Modern Physics
Chapter 24 – Radioactivity and Nuclear Reaction
Class 12 Physics
Nuclear Physics
Legacy Nepal eNotes title • Current Grade 12 syllabus-aligned typed notes
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Current NEB/CDC alignment:
The older Nepal eNotes source page is titled Nuclear Physics. In the current Grade 12 Physics sequence, the exam-aligned chapter is Modern Physics – Chapter 24: Radioactivity and Nuclear Reaction (6 teaching hours). The required core covers natural and artificial radioactivity, alpha/beta/gamma radiation, radioactive disintegration law, half-life/decay constant/mean life, Geiger–Müller tube, medical uses and health hazards, plus numerical and conceptual problems. Radiocarbon dating is retained as a high-value source-note topic. A short legacy Nuclear Physics appendix is placed after the current-syllabus section so older source-PDF material does not get confused with the current Grade 12 requirement.
1. Atomic-Nuclear Basics
An atom contains a tiny, dense nucleus surrounded by electrons. The nucleus contains protons and neutrons, collectively called nucleons.
| Quantity | Meaning | Symbol |
|---|---|---|
| Atomic number | Number of protons in nucleus | Z |
| Mass number | Total protons + neutrons | A |
| Number of neutrons | A − Z | N |
| Nuclide notation | Element X with mass number A and atomic number Z | ᴬZX |
Isotopes
Isotopes are nuclei of the same element having the same atomic number Z but different neutron numbers and therefore different mass numbers.
Diagram 1: Nucleus and standard nuclide notation
2. Radioactivity
Radioactivity
Radioactivity is the spontaneous transformation of an unstable atomic nucleus accompanied by the emission of ionizing radiation.
The original unstable nucleus is called the parent nucleus. The nucleus produced after decay is called the daughter nucleus.
Unstable parent nucleus → daughter nucleus + emitted radiation
Main Features
- Radioactive decay is spontaneous.
- Individual nuclear decay is random; the behavior of a large number of nuclei follows statistical laws.
- The decay rate of a nuclide is characterized by its decay constant λ.
- Ordinary changes in temperature, pressure and chemical state do not significantly alter the nuclear decay constant.
3. Natural and Artificial Radioactivity
3.1 Natural Radioactivity
Natural radioactivity is the spontaneous decay of unstable nuclides occurring in nature.
Example:
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He3.2 Artificial Radioactivity
Artificial radioactivity occurs when a nuclear reaction produces an unstable radioactive nuclide that subsequently decays.
Classic example:
²⁷₁₃Al + ⁴₂He → ³⁰₁₅P + ¹₀n ³⁰₁₅P → ³⁰₁₄Si + ⁰₊₁e + ν| Feature | Natural | Artificial |
|---|---|---|
| Origin | Unstable nuclide occurs naturally | Radioactive nuclide is created by a nuclear reaction |
| Cause of radioisotope production | Natural nuclear instability | Bombardment / induced nuclear transformation |
| After production | Spontaneously decays | Produced nuclide spontaneously decays |
| Example | U-238 decay | P-30 produced from Al-27 |
Diagram 2: Natural and artificial radioactivity
4. Alpha, Beta and Gamma Radiations
| Property | Alpha (α) | Beta (β⁻) | Gamma (γ) |
|---|---|---|---|
| Nature | Helium nucleus, ⁴₂He | High-speed electron emitted in nuclear transformation | High-energy electromagnetic photon |
| Charge | +2e | −e | 0 |
| Rest mass | About 4 u | Electron mass | 0 |
| Speed | High but well below c | Can approach c | c in vacuum |
| Ionizing power | Very high | Moderate | Lower per path than α/β |
| Penetrating power | Low | Moderate | High |
| Deflection in E/B field | Yes; like positive charge | Yes; opposite α and much stronger curvature | No |
| Typical shielding concept | Paper / outer skin stops external α | Thin metal/plastic shielding | Dense thick shielding such as lead/concrete |
4.1 Alpha Decay
ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂HeMass number decreases by 4 and atomic number decreases by 2.
4.2 Beta-Minus Decay
n → p + e⁻ + ν̄ ᴬZX → ᴬZ₊₁Y + ⁰₋₁e + ν̄Mass number remains unchanged while atomic number increases by 1.
4.3 Gamma Decay
ᴬZX* → ᴬZX + γAn excited nucleus loses energy; A and Z remain unchanged.
Diagram 3: Charge behavior of radioactive emissions
Diagram 4: General penetration comparison
5. Radioactive Disintegration
Radioactive disintegration
The spontaneous transformation of an unstable parent nucleus into another nucleus through radioactive decay.
The exact instant at which a particular unstable nucleus will decay cannot be predicted. However, for a very large number of identical radioactive nuclei, the statistical decay rate is predictable.
6. Law of Radioactive Disintegration
The rate of decrease in the number N of undecayed nuclei is directly proportional to N:
−dN/dt = λN
where λ is the decay constant.
6.1 Derivation of Decay Equation
dN/N = −λ dtIntegrating from N = N₀ at t = 0 to N at time t:
∫(N₀→N) dN/N = −λ ∫(0→t) dt ln(N/N₀) = −λtTherefore:
N = N₀e−λt
Since activity A = λN:
A = A₀e−λt
Diagram 5: Exponential radioactive decay
7. Half-Life
Half-life, T1/2
The time required for the number of undecayed nuclei—or the activity—to fall to one-half of its initial value.
At t = T1/2:
N = N₀/2 = N₀e−λT₁/₂ 1/2 = e−λT₁/₂Taking natural logarithm:
ln 2 = λT₁/₂
T1/2 = ln2 / λ = 0.693/λ
7.1 Fraction Remaining after n Half-Lives
N/N₀ = (1/2)n
| Number of half-lives | Fraction remaining | Percentage remaining |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
Diagram 6: Fraction remaining after successive half-lives
8. Decay Constant and Mean Life
8.1 Decay Constant
Decay constant, λ
The probability per unit time that a given radioactive nucleus will decay. Equivalently, A/N = λ.
λ = A/N
Unit of λ = s⁻¹
8.2 Mean Life
Mean life, τ
The average lifetime of nuclei in a radioactive sample.
τ = 1/λ
8.3 Relation between Mean Life and Half-Life
T₁/₂ = 0.693/λ and τ = 1/λ
T₁/₂ = 0.693τ
τ = T₁/₂ / 0.693 ≈ 1.443T₁/₂
| Quantity | Symbol | Relation |
|---|---|---|
| Decay constant | λ | λ = A/N |
| Half-life | T₁/₂ | 0.693/λ |
| Mean life | τ | 1/λ |
| Mean life / half-life | τ/T₁/₂ | 1/0.693 ≈ 1.443 |
9. Activity of a Radioactive Sample
Activity
The number of nuclear disintegrations occurring per unit time.
A = −dN/dt = λN
| Unit | Symbol | Relation |
|---|---|---|
| Becquerel | Bq | 1 Bq = 1 disintegration per second |
| Curie | Ci | 1 Ci = 3.7 × 10¹⁰ Bq |
| Rutherford | Rd | 1 Rd = 10⁶ Bq |
Activity also decays exponentially
Because A = λN and λ is constant for a nuclide, activity follows the same exponential law as the number of undecayed nuclei.
10. Geiger–Müller Tube
Geiger–Müller counter
An instrument used to detect and count ionizing-radiation events by collecting electrical pulses produced through gas ionization in a Geiger–Müller tube.
10.1 Construction
- Cylindrical metal tube acts as cathode.
- Fine axial wire acts as anode.
- Tube contains low-pressure gas and a quenching component.
- High potential difference is applied between anode and cathode.
- A thin window may be provided for radiation that cannot penetrate the tube wall effectively.
- Output pulses are sent to counting electronics.
10.2 Working Principle
- Ionizing radiation enters the tube and ionizes gas atoms/molecules.
- Electrons accelerate toward the positive anode.
- Accelerated electrons create further ionization, producing an avalanche.
- A large electrical pulse is generated.
- The pulse is counted as one detected event.
- Quenching stops the discharge so the tube can detect the next event.
Diagram 7: Construction and working principle of a Geiger–Müller tube
11. Radiocarbon Dating
Radiocarbon dating
A method for estimating the age of once-living carbon-containing material by measuring the remaining amount or activity of carbon-14.
11.1 Formation of Carbon-14
¹⁴₇N + ¹₀n → ¹⁴₆C + ¹₁HCarbon-14 enters atmospheric carbon dioxide and then enters plants and animals through the carbon cycle.
11.2 After Death
When an organism dies, carbon exchange with the environment stops. The existing C-14 continues to decay:
¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄The half-life of C-14 is approximately 5730 years.
t = (1/λ) ln(N₀/N)
or t = (1/λ) ln(A₀/A)
Exam correction
Radiocarbon dating is mainly for once-living organic material. It should not be described as a universal method for determining the age of every rock or geological object.
Diagram 8: Principle of radiocarbon dating
12. Uses of Nuclear Radiation and Radioisotopes
12.1 Medical Uses
| Application | Purpose |
|---|---|
| Diagnostic tracers | Follow the distribution or function of substances/organs using suitable short-lived radioisotopes |
| Nuclear imaging | Detect emitted radiation to form functional images |
| Radiotherapy | Deliver controlled ionizing radiation to destroy or damage malignant cells |
| Thyroid applications | Radioiodine can be used in diagnosis or treatment under specialist control |
| Sterilization | Ionizing radiation can sterilize selected medical products |
12.2 Industrial and Other Uses
- Industrial radiography to inspect internal defects.
- Thickness and level gauges.
- Tracer techniques to study flow and detect leakage.
- Selected food irradiation and sterilization processes.
- Research using radioactive tracers.
Diagram 9: Common controlled applications of radioisotopes
13. Possible Health Hazards of Nuclear Radiation
Ionizing radiation can remove electrons from atoms and molecules and can damage cells, proteins and DNA. The biological effect depends on radiation type, absorbed dose, dose rate, exposure geometry and the tissue involved.
| Possible effect | Description |
|---|---|
| Cell and tissue damage | High exposure can injure or kill cells |
| DNA damage | Ionization can produce mutations or chromosome damage |
| Cancer risk | Long-term risk can increase with cumulative ionizing-radiation dose |
| Acute radiation syndrome | Possible after sufficiently high whole-body dose |
| Eye damage | High dose can damage sensitive tissues such as the lens |
| Reproductive/developmental effects | Depend on dose, timing and exposed tissue |
| Internal contamination | Inhaled/ingested radionuclides can irradiate tissues from inside the body |
Important nuance
Alpha radiation has weak external penetration, but an alpha-emitting radionuclide can be dangerous if inhaled or ingested because the radiation is then delivered directly to internal tissue.
Diagram 10: General biological pathway of ionizing-radiation harm
14. Radiation Safety and Precautions
The simplest protection principles are:
1. Time
Minimize unnecessary time close to a source.
2. Distance
Increase distance from the source whenever practical.
3. Shielding
Use suitable shielding for the radiation type and energy.
4. Contamination Control
Prevent inhalation, ingestion or spread of radioactive material.
- Use remote handling tools where appropriate.
- Store radioactive sources in approved shielded containers.
- Use dosimeters and radiation monitors in controlled workplaces.
- Follow labeling, access-control and regulatory requirements.
- Never handle unknown radioactive material without trained radiation-safety support.
Diagram 11: Time, distance and shielding
15. Worked Numericals
Example 1: Decay Constant from Half-Life
A nuclide has half-life 8 hours. Find λ.
λ = 0.693/T₁/₂ λ = 0.693/8 = 0.0866 h⁻¹
Example 2: Mean Life
For λ = 2.0 × 10⁻⁴ s⁻¹:
τ = 1/λ = 5.0 × 10³ s
Example 3: Remaining Mass after Half-Lives
A 160 mg radioactive sample has half-life 5 days. Find mass after 15 days.
Number of half-lives = 15/5 = 3.
m = 160(1/2)³ = 20 mg
Example 4: Activity
A sample contains 4.0 × 10¹² undecayed nuclei and λ = 2.5 × 10⁻⁶ s⁻¹.
A = λN A = (2.5 × 10⁻⁶)(4.0 × 10¹²) A = 1.0 × 10⁷ Bq
Example 5: Exponential Decay
N₀ = 8.0 × 10⁶ nuclei, λ = 0.20 day⁻¹, t = 5 days.
N = N₀e⁻λt = 8.0 × 10⁶ e⁻¹ N ≈ 2.94 × 10⁶ nuclei
Example 6: Carbon Dating by Simple Half-Life Method
A sample has 25% of the C-14 activity of a comparable living sample.
25% = 1/4 = (1/2)², so two half-lives have elapsed.
Age ≈ 2 × 5730 = 11,460 years
Example 7: Nuclear Equation
Complete the alpha decay:
²²⁶₈₈Ra → ? + ⁴₂HeA: 226 − 4 = 222; Z: 88 − 2 = 86.
²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He16. Supplementary Legacy Nuclear Physics Background
Curriculum note
The following compact section is included because the Nepal eNotes source page is titled Nuclear Physics. These nucleus/binding-energy/fission/fusion concepts are valuable background, but they should not replace the current Grade 12 Chapter 24 radioactivity requirements listed above.
16.1 Nuclear Radius
Nuclear size is approximately described by:
R = R₀A1/3
where R₀ is approximately 1.2 fm and 1 fm = 10⁻¹⁵ m.
Because nuclear mass is roughly proportional to A while volume is proportional to R³ ∝ A, nuclear density is approximately constant across many nuclei.
16.2 Einstein Mass–Energy Relation
E = mc²
A small change in mass corresponds to a large energy change because c² is very large.
16.3 Mass Defect and Binding Energy
Mass defect, Δm
The difference between the total mass of the separate nucleons and the actual mass of the bound nucleus.
Δm = Zmp + (A − Z)mn − Mnucleus
Binding energy = Δmc²
Binding energy per nucleon is the binding energy divided by A and is a useful indicator of relative nuclear stability.
Diagram 12: Why both fusion of light nuclei and fission of heavy nuclei can release energy
16.4 Nuclear Fission
Fission is the splitting of a heavy nucleus into smaller nuclei with energy release and usually neutron emission.
²³⁵₉₂U + ¹₀n → fission fragments + neutrons + energy16.5 Nuclear Fusion
Fusion is the combination of light nuclei to form a heavier nucleus, releasing energy when the products are more tightly bound.
²₁H + ³₁H → ⁴₂He + ¹₀n + energy17. High-Yield Formula Sheet
| Topic | Formula |
|---|---|
| Radioactive decay law | −dN/dt = λN |
| Number remaining | N = N₀e⁻λt |
| Activity | A = λN |
| Activity with time | A = A₀e⁻λt |
| Half-life | T₁/₂ = 0.693/λ |
| Mean life | τ = 1/λ |
| Relation | τ ≈ 1.443T₁/₂ |
| After n half-lives | N/N₀ = (1/2)ⁿ |
| Age from remaining fraction | t = (1/λ)ln(N₀/N) |
| Age from activity | t = (1/λ)ln(A₀/A) |
| Nuclear radius (supplementary) | R = R₀A¹ᐟ³ |
| Mass-energy (supplementary) | E = mc² |
| Binding energy (supplementary) | BE = Δmc² |
18. Common Exam Mistakes
- Confusing the legacy source-page title “Nuclear Physics” with the current Grade 12 chapter title. The exam-aligned chapter is Chapter 24: Radioactivity and Nuclear Reaction.
- Calling radioactivity an electronic-shell phenomenon. It is a nuclear phenomenon.
- Writing that radioactive decay can be stopped by changing temperature or pressure.
- Forgetting the negative sign in dN/dt = −λN.
- Writing N = N₀e+λt. The exponent must be negative for decay.
- Confusing activity A with mass number A in nuclear notation; use context carefully.
- Writing 1 Bq as one decay per minute. It is one disintegration per second.
- Confusing α, β and γ penetrating power with ionizing power.
- Writing alpha decay without reducing A by 4 and Z by 2.
- Writing beta-minus decay as A decreasing by one. Mass number stays unchanged.
- Writing gamma emission as changing the element. A and Z stay unchanged.
- Using T₁/₂ = λ/0.693 instead of 0.693/λ.
- Writing mean life equal to half-life. Mean life is larger: τ ≈ 1.443T₁/₂.
- Halving the original amount every half-life instead of halving the amount that remains.
- Confusing Geiger–Müller tube with a device that precisely measures radiation energy. It mainly counts ionizing events.
- Forgetting the quenching function in a GM tube.
- Claiming radiocarbon dating determines the age of every rock. It is mainly for once-living carbon-containing material.
- Using C-14 half-life as 573 years instead of about 5730 years.
- Calling alpha radiation harmless because it has low penetration; internal alpha contamination can be dangerous.
- Writing generic “lead apron” as the only safety rule. The core protection principles are time, distance, shielding and contamination control.
19. Important Exam Questions
Very Short / Short Questions
- Define radioactivity.
- Differentiate natural and artificial radioactivity.
- What are parent and daughter nuclei?
- State the nature of α, β and γ radiations.
- Compare the penetrating powers of α, β and γ.
- Compare the ionizing powers of α, β and γ.
- How are α, β and γ deflected in an electric field?
- Write the general equation of alpha decay.
- What happens to A and Z during beta-minus decay?
- What changes occur during gamma emission?
- State the radioactive disintegration law.
- Define decay constant.
- Define half-life.
- Define mean life.
- Write the relation between half-life and decay constant.
- Write the relation between mean life and decay constant.
- Define activity and state its SI unit.
- What is a Geiger–Müller tube?
- State the principle of a GM tube.
- What is the purpose of quenching in a GM tube?
- What is radiocarbon dating?
- State the half-life of carbon-14.
- Give three medical uses of nuclear radiation.
- State four harmful effects of ionizing radiation.
- State the three basic radiation-protection principles.
Long Questions / Derivations
- Explain natural and artificial radioactivity with suitable nuclear equations.
- Compare alpha, beta and gamma radiations in tabular form.
- Derive the law N = N₀e⁻λt.
- Derive the relation T₁/₂ = 0.693/λ.
- Derive τ = 1/λ and obtain the relation between mean life and half-life.
- Describe the construction and working of a Geiger–Müller tube with labelled diagram.
- Explain the principle of radiocarbon dating.
- Describe medical uses and possible health hazards of nuclear radiation.
- Explain radiation-safety measures based on time, distance and shielding.
Numerical Practice
- Calculate λ from a given half-life.
- Calculate half-life from λ.
- Calculate mean life from λ or T₁/₂.
- Find the fraction or mass remaining after a given number of half-lives.
- Use N = N₀e⁻λt to find remaining nuclei.
- Use A = λN to calculate activity.
- Calculate time from initial and final activity.
- Solve simple C-14 age problems using the half-life method.
- Balance alpha- and beta-decay equations.
Supplementary Legacy-Source Questions
- State Einstein’s mass–energy relation.
- Define mass defect and binding energy.
- What is binding energy per nucleon?
- Differentiate nuclear fission and fusion.
Exam Strategy
For the current Grade 12 chapter, prioritize: α/β/γ comparison → decay-law derivation → half-life and mean-life relations → GM tube diagram → applications/hazards → numericals. The source page’s broader “Nuclear Physics” material is secondary to these Chapter 24 requirements.
20. One-Minute Revision
- Current Grade 12 Modern Physics Chapter 24 is Radioactivity and Nuclear Reaction.
- Radioactivity is spontaneous transformation of unstable nuclei.
- Natural radioactivity occurs in naturally unstable nuclides.
- Artificial radioactivity involves producing a radioactive nuclide by nuclear reaction.
- α particle = helium nucleus, charge +2e.
- β⁻ particle = electron emitted during nuclear beta transformation.
- γ ray = high-energy electromagnetic photon.
- General penetration: γ > β > α.
- General ionization: α > β > γ.
- Alpha decay: A → A−4, Z → Z−2.
- Beta-minus decay: A unchanged, Z → Z+1.
- Gamma emission: A and Z unchanged.
- Decay law: −dN/dt = λN.
- Number remaining: N = N₀e⁻λt.
- Activity: A = λN.
- 1 Bq = 1 disintegration per second.
- Half-life: T₁/₂ = 0.693/λ.
- Mean life: τ = 1/λ.
- τ ≈ 1.443T₁/₂.
- After n half-lives, fraction remaining = (1/2)ⁿ.
- GM tube detects ionizing-radiation events through gas ionization and avalanche pulses.
- Quenching stops continuous discharge in a GM tube.
- Carbon-14 half-life ≈ 5730 years.
- Radiocarbon dating is mainly used for once-living organic material.
- Nuclear radiation has diagnostic, therapeutic, industrial and research applications.
- Ionizing radiation can damage cells and DNA.
- Protection: minimize time, maximize distance, use proper shielding and contamination control.
21. Diagram Practice
Students should practice these labelled diagrams:
- Nucleus and nuclide notation.
- Natural vs artificial radioactivity.
- Deflection of α, β and γ in electric field.
- Penetrating power of α, β and γ.
- Exponential radioactive decay graph.
- Successive half-life diagram.
- Geiger–Müller tube.
- Radiocarbon-dating cycle.
- Applications of radioisotopes.
- Radiation biological-effect pathway.
- Time–distance–shielding protection.
- Binding-energy-per-nucleon curve (supplementary legacy source background).
Source handling: The original Nepal eNotes Nuclear Physics PDF remains embedded above using the recovered genuine Google Drive file. The source title reflects an older/legacy organization. The main typed section follows the current NEB/CDC Grade 12 Physics structure, where Chapter 24 is Radioactivity and Nuclear Reaction. Broader nucleus, binding-energy, fission and fusion material is clearly marked as supplementary legacy background rather than presented as the main current Grade 12 syllabus. Where the embedded PDF does not expose searchable page text, the typed section is a syllabus-aligned reconstruction and is not claimed to be a word-for-word transcription.
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