have slightly more water content, or slightly weaker casings. Perhaps more significantly, some are heated more than others due to their position in the pan and the position of the heat source, or due differential exposure to the cooking oil… In principle it would be possible to measure relevant variables and model the set up to make good predictions. (In principle, even if in practice a very complex task.) The order of corn popping is no more random than…say…the roll of a dice. That is, physics tells us it follows natural laws, even if we are not in a position to fully model the phenomenon.

(We might suggest that a student who considered the corn popping as a random event because she saw apparently identical kernels all being heated in the same pan at the same time is simply missing certain 'hidden variables'. Einstein wondered if there were also 'hidden variables' that science had not yet uncovered which could explain random events such as why one nucleus rather than another decays at a particular moment.)

On the recoil

Perhaps a more significant difference is what is observed. The corn are observed 'jumping' (more anthropomorphic language?) Physics tells us that momentum must always be conserved, and the kernels act like tiny jet propelled rockets. That is, as steam is released when the kernel bursts, the rest of the kernel 'jumps' in the opposite direction. (That is, by Newton's third law, there is a reaction force to the force pushing the steam out of the kernel. Momentum is a vector, so it is possible for a stationary object to break up into several moving parts with conservation of momentum.)

Something similar happens in radioactive decay. The emitted radiation carries away momentum, and the remaining 'daughter' nucleus recoils – although if the material is in the solid state this effect is dissipated by being spread across the lattice. So, the radioactivity which is detected is not analogous to the jumping corn, but to the steam it has released.

Is this important? That likely depends upon the level being taught. If the topics is being introduced to 14-16 years-olds, perhaps not. If the analogy is being explored with post-compulsory students doing an elective course, then maybe. (If not in chemistry; then certainly in physics, where learners are expected to to apply the principle of conservation of momentum across various scenarios.)

Will this be on the exam?

When I drafted this, I suspected most readers might find my caveats above about the limitations of the analogy, a bit pernickety (the kind of things an academic who's been out of the school classroom too long and forgotten the realities of working with pupils might dream up), but then I found what claims to be an Edexcel GCE Physics paper from 2012 (paper reference 6PH05/01) on line. In this paper, one question begins:

"In a demonstration to her class, a teacher pours popcorn kernels onto a hot surface and waits for them to pop…".

Much to my delight, I found the first part of this question asked learners:

"How realistic is this demonstration as an analogy to radioactive decay?

Consider aspects of the demonstration that are similar to radioactive decay and aspects that are different"

Examination paper asking physics students to identify positive and negative aspects of the analogy.

Classes of radioactivity

One further difference did occur to me that may be important. At some level this analogy works for radioactivity regardless of what is being emitted from an excited nucleus. However, the analogy seems clearer for the emission of an alpha particle, or a beta particle, or a neutron, than in the case of gamma radiation.

Although in gamma decay an excited nucleus relaxes to a lower energy state emitting a photon, it may not be as obvious to learners that the nucleus has changed (arguably, it has not 'substantially' changed as there is no change of substance) – as it has the same mass number and charge as before. This may be a point to be raised if moving on later to discuss different classes of radioactivity.

Or, perhaps, with gamma decay one can use a different version of the analogy?

Another corny analogy

Although I do not think I had never come across this analogy before reading the Education in Chemistry piece (perhaps because I do not make myself popcorn), Richard Gill does not seem to be the only person to have noticed this comparison. (They say 'great minds think alike' – and not just physicist Henri Poincaré thinking like Kryten from'Red Dwarf'). When I looked around the world-wide web I found there were two different approaches to using corn kernels to model radioactivity.

Some people use a similar demonstration to Mr Gill.2 However, there was also a different approach to using the corn. There were variations on this 3, but the gist was that

  • one starts with a large number of kernels
  • they are agitated (e.g., shaken in a box with different cells, poured onto the bench…)
  • then inspected to see which are pointing in some arbitrary direction designated as representing decay
  • the 'decayed' kernels are removed and counted
  • the rest of the sample is agitated again
  • etc.
Choose a direction to represent decay, and remove the aligned kernels as the 'activity' in that interval.
(Original image by Susie from Pixabay)

This lacks the excitement of popping corn, but could be a better model for gamma decay where the daughter nucleus is at a different energy after decay, but is otherwise unchanged.

Perhaps this version of the analogy could be improved by using a tray with an array of small dips (like tiny spot tiles) just the right size to stand corn kernels in the depressions with their points upwards. Then, after a very gentle tap on the bench next to the tile, those which have 'relaxed' from the higher energy state (i.e., fallen onto their sides) would be considered decayed. This would more directly model the change in potential energy and also avoid the need to keep removing kernels from the context (just as daughter atoms usually remain in a sample of radioactive material), as further gentle tapes are unlikely to excite them back to the higher energy state. 4

Or, dear reader, perhaps I've just been thinking about this analogy for just a little too long now.

Sources:
Notes

1 Referring to the nuclei before and after radioactive decay as 'parents' and 'daughters' seems metaphorical, but this use has become so well established (in effect, these are now technical terms) that these descriptors are now what are known (metaphorically!) as 'dead metaphors'.

Read about metaphors in science

2 Here are some examples I found:

Jennifer Wenner, University of Wisconsin-Oshkosh uses the demonstration in undergraduate geosciences:

"I usually perform it after I have introduced radioactive decay and talked about how it works. It only takes a few minutes and I usually talk while I am waiting for the "decay" to happen 'Using Popcorn to Simulate Radioactive Decay'"

https://serc.carleton.edu/quantskills/activities/popcorn.html

The Institute of Physics (IoP) include this activity as part of their 'Modelling decay in the laboratory Classroom Activity for 14-16' but suggest the pan lid is kept on as a safety measure. (Any teacher planing on carrying out any activity in the lab., should undertake a risk assessment first.)

I note the IoP also suggests care in using the term 'random':

Teacher: While we were listening to that there didn't seem to be any fixed pattern to the popping. Is there a word that we could use to describe that?

Lydia: Random?

Teacher: Excellent. But the word random has a very special meaning in physics. It isn't like how we think of things in everyday life. When do you use the word random in everyday life?

Lydia: Like if it's unpredictable? Or has no pattern?

https://spark.iop.org/modelling-decay-laboratory

Kieran Maher and 'Kikibooks contributors' suggests readers of their 'Basic Physics of Nuclear Medicine' could "think about putting some in in a pot, adding the corn, heating the pot…" and indeed their readers "might also like to try this out while considering the situation", but warn readers not to "push this popcorn analogy too far" (pp.20-21).

3 Here are some examples I found:

Florida High School teacher Marc Mayntz offers teachers' notes and student instructions for his 'Nuclear Popcorn' activity, where students are told to "Carefully 'spill' the kernels onto the table".

Chelsea Davis (a student?) reports her results in 'Half Life Popcorn Lab' from an approach where kernels are shaken in a Petri dish.

Redwood High School's worksheet for 'Radioactive Decay and Half Life Simulation' has students work with 100 kernels in a box with its sides labelled 1-4 (kernels that have the small end pointed toward side 1 after "a sharp, single shake (up and down, not side to side)" are considered decayed). Students are told at the start to to "Count the popcorn kernels to be certain there are exactly 100 kernels in your box".

This activity is repeated but with (i) kernels pointing to either side 1 or 2; and in a further run (ii) any of sides 1, 2, or 3; being considered decayed. This allows a graph to be drawn comparing all three sets of results.

The same approach is used in the Utah Education network's 'Radioactive Decay' activity, which specifies the use of a shoe box.

A site called 'Chegg' specified "a square box is filled with 100 popcorn kernels". and asked "What alteration in the experimental design would dramatically change the results? Why?" But, sadly, I needed to subscribe to see the answer.

The 'Lesson Planet' site offers 'Nuclear Popcorn' where "Using popcorn kernels spread over a tabletop, participants pick up all of those that point toward the back of the room, that is, those that represent decayed atoms".

'Anonymous' was set a version of this activity, but could not "seem to figure it out". 'Jiskha Homework Help' (tag line: "Ask questions and get helpful responses") helpfully responded,

"You ought to have a better number than 'two units of shake time…'

Read off the graph, not the data table."

(For some reason this brought to mind my sixth form mathematics teacher imploring us in desperation to "look at the ruddy diagram!")

4 Consider the challenge of developing this model to simulate nuclear magnetic resonance or laser excitation!

How much damage can eight neutrons do?

Scientific literacy and desk accessories in science fiction

Keith S. Taber

Is the principle of conservation of mass that is taught in school science falsified all the time?

I am not really a serious sci-fi buff, but I liked Star Trek (perhaps in part because it was the first television programme I got to see in colour 1) and I did enjoy Blakes7 when it was broadcast by the BBC (from 1978-1981).

Logo of the BBC television series 'Blakes7' with (a) home desk equipment or (b) manual control lever for the most advanced spaceship in the galaxy (Image by Francis Ray from Pixabay).

Blakes7 was made with the same kind of low budget production values of Dr Who of the time. Given that space scenes in early episodes involved what seemed to be a flat image of a spacecraft moving across a star field with no sense of depth or perspective (for later series someone had built a model), and in one early episode the crew were clearly given angle-poise lamps to control the craft, it was certainly not a case of 'no expense spared'. So, it was never quite clear if the BBC budget had also fallen short of a possessive apostrophe in the show title credits or Blakes7 was to be read in some other way.

After all, it was not made explicit who was part of Blake's 7 if that was what the title meant, and no one referred to "Blake's 7" in the script (perhaps reflecting how the doctor in Dr Who was not actually called Dr Who?).

The Blakes7 team on the flight desk of the Liberator – which was the most advanced spaceship in the galaxy (and was, for plot purposes, conveniently found drifting in space without a crew) – at least until they forgot to clean the hull once too often and it corroded away while they were on an away mission.

Blake's group was formed from a kind of prison break and so Blake was something of a 'rough-hero' – but not as much as his sometime unofficial lieutenant, sometime friend, sometime apparent rival, Avon, who seemed to be ruled by self-interest (at least until the script regularly required some act of selfless heroism from him). 'Rough-heroes' are fictional characters presented in the hero role but who have some traits that the audience are likely to find morally questionable if not repugnant.

As well as Blake (a rebel condemned as a traitor, having 'recovered' from brainwashing-supported rehabilitation to rebel again) and Avon (a hacker convicted of a massive computer fraud intended to make himself extremely rich) the rest of the original team were a smuggler, a murderer and a petty thief, to which was added a terrorist (or freedom fighter if you prefer) picked up on an early mission. That aside, they seemed an entirely reasonable and decent bunch, and they set out to rid the galaxy of 'The Federation's tyrannical oppression. At least, that was Blake's aspiration even if most of his companions seemed to see this as a stop-gap activity till they had decided on something with more of a long-term future.

At the end of one season, where the fight with the Federation was temporarily put aside to deal with an intergalactic incursion, Blake went AWOL (well, intergalactic wars can be very disruptive) and was assumed dead/injured/lost/captured/?… for much of the remaining run without affecting the nature of the stories too much.

Among its positive aspects for its time were strong (if not exactly model) roles for women. The main villain, Servalan, was a woman – Supreme Commander of the Federation security forces (and later Federation president).

As the ruthless Supreme Commander of the Federation security forces, Servalan got to wear whatever she liked (a Kid Creole, or Mel and Kim, look comes to mind here) and could insist her staff wore hats that would not upstage hers

In Blake's original team (i.e., 7?), his pilot is a woman. (Reflecting other SciFi series, the spacecraft used by Blakes7 require n crew members to operate effectively, where n is an integer that varies between 0 and 6 depending on the specific plot requirements of an episode.) In a later series, after Avon has taken over the role of 'ipso facto leader-among-equals', the group recruits a female advanced weapons designer/technologist and a female sharpshooter.

The Blakes7 team later in the run. (Presumably they are checking the monitor and having a quick recount.) Was Soolin (played by Glynis Barber, far right) styled as a subtle reference to the 'Seven Samurai'?

When I saw Blakes7 was getting a rerun recently I re-watched the series I had not seen since it was first aired. Despite very silly special effects, dodgy story-lines, and morally questionable choices (the series would make a great focus for a philosophy class) the interactions between the main characters made it an enjoyable watch.

But, it is not science

Of course, the problem with science fiction is that it is fiction, not science. Star Trek may have prided itself on seeking to at least make the science sound feasible, but that is something of an outlier in the genre.

Egrorian and his young assistant Pinder (unfortunately prematurely aged somewhat by a laboratory mishap) show Avon and Vila around their lab.

This is clear, for example, in an episode called 'Orbit' where Avon discuses the tachyon funnel, an 'ultimate weapon', with Egrorian, a renegade scientist. Tachyons are hypothetical particles that travel faster than the speed of light. The theory of special relatively suggests the speed of light is the theoretical maximum speed anything can have, but some other theories suggest tachyons may exist in some circumstances. As always in science, theories that are widely accepted as our current best understanding of some aspect of nature (e.g., relativity) are still open to modification or replacement if new evidence is found that suggests this is indicated.

In the Blakes7 universe, there seemed to be a surprisingly high frequency of genius scientists/engineers who had successfully absconded from the tyrannical and paranoid Federation with sufficient resources to build private research facilities on various obscure deserted planets. Although these bases are secret and hidden away, and the scientists concerned have normally been missing for years or even decades, it usually transpires that the Blakes7 crew and the Federation manage to locate any particular renegade scientist during the same episode.

This is part of the exchange between this particular flawed genius scientist and our flawed and reluctant 'rough hero', Kerr Avon:

Egrorian: You've heard of Hoffal's radiation?

Avon: No.

Ah… Hoffal had a unique mind. Over a century ago he predicted most of the properties that would be found in neutron material.

Neutron material?

Material from a neutron star. That is a… a giant sun which has collapsed and become so tightly compressed that its electrons and protons combine, making neutrons.

I don't need a lecture in astrophysics. [But presumably the scriptwriter felt the audience would need to be told this.]

When neutrons are subjected to intense magnetic force, they form Hoffal's radiation. Poor Pinder [Egrorian's lab. assistant] was subjected for less than a millionth of a second. He aged 50 years in as many seconds. …

So neutrons are part of the tachyon funnel.

Um, eight of them … form the core of the accelerator. 

From the script of 'Orbit' (c) 1981 by the British Broadcasting Corporation – made available 'for research purposes'

Now, for anyone with any kind of science background such dialogue stretches credibility. Chadwick discovered the neutron in everyday matter in 1932, so the neutron's properties could be explored without having to obtain samples from a neutron star – which would certainly be challenging. When bound in nuclei, neutrons (which are electrically neutral, thus the name, and so not usually affected by magnetic fields) are stable.

Thinking at the scale of a neutron

However, any suspension of disbelief (which fiction demands, of course) was stretched past breaking point at the end of this exchange. Not only were the generally inert neutrons the basis of a weapon that could destroy whole worlds – but the core of the accelerator was formed of, not a neutron star, nor a tonne of 'neutron matter', but eight neutrons (i.e., one for each member of Blake's 7 with just a few left over?)

That is, the intensely destructive beam of radiation that could destroy a planet from a distant solar system was generated by subjecting to a magnetic field: a core equivalent to (the arguably less interesting) half of a single oxygen atomic nucleus.

Warning – keep this away from strong magnetic fields if you value your planet! (Image by Gerd Altmann from Pixabay )

Now free neutrons (that is, outside of an atomic nuclei – or neutron star) are unstable, and decay on a timescale of around a quarter of an hour (that is, the half-life is of this order – following the exponential decay familiar with other kinds of radioactivity), to give a proton, an electron and a neutrino. The energy 'released' in this process is significant on the scale of a subatomic particle: 782 343 eV or nearly eight hundred thousand eV.

Eight hundred thousand seems a very large number, but the unit here is electron volt, a unit used for processes at this submicroscopic scale. (An eV is the amount of work that is done when one single electron is moved though a potential difference of 1v – this is about 1.6 x10-19 J). In the more familiar units of joules, this is about 1.25 x 10-13 J. That is,

0.000 000 000 000 125 J

To boil enough water at room temperature to make a single cup of tea would require about 67 200 J. 2 So, if the energy from decaying neutrons were used to boil the water, it would require the decay of about

538 000 000 000 000 000 neutrons.3

That is just to make one cup of tea, so imagine how many more neutrons would have to decay to provide the means to destroy a planet. Certainly, one would imagine,

more than 8.

E=mc2

Now since Einstein (special relativity, again), mass and energy have been considered to have an equivalence. It is commonly thought that mass can be converted to energy and the equation E=mc2 tells you how much of one would be converted to the other: how many J per kg or kg per J. (Spoiler alert – this is not quite right.)

In that way of thinking, the energy released by a free neutron when it decays is due to a tiny part of the neutrino's mass being converted to energy.

The neutron's mass defect

The mass (or so called 'rest mass') of a neutrino is about 1.67 x 10-27 kg. In the usual mode of decay the neutrino gives rise to a proton (which is nearly, but not quite, as heavy as a neutron), an electron (which is much lighter), and a neutrino (which is considered to have zero rest mass.)

Before decayRest mass / 10-31 kgAfter decayRest mass / 10-31 kg
neutron16 749.3proton16 726.2
electron9.1
neutrino–
total16 749.316 735.3
[rest] mass defect in neutrino decay

So, it seems like some mass has disappeared. (And this is the mass sometimes said to have been converted into the released energy.) This might lead us to ask the question of whether Hoffal's discovery was a way to completely annihilate neutrons, so that instead of a tiny proportion of their mass being converted to energy as in neutron decay – all of it was.

Mass as latent energy?

However, when considered from the perspective of special relativity, it is not that mass is being converted to energy in processes such as neutron decay, but rather that mass and energy are considered as being different aspects of something more unified -'mass-energy' if you like. Energy in a sense carries mass, and mass in a sense is a manifestation of energy. The table above may mislead because it only refers to 'rest mass' and that does not tell us all we need to know.

When the neutron decays, the products move apart, so have kinetic energy. According to the principle of mass-energy equivalence there is always a mass equivalence of any energy. So, in relativity, a moving object has more mass than when it is at rest. That is, the 'mass defect' table shows what the mass would be if we compared a motionless neutron with motionless products, not the actual products.

The theory of special relativity boldly asserts that mass and energy are not the independent quantities they were once thought to be. Rather, they are two measures of a single quantity. Since that single quantity does not have its own name, it is called mass-energy, and the relationship between its two measures is known as mass-energy equivalence. We may regard c2 as a conversion factor that enables us to calculate one measurement from the other. Every mass has an energy-equivalent and every energy has a mass-equivalent. If a body emits energy to its surroundings it also emits a quantity of mass equivalent to that energy. The surroundings acquire both the energy and mass in the process.

Treptow, 2005, p.1636

So, rather than thinking mass has been converted to energy, it may be more appropriate to think that the mass of a neutron has a certain (latent) energy associated with it, and that, after decay, most of this energy is divided between products (according to their rest masses), but a small proportion has been converted to kinetic energy (which can be considered to have a mass equivalence).

So, whenever any process involves some kind of energy change, there is an associated change in the equivalent masses. Every time you boil the kettle, or go up in an elevator, there is a tiny increase of mass involved – the hot water is heavier than when it was cold; you are heavier than when you were at a lower level. When you lie down or burn some natural gas, there is a tiny reduction in mass (you weigh less lying down; the products of the chemical reaction weigh less than the reactants).

How much heavier is hot water?

Only in nuclear processes does the energy change involved become large enough for any change in mass to be considered significant. In other processes, the changes are so small, they are insignificant. The water we boiled earlier to make a cup of tea required 67 200J of energy, and at the end of the process the water would not just be hotter, but also heavier by about

0.000 000 000 000 747 kg

0r about 0.000 000 000 75 g. That is easy to calculate 4, but not so easy to notice.

Is mass conserved in chemical reactions?

On this basis, we might suggest that the principle of conservation of mass that is taught in school science is falsified all the time – or at least needs to be understood differently from how it is usually presented.

Type of reactionMass change
endothermicmass of products > mass of reactants
exothermicmass of products < mass of reactants
If we just consider the masses of the substances then mass is not conserved in chemical change

Yet, the discrepancies really are tiny – so tiny that in school examinations candidates are expected to pretend there is no difference. But, strictly, when (as an example) copper carbonate is heated in a crucible and decomposes to give copper oxide and carbon dioxide there is a mass decrease even if you could capture all the CO2. But it would not be measurable with our usual laboratory equipment – so, as far as chemistry is concerned, mass is conserved. 'To all intense and purposes' (even if not absolutely true) mass is always conserved in chemical reactions.

Mass is conserved overall

But actually, according to current scientific thinking, mass is always conserved (not just very nearly conserved), as long as we make sure we consider all relevant factors. The energy that allowed us to boil the kettle or be lifted in an elevator must have been provided from some source (which has lost mass by the same extent). In an exothermic chemical reaction there is an extremely slight difference of mass between the reactants and products, but the surroundings have been warmed and so have got (ever so slightly) heavier.

Type of reactionMass change
endothermicenergy (and equivalent mass) from the surroundings
exothermicenergy (and equivalent mass) to the surroundings
If we just consider the masses of the substances then mass does not seem to be conserved in chemical change

As Einstein himself expressed it,

"The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy."

Einstein, 1917/2015, p.59

Annihilate the neutrons!

So, if we read about how in particle accelerators, particles are accelerated to immense speeds, and collided, and so converted to pure energy we should be suspicious. The particles may well have been destroyed – but something else has now acquired the mass (and not just the rest mass of the annihilated particles, but also the mass associated with their high kinetic energy).

So, we cannot convert all of the mass of a neutron into energy – only reconfigure and redistribute its mass-energy. But we can still ask: what if all the mass of the neutron were to be converted into some kind of radiation that carried away all of its mass as high energy rays (perhaps Hoffal's radiation?)

Perhaps the genius scientist Hoffal, with his "unique mind", had found a way to do this (hm, with a magnetic field?) Even if that does not seem very feasible, it does give us a theoretical limit to the energy that could be produced by a process that converted a neutron into radiation.6 Each neutron has a rest mass of about

1.67 x 10-27 kg

now the conversion factor is c2 (where c is the speed of light, which is near enough 3 x 108 ms-1, so c2 =(3×108ms-1)2 , i.e., about 1017m2s-2), so that mass is equivalent to about 1.50 x 10-10 J 5 or,

0.000 000 000 150 J

Now that is a lot more energy than the 1.25 x 10-13 J released in the decay of a neutron,

0.000 000 000 150 000 J

>

0.000 000 000 000 125 J

and now we could in theory boil the water to make our cup of tea with many fewer neutrons. Indeed, we could do this by annihilating 'only' about 7

448 000 000 000 000 neutrons

This is a lot less neutrons than before, i.e.,

448 000 000 000 000 neutrons

< 538 000 000 000 000 000 neutrons

but it seems fair to say that it remains the case that the number of neutrons needed (now 'only' about 448 million million) is still a good deal more than 8.

448 000 000 000 000 neutrons

> 8 neutrons

So, if over 400 million million neutrons would need to be completely annihilated to make a single cup of tea, how much damage can 8 neutrons do to a distant planet?

A common learning difficulty

In any reasonable scenario we might imagine 8 neutrons would not be significant. This is worth emphasising as it reflates to a common learning difficulty. Quanticles such as atoms, atomic nuclei, neutrons and the like are tiny. Not tiny like specs of dust or grains of salt, but tiny on a scale where specs of dust and grains of salt themselves seem gigantic. The scales involved in considering electronic charge (i.e., 10-19C) or neutron mass (10-27 kg) can reasonably said to be unimaginatively small – no one can readily visualise the shift in scale going from the familiar scale of objects we normally think of as 'small', to the scale of individual molecules or subatomic particles.

Students therefore commonly form alternative conceptions of these types of entities (atoms, electrons, etc.) being too small to see, but yet not being so far beyond reach. And it is not just learners who struggle here. I have even heard someone on a national news programme put forward as an 'expert' make a very similar suggestion to Egrorian, in this case that a "couple of molecules" could be a serious threat to public health after the use of chemical nerve agent. This is a preposterous suggestion to a chemist, but was, I am sure, made in good faith by the international chemical weapons expert.

It is this type of conceptual difficulty which allows scriptwriters to refer to 8 neutrons as being of some significance without expecting the audience to simply laugh at the suggestion (even if some of us do).

It also explains how science fiction writers get away with such plot devices given that many in their audiences will readily accept that a few especially malicious molecules or naughty neutrons is a genuine threat to life.8 But that still does not justify using angle-poise lamps as futuristic spacecraft joysticks.

Jenna pilots the most advanced spacecraft in the galaxy
Works cited:
  • Einstein, A. (1917/2015). Relativity. The special and the general theory. (100th Anniversary ed.). Princeton: Princeton Univerity Press.
  • Treptow, R. S. (2005). E = mc2 for the Chemist: When Is Mass Conserved? Journal of Chemical Education, 82(11), 1636. doi:10.1021/ed082p1636
Notes:

1 To explain: For younger readers, television was first broadcast in monochrome (black and white – in effect shades of grey). My family first got a television after I started primary school – the justification for this luxury was that the teachers sometimes suggested programmes we might watch.

Colour television did not arrive in the UK till 1967, and initially it was only used for selected broadcasts. The first colour sets were too expensive for many families, so most people initially stayed with monochrome. This led to the infamous 'helpful' statement offered by the commentator of the weekly half-hour snooker coverage: "And for those of you who are watching in black and white, the pink [ball] is next to the green". (While this is well known as a famous example of misspeaking, a commentator's blooper, those of a more suspicious mind might bear in mind the BBC chose snooker for broadcast in part because it might encourage more people to watch in colour.)

Snooker – not ideal viewing on 'black and white' television (Image by MasterTux from Pixabay )

My father had a part-time weekend job supervising washing machine rental collections (I kid you not, many people only rented such appliances in those days), to supplement income from his full time job, and this meant on Monday evenings after his day job he had to visit his part-time boss and report and they would go throughout the paperwork to ensure things tallied. I would go with him, and was allowed to watch television whilst they did this – it coincided with Star Trek, and the boss had a colour set!

2 Assuming water had to be heated from 20˚C to 100˚C, and the cup took 200 ml (200 cm3) of tea then the calculation is 4.2 x 80 x 200

4.2 J g-1K-1 is the approximate specific heat capacity of water.

Changing these parameters (perhaps you have a small tea cup and I use a mug?) will change the precise value.

3 That is the energy needed divided by the energy released by each neutron: 67200 J ÷ 1.25 x 10-13 J/neutron = 537 600 000 000 000 000 neutrons

4 E=mc2

so m = E/c2 = 67 200 ÷ (3.00 x 108)2 = 7.47 x 10-13

5 E=mc2 = 1.67 x 10-27 x (3.00 x 108)2 = 1.50 x 10-10

6 Well, we could imagine that somehow Hoffal had devised a process where the neutrons somehow redirect energy provided to initially generate the magnetic field, and perhaps the weapon was actually an enormous field generator producing a massive magnetic field that the funnel somehow converted into a beam (of tachyons?) that could pass across vast amounts of space without being absorbed by space dust, remaining highly collimated, and intense enough to destroy a world.

So, perhaps the neutrons are analogous to the core of a laser.

I somehow think it would still need more than 8 of them.

7 That is the energy needed divided by the energy released by each neutron: 67200 J ÷ 1.50 x 10-10 J/neutron = 4.48 x 1014 neutrons

8 Of course molecules are not actually malicious and neutrons cannot be naughty as they are inanimate entities. I am not anthropomorphising, just alliterating.