Nuclear Decay: The Heart of Unstable Matter

Hello future physicists! Welcome to the fascinating world of Nuclear Decay. This chapter connects the fundamental structure of the atom (the nucleus) with the radiation aspects of our curriculum. Don't worry if the formulas look intimidating at first—we will break them down step-by-step using simple analogies.

Understanding nuclear decay is essential not just for passing the exam, but also for grasping how nuclear power works, how medical imaging saves lives, and even how scientists date ancient artifacts! Let's dive into why some atoms are restless and how they achieve stability.

I. The Unstable Nucleus: Why Atoms Decay

At the core of every atom is the nucleus, made of protons (positive charge) and neutrons (no charge). A specific atom is called a nuclide, defined by its proton number (Z, Atomic Number) and its total nucleon number (A, Mass Number).

Isotopes and Instability
  • Isotopes: Atoms of the same element (same Z) but different numbers of neutrons (different A).
  • If a nucleus is too large (too many protons and neutrons) or has an unstable ratio of neutrons to protons, it becomes radioactive (unstable).
  • To become stable, this nuclide spontaneously emits particles or energy—this process is called nuclear decay.
Quick Review: Notation
A nuclide \(X\) is written as \(^{A}_{Z}X\).

II. The Three Main Types of Decay Radiation

When an unstable nucleus decays, it typically emits one of three types of radiation: Alpha, Beta, or Gamma.

1. Alpha (\(\alpha\)) Decay

An alpha particle is simply a Helium nucleus (\(^{4}_{2}\text{He}\)), consisting of 2 protons and 2 neutrons.

  • What happens? The nucleus ejects a bulky alpha particle.
  • Change in Nucleus: Mass number (A) decreases by 4; Atomic number (Z) decreases by 2.
  • Penetration Power: Very low. They can be stopped by a sheet of paper or skin.
  • Ionizing Power: Very high. Because they are large and charged (+2e), they strip electrons from atoms they pass near easily. (Analogy: A slow, heavy bowling ball knocking pins down easily.)

General Alpha Decay Equation:

\(^{A}_{Z}X \to \quad ^{A-4}_{Z-2}Y \quad + \quad ^{4}_{2}\text{He}\)

2. Beta (\(\beta\)) Decay (Specifically Beta-Minus, \(\beta^-\))

Beta-minus decay occurs when a nucleus has too many neutrons. A neutron converts into a proton and an electron.

  • The Particle: A fast-moving electron (\(^{0}_{-1}e\)).
  • Process: \(n \to p + e^- + \bar{v}\) (Neutron converts to Proton + Electron + Antineutrino \(\bar{v}\)).
  • Change in Nucleus: Mass number (A) stays the same; Atomic number (Z) increases by 1 (since a neutron turned into a proton).
  • Penetration Power: Medium. Stopped by a few millimetres of aluminium.
  • Ionizing Power: Medium. They are lighter and faster than alpha particles, so they are less likely to interact and ionize.
  • Did you know? The antineutrino carries away energy and momentum to ensure conservation laws are obeyed, but it has no charge and almost no mass, making it very hard to detect!

General Beta Decay Equation:

\(^{A}_{Z}X \to \quad ^{A}_{Z+1}Y \quad + \quad ^{0}_{-1}e \quad + \quad \bar{v}\)

3. Gamma (\(\gamma\)) Decay

Gamma radiation is not a particle, but high-energy electromagnetic radiation (a photon).

  • What happens? Gamma decay usually happens immediately after alpha or beta decay. The resulting nucleus is often in an "excited state" (like an electron shell that hasn't fully settled). It releases this excess energy as a gamma ray.
  • Change in Nucleus: Mass number (A) and Atomic number (Z) remain unchanged.
  • Penetration Power: Very high. Requires thick lead or concrete to stop.
  • Ionizing Power: Very low. As they are uncharged photons, they rarely interact with atoms. (Analogy: A silent, high-speed bullet passing through most materials.)
Common Mistake to Avoid!
Students often forget that Gamma decay follows Alpha or Beta. A nucleus rarely decays ONLY by Gamma emission unless it was already excited by some other means. Alpha and Beta change the structure; Gamma just removes excess energy.

III. The Nature of Radioactive Decay

Radioactive decay is characterized by two key properties: it is random and spontaneous.

A. Random Nature

We cannot predict when a specific nucleus will decay.

  • We can know that in a sample of billions of unstable nuclei, half will decay in a certain time (the half-life).
  • However, for any single nucleus, the moment of decay is completely a matter of probability. (Analogy: Imagine a lottery machine with billions of balls. You know a winner will be drawn eventually, but you can’t predict which specific ball it will be.)
B. Spontaneous Nature

The decay process is unaffected by external physical or chemical factors.

  • Changing the temperature, pressure, or chemical state of the substance (e.g., turning solid Uranium into liquid Uranium chloride) does not change the rate of decay.
  • The decay rate depends only on the inherent stability of the nucleus itself.

IV. Quantifying Decay: Activity and Half-Life

Since decay is random, we must use statistics to describe how quickly a sample decays.

A. Activity (\(A\)) and Decay Constant (\(\lambda\))

Activity (\(A\)) is the rate at which nuclei decay.

  • Definition: The number of decays per unit time.
  • Units: The unit for activity is the Becquerel (Bq), where \(1 \text{ Bq} = 1 \text{ decay per second}\).

The Decay Constant (\(\lambda\)) quantifies the probability of decay.

  • Definition: The probability that a specific nucleus will decay per unit time.
  • Units: $\text{s}^{-1}$ (per second).
  • A large \(\lambda\) means the substance decays quickly (high probability of decay).
The Fundamental Decay Relationship

The activity \(A\) is directly proportional to the number of unstable nuclei \(N\) present:

$$A = \lambda N$$

This formula tells us: If we have twice as many unstable atoms (\(N\)), the activity (\(A\)) will be twice as high, because \(\lambda\) is a constant for that specific isotope.

B. Exponential Decay and Half-Life

Because the activity \(A\) decreases over time (as \(N\) decreases), the rate of decay slows down exponentially.

Exponential Decay Equations:

The number of remaining nuclei \(N\) at time \(t\): $$N = N_0 e^{-\lambda t}$$

The activity \(A\) at time \(t\): $$A = A_0 e^{-\lambda t}$$

Where \(N_0\) and \(A_0\) are the initial number of nuclei and initial activity, respectively. \(e\) is the base of natural logarithms (\(e \approx 2.718\)).

What is Half-Life? (\(T_{1/2}\))

The half-life (\(T_{1/2}\)) is the time it takes for half of the unstable nuclei in a sample to decay (or for the activity to fall to half its original value).

Important features of Half-Life:

  • It is a fixed, characteristic value for every isotope (e.g., Carbon-14 has a \(T_{1/2}\) of 5,730 years).
  • After 1 half-life: 50% remains.
  • After 2 half-lives: 25% remains (50% of the remaining 50%).
  • After 3 half-lives: 12.5% remains, and so on.

Relationship between Half-Life and Decay Constant:

Since \(N = N_0/2\) when $t = T_{1/2}$, substituting this into the exponential decay equation allows us to derive a crucial link:

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

(Memory Aid: Since \(\ln(2) \approx 0.693\), you can think of it as \(T_{1/2} \approx 0.7 / \lambda\). If the probability of decay (\(\lambda\)) is high, the half-life is short.)

Using Logarithms in Decay Calculations

To solve for time (\(t\)) or the decay constant (\(\lambda\)) using the exponential equations, you often need to use natural logarithms (\(\ln\)).

Step-by-step for finding \(\lambda\):

  1. Start with the decay equation: \(A/A_0 = e^{-\lambda t}\) (or \(N/N_0\)).
  2. Take the natural logarithm of both sides: \(\ln(A/A_0) = \ln(e^{-\lambda t})\).
  3. Simplify using logarithm rules: \(\ln(A/A_0) = -\lambda t\).
  4. Solve for \(\lambda\): \(\lambda = -\frac{\ln(A/A_0)}{t}\).
Key Takeaway for Calculations
The rate of decay is always proportional to the amount of substance left. This leads to the characteristic exponential curve. Master the relationship \(T_{1/2} = \ln(2)/\lambda\).

V. Background Radiation and Safety

We live in a constantly radioactive world. The radiation we encounter naturally is called background radiation.

Sources of Background Radiation

Background radiation comes from various sources, both natural and artificial:

  • Radon Gas (Natural): Produced by the decay of uranium and thorium in rocks and soil. It is the largest single contributor to background radiation exposure.
  • Cosmic Rays (Natural): High-energy particles from space (mostly from the sun or distant galaxies) that strike the Earth's atmosphere. Exposure increases significantly at higher altitudes (e.g., flying in a plane).
  • Ground and Buildings (Natural): Rocks, bricks, and concrete contain trace amounts of radioactive isotopes.
  • Medical Procedures (Artificial): X-rays and medical tracers used for diagnosis and treatment.
Radiation Safety (Minimizing Exposure)

We need effective methods to minimize the dose received when handling radioactive sources. This usually involves three strategies: Time, Distance, and Shielding.

  • Time: Minimize the duration of exposure. Less time near the source means a smaller dose.
  • Distance: Maximize the distance from the source. Radiation intensity follows the inverse square law, so doubling the distance reduces the dose rate by a factor of four.
  • Shielding: Use appropriate materials to absorb the radiation:
    • Alpha: Stopped by paper or air.
    • Beta: Stopped by aluminum or plastic.
    • Gamma: Requires thick lead or concrete.

Note: Radiation doses are measured using devices like Geiger-Muller tubes, which detect the number of ionizing events (counts per minute) caused by the radiation.