Nuclides with the same number of protons but different numbers of neutrons.

Define mass defect and binding energy.

Mass defect: total mass of separate nucleons minus mass of the nucleus. Binding energy: energy needed to separate a nucleus into its nucleons, $E = \Delta m c^2$.

Which nuclide has (about) the highest binding energy per nucleon?

Iron-56 (and nickel-62), about 8.8 MeV per nucleon. Fusion of lighter and fission of heavier nuclei both release energy.

Compare the penetrating power of alpha, beta and gamma radiation.

Alpha: stopped by paper or a few cm of air. Beta: stopped by a few mm of aluminium. Gamma: reduced but not fully stopped by several cm of lead.

What changes in beta-minus decay?

A neutron changes into a proton: $Z$ increases by 1, $A$ unchanged; an electron and an electron antineutrino are emitted.

The time for half of the undecayed nuclei of a nuclide to decay, or for the activity to fall to half.

E.3 Radioactive decay — IB Physics notes and practice questions

Why some nuclei are unstable, how mass defect gives binding energy, how alpha, beta and gamma decays change a nucleus, and how half-life and activity describe decay; at HL, the decay constant, exponential law and nuclear energy levels. Study guide, 7 practice questions with markschemes and 8 flashcards. Free.

isotopes and binding energy · mass defect and binding energy per nucleon · alpha, beta and gamma decay · neutrinos · half-life and activity · background radiation and safety · (HL) decay constant and exponential decay law · (HL) nuclear energy levels and the strong force evidence

Study guide

Binding energy and mass defect

The mass of a nucleus is always less than the total mass of its separate protons and neutrons. This difference is the mass defect $\Delta m$. By $E = mc^2$, it corresponds to the binding energy: the energy needed to pull the nucleus completely apart into free nucleons (or the energy released when it forms).

In atomic mass units, $1\ \text{u}$ is equivalent to $931.5\ \text{MeV}$, which makes the conversion quick.

The binding energy per nucleon measures stability. Plotted against $A$, it rises steeply for light nuclei, peaks near iron-56 at about $8.8\ \text{MeV}$, then falls slowly for heavy nuclei. As a result:

Fusion of light nuclei moves toward the peak and releases energy.
Fission of heavy nuclei also moves toward the peak and releases energy.

Types of decay

Unstable nuclei decay randomly and spontaneously, emitting radiation:

Decay	Particle	Change to nucleus	Ionising	Stopped by
Alpha ($\alpha$)	$^{4}_{2}\text{He}$ nucleus	$A - 4$, $Z - 2$	strongly	paper or a few cm of air
Beta-minus ($\beta^-$)	electron + antineutrino	$Z + 1$, $A$ same	moderately	a few mm of aluminium
Beta-plus ($\beta^+$)	positron + neutrino	$Z - 1$, $A$ same	moderately	a few mm of aluminium
Gamma ($\gamma$)	high-energy photon	no change in $A$ or $Z$	weakly	reduced by thick lead or concrete

In $\beta^-$ decay a neutron turns into a proton; in $\beta^+$ a proton turns into a neutron. In every nuclear equation both $A$ and $Z$ must balance on each side.

Neutrinos

Beta particles from one nuclide come out with a continuous range of kinetic energies, up to a maximum. If only two particles were produced, energy and momentum conservation would fix a single value. The spread showed that a third, nearly massless, uncharged particle shares the energy: the neutrino (or antineutrino). Alpha particles, by contrast, come out with discrete energies.

Half-life and activity

Decay is random: you cannot predict when a given nucleus will decay, only the probability. With huge numbers of nuclei, the pattern is smooth.

Activity $A$ is the number of decays per second, measured in becquerels (Bq).
Half-life $T_{1/2}$ is the time for the number of undecayed nuclei, or the activity, to halve.

After $n$ half-lives the fraction remaining is $\left(\frac{1}{2}\right)^n$.

Background radiation and safety

Detectors always record a background count from rocks, radon gas, cosmic rays, food and medical sources. Measure it first and subtract it from every reading. Protect yourself by keeping exposure short, distance large and using shielding.

Decay constant and exponential decay (HL)

The decay constant $\lambda$ is the probability per unit time that a nucleus decays. Then:

\[N = N_0 e^{-\lambda t}, \quad A = \lambda N = A_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda}\]

A graph of $\ln A$ against $t$ is a straight line with gradient $-\lambda$.

Nuclear energy levels (HL)

Alpha particles and gamma photons from a given nuclide have discrete energies. This shows that nuclei, like atoms, have quantised energy levels. Gamma emission often follows alpha or beta decay as the daughter nucleus drops from an excited state to its ground state. The very short range of the strong force is supported by the constant nuclear density and the deviations in scattering experiments.

Worked example 1: binding energy of helium-4

Data: proton $1.007276\ \text{u}$, neutron $1.008665\ \text{u}$, helium-4 nucleus $4.001506\ \text{u}$.

Separate nucleons: $2(1.007276) + 2(1.008665) = 4.031882\ \text{u}$
Mass defect: $4.031882 - 4.001506 = 0.030376\ \text{u}$
Binding energy: $0.030376 \times 931.5 = 28.3\ \text{MeV}$
Per nucleon: $\frac{28.3}{4} = 7.07\ \text{MeV}$

Worked example 2: activity after a non-whole number of half-lives (HL)

A source has activity $800\ \text{Bq}$ and half-life $6.0\ \text{h}$. Find its activity after $15\ \text{h}$.

$15\ \text{h}$ is $2.5$ half-lives.
$A = 800 \times \left(\frac{1}{2}\right)^{2.5} = 800 \times 0.177 = 1.4 \times 10^2\ \text{Bq}$ (2 s.f.)

Common mistakes

Saying binding energy is energy "stored" in the nucleus that it gives out; it is the energy needed to separate it.
Forgetting the antineutrino in $\beta^-$ equations or the neutrino in $\beta^+$.
Not subtracting background before calculating half-life.
Thinking a sample is gone after two half-lives; a quarter remains.
Using hours in $\lambda$ but seconds in activity calculations.

Exam tips

Balance top and bottom numbers explicitly when completing a decay equation.
When reading half-life from a graph, use several halvings and average them.
Use the $\left(\frac{1}{2}\right)^n$ shortcut when $n$ is a whole number; use $e^{-\lambda t}$ otherwise.
Keep all six decimal places in mass-defect sums; rounding early can wipe out the answer.

Cheat sheet

Equation	Symbols	Units
$\Delta m = Zm_p + (A-Z)m_n - m_{\text{nucleus}}$	$m_p$, $m_n$ proton and neutron masses	u or kg
$E = \Delta m c^2$	$c = 3.00 \times 10^8$ m s$^{-1}$	J
$1\ \text{u} = 931.5\ \text{MeV}\,c^{-2}$	atomic mass unit	MeV
$N = N_0\left(\frac{1}{2}\right)^{n}$	$n = t/T_{1/2}$ half-lives elapsed	none
(HL) $N = N_0 e^{-\lambda t}$	$\lambda$ decay constant	none
(HL) $A = \lambda N = A_0 e^{-\lambda t}$	$A$ activity	Bq
(HL) $T_{1/2} = \frac{\ln 2}{\lambda}$	$T_{1/2}$ half-life	s

$\alpha$: $A-4$, $Z-2$. $\beta^-$: $Z+1$ plus antineutrino. $\beta^+$: $Z-1$ plus neutrino. $\gamma$: no change.
Binding energy per nucleon peaks near Fe-56; fission and fusion both move toward it.

Key definitions

Practice questions

SLPaper 1AStandard

Question 1

(a) [1]

Carbon-14, $^{14}_{6}\text{C}$, undergoes beta-minus decay. Which are the products?

A. $^{14}_{7}\text{N}$, an electron and an antineutrino
B. $^{14}_{5}\text{B}$, a positron and a neutrino
C. $^{10}_{4}\text{Be}$ and an alpha particle
D. $^{14}_{7}\text{N}$, an electron and a neutrino

Markscheme

1 A. In beta-minus decay a neutron becomes a proton, so $Z$ rises by 1 and $A$ is unchanged; an electron and an electron antineutrino are emitted.

SLPaper 1AStandard

Question 2

(a) [1]

The activity of a radioactive sample falls from $800\ \text{Bq}$ to $50\ \text{Bq}$ in $24\ \text{h}$. What is the half-life?

A. 3 h
B. 6 h
C. 8 h
D. 12 h

Markscheme

1 B. $800 \to 400 \to 200 \to 100 \to 50$ is 4 half-lives in 24 h, so each is 6 h.

SLPaper 1AStandard

Question 3

(a) [1]

The kinetic energy of electrons emitted in beta-minus decay of a given nuclide varies continuously from zero to a maximum value. What does this suggest?

A. Energy is not conserved in beta decay
B. A third particle carries away part of the energy released
C. The nucleus has no discrete energy levels
D. The electrons lose energy in the source material

Markscheme

1 B. The decay energy is fixed, so a continuous electron spectrum means the energy is shared with another particle: the antineutrino.

Equation	Symbols	Units
\(\Delta m = Zm_p + (A-Z)m_n - m_{\text{nucleus}}\)	\(m_p\), \(m_n\) proton and neutron masses	u or kg
\(E = \Delta m c^2\)	\(c = 3.00 \times 10^8\) m s\(^{-1}\)	J
\(1\ \text{u} = 931.5\ \text{MeV}\,c^{-2}\)	atomic mass unit	MeV
\(N = N_0\left(\frac{1}{2}\right)^{n}\)	\(n = t/T_{1/2}\) half-lives elapsed	none
(HL) \(N = N_0 e^{-\lambda t}\)	\(\lambda\) decay constant	none
(HL) \(A = \lambda N = A_0 e^{-\lambda t}\)	\(A\) activity	Bq
(HL) \(T_{1/2} = \frac{\ln 2}{\lambda}\)	\(T_{1/2}\) half-life	s

Decay	Particle	Change to nucleus	Ionising	Stopped by
Alpha (\(\alpha\))	\(^{4}_{2}\text{He}\) nucleus	\(A - 4\), \(Z - 2\)	strongly	paper or a few cm of air
Beta-minus (\(\beta^-\))	electron + antineutrino	\(Z + 1\), \(A\) same	moderately	a few mm of aluminium
Beta-plus (\(\beta^+\))	positron + neutrino	\(Z - 1\), \(A\) same	moderately	a few mm of aluminium
Gamma (\(\gamma\))	high-energy photon	no change in \(A\) or \(Z\)	weakly	reduced by thick lead or concrete

E.3 Radioactive decay

Study guide

Binding energy and mass defect

Types of decay

Neutrinos

Half-life and activity

Background radiation and safety

Decay constant and exponential decay (HL)

Nuclear energy levels (HL)

Worked example 1: binding energy of helium-4

Worked example 2: activity after a non-whole number of half-lives (HL)

Common mistakes

Exam tips

Cheat sheet

Key definitions

Practice questions

Question 1

Question 2

Question 3

Key facts