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After cosmological weak freeze-out, the surviving neutron fraction is $x_n=r_f(1+r_f{)}^{-1}{e}^{-\Delta t/{\tau }_n}$, where $r_f$ is the initial neutron-to-proton ratio and ${\tau }_n=t_{1/2}/\ln 2$ is the mean lifetime. Baryon number is conserved because a decaying neutron becomes a proton. If almost all surviving neutrons enter helium-4, neutron-limited helium synthesis gives the displayed primordial helium mass fraction. Increasing the half-life increases survival. If the longer lifetime also represents uniformly weaker weak interactions, it raises the temperature of cosmological weak freeze-out and the initial neutron fraction, reinforcing the helium increase. That last conclusion requires specifying how the weak rates change, rather than treating lifetime alone as a complete microscopic model.

Escape of neutrons from a specified region before they participate in further reactions there. A leakage fraction depends on the region's geometry, surrounding materials and energy-dependent transport and absorption, rather than its dimensions alone.

Neutron moderation reduces the kinetic energy of fast neutrons through repeated collisions in a neutron moderator. In a simple elastic collision model, hydrogen is especially effective because a proton and a neutron have nearly equal masses: the maximum elastic recoil energy fraction can approach one. The mean logarithmic energy decrement describes typical accumulated slowing, whereas the maximum transfer describes a single endpoint.

Slowing, absorption and escape compete. A material can remove neutrons by neutron capture before they become thermal neutrons, and some neutrons escape its boundaries. Describing a neutron moderator therefore requires more than collision kinematics. Near thermal equilibrium, target thermal motion can also give energy back to a neutron; the initially stationary target approximation must then be replaced.

The number of neutrons in an atomic nucleus. Two isotopes of the same chemical element differ in neutron number while sharing an atomic number.

A constituent that absorbs neutrons and thereby reduces their availability for a nuclear fission chain reaction. Effectiveness depends on isotope, neutron spectrum and spatial distribution, not just one thermal cross section.

Formation of a deuteron from a neutron and a proton with photon emission. For an initial pair at rest, two-body energy and momentum conservation gives $E_{\gamma }=B-B^2/[2(m_p+m_n)]$ in natural units, where $B=m_p+m_n-m_d$. The missing binding energy becomes nuclear recoil.

Neutron scattering measures how a specimen changes the directions and kinetic energies of incident neutrons. Neutron diffraction uses the elastic scattering pattern to investigate crystal structure and magnetic order; inelastic neutron scattering uses energy transfer to investigate excitations such as phonons and magnons. The two measurements ask complementary questions: what arrangement repeats in space, and what disturbances can propagate through it?

The 1994 Nobel Prize in Physics recognized Clifford Glenwood Shull's diffraction work and Bertram Neville Brockhouse's spectroscopy.^[ref]

Nuclear scattering couples to atomic nuclei; magnetic scattering couples the neutron's magnetic moment to electronic magnetic moments. The latter underlies both magnetic neutron diffraction and magnon spectroscopy. An atomic number is not a nuclear neutron scattering weight: the relevant coefficient in the elementary coherent model is the nuclear neutron scattering length.

Treat the incident and outgoing neutrons as free nonrelativistic particles outside the specimen. Their wave vectors, de Broglie wavelengths, momenta and kinetic energies obeyHere $k_i=|{\mathbf{k}}_i|$, $k_f=|{\mathbf{k}}_f|$, $m_n$ is the neutron mass, and $\hslash =h/(2\pi )$ uses the Planck constant. Subtracting the outgoing projectile quantities from the incoming ones gives the momentum transfer and energy delivered to the specimen:Thus $\mathbf{Q}$ has units of inverse length, rather than momentum. A positive $\omega$ means the specimen gains energy; negative $\omega$ means it supplies energy to the neutron. If the angle between incident and outgoing propagation directions is $2\theta$, the law of cosines givesKnowing the detector angle alone therefore leaves $Q$ undetermined unless the outgoing energy is known or an elastic scattering assumption is justified.

For a reproducible unit conversion choose $m_n=1.67492750056\times 10^{-27}\mathrm{kg}$, $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $1\mathrm{eV}=1.602176634\times 10^{-19}\mathrm{J}$. The first is a recommended central value; the latter two conversions are exact.^[ref] With $1$ ångström equal to $10^{-10}\mathrm{m}$ and $1\mathrm{meV}=10^{-3}$ electronvolts, substitution into $E=h^2/(2m_n{\lambda }^2)$ givesSquare brackets here mean the numerical value in the indicated unit. For an original chosen example, ${\lambda }_i=2$ ångströms and ${\lambda }_f=2.5$ ångströms give $E_i\simeq 20.451\mathrm{meV}$, $E_f\simeq 13.089\mathrm{meV}$ and $\hslash \omega \simeq 7.362\mathrm{meV}$. At $2\theta =60^{\circ }$, $Q\simeq 2.879{\mathring{\mathrm{A}}}^{-1}$. These values stipulate an allowed projectile geometry; they do not assert that any particular specimen has an excitation there.

Writing positive nuclear binding energy as $B(Z,N)$ and nuclear mass as $M(Z,N)$,The two-neutron nuclear separation energy is $S_{2n}(Z,N)=B(Z,N)-B(Z,N-2)$. In an independent-particle example, let the last occupied energy level have ${ϵ}_{-}=-8u$ and the next empty level ${ϵ}_{+}=-3u$, for a chosen positive energy unit $u$. With an unchanged core and no residual interactions, removal from the filled level costs $8u$, whereas removing the first particle added above it costs $3u$: the drop $5u$ equals the one-particle gap. These are invented model inputs, not measured nuclear values. Nuclear pairing, rearrangement of the nuclear mean field, and correlations complicate that identification in real masses; a finite difference of nuclear binding energies is not automatically a bare one-particle gap.

An electrically neutral neutron still has a magnetic moment $\mathbf{\mu }={\gamma }_n\mathbf{S}$. A collimated beam passing through an inhomogeneous magnetic field experiences the force derived from $H=-\mathbf{\mu }\cdot \mathbf{B}$. In a region whose field axis is approximately fixed, a spin one-half neutron has two field-axis spin eigenstates, producing two allowed transverse forces and two separated beam components after a drift. A homogeneous field can cause Larmor precession but cannot separate positions through this force.

The neutron's negative gyromagnetic ratio interchanges which spin projection moves toward increasing field; it does not change the number of branches. An unpolarized beam gives equal ideal populations. A beam prepared in one field-axis eigenstate populates only one branch, so splitting into two observed spots presupposes both components and sufficient spatial resolution.

Nickel-56 and iron-54 have the same neutron number and differ by two protons. Dividing their nuclear statistical equilibrium abundance formulas cancels the neutron density and leaves the displayed ratio, with $\Delta B=B_{56}-B_{54}$ and $K$ containing temperature-independent statistical and mass factors in the simplified model. For material with equal total proton and neutron numbers and only these nuclei plus free protons, charge balance gives $n_p=2n_{54}$. Thus $n_{56}/n_{54}^3\propto T^{-3}{e}^{\Delta B/(kT)}$. This has a finite temperature maximum only if the signed energy difference is negative; conventional positive total nuclear binding energies for these two actual isotopes instead give a positive difference.

The nitrogen isotope with seven protons and seven neutrons.

Awarded to James Chadwick for discovering the neutron. The Chadwick neutron discovery experiment follows the recoil measurements and competing interpretations.^[ref]

Awarded to Enrico Fermi for neutron-induced radioactivity and reactions brought about by slow neutrons. The neutron article follows the activation experiments and their interpretation.^[ref]

Awarded in equal thirds to Jerome Isaac Friedman, Henry Way Kendall and Richard Edward Taylor for investigations of deep inelastic scattering of electrons from protons and bound neutrons, essential to developing the quark model.^[ref]

The positive energy needed to separate a bound atomic nucleus into its constituent protons and neutrons. It is the mass defect times $c^2$. In a Skyrmion model of a nucleus, both separated nucleon masses and the multi-soliton state require a consistent quantization and calibration.

A spatial charge density describing the electric charge inside an atomic nucleus. In a static description centered on the nucleus, normalize it bywhere $Z$ is the atomic number and $e>0$ the elementary charge. A nuclear charge distribution is an average over a nuclear quantum state, rather than a photograph of immobile protons. Even when the average has spherical symmetry, individual nucleons need not sit on spherical shells. The proton charge radius and the internal charge density of the neutron also contribute: a distribution of pointlike proton centers is an approximation to a nuclear charge distribution, not its definition.

The experimental attraction is that an electron probes electric charge through the electromagnetic interaction, without itself feeling the nuclear force. Varying its momentum transfer changes the spatial scale tested. Long wavelengths mainly constrain a nuclear charge radius; shorter wavelengths can distinguish a diffuse edge from a sharper one through diffraction. Recovering the entire nuclear charge distribution from a finite range of noisy scattering data remains an inverse problem.

A model with separate filled proton and neutron momentum seas in a nuclear mean field. Each spin-half species has two spin states, so counting momentum-space cells gives $p_{F,i}=\hslash (3{\pi }^2{n}_i{)}^{1/3}$. It describes typical nucleon motion and a kinetic contribution to nuclear symmetry energy, but binding requires additional interactions.

The residual strong interaction between nucleons. Its long-distance part involves pion exchange and depends on spin and isospin. It is distinct from directly exchanging gluons between color-singlet nucleons.

Why do protons and neutrons bind into an atomic nucleus, even though protons repel electrically? An attractive nuclear force can lower the total energy below that of separated nucleons, producing nuclear binding energy. The deuteron, containing one proton and one neutron, is the simplest such bound state.

In Quantum chromodynamics, nucleons are composite colour singlets built from quarks and gluons. Their residual nuclear force has a longest-range strong contribution from one-pion exchange, with additional interactions at shorter distances.^[ref]

The mass deficit of a bound atomic nucleus relative to its separated protons and neutrons. If their rest masses are $m_p$ and $m_n$, the atomic nucleus has rest mass $M_{\mathrm{nuc}}$, and the positive nuclear binding energy is $B$, thenwith $c$ the speed of light. Releasing energy when the nucleons bind reduces the rest mass of the bound system without changing the exact count $A=Z+N$. A neutral atomic mass also includes $Z$ electron rest masses, reduced by the energy required to separate those bound electrons. Consequently $A-m_{\mathrm{a}}/\mathrm{u}$ is not the definition of the nuclear mass defect: the reference unit and constituent rest masses must also be accounted for.