14.1 Atomic Nucleus and Radioactive Decay

Nuclear physics studies the structure of atomic nuclei and the processes that cause unstable nuclei to break down. This topic covers what holds a nucleus together, why some nuclei are unstable, and how radioactive decay transforms one element into another. These ideas form the foundation for understanding nuclear energy, medical imaging, and radiometric dating.

Atomic Structure

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Components of the Atomic Nucleus

The atomic nucleus is the dense central core of an atom, containing nearly all of the atom's mass. It's made of two types of particles, collectively called nucleons:

• Protons carry a positive electric charge. The number of protons defines the element (its atomic number, Z).
• Neutrons have no electric charge. They contribute to the atom's mass but don't change what element it is.

The mass number (A) equals the total count of protons plus neutrons. So a nucleus with 6 protons and 8 neutrons has a mass number of 14.

Isotopes are atoms of the same element that have different numbers of neutrons. Carbon-12 and carbon-14 are both carbon (6 protons each), but carbon-14 has 2 extra neutrons. Isotopes share the same chemical properties, but some isotopes are unstable and radioactive.

What keeps the nucleus from flying apart? Protons are all positively charged, so they repel each other through electrostatic force. The strong nuclear force overcomes this repulsion. It's the strongest of the four fundamental forces, but it only acts over extremely short distances, roughly the size of the nucleus itself. Once nucleons are farther apart than a few femtometers, the strong force drops off rapidly.

For scale, the nucleus is roughly 1 to 10 femtometers across ($1 \text{ fm} = 10^{-15} \text{ m}$). The entire atom is about $10^{-10} \text{ m}$, so the nucleus is roughly 100,000 times smaller than the atom itself.

Nuclear Stability and Binding Energy

Not every combination of protons and neutrons produces a stable nucleus. Stability depends on the proton-to-neutron ratio. For lighter elements (up to about calcium, Z = 20), stable nuclei tend to have roughly equal numbers of protons and neutrons. Heavier elements need proportionally more neutrons to remain stable, because extra neutrons add strong-force attraction without adding electrostatic repulsion.

Binding energy is the energy you'd need to completely pull a nucleus apart into separate protons and neutrons. A higher binding energy per nucleon means a more tightly bound, more stable nucleus.

Where does this energy come from? The assembled nucleus actually has slightly less mass than the sum of its individual protons and neutrons. This difference is called the mass defect. That "missing" mass has been converted into binding energy according to Einstein's equation:

$E = mc^2$

The binding energy per nucleon peaks around iron-56 ($^{56}\text{Fe}$). This is a key fact: nuclei lighter than iron can release energy by fusing together (fusion), and nuclei heavier than iron can release energy by splitting apart (fission). Both processes move toward the iron peak, where nucleons are most tightly bound.

Radioactive Decay

Types of Radioactive Decay

Radioactivity is the spontaneous emission of particles or energy from unstable nuclei as they move toward a more stable configuration. There are three main types:

Alpha decay occurs when a nucleus ejects an alpha particle, which consists of 2 protons and 2 neutrons (essentially a helium-4 nucleus). This reduces the atomic number by 2 and the mass number by 4. Alpha decay is most common in heavy elements like uranium and thorium. Alpha particles are relatively large and slow, so they're stopped easily by a sheet of paper or a few centimeters of air.

Beta decay involves the transformation of one type of nucleon into another inside the nucleus:

• Beta-minus ($\beta^-$) decay: A neutron converts into a proton, emitting an electron and an antineutrino. The atomic number increases by 1.
• Beta-plus ($\beta^+$) decay: A proton converts into a neutron, emitting a positron and a neutrino. The atomic number decreases by 1.

In both cases, the mass number stays the same because the total number of nucleons doesn't change. Beta-minus decay happens in nuclei that have too many neutrons for stability, while beta-plus decay happens in nuclei with too many protons.

Gamma radiation is the emission of high-energy photons from a nucleus. It often accompanies alpha or beta decay, as the daughter nucleus releases excess energy to settle into a lower energy state. Gamma rays change neither the atomic number nor the mass number. Think of it this way: after an alpha or beta decay, the new nucleus can be left in an excited state, and gamma emission is how it dumps that extra energy.

Nuclear Equations and Conservation Laws

Nuclear equations track what happens during decay using element symbols with mass numbers and atomic numbers. Two important examples:

Alpha decay of uranium-238: $^{238}_{92}U \rightarrow ^{234}_{90}Th + ^{4}_{2}He$

Beta-minus decay of carbon-14: $^{14}_{6}C \rightarrow ^{14}_{7}N + e^- + \bar{\nu}_e$

Every nuclear equation must obey these conservation laws:

• Conservation of charge: Total electric charge is the same on both sides. In the uranium example, 92 = 90 + 2.
• Conservation of nucleon number (mass number): Total mass number is the same on both sides. In the uranium example, 238 = 234 + 4.
• Conservation of energy: Total energy (including rest mass energy) is conserved.

To check whether a nuclear equation is balanced, verify that the top numbers (mass numbers) add up equally on both sides, and the bottom numbers (atomic numbers) add up equally on both sides. If either sum doesn't match, the equation is wrong.

The Q-value of a decay is the energy released, calculated from the mass difference between the parent nucleus and all the products. A positive Q-value means the decay is energetically possible and releases kinetic energy to the emitted particles and radiation.

Half-Life and Decay Series

Half-Life and Radioactive Decay Rates

You can't predict when a single radioactive atom will decay, but you can predict the behavior of large numbers of atoms statistically. The half-life ($t_{1/2}$) is the time it takes for half of a radioactive sample to decay. Each radioisotope has its own fixed half-life, ranging from tiny fractions of a second (like polonium-214 at about 164 microseconds) to billions of years (like uranium-238 at 4.5 billion years).

Here's how the pattern works step by step:

1. Start with some number of radioactive atoms, $N_0$.
2. After one half-life, $\frac{N_0}{2}$ atoms remain.
3. After two half-lives, $\frac{N_0}{4}$ remain.
4. After three half-lives, $\frac{N_0}{8}$ remain.
5. The pattern continues: after $n$ half-lives, $\frac{N_0}{2^n}$ atoms remain.

The decay constant ($\lambda$) describes how quickly a particular isotope decays and is related to half-life by:

$t_{1/2} = \frac{\ln(2)}{\lambda}$

The general exponential decay law gives the number of undecayed nuclei at any time $t$:

$N(t) = N_0 e^{-\lambda t}$

Activity (A) measures how many decays happen per second:

$A = \lambda N$

Activity is measured in becquerels (Bq), where 1 Bq = 1 decay per second, or in curies (Ci), where 1 Ci = $3.7 \times 10^{10}$ decays per second. Notice that activity decreases over time just like the number of atoms does, since fewer remaining atoms means fewer decays per second.

Carbon-14 dating is a practical application of half-life. Living organisms constantly take in carbon-14 from the atmosphere, but once they die, the carbon-14 decays with a half-life of 5,730 years. By measuring how much carbon-14 remains in a sample compared to what a living organism would have, scientists can determine when the organism died. This method works for objects up to about 50,000 years old (roughly 8-9 half-lives, after which too little carbon-14 remains to measure reliably).

Decay Series and Radioactive Equilibrium

Many radioactive nuclei don't become stable after just one decay. Instead, they go through a decay series, a chain of successive decays until a stable nuclide is finally produced. The end product is often a stable isotope of lead.

There are three naturally occurring decay series:

• Uranium series: starts with $^{238}U$ and ends at $^{206}Pb$
• Thorium series: starts with $^{232}Th$ and ends at $^{208}Pb$
• Actinium series: starts with $^{235}U$ and ends at $^{207}Pb$

Each step in a series involves either alpha or beta decay (sometimes with accompanying gamma radiation). Branching can occur when a nucleus has more than one possible decay pathway. The probabilities of all branches always add up to 100%.

Secular equilibrium is an important concept in decay chains. When a parent isotope has a much longer half-life than any of its daughter products, the system eventually reaches a state where the activity of every member in the chain becomes equal. At that point, each daughter is being produced at the same rate it's decaying. This principle is used in geological dating and helps explain the levels of natural radioactivity found in rocks and soil.