22.2 Nuclear Forces and Radioactivity

Nuclear forces govern how subatomic particles interact inside the nucleus, determining whether a nucleus holds together or falls apart. Radioactive decay is what happens when an unstable nucleus can't hold together anymore and releases energy as radiation. These concepts connect nuclear structure to real-world phenomena like nuclear energy, medical imaging, and radiometric dating.

Nuclear Forces and Interactions

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Forces within atomic nuclei

Three fundamental forces compete inside every atomic nucleus. The balance between them determines whether a nucleus is stable or radioactive.

Strong nuclear force is the most powerful of all fundamental forces. It binds quarks together inside protons and neutrons, and it also holds protons and neutrons together within the nucleus. This force is mediated by particles called gluons. The catch: it only operates over extremely short distances (about $10^{-15}$ m), roughly the diameter of a nucleus. Beyond that range, it drops to essentially zero.

Electromagnetic force works against nuclear stability by pushing protons apart. Since all protons carry a positive charge, they repel each other according to Coulomb's law:

$F = k \frac{q_1 q_2}{r^2}$

where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the particle charges, and $r$ is the distance between them. This force is weaker than the strong force at nuclear distances, but it has unlimited range. That's why large nuclei with many protons tend to be unstable: the electromagnetic repulsion accumulates across the entire nucleus, while the strong force only acts between nearby neighbors.

Weak nuclear force is responsible for certain types of radioactive decay, most notably beta decay. It's much weaker than both the strong and electromagnetic forces and also operates over very short distances. Despite its weakness, it plays a critical role because it can change one type of quark into another, effectively converting neutrons into protons (or vice versa).

Binding energy

Binding energy is the energy you'd need to supply to completely pull a nucleus apart into its individual protons and neutrons. Higher binding energy per nucleon means a more stable nucleus.

Where does this energy come from? The mass of a nucleus is always slightly less than the total mass of its individual protons and neutrons added up separately. This difference is called the mass defect ($\Delta m$). Einstein's equation converts that "missing" mass into the binding energy:

$\Delta E = \Delta m \, c^2$

where $\Delta E$ is the binding energy, $\Delta m$ is the mass defect, and $c$ is the speed of light. Because $c^2$ is enormous, even a tiny mass defect corresponds to a large amount of energy.

Radioactive Decay and Radiation

Types of nuclear radiation

When an unstable nucleus decays, it can emit several types of radiation. Each type differs in composition, ionizing power, and penetrating ability.

• Alpha ($\alpha$) radiation: Helium-4 nuclei (2 protons + 2 neutrons). These are relatively massive and carry a +2 charge, making them strongly ionizing. However, they're easily stopped by a sheet of paper or even the outer layer of skin. The real danger is internal exposure: inhaling or ingesting alpha emitters like radon gas can cause serious tissue damage.
• Beta ($\beta$) radiation: Energetic electrons ($\beta^-$) or positrons ($\beta^+$). These are much lighter than alpha particles and carry a single charge, so they're moderately ionizing. Beta particles penetrate further than alpha particles but can be stopped by a few millimeters of aluminum or plastic. They can penetrate skin and cause radiation burns.
• Gamma ($\gamma$) radiation: High-energy photons with no mass and no charge. Because they don't interact as readily with matter, they're weakly ionizing but highly penetrating. It takes thick lead or several feet of concrete to effectively block gamma rays. They can damage tissue from a distance, which is why they pose a significant cancer risk.
• Neutron radiation: Free neutrons, typically produced during nuclear fission. Neutrons are highly penetrating because they carry no charge, so electromagnetic forces don't slow them down. They can also make other materials radioactive when absorbed by stable nuclei.

Equations for radioactive decay

Decay equations must conserve both mass number ($A$, total protons + neutrons) and atomic number ($Z$, number of protons). Always check that both sides balance.

Alpha decay:

$^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2 He$

The parent nucleus $X$ ejects an alpha particle ($^4_2 He$), producing a daughter nucleus $Y$. The mass number drops by 4 and the atomic number drops by 2. For example, uranium-238 decays to thorium-234:

$^{238}_{92} U \rightarrow ^{234}_{90} Th + ^4_2 He$

Beta minus ($\beta^-$) decay:

$^A_Z X \rightarrow ^A_{Z+1} Y + e^- + \bar{\nu}_e$

A neutron converts into a proton, emitting an electron ($e^-$) and an electron antineutrino ($\bar{\nu}_e$). The mass number stays the same, but the atomic number increases by 1. Carbon-14 decays to nitrogen-14 this way.

Beta plus ($\beta^+$) decay:

$^A_Z X \rightarrow ^A_{Z-1} Y + e^+ + \nu_e$

A proton converts into a neutron, emitting a positron ($e^+$) and an electron neutrino ($\nu_e$). The mass number stays the same, but the atomic number decreases by 1. This process is used in PET scans, where positron-emitting isotopes like fluorine-18 are injected into the body.

Gamma emission:

$^A_Z X^* \rightarrow ^A_Z X + \gamma$

An excited nucleus (denoted by the asterisk) releases a gamma photon to drop to a lower energy state. Neither the mass number nor the atomic number changes. Technetium-99m is a common gamma emitter used in medical imaging.

Nuclear Reactions and Decay Rates

Nuclear fission is the splitting of a heavy nucleus (like uranium-235) into two lighter nuclei, releasing energy and additional neutrons. Those neutrons can trigger further fission events, creating a chain reaction. This is the principle behind nuclear reactors and nuclear weapons.

Nuclear fusion is the combining of light nuclei (like hydrogen isotopes) into a heavier nucleus, releasing even more energy per unit mass than fission. Fusion powers the Sun, where hydrogen nuclei fuse into helium under extreme temperature and pressure.

Radioactive half-life ($t_{1/2}$) is the time it takes for half the atoms in a radioactive sample to decay. After one half-life, 50% of the original atoms remain. After two half-lives, 25% remain. After three, 12.5%, and so on. The amount remaining after $n$ half-lives is:

$N = N_0 \left(\frac{1}{2}\right)^n$

where $N_0$ is the initial number of atoms and $n$ is the number of half-lives elapsed. Half-lives vary enormously: uranium-238 has a half-life of about 4.5 billion years, while polonium-214 decays in about 164 microseconds.