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Nuclear physics: Class 12 Physics
Nuclear structure and isotopes, mass defect and binding energy, radioactivity and decay law, half life, nuclear reactions, fission and fusion, radiation detectors, biological effects and uses of radioisotopes.
Key points
The nucleus contains protons and neutrons, together called nucleons. The atomic number Z is the number of protons, the mass number A is the number of nucleons, and the neutron number is N = A − Z. A nuclide is written ᴬ_Z X. Isotopes are nuclei of the same element (same Z) with different numbers of neutrons, for example carbon-12 and carbon-14. The size of a nucleus is about 10⁻¹⁵ m (one femtometre), and its radius is R = R₀A^(1/3), with R₀ ≈ 1.2 × 10⁻¹⁵ m.
Atomic masses are measured in the unified atomic mass unit (u), which is 1/12 of the mass of a carbon-12 atom. 1 u = 1.66 × 10⁻²⁷ kg, and by E = mc², 1 u corresponds to 931.5 MeV. Mass of a proton is 1.00728 u, of a neutron 1.00867 u, and of an electron 0.00055 u.
Mass defect: the mass of a stable nucleus is less than the total mass of its separate nucleons. The difference is Δm = Zmp + Nmn − mnucleus. This missing mass appears as the binding energy, which is the energy needed to break the nucleus into free nucleons: BE = Δm c². With Δm in u, BE = Δm × 931.5 MeV.
The binding energy per nucleon is plotted against mass number. It rises steeply for light nuclei, reaches a maximum of about 8.8 MeV near iron (A ≈ 56), and then falls slowly for heavy nuclei. The most stable nuclei have the highest binding energy per nucleon. This curve explains why energy is released when heavy nuclei split (fission) and when light nuclei join (fusion).
The strong nuclear force holds nucleons together. It is much stronger than the electric repulsion between protons but acts only over a very short range, about 10⁻¹⁵ m. It is the same between proton and proton, neutron and neutron, and proton and neutron.
Radioactivity is the spontaneous emission of radiation from an unstable nucleus. It was discovered by Becquerel. The three kinds of radiation are:
- Alpha (α): a helium nucleus ⁴₂He, charge +2e. It is strongly ionising, has a range of a few centimetres in air and is stopped by paper.
- Beta (β): a fast electron (β⁻) or positron (β⁺). It is moderately ionising and is stopped by a few millimetres of aluminium.
- Gamma (γ): high energy photons with no charge or mass. They are weakly ionising and need thick lead or concrete to be reduced.
In alpha decay the parent nucleus loses 2 protons and 2 neutrons: ᴬ_Z X → ᴬ⁻⁴_Z₋₂ Y + ⁴₂He. In beta minus decay a neutron changes to a proton, so Z rises by 1 and A stays the same, with an antineutrino emitted. In both cases charge and mass number are conserved.
The decay law: the number of nuclei decaying per second (the activity) is proportional to the number N present. So N = N₀e^(−λt), where λ is the decay constant. The half life T½ is the time in which half the nuclei decay: T½ = 0.693/λ. After n half lives the number left is N₀/2ⁿ. The SI unit of activity is the becquerel (Bq), which is one decay per second.
A nuclear reaction changes nuclei by bombarding them with particles. Fission is the splitting of a heavy nucleus such as uranium-235 by a slow neutron into two medium nuclei, with 2 or 3 neutrons and about 200 MeV of energy. The neutrons can start a chain reaction. In a reactor, control rods (cadmium or boron) absorb neutrons and a moderator (heavy water or graphite) slows them. Fusion is the joining of light nuclei, such as hydrogen isotopes forming helium, which releases energy. It needs temperatures of millions of kelvin. It is the source of the Sun's energy.
Detectors and uses: the Geiger Müller counter detects radiation by ionisation of a gas and gives a count of pulses. Radioisotopes are used as tracers in medicine, for treating cancer, for sterilising equipment, for checking thickness, and in carbon-14 dating. Radiation damages living cells, so the exposure should be small, with shielding, distance and short time as the main safety rules.
Study tip: Practise writing balanced decay equations, checking that the top and bottom numbers match on both sides. For binding energy, write the masses in a short table and keep four to five decimal places. Learn the shape of the binding energy per nucleon curve and what it tells about fission and fusion.
Definitions and flashcards
Say the meaning first, then open the card to check.
Nucleon
A proton or a neutron in the nucleus.
Isotopes
Atoms of the same element with the same Z but different mass numbers.
Mass defect
The difference between the mass of the separate nucleons and the mass of the nucleus.
Binding energy
The energy needed to separate a nucleus into its nucleons.
Activity
The number of nuclei decaying per second, in becquerel.
Half life
The time taken for half the radioactive nuclei in a sample to decay.
Decay constant
The probability per unit time that a nucleus decays.
Fission
The splitting of a heavy nucleus into two lighter nuclei with release of energy.
Fusion
The joining of two light nuclei into a heavier one with release of energy.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Neutron number | N = A − Z | Mass number minus atomic number. |
| Nuclear radius | R = R₀A^(1/3) | R₀ = 1.2 × 10⁻¹⁵ m. |
| Mass defect | Δm = Zmp + (A − Z)mn − mnucleus | In atomic mass units. |
| Binding energy | BE = Δm × 931.5 MeV | Δm in u. |
| Decay law | N = N₀e^(−λt) | λ is the decay constant. |
| Half life | T½ = 0.693 / λ | Time for half to decay. |
| Remaining after n half lives | N = N₀ / 2ⁿ | n = t / T½. |
| Activity | A = λN | Decays per second. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
What are isotopes? Give an example. (2 marks)
Isotopes are atoms of the same element that have the same atomic number but different mass numbers because they have different numbers of neutrons. Examples are hydrogen-1, hydrogen-2 and hydrogen-3.
Define mass defect and binding energy. (3 marks)
Mass defect is the amount by which the mass of a nucleus is less than the sum of the masses of its separate nucleons. Binding energy is the energy equivalent of this mass, Δm c², and is the energy needed to split the nucleus into free nucleons.
Compare alpha, beta and gamma rays. (3 marks)
Alpha rays are helium nuclei with charge +2e. They ionise strongly and are stopped by paper. Beta rays are fast electrons with charge −e. They ionise moderately and are stopped by thin aluminium. Gamma rays are photons with no charge. They ionise weakly and are only reduced by thick lead.
Define half life and write its relation with decay constant. (2 marks)
Half life is the time in which half of the radioactive nuclei of a sample decay. T½ = 0.693/λ, where λ is the decay constant.
What is the difference between nuclear fission and fusion? (3 marks)
In fission a heavy nucleus such as uranium-235 splits into two smaller nuclei after capturing a neutron, releasing about 200 MeV and further neutrons. In fusion two light nuclei join to form a heavier one. Fusion needs very high temperature to overcome electric repulsion and powers the Sun. Both release energy because the products have a higher binding energy per nucleon.
Describe the chain reaction and the role of control rods and moderator in a reactor. (5 marks)
When uranium-235 captures a neutron it splits and releases two or three more neutrons. Each of these can split another nucleus, and the number of fissions grows rapidly. This is a chain reaction. The fission neutrons are fast, so a moderator such as graphite or heavy water slows them down, as slow neutrons are more likely to cause fission. Control rods of cadmium or boron absorb extra neutrons. Pushing them in reduces the rate of fission and pulling them out raises it, so that exactly one neutron per fission goes on to cause another fission and the reactor runs at a steady power. The heat is carried away by a coolant to produce steam.
Numericals with worked steps
Find the binding energy of the helium nucleus. Masses: mass of ⁴₂He nucleus = 4.0015 u, proton = 1.00728 u, neutron = 1.00867 u.
- Mass of 2 protons = 2 × 1.00728 = 2.01456 u.
- Mass of 2 neutrons = 2 × 1.00867 = 2.01734 u.
- Total = 4.03190 u.
- Δm = 4.03190 − 4.0015 = 0.0304 u.
- BE = 0.0304 × 931.5 = 28.3 MeV.
Answer: BE ≈ 28.3 MeV (about 7.1 MeV per nucleon)
The half life of a sample is 8 days. What fraction remains after 24 days?
- n = t / T½ = 24 / 8 = 3.
- Fraction = 1 / 2³ = 1/8.
Answer: 1/8 (12.5%) remains
Find the decay constant of a nuclide with half life 10 hours, in s⁻¹.
- T½ = 10 × 3600 = 36 000 s.
- λ = 0.693 / 36 000.
- = 1.925 × 10⁻⁵ s⁻¹.
Answer: λ ≈ 1.9 × 10⁻⁵ s⁻¹
A sample of 4.0 × 10⁶ nuclei has a decay constant of 2.0 × 10⁻³ s⁻¹. Find its activity.
- A = λN.
- A = 2.0 × 10⁻³ × 4.0 × 10⁶.
- = 8.0 × 10³.
Answer: A = 8.0 × 10³ Bq
Common mistakes
- Using the mass of the whole atom for one side and the nuclear mass for the other when finding mass defect. Keep the same basis.
- Saying that after two half lives all the nuclei have decayed. One quarter remains.
- Not conserving both A and Z when writing decay equations.
- Confusing fission and fusion, or saying fusion happens at low temperature.
- Using days or hours for T½ without converting to seconds when the unit asked is s⁻¹.