Math Equation of the Radioactive Decay

Decay spectrum Determining the Mathematical Equation
behind the Energy Spectrum
of Radioactive Decay Stream of radio-atoms

After an "introduction", this "applied" mathematical research consists of two parts:

Finding the general equation for the family of curves made by all radioisotopes
Finding the specific equations behind the energy spectrum of individual radioisotopes.

It seems obvious that one should solve the first problem first, and then do whatever mathematical manipulations necessary to derive the answers for the second part. It should be noted, however, that solving the first part will be an important contribution in its own right. Thus you do not need to solve both parts - although doing that would be terrific!

INTRODUCTION

When a radioactive atom's nucleus decays (disintegrates, fissions) it releases energy - if you need a conceptual model to help your thinking, think of it as a sort of "explosion" of energy. As the atoms of the various elements usually come with varying numbers of neutrons in them, these atoms of an element are of different weights and are called "isotopes." Some isotopes are radioactive (prone to eventually decay), and are called radioisotopes. Many isotopes are not radioactive. For example: the element hydrogen always has a single proton in its nucleus, but some nuclei have no neutrons, some have one neutron, and others have two neutrons. Thus the atomic weights of these three isotopes are 1, 2 and 3, and are given the names of "protium", "deuterium" and "tritium." Protium and deuterium are non-radioactive, natural isotopes, while tritium is radioactive, and is hence a radioisotope. In physicists' shorthand, these are denoted as ₁H, ₂H and ₃H. (In the case of hydrogen isotopes, these are frequently further abbreviated to H, D, and T. Thus, what do these mean: H₂O, D₂O, HOT? And what do these mean: ₂H, ₁₂C, ₁₄C, ₃₂S, ₃₅S, ₃₁P, ₃₂P, ₁₃₁I? Which of them are radioactive (those which are radioactive are shown in order of their decay energies - weakest first and strongest last).

What is the energy spectrum of radioactive decay? If you looked individually at the decays of a zillion atoms of a particular radioisotope, you would see that some decays were strong, and others weaker. Were you to graph the numbers of decays of various energies versus the energy of that decay, you would get a graph called the energy spectrum of radioactive decay. It is similar to that shown to the left of this page's title. You will note that very few decays are very weak, but there are a good number of fairly strong decays, and then it drops off so that there are few decays that are very strong. There is a limit-value above which there are no decays. It is that limit value that is usually assigned to radioisotopes in physics book tables. (Does this continuous spectrum violate quantum mechanics?)

While it is easy for a particle/energy counter to accumulate the data ("empirical data") to make a graph of a decay spectrum, the mathematical formula behind that graph is unknown. To be sure, computers can derive formulas that are very close to the "empirical" curve, these are purely trial and error approximations. What is needed is the real equation, since it might reveal something fundamental about radioactive decay. We will call this the "general equation."

PART ONE

Long ago, at what was presumably a dull meeting, a distinguished nuclear physicist was doodling some figures, which he briefly showed to his companions. He explained that he had just solved the general equation that described the curve of the energy spectrum of radioactive decay. He discarded the the paper thinking the calculations would be easily reproducible. Alas, he died soon thereafter. His companions tried to recall what had been scribbled, but their memories had dimmed with the elapsed time. All they could remember is his talking about points moving along lines between integers and their reciprocals. Later one of those onlookers described all this to his laboratory group, of which this author was a member. We all played with this problem for a few minutes before timers called us back to our own research activities. The problem has thus languished for several decades.

Two such sets of parallel lines with moving points are shown here:

Either of these diagrams read as follows: as Y moves a fractional distance between two positive numbers, the value of 1/X is specified by moving that same fractional distance between the reciprocals of the two positive numbers. The only distinguishing item between the two examples is the direction that the fractional distance is measured.

Hidden somewhere in these diagrams, OR SIMILAR ONES!, are a number of relationships. One of those relationships when plotted will give you the perfect shape of the radioactive decay energy spectrum. A hint that might prove worthwhile is to assume that one of the positive numbers is +1. This hints of being handy since its reciprocal is still +1. (But maybe this is an erroneous hint! This author doesn't know if it is or not - just a hunch.) Following that hunch let's look for relationships that give at least - at one fractional distance (say, zero or one) - a value of zero. We need a zero value because the spectrum starts out at zero, at the origin of the graph. Thus we might have a relationship we'll call "Z":

, where

Z equals the difference between Y and 1/X making the sign positive (that's what the two vertical bars are telling you: make it an "absolute number").

Calculus. If you know how to do calculus, you will make many of the analyses you make a bit simpler. But even if you have not mastered calculus and find that a few selected points do match the decay spectrum, you can always "hire" someone to do some calculus on your equation that will result in a really nice formula with some fancy calculus symbols in it (such as dY/df, or even _m∫ⁿ dY/df.

Supposing that you do get a really nice candidate set of points, how do you match them up with an empirical plot of a real decay spectrum? Perhaps the easiest way is by using a transparency projector, which can be used to magnify and project your graph onto the empirical plot by merely moving a transparency film of your plot closer or further from the screen upon which the empirical plot is imaged. If your plot matches, the two graphs should perfectly match - if you are lucky! Some horizontal stretching might be needed. (Does anyone have a rubber transparency?!)

PART TWO

The second part of this project is adapting the "general equation" to fit each of the different spectra of the various radioisotopes, which each have a different limit-energy. For example, the limit energy of ₃₂P is about ten-times higher than that of ₁₄C, which in turn is about ten-times higher than that of tritium (₃H).

Probably - another hunch! - the family of similar curves for real radioisotopes can be made by merely adjusting the numbers and the ends of the lines: instead of going from 1 to 2, you might go from 1 to 3, and so on.

IMPORTANT. Remember that if you have solved part one, but cannot solve part two, the solution to part one is "mighty important" to the nuclear physics world - so publish immediately, and put part two aside for awhile.

Quantum mechanical considerations

On the surface, one would think that radioisotopes should decay with discrete amounts of energy and not a continuous spectrum. You might be familiar with quantum levels with regard to electron orbitals. Specifically, you can see the quantum levels by doing some flame spectroscopy. You will see that different elements emit only specific color bands and not a whole spectrum.

So how is it that nuclear decay occurs in energies that span a broad spectrum? Had you been around a bunch of years ago, and had you thought about this then, you might have been well along the way to a Nobel Prize for the discovery of neutrinos. Let's take a look at tritium decay, for example: the tritium nucleus consists of one proton and two neutrons. When it decays - so it was thought way back when - one of the neutrons decays into a proton and an electron is shot out of the atom (this ejected electron is called a beta-particle, β^-. The energy spectrum that was obtained from the speeds of millions of such beta-particles. But quantum theory said that the disintegration of a neutron MUST have a discrete value.

The only solution to the quandary is that something must be happening that the scientists were looking for. For example, if a nearly weightless, neutral particle was also being ejected along with the beta-particle, perhaps the sum of their two energies would be constant. (Or more correctly, that the sum of all the vectored energies of (1) the recoiling nucleus plus (2) the β^- plus that of this newly imagined particle would all add up to zero. Let's name this imagined particle a "neutrino," and use the Greek symbol "nu": thus - ν^o. So that is exactly what was eventually discovered to be the case.

Thus in some tritium decays the nucleus emits a fast β^- and a slow ν^o, and in other instances a nucleus emits a slow β^- and a fast ν^o; but always the sum of the energies (including that of the recoiling residual nucleus) are the same (zero) - thus quantum mechanics is upheld.