The die of a trillion faces

kw: analysis, radioactivity, quantum physics, chaos

I'm halfway through a book about the edges of scientific knowledge, which I'll review anon. In the meantime, two of the chapters got me thinking: one on mathematical chaos and the other on quantum randomness as it relates to radioactivity.

Mathematical chaos does not refer to utter randomness, but to mathematical process that are completely deterministic but "highly sensitive to initial conditions." Such systems are typically studied by running computer simulations, which brings out an amusing feature: many such systems are also overly prone to amplify rounding errors in the calculations. For example, numerically solving a set of stiff differential equations frequently results in the solution "blowing up" after a certain point, because the rounding errors have accumulated and overwhelm the result.

Natural systems, being analog and not digital, can be described by sets of differential equations. Digital simulations of such systems can proceed only so far before descending into nonsense. The most famous of these is forecasting the weather. Many computer scientists and meteorologists have labored for decades to produce weather models that run longer and longer, farther and farther into the future, before "losing it." So now we have modestly reliable seven-day forecasts (and Accuweather.com has the temerity to show 90-day forecasts); a decade ago or so, no forecast beyond three or four days was any good.

Quantum randomness is a beast of another color, indeed, of a different spectrum of colors! These days the classic illustration is the ultra-low-power two-slit interference pattern. You can produce a visible (and thus moderate-power) pattern with a laser pointer, a pinhole or lens, and a little piece of foil with two narrow slits a short distance apart. The pinhole or lens will spread the beam so you can see it hit both slits. On a screen a few inches behind, a pattern of parallel lines will appear, similar to this image.

The ultra-low-power version is to set this up with the lens/pinhole and the slits and the laser held in stands, and the screen replaced by sensitive photographic film. Then a strong filter is put at the laser's output, calculated to make the beam so weak that no more than one photon will be found in the space between the laser and the film at any one time. Such an arrangement requires an exposure of a few hours to get the beginnings of a record, and several days to get an image like the one above. Whereas this experiment with strong light seems to show the wave nature of light, the ultra-low-power version shows that a photon has a wave nature all by its lonely self!

A "short" exposure of an hour or less will show just a few dots where single photons were captured by the emulsion. They appear entirely random. The longer the exposure, the more a pattern seems to emerge, until a very long exposure will produce a clear pattern. The pattern shows that you can predict with great precision what the ensemble of many photons will do, but you cannot predict where the next photon to pass through the apparatus will strike the film.

Radioactivity also obeys certain quantum regularities (I hesitate to write "laws"). Half-life expresses the activity of a radioactive material in reciprocal terms. A long half life indicates low activity. In the book I was reading the author wrote of a little pot of uranium 238 (U-238) he bought, which contains just enough of the element to experience 766 alpha decays per minute. My first thought was to see how much U-238 he had bought. U-238 has a half life of 4.468 billion years. Working out the math, I determined that he had just over one milligram of uranium. The amount was very close, which made me suspicious that there was a typo: If he actually bought exactly one milligram, the activity would be 746 decays per minute…and that might be the true amount.

What is happening inside a uranium nucleus that leads a certain one to emit a helium (He-4) nucleus (and thus turn into thorium 234, Th-234)? Scattering experiments carried out decades ago showed that although the atomic nucleus is incredibly tiny, it is mostly empty space! I learned this as a physics student in the late 1960's. I had found it hard enough to wrap my mind around the view of an atom as a stadium with a few gnats buzzing around the periphery, centered on a heavy BB. So the protons and neutrons, while not being effectively "dimensionless" like electrons, are still much tinier than the space they can "run around" in. The propensity of proton-heavy elements such as U-238 to decay by emitting helium nuclei indicates that the protons and neutrons "run around" in subgroups.

The standard explanation is that at some point one of the He-4 nuclei "tunnels" through the "strong force barrier", finds itself outside the effective range of the force, and thus is accelerated away by electromagnetic repulsion to an energy of 4.267 MeV. What determines when it tunnels through?

Back in the chapter on chaos, the author spoke of dice with various numbers of faces, though he illustrated the randomness of a die's fall using a "normal" 6-sided die he got in Las Vegas. I guess they make them more accurate there, where large stakes are wagered on their "fairness". But dice with various numbers of faces are produced for board-based role playing games. This illustration, from aliexpress.com/, shows one such set of ten different kinds of die, ranging from 4 to 20 faces.

Put two thoughts together, and you can get some interesting products. Can the randomness of alpha decay be related to the randomness of a tumbling die? We can set up a model system with a box of cubical, 6-sided dice, perhaps 100. Here are the steps:

1. Cast the dice on a table top (with raised sides so none fall off, perhaps).
2. Remove each die that shows a 6.
3. Return the rest to the box.
4. Repeat from step 1.

I did this a few times, stopping each run after 16 trials. Here are two results:

100, 81, 69, 58, 49, 41, 35, 30, 24, 21, 18, 14, 11, 9, 8, 6, 5
100, 90, 78, 64, 53, 46, 37, 31, 26, 22, 18, 15, 12, 10, 9, 8, 7

The calculated half life of these dice, with "activity" of 1/6 per throw, is 4.16 throws. As seen above, small number statistics cause a certain variation, so that after four throws, 49 and 53 are left; after 8 throws, 24 and 26; and so forth. If instead you use 20-sided dice, the half life would be 13.9 throws.

This led me to think of the He-4 (alpha particle) "cores" bouncing around inside the strong-force boundary around a U-238 nucleus as being governed by a die with an immense number of faces, perhaps a trillion. Rather than numbers from one to a trillion on the faces, the only thing that matters is the "get out of here" face, which we might consider to be green (for "go"), the rest being red. On average, once per trillion "bounces" the die momentarily has its green face at the boundary, and the alpha particle flies free. Since the decay constant for U-238 is ln(2)/half life of 4.468 billion years, or one decay yearly per 6.45 billion nuclei, a trillion-sided die would imply a "bounce" time of about two days. The actual transit time for an "orbiting" He-4 is closer to 10^-18 sec, which implies a die with a whole lot more than a trillion faces; say, ten trillion trillion faces.

Can it be that quantum randomness and mathematical chaos are related? Could one cause the other … in either direction?!?

That is as far as I have taken these ideas. I don't know (does anyone?) whether the internal, dynamic structure of a large nucleus is dominated by lone nucleons, by clusters such as He-4 and others, or what. The lack of decay products other than alpha particles, except in cases of spontaneous fission, for nuclei that are proton-rich, indicates that any nucleic clusters don't exceed the He-4 nucleus in size (and beta decay is a subject for another time!).