IB Physics HL Topic 7 — Atomic, Nuclear & Particle Physics Paper 1 & 2 discrete γ energies ~15 min read

Nuclear Energy Levels

You already know electrons sit in discrete energy levels — well, so does the nucleus. After a decay, a daughter nucleus is often left “buzzing” with excess energy, in an excited state. It quickly drops back down to its lowest-energy ground state, and each drop releases a gamma photon of a very precise energy. Because the levels are discrete, the emitted gamma rays come in sharp, specific energies — a nuclear fingerprint you can read straight off an energy-level diagram.

📚 What you need to know

A nucleus can exist in an excited state with extra energy, just like an electron
An excited nucleus drops to its ground state and emits a gamma photon
Gamma emission changes neither the proton number nor the nucleon number — only energy is lost
Nuclear energy levels are discrete, so emitted gamma rays have specific, exact energies
The energy of the photon equals the gap between levels: E = hf = hc/λ
Excited states are usually very short-lived
A “metastable” state (like technetium-99m) is an unusually long-lived excited state

Excited and ground states

When a nucleus decays — say by beta emission — the daughter nucleus is often left with more energy than it can hold in its lowest-energy configuration. It’s in an excited state. It doesn’t stay there long: it drops down to its ground state, either in one jump or via a few steps, releasing the surplus energy as a gamma photon each time.

Crucially, emitting gamma doesn’t change what the nucleus is — the proton and nucleon numbers stay put. Only energy is released:

Gamma emission from an excited nucleus ^A_ZX^* → ^A_ZX + γ the * marks the excited state — same Z and A, just less energy afterwards

The parallel with electrons is exact and worth leaning on. An excited electron drops between shells and emits a photon of light; an excited nucleus drops between nuclear levels and emits a photon of gamma. Same idea, just a million times more energy. If you can picture electron energy-level jumps, you already understand nuclear energy levels.

Discrete levels mean discrete gammas

Because the nuclear energy levels are discrete (only certain values are allowed), the gaps between them are fixed. So when a nucleus drops between two levels, the gamma photon carries an exact energy equal to that gap. This is why gamma spectra show sharp, specific lines rather than a smooth spread.

Photon energy = the energy gap E = hf = hc / λ E = energy gap (J) • h = Planck’s constant • λ = wavelength of the gamma photon

A nucleus drops between discrete levels, emitting a gamma photon each time. The photon’s energy equals the gap, so a bigger drop (γ₃) gives a higher-energy, shorter-wavelength gamma.

Metastable states and technetium-99m

Most excited states last only a tiny fraction of a second. But some, called metastable states, are unusually long-lived. The famous example is technetium-99m (the “m” means metastable), formed when molybdenum-99 undergoes beta decay. It sits in its excited state long enough to be useful, then releases a single, clean gamma photon — which is exactly why it’s a superb medical imaging tracer.

Formation and decay of technetium-99m ⁹⁹₄₂Mo → ^99m₄₃Tc + ⁰₋₁β + v̄_e ^99m₄₃Tc → ⁹⁹₄₃Tc + γ

WE 1

Iron-59 beta-decays into cobalt-59, which can land in one of three excited states, at 2.29, 2.06 and 1.76 (× 10⁻¹³ J) above the ground state. The total energy released is 2.52 × 10⁻¹³ J. (a) Find the maximum kinetic energy of the beta particle, in MeV. (b) State how many discrete gamma wavelengths could be emitted. (1 MeV = 1.6 × 10⁻¹³ J)

(a) Step 1 — the beta gets most energy for the biggest drop Max beta energy comes from decaying to the lowest excited state (1.76). ΔE = (2.52 − 1.76) × 10⁻¹³ = 7.6 × 10⁻¹⁴ J Step 2 — convert to MeV = 7.6×10⁻¹⁴ ÷ 1.6×10⁻¹³ = 0.475 MeV max beta energy = 0.48 MeV (b) counting gamma transitions Between the 3 excited states and the ground state (4 levels), the number of possible drops is 6 discrete wavelengths Max beta energy = total energy minus the smallest excited-state energy (biggest beta share). For gammas, count all possible downward transitions between the four levels: that’s 6.

WE 2

For the cobalt-59 levels above, the longest-wavelength gamma comes from the smallest energy gap, between the 2.29 and 2.06 (× 10⁻¹³ J) levels. Calculate that longest wavelength. (h = 6.63 × 10⁻³⁴ J s, c = 3.0 × 10⁸ m s⁻¹)

Step 1 — smallest energy gap ΔE = (2.29 − 2.06) × 10⁻¹³ = 2.3 × 10⁻¹⁴ J Step 2 — wavelength from E = hc/λ λ = hc/E = (6.63×10⁻³⁴ × 3.0×10⁸) ÷ 2.3×10⁻¹⁴ λ = 8.6 × 10⁻¹² m Longest wavelength = smallest energy gap (since λ and E are inversely related). Rearrange E = hc/λ for λ. Tiny wavelengths like this are typical of high-energy gamma photons.

⚛ Working a nuclear energy level question

Gamma energy? It equals the gap between the two levels.
Wavelength? Use E = hc/λ → λ = hc/E.
Longest wavelength? Comes from the smallest gap.
Max beta energy? Total energy minus the smallest excited-state energy.
Count gammas? Count every possible downward transition between levels.

💡 Top tips

Gamma emission leaves Z and A unchanged — only energy is lost.
Photon energy = the gap between the two levels.
Smallest gap → longest wavelength, lowest energy.
Nuclear levels are discrete, so gammas have exact energies.
Metastable states (like Tc-99m) are unusually long-lived.

⚠ Common mistakes

Thinking gamma emission changes Z or A — it doesn’t
Pairing the largest gap with the longest wavelength — it’s the smallest
Forgetting to convert J to MeV (or vice versa)
Using E = hf when given a wavelength — switch to hc/λ
Miscounting the possible transitions between levels

Quick recap: Nuclei have discrete energy levels, just like electrons. An excited nucleus drops to its ground state and emits a gamma photon whose energy equals the gap: E = hf = hc/λ. Gamma emission leaves Z and A unchanged. Because levels are discrete, gammas have exact energies — and metastable states (like Tc-99m) are unusually long-lived.

Discrete gamma energies come from the nucleus jumping between fixed levels — clean, sharp lines. But when scientists measured beta energies, they found something odd: a whole range of energies, not sharp lines. That puzzle could only be solved by inventing a brand-new, almost-invisible particle. Next page: Evidence for the Neutrino.