Systematics of beta decay

The Feynman diagram for the beta decay of a neutron into a proton, electron, and electron antineutrino via an intermediate heavy W- boson

Beta decay is quite a bit more subtle and complex than the other two common forms of radioactivity—alpha decay and gamma decay. In alpha decay, nucleons fall out of a nucleus because they are electrically repelled by the other protons and can go into a lower energy state by leaving, while in gamma decay nucleons fall down into a lower energy state (under the rules of quantum mechanics) in a similar manner to the drop in electron energy levels in an atom, albeit at a much more energetic level.

Beta decay, unlike the other two, involves actual transmutation of nucleons, typically among protons, neutrons, and electrons/positrons. This has far-reaching consequences involving electric charge and various conservation laws. This article will attempt to peel back the layers of complexity one at a time.

Beta decay involves the "weak force", unlike alpha decay, which involves the "strong force". Nevertheless, beta decay has some extremely far-reaching consequences for radioactivity, nucleosynthesis, and supernovas.

The weak force is one of the four fundamental forces currently believed to exist:

• the gravitational force
• the electromagnetic force
• the weak force
• the strong force

The electric and magnetic forces were once considered to be separate, but they were unified so straightforwardly in the late 19^th century that they are no longer treated as separate. The electromagnetic and weak forces were unified in the late 1960's, into the "electroweak force", but that connection is sufficiently esoteric that it has not supplanted the separate electromagnetic and weak forces. There are "grand unified theories" unifying the last three of these forces, and the physics community continues to search for a "theory of everything" that brings gravity into the picture. For all practical purposes, the four forces listed above are generally considered the fundamental forces.

The first two of these forces are well known without specialized knowledge of nuclear physics, mostly because they operate at macroscopic distances. The last two only operate at microscopic distances (about 10^-15 meters, or the diameter of a proton), so they are limited to subatomic particles.

• Q: Why are they called "forces"? What is doing the pushing?
• A: The term "force", applied to the gravitational and electromagnetic forces, is obviously reasonable. Also, the strong force could sort of be thought of as a force, making nucleons stick together. It takes a certain amount of mechanical pressure to pull nucleons out of a nucleus. But the weak "force" is hard to visualize as a force. Nevertheless, all four of these interactions are called "forces" for historical consistency.

The common transformations

The weak force acts on just abut every particle that exists—baryons (protons and neutrons), hadrons (various other things made of quarks), leptons (electrons, neutrinos, etc.), bosons, and so on. For example, the interaction shown in the above right corner shows weak interactions involving up and down quarks, a "weak vector boson", an electron, and an antineutrino. For all but extremely esoteric analysis of fundamental particle physics, hardly any of that matters. What's important is that a neutron turned into ("decayed into") a proton, and electron and an antineutrino. The various quarks, and the vector boson underwent beta transformations, but, in a simpler sense, we can just say that the neutron underwent a beta transformation. The interaction shown in the figure is the first of these:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} n^0 &\to p^+ + e^- + \bar\nu_e \qquad\textrm{neutron}\ \textrm{decay}\ \textrm{with}\ \textrm{electron}\ \textrm{emission}\\ p^+ &\to n^0 + e^+ + \nu_e \qquad\textrm{proton}\ \textrm{decay}\ \textrm{with}\ \textrm{positron}\ \textrm{emission,}\ \textrm{not}\ \textrm{energetically}\ \textrm{possible}\ \textrm{in}\ \textrm{isolation}\\ n^0 + e^+ &\to p^+ + \bar\nu_e \qquad\qquad\textrm{positron}\ \textrm{capture,}\ \textrm{only}\ \textrm{happens}\ \textrm{if}\ \textrm{a}\ \textrm{positron}\ \textrm{happens}\ \textrm{to}\ \textrm{be}\ \textrm{nearby}\\ p^+ + e^- &\to n^0 + \nu_e \qquad\qquad\textrm{electron}\ \textrm{capture} \end{align}}

The first of these is classical "beta decay", that is, the emission of a "beta ray", first discovered in the earliest days of radioactivity research. The "beta rays" were soon identified as being just electrons.

The second is generally called "beta plus" decay, the positron being considered a "beta plus ray". That, and the fourth one, electron capture, are beta transformations that occur for many isotopes.

It is common to use the Greek letter "β" instead of "e" for the electron, still using the plus or minus superscript. Also, the charge symbol for protons and neutrons is often omitted. So the four equations above become:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} n &\to p + \beta^- + \bar\nu_e \\ p &\to n + \beta^+ + \nu_e \\ n + \beta^+ &\to p + \bar\nu_e \\ p + \beta^- &\to n + \nu_e \end{align}}

The "e" subscript on the neutrino is still used, to distinguish electron-type neutrinos from muon-type and tau-type.

The reverse transformations

All physical processes are supposedly reversible in time, so that, for example, if an alpha particle is thrown at a ²²²Rn₈₆ nucleus with just the right energy, it could undergo "reverse alpha decay", producing a ²²⁶Ra₈₈ nucleus. Or an antineutrino could combine with and electron and a proton with just the right energy to produce a neutron:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} p + \beta^- + \bar\nu_e &\to n \end{align}}

But this is a statistical impossibility—it would require that the particles be in exactly the right place and with exactly the right energy. It is in the same category as heat flowing uphill—a violation of the Second Law of Thermodynamics.

Antiparticles

These transformations work on antiparticles also. Note that the electron ("beta-minus particle") and positron ("beta-plus particle") are antiparticles. So, in principle, we could have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \bar{n} &\to \bar{p} + \beta^+ + \nu_e \end{align}}

Conservation laws

When dealing with beta transformations, one needs to take note of various conservation laws. (For alpha and gamma transformations, particles don't change, so these issues don't come up.)

• Charge - this is the electric charge. So a neutron, with zero charge, can decay into a proton (+1), and electron (-1), and a neutrino (zero).
• Baryon number - Baryons are "big" nucleons. For our purposes, they are protons and neutrons. They can't just go away. Nor can the transform into their antiparticles. (A proton and antiproton annihilating each other would satisfy this, since the baryon numbers are +1 and -1, but that transformation is beyond the scope of this article.)
• Lepton number. The lepton numbers can be thought of as "electron-ness", "mu-ness", and "tau-ness". Electrons and positrons have electron-ness of +1 and -1 respectively (and zero for the other two). Here is where the neutrinos come into play. An electron neutrino has an electron-ness of +1, while and anti-neutrino has -1. That's why the neutron-to-proton-and-electron transformation needs to create the antineutrino to carry off the -1, compensating for the +1 of the electron.

The general principle

With these preliminaries out of the way, the general principle can be stated:

Any beta transformation will happen, with some probability, if the conservation laws are satisfied and the masses of the results add up to less than the masses of the original particle(s). If it can happen, it will happen.

The mass of the neutron is 1.008664916 amu, that of the proton is 1.007276467 amu, and that of the electron is .000548579909070 amu. (The pass of the antineutrino is negligible.) The lost mass is .00083986909093, which is carried off in the kinetic energy of the result particles, about 0.78 MeV, in accordance with E=mc².

• Q: The neutron is unstable and decays? Why don't all atoms fall apart?
• A: Yes, a neutron, standing alone, is unstable. It has a half-life of about 15 minutes. To see why this doesn't happen for neutrons inside a nucleus, keep reading.

The reverse transformation, turning a proton into a neutron, positron, and neutrino, has a negative energy budget, so it does not occur.