Quantum and nuclear physics (A Level)

23. Nuclear Physics

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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

Section 1 of 11Mass–energy equivalence and nuclear equations

Equivalence

Equivalence mass and energy are interchangeable: mass can be converted into energy and energy into mass.

Equation

E = mc², with E in joules, m in kilograms and c = 3.00 × 10⁸ m s⁻¹.
What the symbols mean: E = energy (J) · m or Δm = mass (change) (kg) · c = 3.00 × 10⁸ m s⁻¹

Examples

Examples hydrogen fusing to helium in the Sun, uranium fission in power stations, nuclear weapons and high-energy particle collisions.

Nuclide notation

^A_ZX holds the nucleon number A and proton number Z.

Conservation

Conservation both 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.

Section 2 of 11Mass defect and binding energy

Mass defect Δm

Mass defect Δm the difference between the sum of the masses of the separate protons and neutrons and the mass of the nucleus.

Δm = Zm_p + (A − Z)m_n − m_nucleus.
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.

Binding energy

Binding energy 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.

Calculation

E = Δmc², with 1 u = 1.661 × 10⁻²⁷ kg and 1 MeV = 1.6 × 10⁻¹³ J.

Size

Size 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.

Section 3 of 11Binding energy per nucleon and the curve

1Binding energy per nucleon

Binding energy per nucleon 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.

2Graph

Graph 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

Anomaly 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

Total binding energy binding energy per nucleon × nucleon number.

Section 4 of 11Nuclear fusion

Fusion

Fusion two light nuclei combine into one heavier nucleus, releasing energy.

In stars

In stars four hydrogen nuclei fuse into a helium nucleus, providing fuel for the star to continue burning.
The first step is ¹₁H + ¹₁H → ²₁H + e⁺ + ν_e, where a proton changes to a neutron by β⁺ decay.

On Earth

On Earth research targets deuterium–tritium fusion: ²₁H + ³₁H → ⁴₂He + ¹₀n.

Conditions

Conditions 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.

Section 5 of 11Nuclear fission

Fission

Fission a large nucleus such as uranium splits into smaller daughter nuclei, releasing energy.

Induced fission

Induced fission it is started by a neutron striking the nucleus, and extra neutrons are released.

Chain reaction

Chain reaction released neutrons strike other nuclei, causing a cascade that lasts until the fuel is used up or a moderator halts it.

Uses

Uses controlled in nuclear power stations and uncontrolled in a nuclear bomb.

Section 6 of 11Why fusion and fission release energy

	Fusion (low A)	Fission (high A)
Dominant effect	Attractive nuclear forces beat proton repulsion	Proton repulsion becomes large and tends to break the nucleus
Change	Light nuclei form a more stable nucleus	An unstable nucleus forms more stable nuclei
Energy source	The product has a higher binding energy per nucleon, and the small loss of mass is released as energy	The products have a higher binding energy per nucleon, and the difference is released

1Energy released

− (total binding energy of reactants), or Δmc² from the mass difference.
- (total binding energy of products)

2Both processes move nuclei towards the peak at A

56, where binding energy per nucleon is highest.

Section 7 of 11Random and spontaneous decay

Radioactive 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

Spontaneous not affected by temperature, pressure or chemical conditions.

Random

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.

Evidence

Evidence a Geiger–Müller tube near a source gives irregular, unpredictable counts, though the average over many nuclei is predictable.

Section 8 of 11Activity, decay constant and exponential decay

Activity, decay constant and exponential decay

Decay constant λ

Decay constant λ the probability that an individual nucleus will decay per unit time, in s⁻¹.

Activity A

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 = × N_A.
- (mass ÷ molar mass)

Exponential decay

N = N₀e^−λt, falling rapidly but never reaching zero.
What the symbols mean: N = number of nuclei left after time t · N₀ = number at the start · λ = decay constant (s⁻¹) · t = time (s)
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

Mathematics e ≈ 2.718, and ln is the inverse of e^x.

Section 9 of 11Half-life

Half-life t½

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

Derivation with N = ½N₀ at t = t_½, ½ = e^−λt_½, so ln ½ = −λt_½.

Result

Result t_½ = ln 2/λ = 0.693/λ.
What the symbols mean: t_½ = half-life (s) · λ = decay constant (s⁻¹) · ln 2 ≈ 0.693

Relationship

Relationship half-life and decay constant are inversely proportional, so a short half-life means a large λ and fast decay.

Section 10 of 11Key terms

Term	Meaning
Mass defect	The difference between the total mass of the separate nucleons and the mass of the nucleus.
Binding energy	The energy required to separate a nucleus into its protons and neutrons.
Nuclear fusion	The joining of two light nuclei into a heavier nucleus with release of energy.
Nuclear fission	The splitting of a heavy nucleus into lighter nuclei with release of energy.
Decay constant	The probability of decay per unit time of an individual nucleus.
Activity	The number of nuclei decaying per second, in becquerels.
Half-life	The time for the number of undecayed nuclei to halve.
Spontaneous	Not influenced by external conditions such as temperature or pressure.
Random	The time of decay of an individual nucleus cannot be predicted.

Section 11 of 11Exam technique

Saying binding energy is stored in the nucleus — it is the energy that must be supplied to separate the nucleons.

Confusing mass defect with the mass lost in a decay — mass defect refers to separating all nucleons.

Using the total binding energy to judge stability — use binding energy per nucleon.

Using years for t but seconds for λ — keep the units consistent.

Saying radioactive decay is affected by temperature or pressure — it is spontaneous and random.