Neutron Capture and Nuclear Cross-Sections

Introduction

A neutron can approach a nucleus without the electrostatic repulsion faced by a positively charged projectile. That makes neutron-induced reactions important in reactors, isotope production and analytical chemistry. Yet not every neutron is captured. The capture probability depends on the target isotope and neutron energy and is summarized by an energy-dependent nuclear cross-section, not by the target's geometric size alone.

Core explanation

In a radiative capture reaction A(n,γ)B , the target absorbs a neutron and the excited product emits gamma radiation as it de-excites. The product has one greater mass number and the same proton number. For example, stable ⁵⁹Co can capture a neutron to form radioactive ⁶⁰Co. A written equation identifies a possible channel but says nothing by itself about how likely a neutron is to interact. Competing scattering and reaction channels may also occur.

A cross-section σ has units of area and acts as an effective probability measure for a specified interaction. One barn equals 10⁻²⁴ cm² = 10⁻²⁸ m². The imagery of a target area is useful, but σ need not equal the literal geometric area of a nucleus: quantum resonance effects can make it much larger or smaller. One must state the incident neutron energy and the reaction channel, such as capture rather than total interaction. The IAEA neutron-capture atlas contains energy-dependent (n,γ) evaluations across numerous target isotopes, illustrating that no one cross-section value covers all energies.

Thermal neutrons have been slowed by collisions in a moderator until their energy distribution reflects the surrounding thermal environment; near room temperature, a conventional reference energy is about 0.025 eV. Many capture cross-sections are large at low energies, but this is not a universal 1/v rule over every energy range or nuclide. Nuclear resonances can produce sharp energy dependence, and some reactions have thresholds. IAEA's discussion of capture data illustrates how a particular carbon-13 capture cross-section changes with thermal energy. A value quoted simply as “one barn” without the isotope and neutron spectrum is inadequate for production planning.

For a thin, weakly absorbing target in approximately uniform flux Φ, the reaction rate is R ≈ ΦσN, where Φ is neutrons per area per time, σ is area and N is number of target atoms. Units cancel to reactions per time. If neutrons are spread across energies, use an appropriate spectrum-weighted cross-section or integrate Φ(E)σ(E) over energy. If a thick target significantly attenuates the beam, flux varies through it and the simple product with one uniform Φ may overestimate reactions.

When the capture product is radioactive, its inventory N grows as dN /dt = R − λN if R is approximately constant and further reactions are negligible. Starting with no product, N (t) = (R/λ)(1 − e^(−λt)), so product activity A (t) = λN (t) = R(1 − e^(−λt)). The saturation activity approaches R for a constant irradiation source. That does not mean all target atoms are converted; it means newly produced radioactive atoms are, on average, decaying as fast as new ones are made. After irradiation stops, product activity falls according to its own half-life.

Neutron activation analysis uses this principle to identify or quantify elements. A sample is irradiated; capture creates characteristic radionuclides whose gamma emissions and half-lives can be measured. The observed gamma count rate depends on product activity, gamma emission probability, detector efficiency and irradiation/cooling/counting times. Interfering reactions and isotopic abundances need attention. The method is analytical because nuclear identity and decay signatures can distinguish elements, not because every neutron is captured.

Cross-section and half-life describe different processes. Cross-section tells how readily an incident neutron induces a reaction under the specified energy conditions; half-life tells how quickly the product later decays. A product can form efficiently but decay slowly, or form rarely and decay quickly. Neutron flux sets the attempted interaction rate, while target inventory and σ govern production. Keeping these factors separate prevents the common error of interpreting a large σ as a short product half-life.

Step-by-step reasoning

Write the capture reaction and identify the target isotope, not just the element. Specify neutron energy or spectrum and choose the relevant capture cross-section. Convert Φ, σ and N to compatible units, then calculate initial production rate. If the product is radioactive, account for simultaneous decay during irradiation with the growth equation; apply exponential decay after irradiation ends. Before converting detector counts to amount, include emission probability, efficiency and timing. Check target depletion or beam attenuation if irradiation is long or the target is thick.

Visual explanation

Draw a neutron beam entering a target foil. Some arrows pass through, some scatter and some terminate in capture events followed by gamma arrows. Beneath the foil put Φ × σ × N → production rate. At right, plot product activity rising toward a horizontal saturation line while the beam is on, then falling exponentially after the beam turns off. This separates collision probability, radioactive growth and later decay.

Real-world analogy

Rainfall onto a field can be described by drops per area per second, while a particular collector has an effective catch area. The expected collection rate is rainfall flux times effective area. A nuclear cross-section plays that mathematical role for interactions, though it can depend sharply on neutron energy and quantum state rather than on a fixed physical rim.

Real-world example

An analyst irradiates a trace-containing specimen with a known neutron spectrum. A stable isotope captures some neutrons and becomes radioactive. After a controlled cooling period, its characteristic gamma peak is counted. If the analyst knows isotope abundance, cross-section, irradiation time, decay constant and detector efficiency, the original element amount can be inferred. A delay that is harmless for a long-lived product may greatly reduce counts from a short-lived one.

Why?

Why can slowing neutrons change capture yield? The neutron–nucleus interaction probability can vary with energy; for some targets, lower energies increase capture cross-section, and resonances can create additional peaks. A moderator changes the neutron spectrum reaching the target, changing the weighted interaction rate even if the total number of incident neutrons were unchanged. The direction and size of the effect must be checked for the specific isotope.

Common misconception

“A barn is the physical area of every nucleus.” It is a unit for effective cross-section; measured σ can differ greatly from geometric area. Another error says saturation means every stable target atom has turned radioactive. Saturation instead balances product formation and decay under the idealized constant-rate model. A third mistake uses a thermal capture σ for a fast-neutron beam without spectrum correction.

Worked example

A thin target contains N = 1.0 × 10¹⁸ atoms of an isotope. It receives a uniform neutron flux Φ = 1.0 × 10¹² cm⁻² s⁻¹, and its capture cross-section at that energy is hypothetically σ = 1.0 barn = 1.0 × 10⁻²⁴ cm². The initial capture rate is R = ΦσN = (10¹²)(10⁻²⁴)(10¹⁸) = 1.0 × 10⁶ captures s⁻¹ . This assumes negligible target depletion, no flux attenuation and one applicable σ. It is not an activity unless production and decay have reached the corresponding balance.

Quick check

1. Does an (n,γ) reaction change the proton number of the target? Answer: No. The captured neutron adds one to mass number A but zero electric charge, so Z remains the same element.

Exam focus

State 1 barn = 10⁻²⁴ cm² and keep area units compatible with flux. Distinguish capture, scattering and total cross-sections, and always identify neutron energy. For activation, combine ΦσN with product decay during and after irradiation. Explain saturation as equal rates of formation and decay, not conversion of all target atoms.

Advanced insight

In a real neutron field, flux varies with energy and position, so R = N∫Φ(E)σ(E)dE for a uniform target with suitable units. Self-shielding occurs when the target itself removes neutrons, lowering flux inside it. Product atoms may also capture further neutrons, and irradiation may alter target composition. These effects make a calibrated activation analysis more complex than the thin-target formula, but the simple expression remains a useful first estimate and dimensional check.

Summary

Neutron capture can turn a stable isotope into a radioactive product while leaving Z unchanged. Cross-section measures an energy- and channel-dependent interaction probability and is expressed in barns. In a thin target, ΦσN estimates production rate. A radioactive product builds toward a formation–decay balance during irradiation and decays afterward, so activation calculations need both reaction and decay data.

Practice questions

1. Convert 2 barns to square centimetres. Answer: 2 barns = 2 × 10⁻²⁴ cm².

2. Why is product activity initially less than production rate in a new activation experiment? Answer: Few radioactive product atoms exist at first, so their decay rate λN starts low while capture produces them.

3. What additional data are needed to use a thermal cross-section in a mixed-energy neutron field? Answer: The neutron energy spectrum and the cross-section versus energy, or a validated spectrum-averaged cross-section.

4. If irradiation stops at saturation, does activity remain constant? Answer: No. Production stops, so activity then decreases according to the product's radioactive half-life.