Nuclear physics and radioactivity

Almost all of an atom’s mass is concentrated in a nucleus about ~10⁻¹⁵ m across. Nuclear physics studies the composition of the nucleus, the forces that hold it together, radioactive decay and nuclear reactions. Nuclear power stations, medical radioisotopes, radiocarbon dating and the Sun’s radiation all belong to this topic.

§1. Composition of the nucleus and nuclear forces

A nucleus is made of nucleons — protons (p, charge +e) and neutrons (n, uncharged). Notation ᴬ_Z X:

• Z — number of protons (atomic number, position in the periodic table);
• A — number of nucleons (mass number);
• number of neutrons N = A − Z.

Isotopes are nuclei with the same Z but different A (e.g. hydrogen: protium ¹H, deuterium ²H, tritium ³H). Their chemistry is the same, their nuclear properties differ.

Protons repel each other, yet the nucleus is stable — it is held by nuclear (strong) forces. They are very strong (about 100 times stronger than electric forces), short-range (~10⁻¹⁵ m) and charge-independent (p–p, p–n, n–n are alike).

Worked example

How many protons and neutrons are in ²⁰⁹₈₄Po?

1. Protons: Z = 84.
2. Neutrons: N = 209 − 84 = 125.

§2. Mass defect and binding energy

The mass of a nucleus is less than the sum of the masses of its free nucleons. The difference is the mass defect:

The binding energy per nucleon E_b/A is largest near iron (~8.8 MeV). That is why both the fission of heavy nuclei and the fusion of light ones release energy.

Worked example

Binding energy of ⁴₂He (m_p = 1.00728, m_n = 1.00866, M = 4.00151 u)?

1. Δm = 2·1.00728 + 2·1.00866 − 4.00151 = 4.03188 − 4.00151 = 0.03037 u.
2. E = 0.03037·931.5 ≈ 28.3 MeV.
3. Per nucleon: 28.3/4 ≈ 7.1 MeV.

§3. Radioactivity

Radioactivity is the spontaneous transformation of unstable nuclei into others with the emission of radiation (Becquerel, 1896; Pierre and Marie Curie).

• α-decay — a helium nucleus ⁴₂He is emitted: A − 4, Z − 2. Stopped by a sheet of paper.
• β⁻-decay — a neutron in the nucleus turns into a proton, and an electron (plus an antineutrino) is emitted: A unchanged, Z + 1. Stopped by a few mm of aluminium.
• γ-radiation — short-wavelength electromagnetic radiation; the composition does not change. The most penetrating; thick lead or concrete is needed.

Worked example

²³⁸₉₂U turns into ²⁰⁶₈₂Pb through a chain of decays. How many α and β decays occurred?

1. Only α-decay changes A: (238 − 206)/4 = 8 α-decays.
2. After 8 α-decays Z = 92 − 16 = 76, but 82 is needed: 82 − 76 = 6 β⁻-decays.

§4. The law of radioactive decay

When a single nucleus decays is random, but for many nuclei the pattern is exact. The half-life T is the time for half of the nuclei to decay.

• After T, 1/2 remains; after 2T, 1/4; after 3T, 1/8.
• The number decayed is N₀ − N.
• T does not depend on external conditions (temperature, pressure).

Worked example

T = 10 days. What fraction of nuclei remains after 30 days, and what fraction decays?

1. t/T = 30/10 = 3.
2. Remaining: 2⁻³ = 1/8; decayed: 1 − 1/8 = 7/8 (87.5 %).

§5. Nuclear reactions and nuclear energy

In nuclear reactions both charge number and mass number are conserved. The first artificial reaction (Rutherford, 1919): ¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁H. Energy released: Q = (Σm_before − Σm_after)·c².

Fission: a ²³⁵U nucleus absorbs a neutron and splits into two fragments, releasing ~200 MeV and 2–3 neutrons. These cause further fissions — a chain reaction. With multiplication factor k = 1 the reaction is controlled (a power reactor: fuel, moderator, control rods); with k > 1 it is explosive. A chain reaction needs a critical mass.

Fusion (thermonuclear reaction): ²₁H + ³₁H → ⁴₂He + ¹₀n + 17.6 MeV. It requires extremely high temperatures (~10⁷ K); it powers the Sun and stars.

Worked example

²⁷₁₃Al + ¹₀n → ²⁴₁₁Na + X. What is particle X?

1. Mass number: 27 + 1 = 24 + A → A = 4.
2. Charge: 13 + 0 = 11 + Z → Z = 2. So X is an α-particle (⁴₂He).

§6. Radiation dose and the energy of nuclear fuel

Ionising radiation damages molecules in living tissue, so its effect must be measured. The absorbed dose is the radiation energy absorbed per kilogram of material:

For the same absorbed dose, different radiations do different harm: α-particles create a great many ions over a short path in tissue. To allow for this the equivalent dose is introduced:

Natural background radiation (cosmic rays, radon in soil and buildings) gives a person roughly 2–3 mSv a year. The three principles of protection are: stay further from the source, spend less time exposed, and use shielding (paper for α, aluminium for β, lead or concrete for γ). Note that an α-source outside the body is almost harmless, but very dangerous if breathed in or swallowed.

The figures in §5 show why nuclear fuel is so effective: one fission releases about 200 MeV, while burning releases a few eV per molecule — a difference of tens of millions of times.

Worked example

A 70 kg person absorbs 0.14 J of γ-radiation energy. Absorbed and equivalent dose?

1. D = 0.14/70 = 0.002 Gy = 2 mGy.
2. For γ, w = 1: H = 2 mSv — roughly one year of natural background.

Worked example

How much energy is released if 1 g of uranium-235 undergoes complete fission, and how much coal (q = 29 MJ/kg) is that equivalent to?

1. Number of nuclei: N = 6.02·10²³/235 ≈ 2.56·10²¹.
2. Energy: E = 2.56·10²¹·200·1.6·10⁻¹³ J ≈ 8.2·10¹⁰ J.
3. Coal: m = 8.2·10¹⁰/2.9·10⁷ ≈ 2.8·10³ kg — almost 3 tonnes!

Check yourself

1. How many neutrons are in ²³⁵₉₂U?

2. ²²⁶₈₈Ra undergoes α-decay. What nucleus forms?

3. ¹⁴₆C undergoes β⁻-decay. Product?

4. For carbon-14 T = 5730 years. A sample keeps 1/16 of its original amount. Its age?

5. The mass defect is 0.1 u. Binding energy?

6. ⁹₄Be + ⁴₂He → ¹²₆C + X. What is X?

7. Tissue receives an absorbed dose of 1 mGy from α-radiation. Equivalent dose?

8. How many joules is the 200 MeV released in one fission?

Common mistakes

• Thinking the nucleus contains electrons: the β-electron is created from a neutron at the moment of decay.
• Treating half-life as linear (after 2 half-lives not everything has decayed — 1/4 remains).
• Confusing the fraction "remaining" with the fraction "decayed".
• Using the atomic mass (with electrons) instead of the nuclear mass in the mass defect without accounting for the electrons.
• Not checking the balance of charge and mass numbers in a reaction equation.
• Treating absorbed dose (Gy) and equivalent dose (Sv) as the same: for α-radiation the equivalent dose is 20 times larger.