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Identify what is being counted, what is actually detected and which energy account is complete. Those decisions come before substituting numbers.

Structure and evidence

Most alpha particles passing a thin foil are little deflected; rare large deflections support a small concentration of positive charge and mass. A nucleus has Z protons and A - Z neutrons. A neutral atom has Z electrons. Isotopes have the same Z and different neutron counts; ionisation changes electrons without changing the nucleus.

Radiation and detection

Alpha is a helium-4 nucleus, beta-minus an electron from a nuclear process, and gamma an electromagnetic photon. Alpha and beta have opposite electric-force directions; gamma is uncharged. Penetration and ionisation comparisons need an energy and absorber context. Dense internal ionisation can be hazardous even for short-range radiation.

Count rate = recorded count / duration
Net source rate = total rate - background rate

Activity in Bq counts decays per second. Detector rate counts recorded events per second. They are related only through the stated emission and detection conditions. For unequal durations in a stable-rate series, pool counts over the total time; do not give every interval equal weight.

Decay is spontaneous and random for an individual nucleus. A large population has a predictable statistical trend, with fluctuations in finite counts. Repeats do not remove an incorrect background or changing geometry.

Half-life and time dependence

After one, two and three half-lives, the expected parent population, activity or proportional corrected rate is 1/2, 1/4 and 1/8 of its initial value. Halve a background-subtracted rate. The raw total tends towards background rather than zero.

A = λN
N = N₀e^-λt
A = A₀e^-λt
λt_1/2 = ln(2)

For fixed detection conditions, ln(r/r₀) versus t has gradient -λ. Preserve zero and negative corrected records but do not logarithmically fit them as zero. Near-background positive points also need an uncertainty judgement. A fitted half-life from constructed values checks a model calculation; it does not establish experimental agreement.

Applications need two kinds of reasoning

Explain the penetration or ionisation that makes the task possible, then relate half-life to its preparation, measurement or operating interval. A thickness gauge needs partial transmission over the useful range. An externally detected internal tracer needs radiation that escapes tissue. Treatment and sterilisation use damage from ionisation, with suitable penetration and controlled exposure.

Longer half-life is not universally preferable. Physical half-life is distinct from biological clearance. Irradiation is exposure to radiation; contamination is the presence of radioactive material.

Conservation and binding

Balance total nucleon number and charge across all particles. Alpha emission changes the daughter by ΔA = -4 and ΔZ = -2. Beta-minus leaves A unchanged and raises daughter Z by one. Gamma leaves A and Z unchanged while reducing nuclear energy.

For a fixed beta-decay branch, a two-product model would fix the electron energy. Its continuous energy spectrum and a missing momentum vector support an additional emitted particle. Energy and momentum conservation apply to the complete account.

Mass defect = Zm_p + (A - Z)m_n - m_nucleus
Binding energy = Δmc²
Binding energy per nucleon = binding energy / A
Released energy Q = (m_initial - m_final)c²

Keep atomic and nuclear mass conventions consistent and retain digits until after subtraction. Binding energy is the positive energy required for complete separation. A fusion or fission energy account must include every reactant and product, including free neutrons.

The binding-energy-per-nucleon curve rises steeply across light nuclei, with helium-4 (about 7.1 MeV) above its neighbours. It peaks at about 8.8 MeV near A = 56 (iron and nickel) and falls slowly to about 7.6 MeV for uranium. Fusion of light nuclei and fission of heavy nuclei both move towards the peak. The products then have greater total binding energy and less rest mass, and the mass difference is released as energy. Compare total binding energies for a reaction by multiplying each per-nucleon value by its own nucleon count.

Nuclear quantities, symbols and units
Quantity	Symbol	Unit or meaning
Nucleon, proton and neutron numbers	A, Z, N	Dimensionless counts; neutron N = A - Z
Number of specified particles or parent nuclei	N	Dimensionless count
Amount of substance	n	mol
Avogadro constant	N_A	mol^-1
Molar mass	M	kg/mol; g/mol when paired with g
Sample or particle mass	m	kg; u for a specified particle
Atomic, electron, proton and neutron masses	m_a, m_e, m_p, m_n	kg or u; state which entity
Activity	A	Bq = decays per second
Detected count rate	r	counts/s or counts/min
Time and half-life	t, t_1/2	s; convert min consistently
Decay constant	λ	s^-1 for t in s
Energy, including binding energy	E	J, eV or MeV
Energy released per reaction	Q	J or MeV
Vacuum light speed	c	m/s

A denotes a nucleon count in nuclide notation and activity in a decay-rate equation; identify the context and unit. N can denote neutron number A - Z or the remaining parent population. Dimensionless counts can also be labelled n or m under a stated convention; in this chapter n denotes amount in mol and m denotes mass. Read the definition and unit before substituting.

Supplied rounded conversions:
1 u = 1.66 × 10^-27 kg
1 u c² = 931.5 MeV
1 eV = 1.60 × 10^-19 J
1 MeV = 1.60 × 10^-13 J

Before reporting an answer, check the named entity, time unit, background correction, mass convention and whether the requested energy is total or per nucleon.

Practise exam questions

Write your answer first, then tick the mark points it makes.

Exam-style question

Define the binding energy of a nucleus. [2]

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Exam-style question

A helium-4 nucleus has a mass of 4.00151 u. A proton has a mass of 1.00728 u and a neutron 1.00867 u. Taking 1 u as equivalent to 931.5 MeV, calculate the binding energy per nucleon of helium-4. [4]

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Exam-style question

Use the variation of binding energy per nucleon with nucleon number to explain why energy is released both when light nuclei fuse and when a heavy nucleus splits. [3]

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Exam-style question

Sketch a graph of binding energy per nucleon against nucleon number A for A from 1 to 240. Label the values and features an examiner would look for. [4]

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Exam-style question

Data-based question: A radioisotope generator on a space probe [20]

Space probes sent far from the Sun cannot rely on solar panels. Instead they carry a radioisotope generator. It contains plutonium-238 (proton number 94), which decays by alpha emission to uranium-234 with a half-life of 87.7 years. The alpha particles are absorbed inside the fuel and its casing, which become hot. Thermocouples convert part of this heat into electrical energy; the rest is radiated away by cooling fins.

Table 1. Generator data
mass of plutonium-238 at launch	4.0 kg
molar mass of plutonium-238	0.238 kg mol^-1
kinetic energy of each alpha particle	5.59 MeV
efficiency of conversion from heat to electrical energy	6.0%
output voltage	28 V
1 year	3.16 x 10⁷ s

Table 2. Measured electrical power output during the mission
time after launch / years	0	5	10	15	20
power / W	136	122	110	99	89

(a) Write the nuclear equation for the decay of plutonium-238. [2]
(b) Explain why an alpha emitter with a half-life of several decades is chosen for this use. [3]
(c) Show that the decay constant of plutonium-238 is about 2.5 x 10^-10 s^-1. [1]
(d) Calculate the number of plutonium-238 nuclei at launch. [1]
(e) Calculate the activity at launch and the rate at which heat is produced. [3]
(f) State one assumption you made in (e). [1]
(g) Calculate the electrical power at launch and the current supplied at 28 V. [2]
(h) Calculate the electrical power expected after 20 years if only radioactive decay reduced it. [2]
(i) Use Table 2 to evaluate whether radioactive decay alone explains the fall in output. [3]
(j) If the cooling fins failed, all the heat would go into the 10 kg fuel capsule, of average specific heat capacity 500 J kg^-1 K^-1. Calculate the initial rate of rise of its temperature. [2]

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Quick recall

Card 1 of 10. Answer in your head, then check.

Most alpha particles pass through a thin foil with little deflection, while very few are scattered through large angles or backwards. Which inference is supported?

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