E.1 Energy levels and radioactivity

IB Physics HL· Theme E: Nuclear and Quantum Physics, Topic E.1· 15 min read

1. Discrete Nuclear Energy Levels★★☆☆☆⏱ 4 min

📘 Definition

Discrete nuclear energy levels

Nuclei can only exist in specific, quantized energy states, rather than any arbitrary energy. The lowest energy state is the ground state, all higher states are excited states.

Example:

A carbon-14 nucleus in its ground state has a fixed energy, any excited state has a specific, higher fixed energy.

Quantization of nuclear energy levels arises from the wave nature of nucleons confined within the small nuclear radius, similar to how electron energy levels in atoms are quantized. Transitions between levels always involve absorption or emission of energy equal to the difference between levels, $Δ E = E_{f ina l} - E_{ini t ia l}$ .

📐 Worked Example

A nuclear transition between two levels emits a photon of wavelength $1.2 \times 1 0^{- 12}$ m. Calculate the energy difference between the two levels in MeV.

1
Recall photon energy is $E = \frac{h c}{λ}$ . For IB calculations, $h c = 1240$ MeV fm, and $1$ fm $= 1 0^{- 15}$ m:
2
$\lambda = 1.2 \times 10^{-12} \text{ m} = 1200 \text{ fm}$
3
Substitute into the energy difference formula ( $Δ E = E_{p h o t o n}$ , recoil energy is negligible):
4
$\Delta E = \frac{1240 \text{ MeV fm}}{1200 \text{ fm}} \approx 1.03 \text{ MeV}$
5
The energy difference between the levels is approximately 1.0 MeV (2 significant figures).

2. Alpha Decay and Energy Levels★★☆☆☆⏱ 4 min

✓ Calculator OK

📘 Definition

Alpha decay

A spontaneous decay where an unstable parent nucleus emits an alpha particle ( $^{4}$ He nucleus), producing a lighter daughter nucleus. The total energy released is the Q-value of the decay.

Alpha decay occurs when the total energy of the parent is higher than the sum of the energies of the daughter and alpha. Discrete energy levels of the daughter mean emitted alpha particles are always monoenergetic (have one fixed energy) for each transition.

📐 Worked Example

Uranium-238 decays to thorium-234. The total Q-value for decay to the ground state of thorium is 4.27 MeV. What is the energy of the alpha emitted when decaying to an excited state 0.41 MeV above the ground state?

1
The Q-value equals the total kinetic energy available for decay products. Excitation energy of the daughter is internal energy, so we subtract it from the total Q-value:
2
$\text{Available KE} = Q_{\text{ground}} - \Delta E_{\text{excited}}$
3
Substitute values, and note almost all kinetic energy goes to the lighter alpha particle (conservation of momentum):
4
$E_\alpha = 4.27 \text{ MeV} - 0.41 \text{ MeV} = 3.86 \text{ MeV}$

3. Beta Decay and Neutrino Energy★★★☆☆HL only⏱ 4 min

✓ Calculator OK

Beta decay (beta-minus and beta-plus) produces three particles after decay: the daughter nucleus, the beta particle (electron or positron), and a neutrino/antineutrino respectively. This is what causes the key difference between beta decay and alpha/gamma decay.

📘 Definition

Beta energy spectrum

The total decay energy Q is randomly shared between the beta particle and the neutrino, so beta particles have a continuous range of energies from 0 up to a maximum value equal to Q.

📐 Worked Example

A beta-minus decay has a maximum beta energy of 1.71 MeV. What is the total decay energy Q, and what is the antineutrino energy when the beta has 0.62 MeV?

1
The maximum beta energy occurs when the antineutrino carries almost zero energy, so Q equals the maximum beta energy:
2
$Q = E_{\beta, \text{max}} = 1.71 \text{ MeV}$
3
Total energy is conserved, so the sum of beta energy and antineutrino energy equals Q (daughter KE is negligible):
4
$E_\nu = Q - E_\beta = 1.71 \text{ MeV} - 0.62 \text{ MeV} = 1.09 \text{ MeV}$

4. Gamma Decay and Decay Schemes★★★☆☆⏱ 5 min

Gamma emission occurs when a nucleus is left in an excited state after alpha or beta decay, and transitions to a lower energy state by emitting a high-energy photon (gamma ray). The energy of the gamma photon exactly equals the difference between the two nuclear energy levels.

📘 Definition

Decay scheme

A diagram that shows energy levels of parent and daughter nuclides, with arrows representing decay transitions between levels. The vertical axis represents increasing nuclear energy, with the ground state set to 0 MeV.

📐 Worked Example

Cobalt-60 beta decays to an excited state of nickel-60 2.50 MeV above the nickel ground state. Nickel-60 then decays to ground via two sequential transitions: 1.17 MeV and 1.33 MeV. Find the total energy released by the gamma transitions.

1
Set the nickel ground state energy to 0 MeV. The higher excited state after beta decay is 2.50 MeV, and the intermediate excited state between the two transitions is 1.33 MeV.
2
Sum the energies of the two emitted gamma photons:
3
$E_{\text{total}} = 1.17 \text{ MeV} + 1.33 \text{ MeV} = 2.50 \text{ MeV}$
4
This matches the initial excitation energy, as expected from conservation of energy.

5. Common Pitfalls

Wrong move:

Assuming beta particles have discrete energies like alpha particles

Why:

Only total decay energy is fixed; energy is shared randomly between beta and neutrino, so energies are continuous

Correct move:

Remember beta spectra are continuous (maximum energy = Q), alpha spectra are discrete

Wrong move:

Forgetting nuclear recoil energy is negligible for gamma calculations

Why:

Students often unnecessarily adjust the photon energy for nuclear recoil, which is not required for IB exams

Correct move:

Use $Δ E = \frac{h c}{λ}$ directly for gamma energy calculations

Wrong move:

Adding daughter excitation energy to the Q-value when calculating alpha energy

Why:

Excitation energy is internal energy of the daughter, not kinetic energy available to decay products

Correct move:

Subtract the excitation energy from the total Q-value to get the alpha particle energy

Wrong move:

Confusing nuclear energy level and atomic electron energy level differences

Why:

Students often mix up the energy scale of nuclear vs atomic transitions

Correct move:

Nuclear transitions are MeV-scale (gamma rays), atomic transitions are eV/keV-scale (visible/X-rays)

6. Quick Reference Cheatsheet

Concept	Key Property	Key Formula
Discrete nuclear levels	Only specific fixed energies allowed	$Δ E = E_{2} - E_{1}$
Alpha decay	Monoenergetic alpha particles	$E_{α} \approx Q - Δ E_{excited}$
Beta decay	Continuous energy spectrum	$Q = E_{β, max} = E_{β} + E_{ν}$
Gamma decay	Photon energy = level difference	$Δ E = 1240 MeV fm / λ (fm)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 1
Gamma photon energy calculation
2022 · Paper 2
Decay scheme energy analysis
2021 · Paper 1
Beta energy spectrum origin

Going deeper

IB SyllabusIB Physics HL Guide 2025Official syllabus reference for E.1

What's Next

Understanding discrete nuclear energy levels is the foundation for all further topics in nuclear physics, from radioactive decay series to nuclear reactions and binding energy. The concepts of quantized energy levels you learned here also connect back to core quantum physics principles of wave-particle duality and energy quantization that you explored earlier in the course. Next, you will build on this understanding to explore radioactive decay half-lives and decay chains, then move on to nuclear fission and fusion, which rely on the same energy conservation and energy level principles you have mastered here.