4.3: Radioactive Decay

Radioactive decay is a process in which a nucleus of an unstable isotope, called the “parent nucleus”, loses energy by emitting radiation, and trans- forms itself into a nucleus of a “daughter isotope”. There are several types of radioactive decay, listed below.

One important case, not shown in the Fig. \(\PageIndex{1}\), is the decay of a neutron. The neutron is a single particle, not a nucleus, but an extremely important component of all atomic nuclei, with the exception of the simplest one, the nucleus of ordinary hydrogen, which is a single proton. Protons repel each other due to their electric charge +e. All nuclei with two or more protons do contain neutrons, so that – in a naive picture – they may be thought of as “glue particles” that neutralize the repulsive forces between protons and hold the nucleus together. In fact, the nucleus is held together by the “strong force”, an interaction much stronger than the repulsive Coulomb force, but of a very short range.

Figure \(\PageIndex{1}\): The three most often occuring modes of nuclear decay: α, β minus and γ. In the case of the first two the decay produces a new nucleus (see the text for more details), but in γ decay the number of neutrons and protons is the same before and after the act – the transition is from an excited state with higher energy to a state with lower energy. However, a question arises, how to depict an “excited state” of the atomic nucleus in the figure? We decided to use a metaphorical comparison with a balloon: by inflating a balloon, we increase its energy and size; and if we release the air, the energy decreases and the size shrinks. but it remains the same balloon! So the larger nucleus in the figure symbolizes one in the state of a higher energy – and the smaller is the same nucleus in a lower energy state, after a portion of energy is “released” – or jettisoned – in the form of a quantum of electromagnetic radiation, called a “gamma quantum”. But in real gamma decay it doesn’t have to be that the nucleus “shrinks”!.

A free neutron is not stable, it decays into a proton, an electron and a neutral particle called neutrino¹ (symbol: ν), with an average lifetime of about 15 minutes (it's called a "beta-minus", or a \(\beta^{-}\)decay). However, in stable nuclei the neutrons never decay. In contrast, in some unstable nuclei they do decay - see the list below - but their average lifetime in the nucleus may be either shorter, or longer than in the free state (sometimes, as long as many years).

As far as nuclei are concerned, there is a number of possible decay modes:

• The Alpha (or a) decay - mostly in heavy nuclei. The nucleus emits an \(a\)-particle, which is the same as the nucleus of the light element Helium\(4\left({ }_{2}^{4} \mathrm{He}\right)\). It means that the parent nucleus transforms into a daughter nucleus with 4 nucleons less, and 2 protons less. It can be written in the form of an equation, with \(X\) and Y symbolizing, respectively, the parent and the daughter nucleus:
  
  \[ { }_{Z}^{A} \mathrm{X} \rightarrow{ }_{Z-2}^{A-4} \mathrm{Y}+{ }_{2}^{4} \mathrm{He}+\text { kinetic energy } \]
  In the Fig. \(\PageIndex{1}\) we drew the "daughter nucleus" as if the detached alpha particle left a "pothole" on its surface. Of course, in a real alpha decay such a pothole is unlikely to be created - and we only drew it so that it could be clearly seen that the "daughter nucleus" is the "parent" one minus four nucleons.
• The beta decays (there are more than one possible modes:

* The most common is the beta-minus \(\left(\beta^{-}\right)\)decay: The nucleus emits an electron, \({ }^{0} e\), and a neutrino \(v\). The number of nucleons \((A)\) does not change, but the number of protons increases by 1 , and the number of neutrons \((\mathrm{N})\) decreases by 1 :

\[{ }_{Z}^{A} \mathrm{X} \rightarrow \underset{Z+1}{A} \mathrm{Y}+{ }_{-1}^{0} e+{ }_{0}^{0} \bar{\nu}+\text { kinetic energy }\]

It means that one of the constituent neutrons undergoes a \(\beta^{-}\) decay the same way as a free neutron does. Note that in the illustration of a beta minus decay in Fig. \(\PageIndex{1}\) a nucleon that is a neutron in the "parent" nucleus (right in its center) is a proton in the "daughter" nucleus. By the way, the neutrino is not shown in Fig. \(\PageIndex{1}\).

*The less common beta-plus \(\left(\beta^{+}\right)\)decay. Now the nucleus emits a positron - a particle which is the "twin brother" of the electron: it has the same mass, but a positive charge of the same magnitude as that of the electron. It means that one of the constituent protons undergoes a change to a neutron:

\[ { }_{Z}^{A} \mathrm{X} \rightarrow{ }_{Z-1}^{A} \mathrm{Y}+{ }_{+1}^{0} e+{ }_{0}^{0} \nu+\text { kinetic energy } \]

* A pretty exotic \(\beta\) process is the electron capture: the nucleus "captures" one electron from its electronic shell, and uses it to convert one
of the constituent protons into a neutron:

\[ { }_{Z}^{A} \mathrm{X}+{ }_{-1}^{0} e \rightarrow \underset{Z-1}{A} \mathrm{Y}+{ }^{0}{ }_{0}^{\bar{\nu}} \]

All energy released in the process is carried away by the antineutrino - there is no practical way to detect this elusive particle, so that the process can be detected only by indirect methods.

• The gamma decay. Let's talk first about the well-known Bohr Model of Hydrogen Atom - because it's good aid for explaining what the gamma-decay is.

In the Bohr Model an electron travels around a single proton in circular "orbits" corresponding to specific "allowed" energies \(E_{1}, E_{2}, E_{3}, \ldots\)., where \(E_{1}<E_{2}, E_{2}<E_{3}\), and so on. When on the orbit with the lowest energy \(E_{1}\), such a state may last forever, and it is referred to as the "ground state". On any orbit with higher energy the electron can stay only for a very short time, and then "jumps" to an orbit with lower energy - and such jump is associated with an emission of a photon of light of energy equal to to the difference between the energies of the two orbits. The frequency \(f\) of a photon is related to its energy by the Planck Constant: \(E_{\mathrm{ph}}=h f\), so the energy and the frequency of the emitted photon is:

\[ E_{\mathrm{ph}}=h f=E_{\mathrm{n}}-E_{\mathrm{m}} \quad \text { where } \quad n>m \geq 1 \]

The Bohr Model is now considered obsolete - it has been replaced by a much better theory, in which there are no longer orbits, but discrete states. Anyway, the rules are still the same: there is the ground state \(E_{1}\) and a number of states with higher energies - called excited states. The electron "jumps" from an excited state to another excited state with lower energy, or at once to the ground state, and the Eq. \(1\) is still valid in the modern theory.

Let's go back to nuclear \(Y\) decay. There is one important difference between this decay and the decays discussed before: namely, thereis no such thing as an "isolated" \(\gamma\) decay in a parent nucleus. There is always a preceding \(a\) or \(\beta\) decay which leaves the "daughter" nucleus in an excited state. Nuclei do have excited states, in analogy to the excited states of electrons in atoms. Likewise, nuclei cannot remain in their excited states for a long time - so they "jump down" to the ground state, "jettisoning" the excitation energy - again, in analogy to electronic "jumps" in atoms - in the form of one or more photons of electromagnetic radiations. Yes, gamma radiation, \(\mathrm{X}\)-rays, ultraviolet light, visible light, infrared light, microwaves, and radio waves areall forms of electromagnetic radiation. The only difference between photons of visible light and \(\gamma\) photons is in their energies: typically, a Y photon carries \(10^{4}-10^{6}\) times more energy than a visible light photon.

• An exotic mode of decay, occurring only in some very heavy nuclei is fission: the nucleus "splits" into two, sometimes releasing a few free neutrons. Fission may occur spontaneously, but it is very rare phenomenon-however, it may be stimulated by an absorption of a free neutron by the nucleus. We will discuss fission in greater detail later on, because essentially all electric power generated in nuclear power plants comes from fission reactions in uranium isotopes and some other heavy elements.

Figure \(\PageIndex{2}\): A fission of a heavy nucleus stimulated by absorption of a neutron. In the 235 isotope of Uranium such a process produces, in addition to the two “daughter” nuclei, two or more free neutrons which may trigger a “chain reaction” in other Uranium nuclei. A heavier Uranium isotope 238 also undergoes fission triggered by impinging neutrons, but it produces the “daughters” only, no free neutrons. The next panel show the less common beta decay types: the so-called “beta plus”, in which not en electron, but a positron is emitted, and one of the constituent protons in the nucleus changes into a neutron. And the third panel depicts the “electron capture” – a process in which the nucleus “captures” one of the electrons orbiting it. Neutrinos are emitted in both those processes, but are not shown in the plots.

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1Neutrinos are elementary particles of extremely small mass. There are three kinds ("flavors") of neutrinos, in each kind there is a "partner" and "anti-partner", making six different neutrinos altogether. They interact extremely weakly with matter, therefore they are very hard to detect. The neutrinos created in \(\beta\) decay processes are either the "electron neutrinos" \(\left(V_{e}\right)\) or "electron anti-neutrinos" \(\left(\bar{V}_{e}\right)\). In this text, we skip the "e" subscript because neutrinos created in \(\beta\) decay processes are always of the "electron" flavor.