Welcome to Nuclear Decay!
Welcome to one of the most fascinating topics in A2 Physics! In this chapter, we explore what happens deep inside unstable atomic nuclei. You will learn why certain nuclei break apart, how we model this process mathematically using the concept of half-life and exponential decay, and how to analyse real experimental data using graphs.
Don't worry if the mathematics looks intimidating at first glance! Radioactive decay follows neat, predictable patterns that we can easily crack with a few simple rules, step-by-step methods, and clear definitions.
1. The Nature of Radioactive Decay
At the core of nuclear physics is a fundamental rule: some combinations of protons and neutrons are simply unstable. To achieve stability, the nucleus ejects particles or electromagnetic energy. This process is known as radioactive decay.
Nuclear decay has two essential characteristics that you must be able to define in your exams:
• Spontaneous: The decay process is completely unaffected by external physical conditions such as temperature, pressure, chemical reactions, or magnetic fields. It happens entirely due to internal nuclear instability.
• Random: It is impossible to predict which individual nucleus will decay next, or when a specific nucleus will decay. However, across a very large sample of nuclei, the overall probability of decay per unit time remains constant.
Analogy Time: The Popcorn Popper
Imagine making popcorn in a hot pan. You cannot point to one single kernel and predict the exact second it will pop (it is random), and you cannot force an individual kernel to pop sooner just by shaking it gently (it is spontaneous). Yet, you know with great certainty how long the entire batch takes to finish popping!
Key Takeaway: Radioactive decay is spontaneous (independent of external conditions) and random (individual decays cannot be predicted, but large collections follow strict statistical laws).
2. Activity and the Decay Constant
Activity (\(A\))
Activity is defined as the rate at which nuclei decay (or the number of disintegrations per unit time) in a given radioactive sample.
Mathematically, it is written as:
\(A = -\frac{\Delta N}{\Delta t}\)
• The unit of activity is the Becquerel (\(\text{Bq}\)), where \(1\text{ Bq} = 1\text{ decay per second}\) (\(1\text{ s}^{-1}\)).
• The negative sign simply indicates that the number of undecayed nuclei, \(N\), decreases over time.
The Decay Constant (\(\lambda\))
The decay constant (\(\lambda\)) represents the probability per unit time that an individual nucleus will decay. Its standard unit is \(\text{s}^{-1}\) (per second), though it can also be expressed in \(\text{min}^{-1}\), \(\text{hour}^{-1}\), or \(\text{year}^{-1}\).
Linking Activity, Decay Constant, and Number of Nuclei
Because each undecayed nucleus has an equal probability \(\lambda\) of decaying per unit time, the total activity of a sample containing \(N\) undecayed nuclei is directly proportional to \(N\):
\(A = \lambda N\)
Corrected Count Rate and Background Radiation
In a laboratory, radiation detectors (like a Geiger-Müller tube) measure count rate (\(C\)) rather than absolute activity (\(A\)). Because detectors do not catch 100% of the emissions, \(C\) is proportional to \(A\).
Furthermore, radiation is naturally present all around us from rocks, cosmic rays, and radon gas—this is called background radiation. To find the true rate of decay from your source, you must always calculate the corrected count rate:
\(\text{Corrected Count Rate} = \text{Measured Count Rate} - \text{Background Count Rate}\)
Key Takeaway: \(A = \lambda N\). A higher decay constant means a nucleus is more unstable and decays more rapidly. Always remember to subtract background radiation in experimental questions!
3. The Mathematics of Exponential Decay
Because the rate of decay is proportional to the number of remaining nuclei (\(\frac{\Delta N}{\Delta t} \propto -N\)), the decay of radioactive nuclei follows an exponential decay law.
The Exponential Decay Equations
The number of undecayed nuclei \(N\) remaining after a time \(t\) is given by:
\(N = N_0 e^{-\lambda t}\)
Where:
• \(N_0\) = initial number of undecayed nuclei at \(t = 0\)
• \(N\) = number of undecayed nuclei remaining at time \(t\)
• \(\lambda\) = decay constant (\(\text{s}^{-1}\))
• \(t\) = time elapsed (\(\text{s}\))
• \(e\) = the base of the natural logarithm (\(\approx 2.718\))
Since activity \(A\) and corrected count rate \(C\) are directly proportional to \(N\), these quantities follow the exact same exponential form:
\(A = A_0 e^{-\lambda t}\)
\(C = C_0 e^{-\lambda t}\)
Step-by-Step Example Calculation
Problem: A sample contains \(6.0 \times 10^{18}\) unstable nuclei with a decay constant \(\lambda = 0.035\text{ s}^{-1}\). How many undecayed nuclei remain after \(40\text{ seconds}\)?
Step 1: Identify the known values.
\(N_0 = 6.0 \times 10^{18}\)
\(\lambda = 0.035\text{ s}^{-1}\)
\(t = 40\text{ s}\)
Step 2: Calculate the exponent \(-\lambda t\).
\(-\lambda t = -(0.035 \times 40) = -1.40\)
Step 3: Apply the exponential equation.
\(N = N_0 e^{-\lambda t} = 6.0 \times 10^{18} \times e^{-1.40}\)
\(N = 6.0 \times 10^{18} \times 0.2466 = 1.48 \times 10^{18}\text{ nuclei}\)
Key Takeaway: Radioactive decay reduces quantities by equal fractions over equal intervals of time, governed by the exponential function \(e^{-\lambda t}\).
4. Half-Life (\(t_{1/2}\))
Definition of Half-Life
The half-life (\(t_{1/2}\)) of a radioactive isotope is the time taken for half of the undecayed nuclei in a sample to decay (or the time taken for the activity to fall to half of its initial value).
Deriving the Relationship Between \(t_{1/2}\) and \(\lambda\)
Let's see how half-life is directly connected to the decay constant:
1. Start with the exponential decay equation: \(N = N_0 e^{-\lambda t}\)
2. When time \(t = t_{1/2}\), the number of nuclei remaining is \(N = \frac{N_0}{2}\):
\(\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}\)
3. Divide both sides by \(N_0\):
\(\frac{1}{2} = e^{-\lambda t_{1/2}}\)
4. Take the natural logarithm (\(\ln\)) of both sides:
\(\ln\left(\frac{1}{2}\right) = -\lambda t_{1/2}\)
\(-\ln(2) = -\lambda t_{1/2}\)
5. Rearrange to get the formula:
\(t_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}\)
Quick Rule of Thumb for Calculations
After \(n\) half-lives have elapsed:
• The fraction of undecayed nuclei remaining is \(\left(\frac{1}{2}\right)^n\)
• For example, after \(1\text{ half-life} \rightarrow \frac{1}{2}\) remaining
• After \(2\text{ half-lives} \rightarrow \frac{1}{4}\) remaining
• After \(3\text{ half-lives} \rightarrow \frac{1}{8}\) remaining
• After \(4\text{ half-lives} \rightarrow \frac{1}{16}\) remaining
Key Takeaway: A shorter half-life means a larger decay constant and a faster, more intense decay process (\(t_{1/2} = \frac{\ln 2}{\lambda}\)).
5. Linearising Decay Graphs using Natural Logarithms
Exponential decay curves are helpful, but reading precise values from a curved graph can be tricky. In A2 Physics, we transform the curve into a straight line using natural logarithms.
From Exponential to Straight Line (\(y = mx + c\))
Taking natural logarithms of both sides of \(N = N_0 e^{-\lambda t}\):
\(\ln(N) = \ln(N_0 e^{-\lambda t})\)
\(\ln(N) = \ln(N_0) + \ln(e^{-\lambda t})\)
\(\ln(N) = -\lambda t + \ln(N_0)\)
Compare this directly with the straight-line equation \(y = mx + c\):
• \(y\)-axis: \(\ln(N)\) (or \(\ln(A)\), or \(\ln(C)\))
• \(x\)-axis: Time \(t\)
• Gradient (\(m\)): \(-\lambda\) (a negative slope!)
• \(y\)-intercept (\(c\)): \(\ln(N_0)\) (or \(\ln(A_0)\))
How to determine \(\lambda\) and \(N_0\) from experimental data:
1. Calculate \(\ln(\text{corrected count rate})\) for each time data point.
2. Plot \(\ln(C)\) against \(t\).
3. Draw the line of best fit.
4. Calculate the gradient: \(\text{Gradient} = \frac{\Delta \ln(C)}{\Delta t}\). The decay constant is \(\lambda = -\text{Gradient}\).
5. Find the intercept on the vertical axis: \(\text{Intercept} = \ln(C_0)\), so \(C_0 = e^{\text{Intercept}}\).
Key Takeaway: A plot of \(\ln(N)\) or \(\ln(A)\) against time yields a straight line with \(\text{Gradient} = -\lambda\) and \(y\text{-intercept} = \ln(N_0)\).
6. Practical Applications of Radioactive Decay
1. Radioactive Dating (e.g., Carbon-14 Dating)
Living organisms constantly absorb Carbon-14 through respiration and food. When an organism dies, it stops taking in Carbon-14. The remaining Carbon-14 decays with a known half-life of roughly \(5730\text{ years}\). By measuring the remaining ratio of Carbon-14 to Carbon-12, scientists can calculate how long ago the organism died.
2. Medical Tracers
Radioisotopes injected into the body allow doctors to image organs or detect blockages.
• Ideal half-life: Must be a few hours long. If it is too short (seconds), it decays before imaging is complete. If it is too long (months/years), it exposes the patient to dangerous residual radiation.
• Common example: Technetium-99m (\(t_{1/2} \approx 6\text{ hours}\)), which emits penetrative, low-ionising gamma rays.
7. Common Exam Traps & How to Avoid Them
• Trap 1: Forgetting Background Radiation. Always check if the question gives a background count. If it does, subtract it from every reading before doing any logarithm or half-life calculation!
• Trap 2: Mismatched Time Units. If \(\lambda\) is given in \(\text{s}^{-1}\), your time \(t\) must be in seconds. If half-life is in years, convert to seconds when calculating activity in \(\text{Bq}\) (\(1\text{ year} \approx 3.16 \times 10^7\text{ s}\)).
• Trap 3: Natural Log (\(\ln\)) vs Log Base 10 (\(\log_{10}\)). Always use the \(\ln\) button on your calculator for nuclear decay equations, not \(\log\).
• Trap 4: "Decayed" vs "Remaining". Read carefully! If a question asks for the number of nuclei that have decayed, you must calculate \(N_{\text{decayed}} = N_0 - N\).
Summary Checklist
Before moving on to practice exam questions, ensure you can:
• State the meaning of random and spontaneous decay.
• Define activity (\(A\)) and its unit, the Becquerel (\(\text{Bq}\)).
• Define the decay constant (\(\lambda\)) and write \(A = \lambda N\).
• Apply the exponential equations \(N = N_0 e^{-\lambda t}\) and \(A = A_0 e^{-\lambda t}\).
• State the definition of half-life (\(t_{1/2}\)) and use \(t_{1/2} = \frac{\ln 2}{\lambda}\).
• Plot and interpret \(\ln(A)\) vs \(t\) graphs to determine \(\lambda\) and initial activity.