Welcome to Radioactive Decay!
Hello! This chapter dives into one of the most fundamental processes in nuclear physics: radioactive decay. Don't worry if the maths seems intimidating—we're going to break down the exponential nature of decay step-by-step.
Understanding radioactive decay is crucial, not just for exams, but because it underpins things like carbon dating, nuclear energy, and medical imaging (like PET scans, which we discussed earlier in the course!). Let's get started on understanding how unstable nuclei decide to break apart.
1. The Essential Nature of Radioactive Decay
The syllabus requires us to understand two key characteristics of radioactive decay: it is random and spontaneous.
What does 'Random' mean?
Random means we cannot predict when a specific unstable nucleus (or nuclide) will decay.
- If you have a million uranium atoms, you know how many will decay in the next hour (on average), but you cannot point to a single atom and say, "That one will decay in 5 minutes."
- Evidence for Randomness: If you measure the count rate (the number of decays per second) using a Geiger counter, you will observe fluctuations. These slight, random variations in the count rate prove that the decay events happen unpredictably in time.
What does 'Spontaneous' mean?
Spontaneous means the decay process is unaffected by external conditions.
- It doesn't matter what you do to the sample—heating it up, cooling it down, crushing it, or placing it under huge pressure.
- The probability of a nucleus decaying is determined entirely by the unstable nucleus itself and the fundamental forces within the atom, not by its environment.
Analogy: The Popcorn Machine
Imagine you put a bag of popcorn kernels in a microwave. Decay is like the kernels popping.
Spontaneous: The popping rate isn't affected by the weather outside the kitchen.
Random: You know half the kernels will pop in the first minute (if that's the half-life), but you can't tell which specific kernel will pop next!
Key Takeaway: The fundamental process of radioactive decay is unpredictable (random) and unavoidable by external means (spontaneous).
2. Quantifying Decay: Activity and the Decay Constant
Since we can't predict individual decays, we work with statistics and rates. We need specific quantities to measure how fast a sample is decaying.
2.1 Activity (\(A\))
Activity (\(A\)) is the rate at which unstable nuclei decay.
- Definition: Activity is the number of disintegrations (decays) occurring per unit time.
- SI Unit: The becquerel (Bq), where \(1 \text{ Bq} = 1 \text{ decay per second}\) (\(1 \text{ s}^{-1}\)).
2.2 The Decay Constant (\(\lambda\))
The decay constant (\(\lambda\)) is the key link that connects the random nature of decay to a measurable rate.
- Definition: \(\lambda\) is the probability that an individual nucleus will decay per unit time.
- Unit: \(\text{s}^{-1}\) (or \(\text{min}^{-1}\), \(\text{year}^{-1}\), etc., depending on the time unit used).
Think of it like this: If \(\lambda = 0.01 \text{ s}^{-1}\), this means there is a 1% chance that any given nucleus will decay in the next second.
2.3 The Fundamental Decay Equation: \(A = \lambda N\)
The rate of decay (Activity, \(A\)) is directly proportional to the number of undecayed nuclei (\(N\)) present.
$$A = \lambda N$$
- \(A\): Activity (Bq).
- \(\lambda\): Decay constant (\(\text{s}^{-1}\)).
- \(N\): Number of undecayed nuclei remaining at that time (unitless count).
Why is this equation so important?
The more unstable atoms you have (\(N\)), the higher the overall rate of decay (\(A\)). As the sample decays, \(N\) decreases, and therefore \(A\) also decreases over time. This leads directly to the exponential nature of decay.
Key Takeaway: Activity \(A\) tells us how fast the sample is decaying, and the Decay Constant \(\lambda\) represents the intrinsic probability of decay for one nucleus. They are linked by the number of undecayed nuclei \(N\).
3. Half-Life (\(t_{1/2}\))
The most common way to describe how quickly a radioactive source decays is its half-life. This concept makes the exponential process easy to grasp.
3.1 Defining Half-Life
Half-life (\(t_{1/2}\)) is defined as the time taken for the activity of a radioactive sample to halve, or for the number of undecayed nuclei to halve.
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If a sample has an initial activity of \(100 \text{ Bq}\) and a half-life of 2 hours:
- After 2 hours, the activity is \(50 \text{ Bq}\).
- After 4 hours (two half-lives), the activity is \(25 \text{ Bq}\).
- After 6 hours (three half-lives), the activity is \(12.5 \text{ Bq}\).
- Half-life can range dramatically, from microseconds (for highly unstable elements) to billions of years (like Uranium-238).
Students often think that after two half-lives, the sample is gone. This is false! After two half-lives, only 75% has decayed (50% + 25%), and 25% remains. Because the decay is exponential, you never fully reach zero, although the activity will drop to background levels quickly for short half-lives.
3.2 Relating Decay Constant and Half-Life
Since both \(\lambda\) and \(t_{1/2}\) describe the rate of decay, they are related by a simple formula:
$$ \lambda = \frac{0.693}{t_{1/2}} $$
(The value 0.693 is mathematically derived as the natural logarithm of 2, i.e., \(\ln 2\)).
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Interpretation:
A large \(\lambda\) (high probability of decay) means a very short half-life (\(t_{1/2}\)).
A small \(\lambda\) (low probability of decay) means a very long half-life. - Memory Aid: If you need to find the decay constant \(\lambda\), you always divide the "natural constant" (0.693) by the half-life.
Quick Review Box
To find Half-Life: Use graphs (find the time taken for activity/nuclei/count rate to fall to half its initial value).
To find Decay Constant: Use the formula \(\lambda = 0.693 / t_{1/2}\).
4. The Exponential Decay Law
Because the rate of decay is proportional to the amount remaining (\(A \propto N\)), the quantity remaining decreases exponentially over time. This leads to the all-important exponential decay equation.
4.1 The Decay Equation
The relationship showing how a quantity \(x\) changes over time \(t\) is:
$$x = x_0 e^{-\lambda t}$$
Let's break down the symbols:
- \(x\): The quantity remaining at time \(t\).
- \(x_0\): The initial quantity (at \(t=0\)).
- \(e\): The base of natural logarithms (approximately 2.718...).
- \(\lambda\): The decay constant (\(\text{s}^{-1}\)).
- \(t\): The time elapsed.
What can \(x\) represent?
This formula is versatile! In the Cambridge syllabus, \(x\) can represent three different, but related, quantities:
- Number of Undecayed Nuclei: \(N = N_0 e^{-\lambda t}\)
- Activity: \(A = A_0 e^{-\lambda t}\)
- Received Count Rate: \(C = C_0 e^{-\lambda t}\)
Note: The count rate \(C\) is proportional to the activity \(A\), so it follows the same exponential law.
4.2 Sketching the Decay Curve
When asked to sketch the variation of \(x\) (Activity, \(N\), or Count Rate) with time, remember these features:
- The curve must start at the maximum value (\(x_0\)) at \(t=0\).
- The gradient (rate of decay) is steepest at the start because the most nuclei are present then.
- The curve decreases smoothly and never actually reaches zero (it is asymptotic to the time axis).
- Show the half-life points clearly: \(x\) drops to \(x_0/2\) after \(t_{1/2}\), and to \(x_0/4\) after \(2t_{1/2}\), and so on.
Did you know? This mathematical form, where the rate of decrease is proportional to the amount present, appears in many areas of science, including capacitor discharge and Newton's law of cooling! Physics likes simple mathematical relationships!
4.3 Using the Exponential Equation in Calculations
You will often be asked to use logarithms (specifically natural log, \(\ln\)) to solve for \(t\) or \(\lambda\).
Example process to find time \(t\):
- Start with: \(A = A_0 e^{-\lambda t}\)
- Rearrange to get the exponential term alone: \(\frac{A}{A_0} = e^{-\lambda t}\)
- Take the natural logarithm of both sides: \(\ln \left(\frac{A}{A_0}\right) = -\lambda t\)
- Solve for \(t\): \(t = -\frac{1}{\lambda} \ln \left(\frac{A}{A_0}\right)\)
Helpful tip: Remember that \(\ln(1/2) = -0.693\). This is where the relation \(\lambda t_{1/2} = 0.693\) comes from, providing a good self-check for your calculations!