Physics (9702) | Nuclear physics

A Level Physics (9702) Study Notes: Nuclear Physics

Hello future physicist! This chapter, Nuclear Physics, is where we zoom in on the tiny, dense core of the atom—the nucleus. Understanding the nucleus is critical, not just for passing exams, but because it explains the energy that powers the Sun (fusion) and the technology behind nuclear power stations and medical imaging (fission and decay).

Don't worry if the concepts of mass converting to energy seem strange; we'll break them down using clear explanations and the famous equation that changed the world: \(E = mc^2\).

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23.1 Mass Defect and Nuclear Binding Energy

Mass-Energy Equivalence: \(E = mc^2\)

This is arguably the most famous equation in science, developed by Albert Einstein. It states that mass and energy are fundamentally interchangeable. They are two different forms of the same thing.

• \(E\) is the energy (J)
• \(m\) is the mass (kg)
• \(c\) is the speed of light in a vacuum (\(3.00 \times 10^8 \text{ m s}^{-1}\))

Because \(c^2\) is a huge number (about \(9 \times 10^{16}\)), even a tiny change in mass (\(\Delta m\)) results in an enormous amount of energy (\(\Delta E\)).

Key Concept: Mass Defect

If you take a nucleus apart into its individual protons and neutrons (called nucleons), and measure the total mass of those separate components, you find something surprising:

The total mass of the separate nucleons is always greater than the mass of the nucleus when they are bound together.

This missing mass is called the mass defect (\(\Delta m\)).

The mass defect \(\Delta m\) is calculated as:

$$ \Delta m = (\text{Total mass of separate nucleons}) - (\text{Mass of nucleus}) $$

Binding Energy (BE)

The mass defect hasn't disappeared—it has been converted into energy, known as the binding energy (BE).

Definition: The binding energy of a nucleus is the energy required to completely separate a nucleus into its constituent protons and neutrons.

When nucleons come together to form a stable nucleus, this energy is released. If you want to break the nucleus apart, you must put that energy back in.

Calculation of Energy Released:

$$ E = c^2 \Delta m $$

Unit of Mass: The Unified Atomic Mass Unit (u)

Atomic masses are tiny, so we use the unified atomic mass unit (u). It is defined as one-twelfth of the mass of a carbon-12 atom.

Memory Aid: You often need to convert 'u' into energy. You must use \(E = mc^2\) where $m$ is in kg, or use the standard conversion factor provided in your data booklet (usually relating 1 u to MeV or J).

Binding Energy Per Nucleon (BEPN)

The total binding energy tells you how much energy holds the nucleus together. The binding energy per nucleon (BEPN) tells you how tightly each individual proton or neutron is held.

Definition: BEPN is the total binding energy divided by the nucleon number ($A$).

$$ \text{BEPN} = \frac{\text{Binding Energy}}{\text{Nucleon Number (A)}} $$

The higher the BEPN, the more stable the nucleus is.

The Stability Curve

If you sketch a graph of BEPN against Nucleon Number ($A$):

Sketch Requirement: Candidates must be able to sketch and explain this graph.

1. The curve rises sharply for small $A$.
2. It reaches a maximum peak.
3. It drops slowly for large $A$.

The peak of the curve occurs around a nucleon number of 56 (Iron-56, \(^{56}\text{Fe}\)). This element is the most stable element in the Universe!

Nuclear Fusion and Fission

The BEPN curve explains why we can get energy from two key nuclear processes:

1. Nuclear Fusion

Definition: Nuclear fusion is the process where two light nuclei join together to form a heavier, more stable nucleus.

• Process: Two light nuclei (low A, low BEPN) combine to form a nucleus closer to the peak (higher A, higher BEPN).
• Energy Release: Since the product nucleus has a higher BEPN, the final nucleus is more stable. The difference in binding energy (or mass defect) is released as massive amounts of energy.
• Example: Fusion powers the Sun. \(^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n} + \text{Energy}\)

2. Nuclear Fission

Definition: Nuclear fission is the process where a heavy nucleus splits into two smaller, more stable nuclei.

• Process: A heavy nucleus (high A, low BEPN, e.g., Uranium) splits into intermediate nuclei (mid-range A, higher BEPN).
• Energy Release: The product fragments have higher BEPN than the original nucleus, meaning the overall process releases energy.
• Example: Used in nuclear power plants.

The Relevance of BEPN: Both fission and fusion release energy because the resulting products have greater binding energy per nucleon than the original reactants, meaning they have become more tightly bound and therefore more stable.

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23.2 Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation (\(\alpha, \beta, \gamma\)).

The Nature of Radioactive Decay

Random and Spontaneous

The syllabus requires you to understand two crucial characteristics of decay:

1. Random: You cannot predict when a specific nucleus will decay. It's totally random, like winning the lottery—you know someone will win eventually, but not who or when.

• Evidence: Fluctuations in the observed count rate over short time periods provide evidence for this random nature.

2. Spontaneous: The decay is unaffected by external physical or chemical conditions (like temperature, pressure, or chemical bonding).

• Analogy: Radioactive decay is like a bag of magical popcorn. Whether you put the bag in the fridge or boil it, the popcorn will still pop randomly, independent of the external conditions.

Activity and the Decay Constant

Activity (A)

Definition: Activity ($A$) is the rate at which nuclei decay. It is the number of disintegrations per unit time.

$$ A = -\frac{dN}{dt} $$

The SI unit for activity is the Becquerel (Bq), where $1 \text{ Bq} = 1 \text{ decay s}^{-1}$.

Decay Constant ($\lambda$)

Definition: The decay constant ($\lambda$) is the probability that an individual nucleus will decay per unit time.

• It has units of $s^{-1}$.
• A large \(\lambda\) means the substance decays quickly (high probability of decay).

Activity, Decay Constant, and Nuclei Relationship

The activity ($A$) is directly proportional to the number of undecayed nuclei ($N$) present.

$$ \mathbf{A = \lambda N} $$

Don't confuse $N$ (number of nuclei) with $A$ (activity). A large sample ($N$) will have high activity ($A$) even if the decay constant ($\lambda$) is small.

The Exponential Decay Law

Because the activity ($A$) depends on the number of undecayed nuclei ($N$), and $N$ decreases over time, the rate of decay also decreases over time. This leads to exponential decay.

The Decay Equation

The number of undecayed nuclei ($N$) remaining after time $t$ is given by:

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

Where:

• $N_0$ is the initial number of nuclei.
• $N$ is the number of nuclei remaining at time $t$.
• $e$ is the base of natural logarithms (\(\approx 2.718\)).
• $\lambda$ is the decay constant.

Since activity ($A$) and count rate ($C$) are both proportional to $N$, they follow the same exponential law:

$$ \mathbf{A = A_0 e^{-\lambda t}} \quad \text{and} \quad \mathbf{C = C_0 e^{-\lambda t}} $$

Candidates must be able to sketch graphs showing the exponential decrease of $N$, $A$, or $C$ over time $t$. The graph starts high and curves down, approaching zero but never quite reaching it.

Half-Life ($t_{1/2}$)

Dealing with the exponential equation can be tricky, so physicists often use the concept of half-life to simplify decay calculations.

Definition: The half-life ($t_{1/2}$) is the time taken for the number of undecayed nuclei (or activity, or count rate) to fall to half of its original value.

Example: If a substance has a half-life of 5 hours, then after 5 hours, 50% remains. After 10 hours (two half-lives), 25% remains. After 15 hours (three half-lives), 12.5% remains, and so on.

Relationship between Half-Life and Decay Constant

We can derive a simple relationship between $t_{1/2}$ and $\lambda$ using the decay equation. We set $N = N_0/2$ and $t = t_{1/2}$:

$$ \frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}} $$

This simplifies to:

$$ \mathbf{\lambda = \frac{0.693}{t_{1/2}}} \quad \text{or} \quad \mathbf{t_{1/2} = \frac{\ln(2)}{\lambda}} $$

This formula is essential for converting between the decay constant (used in exponential equations) and the half-life (often given in problem questions).

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Did You Know?

Nuclear fusion reactions require incredibly high temperatures (millions of degrees Celsius) to overcome the electrostatic repulsion between the positively charged nuclei. This is why fusion reactors are so difficult to build on Earth, but when successful, they promise virtually limitless, clean energy!