# E.1 Energy levels and radioactivity

> IB Physics HL · IB Physics HL 2025 Syllabus
> Source: https://www.owlsprep.com/study/ib-physics-hl-u5-e-1-energy-levels-and/

This sub-topic covers discrete quantized energy levels in atomic nuclei, the origin of the three main types of radioactive decay, and how to interpret nuclear decay diagrams. You will learn to calculate gamma photon energies and relate decay processes to energy level transitions.

**Prerequisites:** [Atomic structure and fundamental particles](https://www.owlsprep.com/study/ib-physics-hl-atomic-structure-fundamental-particles/); [Basic properties of ionizing radiation](https://www.owlsprep.com/study/ib-physics-intro-radiation-properties/)

## Learning objectives

- Distinguish between discrete and continuous energy levels in nuclear systems
- Explain the origin of alpha, beta, and gamma decay in terms of nuclear energy levels
- Calculate energy differences between nuclear levels from photon wavelengths
- Interpret nuclear decay diagrams and decay schemes

## Discrete Nuclear Energy Levels

**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, $\Delta E = E_{final} - E_{initial}$.

**Worked example:** A nuclear transition between two levels emits a photon of wavelength $1.2 \times 10^{-12}$ m. Calculate the energy difference between the two levels in MeV.

1. Recall photon energy is $E = \frac{hc}{\lambda}$. For IB calculations, $hc = 1240$ MeV fm, and $1$ fm $= 10^{-15}$ m:
2. $$\lambda = 1.2 \times 10^{-12} \text{ m} = 1200 \text{ fm}$$
3. Substitute into the energy difference formula ($\Delta E = E_{photon}$, 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).

## Alpha Decay and Energy Levels

**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. $$E_\alpha = 4.27 \text{ MeV} - 0.41 \text{ MeV} = 3.86 \text{ MeV}$$

*Calculator:* allowed

## Beta Decay and Neutrino Energy

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.

> **info**
>
> The continuous energy spectrum of beta particles was a long-standing puzzle: energy appeared not to be conserved, leading Wolfgang Pauli to hypothesize the neutrino in 1930, which was detected experimentally 26 years later.

**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. $$E_\nu = Q - E_\beta = 1.71 \text{ MeV} - 0.62 \text{ MeV} = 1.09 \text{ MeV}$$

*Calculator:* allowed

## Gamma Decay and Decay Schemes

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.

**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.

## Common pitfalls

- **Wrong:** Assuming beta particles have discrete energies like alpha particles
  - Why it fails: Only total decay energy is fixed; energy is shared randomly between beta and neutrino, so energies are continuous
  - Correct: Remember beta spectra are continuous (maximum energy = Q), alpha spectra are discrete
- **Wrong:** Forgetting nuclear recoil energy is negligible for gamma calculations
  - Why it fails: Students often unnecessarily adjust the photon energy for nuclear recoil, which is not required for IB exams
  - Correct: Use $\Delta E = \frac{hc}{\lambda}$ directly for gamma energy calculations
- **Wrong:** Adding daughter excitation energy to the Q-value when calculating alpha energy
  - Why it fails: Excitation energy is internal energy of the daughter, not kinetic energy available to decay products
  - Correct: Subtract the excitation energy from the total Q-value to get the alpha particle energy
- **Wrong:** Confusing nuclear energy level and atomic electron energy level differences
  - Why it fails: Students often mix up the energy scale of nuclear vs atomic transitions
  - Correct: Nuclear transitions are MeV-scale (gamma rays), atomic transitions are eV/keV-scale (visible/X-rays)

## Cheatsheet

| Concept | Key Property | Key Formula |
| --- | --- | --- |
| Discrete nuclear levels | Only specific fixed energies allowed | $\Delta E = E_2 - E_1$ |
| Alpha decay | Monoenergetic alpha particles | $E_\alpha \approx Q - \Delta E_{\text{excited}}$ |
| Beta decay | Continuous energy spectrum | $Q = E_{\beta,\text{max}} = E_\beta + E_\nu$ |
| Gamma decay | Photon energy = level difference | $\Delta E = 1240 \text{ MeV fm} / \lambda (\text{fm})$ |

## 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.

- [E.2 Nuclear structure and reactions](https://www.owlsprep.com/study/ib-physics-hl-u5-e-2-nuclear-structure-and/)
- [E.3 Quantum physics: photons and matter waves](https://www.owlsprep.com/study/ib-physics-hl-u5-e-3-quantum-physics-photons/)
- [E.4 The nuclear atom](https://www.owlsprep.com/study/ib-physics-hl-u5-e-4-the-nuclear-atom/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-physics-hl-u5-e-1-energy-levels-and/