🔬 Your Guide to Atoms, Nuclei, and Radiation ☢️
Welcome to one of the most fascinating (and sometimes challenging!) chapters in Physics: the world inside the atom. Here, we move beyond forces and circuits to study the nucleus, the fundamental particles that make up matter, and the amazing energy locked within. Don't worry if these ideas seem abstract; we will break them down using clear steps and everyday analogies!
Key Takeaway Goal: By the end of this chapter, you will understand the structure of the nucleus, why certain nuclei are unstable, and how mass and energy are related in nuclear reactions.
1. The Structure of the Atom and Nucleus (Syllabus 11.1)
1.1 The Nuclear Atom Model (Rutherford Scattering)
Before 1911, scientists thought the atom was a big mush of positive charge with electrons scattered throughout (the plum pudding model). Ernest Rutherford proved this wrong with the famous alpha-particle scattering experiment.
- Experiment: Alpha particles (positively charged) were fired at a very thin gold foil.
- Observations & Inferences:
- Most alpha particles passed straight through. Inference: The atom is mostly empty space.
- A small fraction were deflected at large angles (>90°), some even bounced back. Inference: There must be a tiny, dense, positively charged centre, which we call the nucleus.
Analogy: Imagine shooting a cannonball at a tissue paper target. Most go straight through. But if you hit a tiny, hard marble hidden in the middle, the cannonball might bounce back! The marble is the nucleus.
1.2 Describing the Nucleus
The nucleus is made up of particles called nucleons. These are protons and neutrons.
- Proton Number (\(Z\)): The number of protons. This defines the element. (Also called Atomic Number).
- Nucleon Number (\(A\)): The total number of protons and neutrons. (Also called Mass Number).
- Neutron Number (\(N\)): Calculated as \(N = A - Z\).
Nuclide Notation and Isotopes
We represent a specific nucleus (a nuclide) using the notation:
$$ {}_Z^A \text{X} $$
An Isotope refers to atoms of the same element (same \(Z\)) but with different numbers of neutrons (different \(N\) and thus different \(A\)).
Example: Carbon-12 (\({}_6^{12}\text{C}\)) and Carbon-14 (\({}_6^{14}\text{C}\)). They both have 6 protons, but C-14 has 8 neutrons while C-12 has 6.
Quick Review: \(A\) is the total count of particles in the nucleus; \(Z\) is the proton ID badge.
2. Radioactive Emissions and Conservation Laws (Syllabus 11.1)
2.1 Nuclear Conservation Rules
In all nuclear processes (like radioactive decay or reactions), two key quantities must be conserved (kept the same) on both sides of the reaction equation:
- Conservation of Nucleon Number (\(A\)): The total mass number must be equal.
- Conservation of Charge (Proton Number, \(Z\)): The total charge (atomic number) must be equal.
2.2 Types of Radiation
Radiation comes in three main types, all emitted from an unstable nucleus:
| Radiation | Composition/Identity | Charge | Mass | Energy Spectrum |
|---|---|---|---|---|
| Alpha (\(\alpha\)) | Helium nucleus (\({}_2^4 \text{He}\)) | +2e | \(\approx\) 4u (large) | Discrete (single energy values) |
| Beta minus (\(\beta^-\)) | Fast electron (\({}_{-1}^0 \text{e}\)) | -1e | Negligible | Continuous range |
| Beta plus (\(\beta^+\)) | Positron (\({}_{+1}^0 \text{e}\)) | +1e | Negligible | Continuous range |
| Gamma (\(\gamma\)) | High-energy electromagnetic photon | 0 | 0 | Discrete (single energy values) |
Important Note on Beta Decay Energies:
Alpha and gamma decay release particles or photons with discrete (fixed) energies. However, beta particles are emitted with a continuous range of energies. Why?
This is because another small, neutral particle is emitted along with the beta particle:
- In \(\beta^-\) decay, an (electron) antineutrino (\(\bar{\nu}\)) is also produced.
- In \(\beta^+\) decay, an (electron) neutrino (\(\nu\)) is also produced.
The total energy released in the decay is shared between the beta particle and the (anti)neutrino, leading to a continuous spectrum of kinetic energy for the beta particle.
2.3 Nuclear Decay Equations (Syllabus 11.1 Example)
We use the conservation laws to complete decay equations.
Alpha Decay Example: Uranium-238 decays into Thorium-234.
$$ {}_{92}^{238} \text{U} \rightarrow {}_{90}^{234} \text{Th} + {}_{2}^{4} \alpha $$
Check: Nucleons: \(238 = 234 + 4\) (Conserved). Charge: \(92 = 90 + 2\) (Conserved).
Unified Atomic Mass Unit (\(u\)):
The mass of atoms is tiny, so we use the unified atomic mass unit (\(u\)). This is defined as 1/12th the mass of a nucleus of carbon-12.
Do not confuse \(\beta^-\) (electron) with \(\beta^+\) (positron). They have opposite charges! In equations, an electron is \({}_{-1}^0 \text{e}\) and a positron is \({}_{+1}^0 \text{e}\).
3. Fundamental Particles (Syllabus 11.2)
Are protons and neutrons truly the smallest bits of matter? No! We dive deeper into the Standard Model of particle physics.
3.1 Quarks and Hadrons
A quark is a truly fundamental particle. Protons and neutrons are made up of quarks.
- There are six "flavours" of quark: up (u), down (d), strange (s), charm (c), top (t), and bottom (b).
- We only focus on up and down quarks for nuclear structure.
| Quark (Flavour) | Charge | Antiquark (Flavour) | Charge |
|---|---|---|---|
| up (u) | \(+2/3 \text{e}\) | anti-up (\(\bar{\text{u}}\)) | \(-2/3 \text{e}\) |
| down (d) | \(-1/3 \text{e}\) | anti-down (\(\bar{\text{d}}\)) | \(+1/3 \text{e}\) |
Particles made of quarks are called Hadrons. Hadrons are split into two groups:
- Baryons (3 quarks):
- Proton: Composition is uud. Total charge: \((+2/3) + (+2/3) + (-1/3) = +1 \text{e}\).
- Neutron: Composition is udd. Total charge: \((+2/3) + (-1/3) + (-1/3) = 0\).
- Mesons (1 quark and 1 antiquark):
- These are unstable, short-lived particles. Example: A pion (\(u\bar{d}\)).
3.2 Leptons
Leptons are fundamental particles that do not feel the strong nuclear force. The key leptons in this syllabus are:
- The electron (\(\text{e}^-\)) and its antiparticle, the positron (\(\text{e}^+\)).
- The neutrino (\(\nu\)) and its antiparticle, the antineutrino (\(\bar{\nu}\)).
3.3 Quark Changes during Beta Decay
Beta decay happens when a particle within the nucleus changes its flavour:
1. Beta-minus (\(\beta^-\)) Decay: A neutron decays into a proton, an electron, and an antineutrino.
- Quark Change: A down quark (d) changes into an up quark (u).
- $$ \text{n} (\text{udd}) \rightarrow \text{p} (\text{uud}) + \text{e}^- + \bar{\nu} $$
2. Beta-plus (\(\beta^+\)) Decay: A proton decays into a neutron, a positron, and a neutrino.
- Quark Change: An up quark (u) changes into a down quark (d).
- $$ \text{p} (\text{uud}) \rightarrow \text{n} (\text{udd}) + \text{e}^+ + \nu $$
Every particle has an antiparticle with the exact same mass but opposite charge (and opposite values for other properties like quark flavour). For example, the antiparticle of a proton (uud) is an antiproton (\(\bar{u}\bar{u}\bar{d}\)).
4. Mass, Energy, and Nuclear Stability (Syllabus 23.1)
4.1 Mass-Energy Equivalence: \(E = mc^2\)
One of the most profound ideas in physics is that mass and energy are interchangeable. They are just two different forms of the same thing. This relationship is quantified by Einstein's famous equation:
$$ E = mc^2 $$
where:
- \(E\) 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, even a tiny amount of mass is equivalent to an enormous amount of energy!
4.2 Mass Defect and Binding Energy
If we weigh the individual protons and neutrons (nucleons) that make up a nucleus and compare that to the mass of the final assembled nucleus, we find a strange result:
The mass of the assembled nucleus is always less than the total mass of its separate constituent nucleons.
- Mass Defect (\(\Delta m\)): This is the difference in mass:
$$ \Delta m = (\text{Total mass of separate nucleons}) - (\text{Mass of nucleus}) $$
Where did this mass go? It was converted into the energy needed to hold the nucleus together! This energy is called the Binding Energy (\(E_{\text{B}}\)).
- Binding Energy: The energy required to completely separate a nucleus into its individual protons and neutrons. (It is also the energy released when the nucleus is formed from its separate parts).
- Calculation: We use the mass defect and the mass-energy equation: $$ E_{\text{B}} = (\Delta m) c^2 $$
Analogy: Imagine building a Lego house. The total mass of the separate bricks (nucleons) is slightly more than the mass of the final, locked-together house (nucleus). The missing mass was released as energy when the strong forces locked the pieces together.
4.3 Binding Energy per Nucleon
To compare the stability of different nuclei, we calculate the Binding Energy per Nucleon. This is the average energy required to remove a single nucleon from the nucleus.
$$ \text{Binding Energy per Nucleon} = \frac{\text{Total Binding Energy } (E_{\text{B}})}{\text{Nucleon Number } (A)} $$
- The higher the binding energy per nucleon, the more stable the nucleus.
- The peak stability occurs around a nucleon number of 56 (Iron, \({}^{56}\text{Fe}\)).
Sketching the Graph:
If you sketch a graph of Binding Energy per Nucleon vs. Nucleon Number (\(A\)), it starts low, rises sharply, peaks near \(A=56\), and then drops off slowly for heavy nuclei.
4.4 Fission and Fusion Explained
Nuclear reactions release energy when the products are more stable (have a higher binding energy per nucleon) than the reactants.
- Nuclear Fusion:
- Process: Two small, light nuclei join together to form a single, heavier nucleus.
- Relevance to BE/Nucleon: Light nuclei (left side of the graph) increase their stability dramatically by fusing and moving up the curve toward \(A=56\).
- Example: The power source of the Sun.
- Nuclear Fission:
- Process: A heavy nucleus splits into two smaller, lighter nuclei.
- Relevance to BE/Nucleon: Heavy nuclei (right side of the graph) increase their stability by splitting and moving up the curve toward \(A=56\).
- Example: Used in nuclear power plants (e.g., Uranium-235).
Summary Key Takeaway: Energy is released in both fission and fusion because the resulting nuclei are more tightly bound (more stable) than the initial nuclei, meaning energy had to be released during their formation.
5. Radioactive Decay (Syllabus 23.2)
5.1 The Nature of Decay
Radioactive decay is the process by which an unstable nucleus transforms into a more stable nucleus by emitting radiation (\(\alpha, \beta, \gamma\)).
Decay has two key properties:
- Spontaneous: The decay cannot be influenced by external physical or chemical conditions (like temperature, pressure, or chemical state).
- Random: It is impossible to predict when any one nucleus will decay. We can only predict the probability of decay over a time period, especially for a large sample.
Proof of Randomness: Observing the count rate of a sample shows small, unpredictable fluctuations over short time periods.
5.2 Quantifying Decay: Activity and Constant
- Activity (\(A\)): Defined as the rate of decay (the number of nuclei decaying per unit time). Unit is Becquerel (Bq), where \(1 \text{ Bq} = 1 \text{ decay s}^{-1}\).
- Decay Constant (\(\lambda\)): The probability that an individual nucleus will decay per unit time. Unit is \(\text{s}^{-1}\).
The Activity is proportional to the number of undecayed nuclei (\(N\)) present:
$$ A = \lambda N $$
5.3 Half-Life (\(T_{1/2}\))
The half-life is the time taken for the mass of the substance, the number of undecayed nuclei (\(N\)), or the activity (\(A\)) to halve.
The decay constant and half-life are linked by a simple formula derived from the exponential decay relationship:
$$ \lambda = \frac{\ln 2}{T_{1/2}} $$
Using \(\ln 2 \approx 0.693\):
$$ T_{1/2} = \frac{0.693}{\lambda} $$
5.4 Exponential Decay
Since the rate of decay (\(A\)) depends on how many nuclei are left (\(N\)), the decay process follows an exponential relationship. The amount of radioactive material decreases rapidly at first and then slows down.
The general decay equation is:
$$ x = x_0 e^{-\lambda t} $$
Where:
- \(x\) can be the number of undecayed nuclei (\(N\)), the Activity (\(A\)), or the received count rate.
- \(x_0\) is the initial value of \(x\) at \(t=0\).
- \(e\) is the base of natural logarithms (\(\approx 2.718\)).
- \(\lambda\) is the decay constant.
To sketch the graph: Draw a curve starting at \(x_0\) at \(t=0\). At \(t=T_{1/2}\), the value should be \(x_0/2\). At \(t=2T_{1/2}\), the value should be \(x_0/4\), and so on. The curve never actually reaches zero.
\(\lambda\) is the probability of *instantaneous* decay. If \(\lambda\) is large, the substance decays fast, and the half-life (\(T_{1/2}\)) is short. They are inversely related.
Final Review Takeaway: Nuclear stability is governed by the binding energy per nucleon. Decay is quantified by activity and half-life, which follow the laws of exponential decay.