E.3 Radioactive Decay: Unstable Atoms Finding Peace
Hello future Nuclear Physicists! Welcome to one of the most intriguing chapters in physics: Radioactive Decay. This topic explores why some atomic nuclei are unstable, how they break apart, and the beautiful mathematical patterns governing this seemingly random process.
Understanding decay is crucial, not just for exams, but because it underpins everything from carbon dating ancient artifacts to powering medical imaging devices. Don't worry if the equations look complex; we will break down the concepts one step at a time!
1. The Fundamentals of Radioactive Decay
1.1 The Nature of Decay: Spontaneous and Random
Radioactive decay is the process by which an unstable atomic nucleus (called a radionuclide) transforms into a more stable nucleus by emitting radiation (particles or energy).
- Spontaneous: The decay happens without any external influence. You cannot speed it up or slow it down by changing temperature, pressure, or chemical bonds.
- Random: We cannot predict when a specific nucleus will decay. We can only talk about the probability of decay over time.
Analogy: Popcorn
Imagine a bag of popcorn. You can predict that after 3 minutes in the microwave, about half the kernels will have popped. But you can never predict exactly which kernel will pop next. Decay works the same way: we track the behavior of the large group, not the individual nucleus.
1.2 Types of Decay and Conservation Rules
When a nucleus decays, both mass number (\(A\)) and charge number (\(Z\)) must be conserved. This means the total number of protons and neutrons (A) and the total charge (Z) before and after the decay must be equal.
A. Alpha Decay (\(\alpha\))
An alpha particle is a helium nucleus (\(_{2}^{4}\text{He}\)). It consists of two protons and two neutrons. This type of decay usually occurs in very heavy, proton-rich nuclei.
- Emission: \(\alpha\) particle (\(_{2}^{4}\text{He}\)).
- Effect on Parent Nucleus: \(A\) decreases by 4, \(Z\) decreases by 2.
- General Equation: \(\text{X}_{Z}^{A} \rightarrow \text{Y}_{Z-2}^{A-4} + _{2}^{4}\text{He} + \text{energy}\)
B. Beta Decay (\(\beta\))
Beta decay occurs when the nucleus needs to adjust its neutron-to-proton ratio.
1. Beta-Minus Decay (\(\beta^{-}\))
A neutron turns into a proton, an electron (the \(\beta^{-}\) particle), and an electron antineutrino (\(\bar{\nu}_e\)).
- Emission: Electron and antineutrino.
- Effect on Parent Nucleus: \(A\) is unchanged, \(Z\) increases by 1.
- General Equation: \(\text{X}_{Z}^{A} \rightarrow \text{Y}_{Z+1}^{A} + _{-1}^{0}\text{e} + \bar{\nu}_e + \text{energy}\)
2. Beta-Plus Decay (\(\beta^{+}\)) (HL Extension)
A proton turns into a neutron, a positron (the \(\beta^{+}\) particle), and an electron neutrino (\(\nu_e\)).
- Emission: Positron and neutrino.
- Effect on Parent Nucleus: \(A\) is unchanged, \(Z\) decreases by 1.
- General Equation: \(\text{X}_{Z}^{A} \rightarrow \text{Y}_{Z-1}^{A} + _{+1}^{0}\text{e} + \nu_e + \text{energy}\)
C. Gamma Decay (\(\gamma\))
Gamma radiation is a high-energy photon (electromagnetic wave). It is often emitted after an \(\alpha\) or \(\beta\) decay when the daughter nucleus is left in an excited, high-energy state. It is the nucleus "relaxing."
- Emission: Photon (no mass, no charge).
- Effect on Parent Nucleus: No change in \(A\) or \(Z\).
Quick Review: The Three Musketeers
Alpha: Heavy, large change in nucleus.
Beta: Light, changes neutron/proton balance.
Gamma: Pure energy, nucleus relaxes.
2. The Concept of Half-Life (\(T_{1/2}\))
Since decay is random, we use the statistical measure half-life (\(T_{1/2}\)) to describe the rate of decay for a large sample.
2.1 Defining Half-Life
The half-life (\(T_{1/2}\)) is the time required for the number of radioactive nuclei in a sample to decrease to half its initial value.
This is an exponential process. If you start with 100 g of an isotope:
- After \(1 \times T_{1/2}\), you have 50 g remaining.
- After \(2 \times T_{1/2}\), you have 25 g remaining (half of 50 g).
- After \(3 \times T_{1/2}\), you have 12.5 g remaining (half of 25 g), and so on.
Key Takeaway for SL: If \(n\) is the number of half-lives elapsed, the fraction of the substance remaining is \(\left(\frac{1}{2}\right)^n\).
This principle applies equally to the number of nuclei (\(N\)), the mass (\(m\)), and the Activity (\(A\)) of the sample.
2.2 Calculating the Number of Half-Lives
Don't worry if the mathematics seems tough, the concept is simple!
If the total time elapsed is \(t\) and the half-life is \(T_{1/2}\):
$$n = \frac{t}{T_{1/2}}$$
Example: Iodine-131 has a half-life of 8 days. After 24 days, how much remains? \(n = 24 / 8 = 3\) half-lives. Fraction remaining = \((1/2)^3 = 1/8\).
3. The Mathematics of Exponential Decay (HL Focus)
For HL students, we move from the simple half-life concept to the underlying differential and exponential equations that govern the decay process. SL students should still be familiar with the final exponential forms.
3.1 The Decay Constant (\(\lambda\))
The decay constant (\(\lambda\)) is the probability that any single nucleus will decay per unit time. It dictates how quickly an isotope decays.
- Units: \(\text{s}^{-1}\) (or other units of inverse time, like \(\text{days}^{-1}\)).
- A large \(\lambda\) means a high probability of decay, resulting in a short half-life.
The relationship between the decay constant and the half-life is:
$$T_{1/2} = \frac{\ln 2}{\lambda}$$ $$T_{1/2} = \frac{0.693}{\lambda}$$
Memory Aid: "The half-life uses the natural log of 2 (0.693) over lambda."
3.2 Activity (\(A\))
Activity (\(A\)) is the rate at which nuclei decay (the number of decays per unit time).
- Definition: \(A = \frac{\Delta N}{\Delta t}\) (or \(-\frac{dN}{dt}\)).
- Unit: The Becquerel (Bq), where \(1 \text{ Bq} = 1 \text{ decay per second}\).
- Activity is proportional to the number of radioactive nuclei remaining: \(A \propto N\).
The relationship is given by: $$A = \lambda N$$
3.3 The Exponential Decay Law
The fundamental equation describing exponential decay is derived from the fact that the rate of decay is proportional to the number of nuclei present.
1. Number of Nuclei Remaining (\(N\))
The number of nuclei remaining at time \(t\) is given by:
$$N = N_0 e^{-\lambda t}$$
Where:
- \(N\) is the number of radioactive nuclei remaining at time \(t\).
- \(N_0\) is the initial number of radioactive nuclei at \(t=0\).
- \(e\) is the base of the natural logarithm (approximately 2.718).
- \(\lambda\) is the decay constant.
2. Activity Remaining (\(A\))
Since Activity \(A\) is directly proportional to \(N\) (\(A = \lambda N\)), the activity follows the same exponential law:
$$A = A_0 e^{-\lambda t}$$
Where \(A_0\) is the initial activity.
Graphing the Decay:
If you plot \(N\) vs. \(t\) or \(A\) vs. \(t\), you get a classic exponential decay curve that approaches zero asymptotically (never quite reaching it in theory).
If you plot \(\ln(N)\) vs. \(t\), you get a straight line with a gradient equal to \(-\lambda\). This is a common way physicists determine the decay constant in the laboratory.
3.4 Common Mistakes to Avoid
1. Unit Confusion: Always ensure the units of time used for \(t\) and for \(\lambda\) or \(T_{1/2}\) are consistent (e.g., if \(T_{1/2}\) is in years, \(t\) must be in years).
2. Activity vs. Number: Remember that \(A = \lambda N\). You must use the decay constant \(\lambda\) to link the number of nuclei to the measured activity.
3. Half-life vs. Decay Constant: A shorter \(T_{1/2}\) means a faster decay, which corresponds to a larger \(\lambda\). They are inversely related!
Did you know? Half-lives can range dramatically. Uranium-238 has a half-life of 4.5 billion years, while Polonium-213 has a half-life of 0.16 microseconds!
Key Takeaway Summary
- Decay is random and spontaneous, governed by probability.
- \(\alpha\), \(\beta\), and \(\gamma\) decays conserve both mass number (\(A\)) and charge number (\(Z\)).
- Half-life (\(T_{1/2}\)) is the time for half the nuclei (or half the activity) to disappear.
- The Decay Constant (\(\lambda\)) is the probability of decay per unit time, related to half-life by \(T_{1/2} = \frac{\ln 2}{\lambda}\).
- The decay process is exponential: \(N = N_0 e^{-\lambda t}\) and \(A = A_0 e^{-\lambda t}\).