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IB DP Physics · Theme E Nuclear and quantum physics · E.3 · ~7 min read

E.3 Radioactive decay

Isotopes and nuclear stability, the strong force, mass defect and binding energy, the binding-energy-per-nucleon curve, alpha, beta and gamma decay, neutrinos, half-life and activity, background radiation, and (HL) the decay constant and exponential decay.

Nuclear stabilityMass defect & binding energyBE per nucleonα, β, γ decayNeutrinosHalf-lifeActivityDecay constant (HL)
RSby Riyaz Sir
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Protons repel each other, but the strong nuclear force (attractive, very short range) holds the nucleus together. Too many protons or neutrons makes a nucleus unstable.
  • A nucleus has less mass than its separate nucleons. This mass defect Δm is the binding energy E = Δmc2. Iron-56 has the highest binding energy per nucleon: the most stable.
  • Unstable nuclei decay randomly and spontaneously by α, β−, β+ or γ emission. Half-life is the time for half the nuclei (or the activity) to go.
  • HL: N = N0e−λt, A = λN and T½ = ln 2/λ; the β energy spectrum as evidence for neutrinos.

By the end you can

  • Use nuclear notation to write balanced decay equations for α, β−, β+ and γ.
  • Describe the strong force and use an N–Z chart to predict decay type.
  • Calculate mass defect and binding energy, and interpret the binding-energy-per-nucleon curve.
  • Compare the properties of α, β and γ radiation and describe background radiation.
  • Use half-life with graphs and calculations.
  • Use the decay constant, N = N0e−λt and A = λN. HL
  • Explain how the β energy spectrum shows that neutrinos exist. HL

1Nuclear stability

ZNN = Zstable nucleitoo many neutrons: β⁻too many protons: β⁺very heavy: α
Stable nuclei lie on a band that bends away from N = Z, because large nuclei need extra neutrons to dilute the proton repulsion.
  • Strong nuclear force: attractive between all nucleons, about 10−15 m range, repulsive at even shorter distances. Light nuclei are stable with N ≈ Z; heavy nuclei need N > Z.

2Mass defect and binding energy

Binding energy — the energy needed to separate a nucleus completely into its individual protons and neutrons (equivalently, the energy released when it forms).

E = mc2 DATA BOOKLET
1 u = 931.5 MeV c−2
Worked example

Find the binding energy per nucleon of helium-4 (nuclear mass 4.001506 u; mp = 1.007276 u, mn = 1.008665 u).

  1. mass of nucleons = 2(1.007276) + 2(1.008665) = 4.031882 u M1
  2. Δm = 4.031882 − 4.001506 = 0.030376 u A1
  3. E = 0.030376 × 931.5 = 28.3 MeV, so 7.07 MeV per nucleon A1
ABE per nucleon / MeV⁴He⁵⁶Fe: most stable246850100150200fusionfission²³⁵U
Moving towards the peak (by fusion from the left or fission from the right) releases energy.
Common mistakeThinking a larger total binding energy means a "heavier = less stable" nucleus. Stability is about binding energy per nucleon.

3Types of decay

TypeWhat leavesChangeExample
alpha α42He nucleusA − 4, Z − 223892U → 23490Th + 42α
beta-minus β−electron + antineutrinon → p: A same, Z + 1146C → 147N + 0−1e + ν̄
beta-plus β+positron + neutrinop → n: A same, Z − 12211Na → 2210Ne + 0+1e + ν
gamma γphotonno change in A or Zafter α or β, the excited nucleus drops to its ground state
  • Balancing: both A and Z must add up on each side. Charge, nucleon number and (with the neutrino) lepton number are conserved.
paperAl, few mmlead, cmαβγreduced
Each radiation stopped by a different thickness of material.
αβγ
ionisingstronglymoderatelyweakly
range in aira few cmabout 1 mvery long (intensity falls with distance)
stopped bypaper or skina few mm of aluminiumreduced by cm of lead
deflected by fieldsyes (positive)yes (opposite way, more)no
  • Background radiation comes from rocks (radon gas is the largest part), cosmic rays, food, medical uses and nuclear industry. Subtract it from every count rate you measure.

4Half-life

Half-life T½ — the time for half of the undecayed nuclei in a sample to decay, or for the activity to halve.

Activity A — the number of decays per second, in becquerels (Bq).

t / T½N or A1T½2T½3T½100%50%25%after each half-life,half of what is leftremains: never zero
Random decay of many nuclei gives a smooth exponential curve.
Worked example

A sample of 80 mg has a half-life of 6.0 hours. How much is left after one day?

  1. 24 h = 4 half-lives M1
  2. 80 → 40 → 20 → 10 → 5.0 mg A1
  • Random: you cannot predict when a given nucleus decays, only the probability. Spontaneous: temperature, pressure and chemistry do not change it.
Exam tipCount rates on graphs include background. Subtract it before reading a half-life off the curve.

5The decay constant HL

N = N0e−λt DATA BOOKLET
λ = decay constant, the probability per second that a nucleus decays
A = λN = λN0e−λt DATA BOOKLET
activity
T½ = ln 2λ DATA BOOKLET
Worked example

A wooden object has 40 % of the carbon-14 activity of living wood (T½ = 5730 years). How old is it?

  1. λ = ln 2/5730 = 1.21 × 10−4 yr−1 M1
  2. 0.40 = e−λt ⇒ t = ln(1/0.40)/λ M1
  3. t = 7.6 × 103 years A1
  • Very long half-lives (uranium, about 109 years) are found from A = λN: count the activity and the number of nuclei (from the mass), then λ = A/N.
β kinetic energynumberEmaxA continuous spread: the missing energy is carried by a neutrino.
If β decay made only two particles, every electron would have the same energy. It doesn't.
  • Neutrino evidence: β particles have a continuous range of energies up to a maximum. Energy and momentum are conserved only if a third, almost undetectable particle (the neutrino, or antineutrino in β−) takes the rest.
Linking questionBinding energy explains why both fission (E.4) and fusion (E.5) release energy. Radioactive decay is also a source of the Earth's internal heat.
Idea for your IA / explorationModel radioactive decay with dice (or a spreadsheet), or measure the "half-life" of beer froth or a discharging capacitor, and compare with the exponential model.

Test yourself — cover the answers first

Quick check

  1. What force holds the nucleus together?
  2. What changes in β− decay?
  3. Which radiation is most ionising?
  4. A count rate falls from 800 to 100 in 30 minutes. What is the half-life?
  5. Which nucleus has the highest binding energy per nucleon?
  6. HL Find λ for a half-life of 8.0 days, in s−1.

Exam-style

  1. Complete: 22688Ra → ? + 42He, and 9038Sr → ? + 0−1e + ν̄.
  2. HL A source has activity 4.0 × 105 Bq and half-life 30 years. How many radioactive nuclei does it contain?
Show answers
  1. the strong nuclear force
  2. a neutron becomes a proton: Z rises by 1, A stays the same
  3. alpha
  4. 3 half-lives, so 10 minutes
  5. iron-56 (or nickel-62)
  6. ln 2/(8.0 × 86 400) = 1.0 × 10−6 s−1
  7. 22286Rn
    9039Y
  8. λ = ln 2/(30 × 3.15 × 107) = 7.3 × 10−10 s−1
    N = A/λ = 4.0 × 105/7.3 × 10−10
    = 5.5 × 1014 nuclei
Practise this chapter
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Next chapterE.4 Fission →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.