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Nuclear Physics

23. Nuclear Physics

5 min read 9 parts Lesson 1 of 2

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23. Nuclear Physics — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use mass–energy equivalence, mass defect and binding energy to explain fission and fusion, and describe radioactive decay using activity, the decay constant and…

  • Nuclide notation
  • Conservation
  • Mass defect Δm
  • Calculation
  • Binding energy per nucleon
  • Total binding energy

Part 1 of 9Mass–energy equivalence and nuclear equations

  • Equivalencemass and energy are interchangeable: mass can be converted into energy and energy into mass.
  • EquationE = mc2, with E in joules, m in kilograms and c = 3.00 × 108 m s−1.
  • Exampleshydrogen fusing to helium in the Sun, uranium fission in power stations, nuclear weapons and high-energy particle collisions.
  • Nuclide notationAZX holds the nucleon number A and proton number Z.
  • Conservationboth A and Z are conserved in a nuclear reaction, and the sums on each side are equated to find an unknown such as the number of neutrons released.

Part 2 of 9Mass defect and binding energy

1Mass defect Δm3 points
= the difference between the sum of the masses of the separate protons and neutrons and the mass of the nucleus.
  • Δm = Zmp + (A − Z)mn − mnucleus.
  • A bound system has less mass than separated nucleons, so energy is released when a nucleus forms.
  • The term applies to separating all nucleons, not to the mass lost in a radioactive decay.
2Binding energy1 point
= the energy required to separate a nucleus completely into its individual protons and neutrons.
  • It is not energy stored in the nucleus but energy that must be put in to pull the nucleons apart.
3Calculation
— E = Δmc2, with 1 u = 1.661 × 10−27 kg and 1 MeV = 1.6 × 10−13 J.
4Size
— nuclear binding energies are in MeV, far greater than the few eV that bind electrons, so nuclear reactions release much more energy than chemical ones.

Part 3 of 9Binding energy per nucleon and the curve

1Binding energy per nucleon1 point
= the binding energy of a nucleus divided by its number of nucleons.
  • A higher value means a more stable nucleus, because more energy per nucleon is needed to break it apart.
2Graph3 points
— binding energy per nucleon plotted against nucleon number A.
  • At low A there is a steep rise, so light nuclei have low values and can fuse, releasing a large energy per nucleon.
  • The peak is at iron-56 (A = 56), the most stable nucleus.
  • At high A there is a gentle fall, so the heaviest nuclei are less stable and can undergo fission, releasing less energy per nucleon than fusion.
3Anomaly
— helium-4 is unusually stable and lies above the curve, and carbon-12 and oxygen-16 behave like three and four helium nuclei.
4Total binding energy
— binding energy per nucleon × nucleon number.

Part 4 of 9Nuclear fusion

1Fusion
= two light nuclei combine into one heavier nucleus, releasing energy.
2In stars1 point
— four hydrogen nuclei fuse into a helium nucleus, providing fuel for the star to continue burning.
  • The first step is 11H + 11H → 21H + e+ + νe, where a proton changes to a neutron by β+ decay.
3On Earth
— research targets deuterium–tritium fusion: 21H + 31H → 42He + 10n.
4Conditions3 points
— nuclei need very high kinetic energy to get close enough.
  • Electrostatic repulsion between the protons must be overcome.
  • The strong nuclear force acts only over very short distances, so the nuclei must almost touch.
  • This needs extremely high temperatures, such as in a star's core.

Part 5 of 9Nuclear fission

  • Fission a large nucleus such as uranium splits into smaller daughter nuclei, releasing energy.
  • Induced fissionit is started by a neutron striking the nucleus, and extra neutrons are released.
  • Chain reactionreleased neutrons strike other nuclei, causing a cascade that lasts until the fuel is used up or a moderator halts it.
  • Usescontrolled in nuclear power stations and uncontrolled in a nuclear bomb.

Part 6 of 9Why fusion and fission release energy

Fusion (low A)Fission (high A)
Dominant effectAttractive nuclear forces beat proton repulsionProton repulsion becomes large and tends to break the nucleus
ChangeLight nuclei form a more stable nucleusAn unstable nucleus forms more stable nuclei
Energy sourceThe product has a higher binding energy per nucleon, and the small loss of mass is released as energyThe products have a higher binding energy per nucleon, and the difference is released
  • Energy released — (total binding energy of products) − (total binding energy of reactants), or Δmc2 from the mass difference.
  • Both processes move nuclei towards the peak at A 56, where binding energy per nucleon is highest.

Part 7 of 9Random and spontaneous decay

  • Radioactive decay the spontaneous disintegration of an unstable nucleus into a more stable one, with emission of an α, β or γ particle.
  • Spontaneous not affected by temperature, pressure or chemical conditions.
  • Random the time at which a given nucleus decays cannot be predicted, because each nucleus has the same constant probability of decay per unit time.
  • Evidencea Geiger–Müller tube near a source gives irregular, unpredictable counts, though the average over many nuclei is predictable.

Part 8 of 9Activity, decay constant and exponential decay

  • Decay constant λ = the probability that an individual nucleus will decay per unit time, in s−1.
  • Activity A = the number of decays per unit time, in becquerels (1 Bq = 1 decay per second).
    • A = ΔN/Δt = −λN, where N is the number of undecayed nuclei, and the minus sign can be dropped in calculations.
    • A larger λ means a greater activity for the same number of nuclei, and activity depends on the undecayed nuclei remaining.
  • Number of nuclei in a sample — N = (mass ÷ molar mass) × NA.
  • Exponential decay — N = N0e−λt, falling rapidly but never reaching zero.
    • A = A0e−λt since A ∝ N, and C = C0e−λt for count rate.
    • All decay curves start at N0 on the y-axis, and a steeper curve means a larger λ.
  • Mathematics — e ≈ 2.718, and ln is the inverse of ex.

Part 9 of 9Half-life

  • Half-life t½ = the time taken for the number of undecayed nuclei (and the activity) to fall to half.
    • After two half-lives the activity is a quarter of the original, and after three an eighth.
  • Derivation — with N = ½N0 at t = t½, ½ = e−λt½, so ln ½ = −λt½.
  • Result — t½ = ln 2/λ = 0.693/λ.
  • Relationship — half-life and decay constant are inversely proportional, so a short half-life means a large λ and fast decay.

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