Nuclear Reactions: Q-Value and Cross-Section Simple Notes

A nuclear reaction happens when a fast particle — the projectile — strikes a target nucleus and the nucleons rearrange into new nuclei. The first, seen by Rutherford in 1919, fired alpha particles at nitrogen and yielded oxygen-17 plus a proton: 14N(α, p)17O — the first artificial transmutation of an element.

Two numbers describe almost every such reaction. The Q-value says whether it releases energy or swallows it. The cross-section says how likely it is — the effective target size the projectile "sees". Reactor designers, fusion scientists, and isotope makers all live by these two numbers.

Key Concepts

• Q-value: net energy released or absorbed, Q = (mass of reactants − mass of products)c², in MeV.
• Exothermic: Q > 0 — products weigh less; the missing mass becomes kinetic energy.
• Endothermic: Q < 0 — energy must be supplied; impossible below a threshold energy.
• Cross-section (σ): reaction probability as an effective area; unit barn = 10⁻²⁸ m².
• Reaction rate: R = φσN, with particle flux φ and target count N.

Formulas

Q = [m(a) + m(A) − m(b) − m(B)]c², with a the projectile, A the target, and b, B the products. Convert with 1 u = 931.5 MeV/c²; atomic masses may be used on both sides since electron masses cancel.

For Q < 0 with the target at rest, the minimum projectile energy is the threshold energy E_th = −Q(1 + m_a/m_A) (nonrelativistic; ground-state products).

Rates follow R = φσN; with 1 barn = 10⁻²⁴ cm², flux in cm⁻²s⁻¹ times barns times target count gives reactions per second.

Worked Example

Q-value of D–T fusion: ²H + ³H → ⁴He + n.

1. Masses: ²H = 2.014102 u, ³H = 3.016049 u, ⁴He = 4.002603 u, n = 1.008665 u.
2. Reactants: 2.014102 + 3.016049 = 5.030151 u.
3. Products: 4.002603 + 1.008665 = 5.011268 u.
4. Difference: 0.018883 u.
5. Q = 0.018883 × 931.5 ≈ +17.6 MeV — exothermic, which is why fusion is pursued for power.

Common Misconceptions

• "Q > 0 means it happens by itself." Wrong — charged nuclei must climb electrostatic repulsion first, so D–T needs ~100 million K despite its +17.6 MeV.
• "Cross-section is the nucleus's physical size." No, it is a probability: cadmium-113 captures slow neutrons with σ ≈ 20,000 barns, thousands of times its geometric area.
• "Negative Q means impossible." It only means energy must be supplied; above threshold the reaction proceeds normally.

Real-World Applications

• Fission power: uranium-235 has σ ≈ 585 barns for slow neutrons — the number reactor cores are designed around.
• Medicine: ⁹⁸Mo(n,γ)⁹⁹Mo breeds the parent of technetium-99m, the most used imaging isotope.
• Fusion: the D–T cross-section peaks near 64 keV deuteron energy, setting ITER's target plasma conditions.
• Archaeology: ¹⁴N(n,p)¹⁴C from cosmic-ray neutrons underlies radiocarbon dating.

The Q-value is a reaction's energy ledger — lost mass becomes released energy — and the cross-section is its probability dial. With those two numbers you can read any nuclear reaction: what goes in, what comes out, and how likely it is.