Nuclear physics

Revision notes

A nucleus contains protons and neutrons (nucleons). The mass of a bound nucleus is less than the total mass of its separated nucleons; this difference is the mass defect. The corresponding binding energy is E = Δmc². Binding energy per nucleon is useful when comparing nuclear stability.

Radioactive decay is random for an individual nucleus but statistically predictable for a large population. The activity A is proportional to the number N of undecayed nuclei: A = λN. The number remaining follows N = N0\(e^{-\lambda t}\), and half-life is related to decay constant by \(\dfrac{t1}{2}\) = \(\dfrac{\ln 2}{\lambda }\).

In fission, a heavy nucleus splits into smaller nuclei and releases energy, often with neutrons that can sustain a chain reaction. In fusion, light nuclei combine to form a more tightly bound nucleus. Both processes can release energy because the products have greater binding energy per nucleon; conservation of mass-energy accounts for the released energy.

Radioactive decay

Radioactive decay is random for an individual nucleus but statistically predictable for a large population. Activity A is decay rate, measured in becquerels. The number of undecayed nuclei N and activity decrease exponentially.

Half-life is the time for N or A to fall to half its value. The decay constant λ is linked by \(T_{1/2}\) = \(\dfrac{\ln 2}{\lambda }\). The equations N = N0\(e^{-\lambda t}\) and A = A0\(e^{-\lambda t}\) describe ideal exponential decay.

Nuclear equations and decay modes

Nuclear equations conserve nucleon number and charge. Alpha decay reduces nucleon number by 4 and proton number by 2. Beta-minus decay changes a neutron into a proton with electron and antineutrino emission. Beta-plus decay changes a proton into a neutron with emission of a positron and an electron neutrino. In electron capture, a proton captures an inner electron and becomes a neutron, emitting a neutrino. Gamma emission releases a photon from an excited nucleus, with no change in proton or nucleon number.

Individual decay cannot be predicted from the age of one nucleus. The probability per unit time is constant for a given isotope in the simple model.

Mass defect and binding energy

The mass of a bound nucleus is less than the total mass of its separated nucleons. The difference is the mass defect. Binding energy is ΔE = Δmc² and represents the energy required to separate the nucleus completely into free nucleons.

Binding energy per nucleon helps compare nuclear stability. Energy can be released when a nuclear process moves nuclei towards more tightly bound configurations.

Fission

In nuclear fission, a heavy nucleus splits into smaller nuclei, usually after neutron absorption, releasing energy and additional neutrons. These neutrons can sustain a chain reaction. A reactor controls the neutron population while transferring released energy to a thermal system.

Fusion

Fusion combines light nuclei and can release energy because the products have greater binding energy per nucleon. Very high temperatures are required to give nuclei sufficient kinetic energy for close approach, while confinement is needed to sustain useful reaction rates.

Radiation measurements

Measured count rate can include background. Corrected activity or count rate should be used when analysing source decay. Statistical fluctuations are expected because decay is random; uncertainty becomes proportionally smaller when more counts are collected.

Exam technique

Separate activity from number of nuclei, even though they are proportional for one isotope. Use natural logarithms carefully when rearranging exponential decay equations. In binding-energy calculations, keep mass units and c² conversion consistent.

Atomic mass units and binding energy

Nuclear masses are given in unified atomic mass units: 1 u = 1.661 × 10⁻²⁷ kg, equivalent to 931.5 MeV. A mass defect in u multiplied by 931.5 gives the binding energy in MeV; 1 MeV = 1.60 × 10⁻¹³ J.

Binding energy per nucleon rises steeply for light nuclei, peaks at about 8.8 MeV near iron-56, and falls slowly for heavier nuclei. Fusion of light nuclei and fission of heavy nuclei both move towards the peak, so both release energy.

Nuclear size and Rutherford scattering

In Rutherford's alpha-scattering experiment, most alpha particles passed straight through gold foil, but a few were deflected through large angles. This showed that an atom has a tiny, dense, positively charged nucleus.

Nuclear radius depends on nucleon number: R = R₀\(A^{1/3}\), with R₀ ≈ 1.2 fm, so all nuclei have roughly the same density.

Radiation and reactors

Alpha particles are helium nuclei, stopped by paper; beta particles are fast electrons or positrons, stopped by a few millimetres of aluminium; gamma rays are high-energy photons, reduced by thick lead.

Gamma intensity from a point source follows an inverse-square law, I ∝ \(\dfrac{1}{r^{2}}\).

In a thermal fission reactor, a moderator such as water or graphite slows neutrons so they are more likely to cause fission of uranium-235. Control rods (boron or cadmium) absorb neutrons to keep the chain reaction steady. Fuel must exceed a critical mass for a chain reaction to be self-sustaining. A coolant transfers the energy released.