Nuclear Reactions and Binding Energy

01 / UNDERSTAND & APPLY

1. Count nucleons before balancing a nuclear change

A nuclide is specified by proton number Z and mass number A. It contains Z protons and N = A − Z neutrons. The chemical element is fixed by Z, while isotopes share Z but differ in neutron count. Isobars share A; isotones share N. The number of atomic electrons depends on ionisation and must not be confused with the nuclear proton count.

Mass number is an integer count, not a measured mass in atomic mass units. A nucleus with A nucleons does not generally have mass exactly A u. The atomic mass unit u is defined using neutral carbon-12 atoms; nuclear reaction calculations must use the actual supplied masses consistently rather than replace each mass by its mass number.

An empirical nuclear-radius estimate is R = r₀A^(1/3), with r₀ about 1.2 fm and 1 fm = 10⁻¹⁵ m. Volume then scales approximately as A, giving roughly similar nuclear mass density across many nuclides. The nucleus has a diffuse quantum boundary, so this radius law is a useful scale estimate rather than a rigid-ball geometry valid to unlimited precision.

The residual strong interaction between nucleons is short-ranged and approximately saturating: a nucleon interacts strongly mainly with nearby nucleons. Proton–proton electric repulsion is long-ranged. Their competition helps explain why heavy stable nuclei generally need more neutrons than protons and why there is no unlimited progression of stable heavier nuclei.

02 / UNDERSTAND & APPLY

2. A bound nucleus has less rest energy than separated nucleons

The binding energy B is the positive energy required to separate a nucleus completely into free protons and neutrons at rest, with all particles far apart. A bound system has lower total rest energy than those separated constituents. Thus the mass defect is Δm = Zmₚ + Nmₙ − M_nucleus and B = Δmc². Mass has not disappeared from conservation of total energy; energy released during formation leaves the system.

For mass differences expressed in u, use 1 u c² ≈ 931.5 MeV. If a mass is already given in MeV/c², multiplying by c² leaves MeV. Multiplying an energy expressed in MeV by c² a second time is a dimensional mistake. Nuclear energies are commonly millions of electron volts per event, but each event still involves a very small number of joules.

Tables often provide neutral atomic masses rather than bare nuclear masses. For binding-energy work, the convenient approximate atomic-mass expression is B = [Z m_H + N mₙ − M_atom]c², where m_H is the neutral hydrogen-atom mass. Electron rest masses then cancel. Small electron binding-energy corrections are usually negligible at the stated nuclear precision but matter in high-precision work.

Total binding energy and binding energy per nucleon answer different questions. B/A is an average measure of binding; the energy to remove one specified neutron is a separation energy obtained from a difference of actual neighbouring masses. It is not generally equal to B/A. A larger nucleus can have greater total B yet a lower average binding per nucleon.

03 / UNDERSTAND & APPLY

3. Read the binding curve as an energy comparison, not a stability certificate

The broad binding-energy-per-nucleon curve rises rapidly among light nuclei, reaches a high region around iron and nickel, and decreases slowly toward very heavy nuclei. Individual nuclei show structure around this smooth trend. Avoid treating a rough classroom curve as exact isotope data or assuming one isotope must lead every possible measure of stability.

If a reaction conserves the same total numbers of constituent protons and neutrons and increases total binding energy, it releases energy. Fusion of suitable light nuclei and fission of suitable heavy nuclei can move products toward more tightly bound arrangements. The released energy is the increase in total binding, not merely a difference between two unweighted B/A values.

For example, when one A-nucleon parent splits into fragments with A₁ and A₂, compare A₁(B/A)₁ + A₂(B/A)₂ with A(B/A)_parent, including any other products. Averaging the two fragment B/A values without weighting by their nucleon counts is generally wrong. Full reaction masses are the more universal method when particle species change.

An energetically favourable rearrangement is not necessarily fast. Coulomb barriers, quantum tunnelling, angular-momentum constraints and available decay pathways affect rates. Binding per nucleon alone cannot determine half-life, and a positive Q-value is not a guarantee of an easily initiated reaction. Energy accounting and reaction probability are separate parts of the physics.

04 / UNDERSTAND & APPLY

4. Q-values compare complete initial and final states

For a nuclear reaction, Q = (sum of initial rest masses − sum of final rest masses)c². Positive Q means rest energy can become kinetic energy or radiation; negative Q means energy must be supplied. In a closed reaction, total energy, momentum, electric charge and the relevant conserved quantum numbers must balance. Merely balancing mass numbers does not finish the problem.

For many reactions written using neutral atoms, electron rest masses cancel when the total atomic charge numbers balance and no explicit electron or positron is mishandled. Otherwise account for electrons separately. In particular, using neutral atomic masses gives Q for beta-minus decay approximately [M_parent − M_daughter]c², but beta-plus decay requires subtracting 2mₑc². Electron capture has different electron bookkeeping from positron emission.

An endothermic reaction on a target initially at rest generally needs projectile kinetic energy greater than |Q| because momentum conservation leaves kinetic energy in the products. For a nonrelativistic two-body threshold with negative Q small compared with rest energies, a common approximation is K_threshold ≈ |Q|(1 + m_projectile/M_target). This approximation is not a universal threshold formula for arbitrary relativistic or multiparticle events.

In a two-body decay of a parent initially at rest, daughter momenta are equal and opposite. For nonrelativistic products, kinetic energies are inversely proportional to their masses: K₁/K₂ = m₂/m₁. A light alpha particle therefore receives most, but not all, of the available decay energy. Neglecting the recoil may be acceptable only when the requested precision permits it.

05 / UNDERSTAND & APPLY

5. Identify what changes in alpha, beta and gamma emission

Alpha emission ejects a helium-4 nucleus: the daughter has A − 4 and Z − 2. In beta-minus decay a neutron converts into a proton while an electron and electron antineutrino are emitted; A is unchanged and Z increases by one. The emitted electron is created in the weak process, rather than being a pre-existing nuclear constituent shaken loose.

In beta-plus decay a proton converts into a neutron with a positron and electron neutrino emitted; A is unchanged and Z decreases by one. Electron capture also reduces Z by one at fixed A by absorbing an atomic electron and emitting a neutrino. Whether a channel is allowed depends on actual masses and energy requirements, not just whether charge numbers can be balanced.

Gamma emission lowers the excitation energy of a nucleus without changing A or Z. The atom may also produce characteristic X-rays after electron-shell rearrangements, but X-rays and gamma rays are distinguished by their origin rather than by a perfectly nonoverlapping energy boundary. Recoil gives a small correction to the photon energy relative to the full level gap.

Alpha decay often has discrete characteristic energies because it is essentially a two-body transition to particular daughter levels. Beta electrons or positrons have a continuous energy distribution because energy and momentum are shared among the charged particle, neutrino and recoiling daughter. A listed beta endpoint is therefore not the energy carried by every emitted electron.

After a sequence of decays, balance changes in A first to determine the number of alpha events, then use changes in Z to infer beta events under the specified channels. Do not infer a unique physical chain from algebra alone: gamma steps and alternative branches can occur without changing the final A and Z labels.

06 / UNDERSTAND & APPLY

6. Constant decay probability produces an exponential population

For a single independently decaying nuclide with constant decay probability per unit time λ, the expected number of decays in a short interval is λN dt. Thus dN/dt = −λN and N(t) = N₀e^(−λt). λ is positive and has inverse-time units. The minus sign belongs to the decrease of the undecayed population; the activity is the positive rate A_activity = λN.

Individual decay times are unpredictable in this model. Exponential decay describes an expected population or an ensemble average, not a clock forcing every nucleus to decay at exactly one half-life. Radioactive nuclei do not grow deterministically old until they reach a fixed expiry time. For constant λ, the conditional probability of survival over a further interval is independent of how long a nucleus has already survived.

The half-life T₁/₂ satisfies e^(−λT₁/₂) = 1/2, giving T₁/₂ = ln 2/λ. The mean lifetime is τ = 1/λ, so τ = T₁/₂/ln 2 and is longer than the half-life. After n half-lives, the expected surviving fraction is 2^(−n); the decayed fraction is 1 − 2^(−n). These two fractions are often confused.

Activity is measured in becquerels, with 1 Bq = one decay per second. It depends both on how much material is present and on λ. Two samples with equal numbers of nuclei need not have equal activity if their half-lives differ. Equal masses do not imply equal numbers of nuclei when molar masses differ.

The simple law assumes no additional production or removal and a constant decay constant. Ordinary temperature and pressure changes usually have negligible effects on common nuclear decay rates, but blanket statements of perfect environmental independence are too strong: electron-capture rates can depend on electronic conditions. State the isolated constant-rate model used in a calculation.

07 / UNDERSTAND & APPLY

7. Separate physical decay from what a detector records

A detector records only some emitted radiation. In a simple model, measured count rate C = εA_activity + C_background, with constant detection efficiency ε. Subtract background before using count-rate ratios to infer a half-life. Counts are not automatically the total number of nuclear decays, especially when emission branching fractions and geometry matter.

For independent counts with Poisson statistics, the standard deviation of a large count n is approximately √n. Relative counting uncertainty is therefore about 1/√n. Counting for four times as long reduces relative statistical uncertainty by roughly a factor of two under constant-rate conditions. This is a statistical estimate, not a bound guaranteeing every observed count lies within one standard deviation.

If a detector sees a mixture of independent nuclides, activity is a sum Σ λᵢNᵢ₀e^(−λᵢt), generally not a single exponential. A parent producing a radioactive daughter requires production as well as decay: dN_d/dt = λ_pN_p − λ_dN_d when the branch is fully to that daughter. Daughter activity can initially rise even though both species are radioactive.

Dating from a surviving fraction uses t = ln(N₀/N)/λ under an appropriate closed-system model. The starting abundance, contamination, replenishment and environmental history can matter more than arithmetic. A numerical decay-age exercise provides these assumptions; an actual sample cannot be dated reliably from an unqualified activity ratio alone.

08 / UNDERSTAND & APPLY

8. Original practice with complete reasoning

These problems are original exercises in nuclear bookkeeping and exponential decay. Use supplied masses or energy differences; do not reconstruct precise isotope data from the smooth binding-energy trend.

09 / UNDERSTAND & APPLY

9. Audit energy signs, mass conventions and decay assumptions

Begin by distinguishing an atom from its nucleus, a mass number from a measured mass, and a population from an activity. Then select the process: separating a nucleus calls for binding energy; transforming initial particles into final particles calls for Q; waiting for an isolated radioactive sample calls for exponential decay. These operations use related ideas but different inputs.

Check the physical conclusion after the arithmetic. Positive binding energy means energy must be supplied for complete separation. Positive Q means the chosen reaction releases rest energy, without guaranteeing a high reaction rate. A half-life divides the expected remaining population, not the original population anew each time. The same check prevents most apparent paradoxes in this chapter.

• Balance charge, nucleon labels and all particles required by the decay channel.
• Never mix nuclear and atomic masses without electron bookkeeping.
• Use total binding energies, not unweighted averages, to compare reaction states.
• Account for recoil whenever energy is shared between products.
• Convert half-life to seconds before reporting activity in Bq.
• Subtract background and state efficiency assumptions for detector ratios.
• A mixture or replenished daughter need not follow one exponential.