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A reaction coefficient formed from target number densities and microscopic nuclear reaction cross-sections. It has inverse-length units. Multiplying it by the scalar neutron flux gives the corresponding reaction rate per volume, under the stated energy averaging.

Neutron activation is the production of radioactivity by irradiating a material with neutrons. One route is neutron capture producing an unstable isotope; other nuclear reactions can also produce unstable atomic nuclei. A specimen's later radioactive decay provides a way to detect that a transformation occurred during irradiation. It does not by itself identify the product or reconstruct the incoming neutron energy distribution.

This makes neutron activation a useful bridge between an invisible radiation field and a delayed measurement. The neutron flux and collision cross-section determine production; radioactive decay and detection efficiency determine what remains observable later. Neutron activation buildup and decay keeps these stages separate.

The coefficient in $\mathbf{J}=-D\nabla \varphi$ for the scalar neutron flux convention. It has dimensions of length, unlike a particle-density diffusion coefficient with dimensions length squared per time. For a single fixed speed $v$, the corresponding density diffusion coefficient is $vD$.

The time integral of scalar neutron flux, with inverse-area units. Multiplication by a constant microscopic reaction collision cross-section gives a dimensionless exposure parameter. A single integrated exposure does not specify how the neutron energy distribution varied during irradiation.

Let $n(E)dE$ be the number of neutrons per unit volume in an energy interval. The scalar neutron flux per unit energy is $\varphi (E)=v(E)n(E)$, with $v(E)$ the speed. Its integral has units of inverse area per unit time. It counts path length traversed per volume per time, including all directions; it is not the signed current through an oriented surface.

For a small uniform specimen containing $N_T$ target atomic nuclei, a reaction-specific collision cross-section ${\sigma }_a(E)$ gives the expected production rateAssume the incident field is little changed within the specimen, target depletion is negligible and each counted reaction creates the product of interest. If that product has a production branching fraction below one, include it in ${\sigma }_a$ or as an additional factor. A single neutron activation measurement supplies one weighted integral of an energy distribution, not a unique reconstruction of that distribution.

For an original two-group example, stipulate response collision cross-sections ${\sigma }_s=10{\sigma }_f$. With the same total scalar neutron flux $\Phi$, changing from all fast neutrons to half slow and half fast changes $R/(N_T{\sigma }_f\Phi )$ from $1$ to $5.5$. No additional neutrons need be emitted by the source. The chosen factor ten is not a historical silver measurement.

For some capture reactions over a limited low-energy interval, ${\sigma }_a\propto 1/v$ is a useful approximation. At fixed neutron flux, it gives increasing reaction probability per path length as speed decreases. At fixed neutron number density, ${\sigma }_{a}v$ is constant in this approximation, so the rate per target is unchanged. A real neutron moderator changes the local density, energy distribution, escape and absorption together. Neither all collision cross-sections nor all neutron activation rates vary monotonically as $1/v$.

A diffusion approximation for a scalar neutron flux, with one prescribed energy group. Its terms balance current loss, absorption, nuclear fission production and a source. A solution also requires appropriate boundary conditions.

One energy group and a nearly isotropic local neutron distribution are the main approximations. The one-group neutron diffusion equation replaces the full energy-dependent transport problem by one scalar neutron flux and averaged coefficients. A single group cannot resolve spectral changes, resonance absorption or differences between rapidly moving and thermal neutrons; its effective parameters depend on the spectrum used for averaging.

The diffusion approximation also replaces angular transport by a current proportional to the negative neutron flux gradient. It works best where repeated scattering makes the angular distribution nearly isotropic over distances comparable to the field's variation length. It is less reliable close to a sharply localized source, a vacuum boundary, a strong absorber or an abrupt material interface, and in weakly scattering regions. Approximate boundary extrapolation does not restore the lost angular information.

Thus an apparently smooth solution of the one-group neutron diffusion equation need not accurately represent a transport field outside the assumptions that produced its coefficients and boundary conditions.

The one-group particle balance with a prescribed external volume source $S$ can be writtenHere $\varphi$ is scalar neutron flux, $D$ is the neutron diffusion coefficient, and ${\Sigma }_a,{\Sigma }_f$ are macroscopic cross sections. The terms describe accumulation, current divergence, absorption, nuclear fission production and external input.

For a steady state, set ${\partial }_t\varphi =0$; source-free means $S=0$. In a homogeneous medium $D$ is spatially constant, so substitution givesThe supplied notation defines $\eta$ as the effective production per absorption, $\eta =\nu {\Sigma }_f/{\Sigma }_a$. ThereforeThis identification uses the absorption accounting for the homogeneous model; a fuel-only reproduction factor with other absorbers excluded would not automatically equal this $\eta$. The derivation assumes one prescribed energy group, constant effective coefficients and the diffusion approximation; all neutron production included here is represented by the steady nuclear fission term.

The fundamental nonnegative mode is ${\varphi }_0{J}_0(2.405r/R)\cos (\pi z/H)$. Put the origin at the centre of the cylinder, so $-H/2\le z\le H/2$. Neglecting extrapolation imposes zero neutron flux at $r=R$ and $z=\pm H/2$. WriteFor an axisymmetric field, separation of variables in cylindrical coordinates givesChoose the axial separation constant so $Z^{\prime \prime }+{\beta }^{2}Z=0$. Midplane symmetry and the lowest positive axial mode give $Z(z)=\cos (\beta z)$, withThe radial equation isThe solution regular on the axis is the zero-order Bessel function of the first kind, $F(r)\propto J_0(\alpha r)$; the independent second-kind solution is singular there. The first radial zero gives $\alpha R=j_{0,1}\simeq 2.405$. Higher zeros would introduce sign changes and do not describe the fundamental nonnegative neutron flux. Normalize by the central value, using $J_0(0)=1$:The printed $2.405$ is a rounded Bessel function zero, rather than an exact root.

Xenon poisoning is neutron removal by xenon-135, which reduces reactor reactivity. Its important precursor is iodine-135. At a long-established constant neutron flux, production through precursor decay is balanced by xenon radioactive decay and neutron capture. A steady model must account for that additional absorber in its neutron balance.

After a rapid removal of the imposed neutron flux, capture removal becomes small immediately, while the existing iodine population continues supplying xenon. The xenon population can therefore rise before eventually falling as the precursor reservoir is exhausted. The resulting extra negative reactor reactivity is a delayed effect, rather than an instantaneous disappearance of the absorber when nuclear fission stops.

An intermediate post-shutdown maximum is possible. This explains why the reactivity associated with a previously operating system can differ substantially from its long-time shutdown value. The population model below quantifies that mechanism; a population maximum is not by itself an operating or restart prescription.

Let $G={\gamma }_I{\Sigma }_f\varphi$ be the constant precursor production rate per volume. Here ${\gamma }_I$ is the precursor yield per nuclear fission, and $I$ and $X$ are number densities. At a steady state,For fixed homogeneous coefficients and spatial shape, the reactivity contribution of an added absorption coefficient ${\Sigma }_{aX}={\sigma }_{aX}{X}_0$ is $-{\Sigma }_{aX}/(\nu {\Sigma }_f)$. Thus the model givesWhen capture removal dominates xenon radioactive decay, $\varphi {\sigma }_{aX}\gg {\lambda }_X$, and henceThe limit is independent of the xenon neutron absorption cross section: at high neutron flux, nearly every xenon atom supplied by the precursor ultimately consumes a neutron. The assumptions include constant pre-shutdown neutron flux, sufficient time for precursor and xenon equilibrium, the supplied single-precursor model with no direct xenon fission source, and fixed parameters for translating absorption into reactor reactivity.

Irradiation creep adds stress-directed, time-dependent strain driven by the irradiation-generated defect population. It can be appreciable where ordinary thermal material creep would be slow, because irradiation maintains mobile point defects far above their equilibrium populations.

Absorption or emission of vacancies and self-interstitials permits dislocation climb. Applied stress can make differently oriented dislocations or dislocation loops absorb defects at different rates: stress-induced preferential absorption then produces a directional strain. Climb can also allow a pinned dislocation to bypass an obstacle and continue dislocation glide, providing a second route to strain.

A useful small-strain empirical form, over a limited range of conditions, isfor constant stress $\sigma$ and approximately constant irradiation-creep compliance $B$. Here $d$ is the accumulated displacements per atom and $\dot{d}$ the displacement dose rate; $B$ is measured per unit stress and per unit dpa. This relation explains why creep can scale approximately with both stress and neutron flux when the damage spectrum is fixed. It is not a constitutive law valid at every dose or temperature.

The total strain can also include thermal material creep, void swelling, irradiation growth and changes due to precipitation or recovery. Stress-free swelling is not itself irradiation creep. Experiments on pressurized type316 tubes separated these contributions and found that correcting stress-enhanced swelling materially affected comparisons between annealed and specimens strengthened by cold working.^[ref]

At sufficiently high temperature, thermal material creep can dominate. Changes in microstructure caused by irradiation hardening, softening, bubbles and cavities can either alter the creep rate or shorten rupture life. Thus enhanced defect-driven creep is the central effect, while the net rate and lifetime remain material- and history-dependent.

The first equation is a prescribed first-order removal law. For a constant absorption coefficient,Integrating from the initial condition givesand henceThe neutron fluence $\Phi$ is the time integral of scalar neutron flux; it has inverse-area units, making $\sigma \Phi$ dimensionless. Constant flux is not needed for this integration, but a constant effective coefficient is.

Smooth-model fission energy. Use natural units $c=1$, so the quoted nuclear masses are expressed as rest energies. Conservation of mass number and atomic number fixes the second fragment to $(A_2,Z_2)=((1-y)A,(1-y)Z)$. First omit nuclear pairing to compare a continuous range of $y$. Pairing can subsequently add ${\delta }_1+{\delta }_2-{\delta }_{\mathrm{parent}}$ for integer fragments; it has no smooth parity assignment for arbitrary real $y$.

In the parent-minus-daughters mass difference, the nucleon rest energy and volume terms cancel. The nuclear symmetry energy also cancels because both fragments have the same $Z/A$:The remaining liquid-drop fission energy isSplitting increases the surface energy cost, but reduces nuclear Coulomb energy. Both contributions are symmetric under $y\leftrightarrow 1-y$, so the symmetric split is stationary. Symmetry alone would not establish a maximum, since the two contributions compete.

For a full check, put $S=a_S{A}^{2/3}$, $x=a_C{Z}^2/(a_{S}A)$ and $t=y^{1/3}+(1-y{)}^{1/3}$. On $0\le y\le 1/2$, $t$ increases from $1$ to $2^{2/3}$. Expressing the two fractional-power sums through $t$ givesThe derivative can change sign only from negative to positive, so any interior stationary point in $t$ is a minimum. The largest value is therefore either the no-split endpoint $Q=0$ or the symmetric split. Thus symmetric fission maximizes the positive liquid-drop energy release: whenever this smooth model permits an exothermic split, its largest release occurs at $y=1/2$.

There is a qualification to the paper's unqualified wording. For arbitrary low $Z^2/A$, the symmetric split need not maximize $Q$; indeed it is a local minimum when $Z^2/A<(2/5)a_S/a_C=10$. The symmetric-maximum conclusion is valid in the heavy, fission-favorable regime relevant here, rather than a universal property of the formula.

At $y=1/2$,Requiring $Q_{\mathrm{sym}}>0$ gives

Barrier and fragment yields. Positive final-state Q value does not remove the fission barrier along a deformation path. Surface energy initially rises as the nucleus deforms; spontaneous fission then requires collective quantum tunnelling unless enough excitation is present. Its rate is exponentially sensitive to barrier height, width and collective inertia. Thus nuclei close to the exothermicity threshold can have immeasurably long spontaneous-fission mean lifetimes, while sufficiently heavier nuclei have smaller effective fission barriers or larger quantum tunnelling probabilities.

The fission-fragment mass distribution of low-excitation actinide fission is usually double peaked: complementary fragments have mass fractions roughly $0.4$ and $0.6$ rather than both exactly $1/2$. Nuclear shell model effects and the fission dynamics help favor asymmetric fragments. Maximizing the final smooth-model energy release is not a calculation of the fragment yield. The following yield curve is schematic, with no measured counts implied:

Capture excitation energies. Capturing a negligible-energy neutron forms a compound nucleus with the rest energy of the target plus the free neutron. Relative to the compound ground state, the neutron-capture excitation energy is thereforewhere $B$ is positive nuclear binding energy and $S_n$ the compound nucleus's neutron separation energy. The free neutron rest energy cancels in the binding-energy difference. With the supplied semi-empirical mass formula,At $Z=92$, uranium-235 has odd $A$, while uranium-236 is even-even. The pairing contribution is $+33.5(236{)}^{-3/4}=+0.55637\mathrm{MeV}$. By contrast uranium-238 is even-even and uranium-239 has odd $A$, giving $-33.5(238{)}^{-3/4}=-0.55286\mathrm{MeV}$. The individual contributions areThus the model predictsOf the $1.48305\mathrm{MeV}$ predicted difference, $1.10922\mathrm{MeV}$ comes from the change in nuclear pairing. It is the primary reason the even-even capture product is more strongly excited; the smooth terms account for the remaining $0.37383\mathrm{MeV}$.

Thermal versus fast-neutron fission. The last comparison uses the supplied observed capture excitations, $6.5$ and $4.8\mathrm{MeV}$, rather than replacing them by the preceding smooth-model estimates. Interpreting the quoted $6.2$ and $6.6\mathrm{MeV}$ activation energies as the barriers in the respective capture channels givesA simple neutron-induced fission threshold estimate for the second channel requires about $6.6-4.8=1.8\mathrm{MeV}$ additional centre-of-mass kinetic energy. Sub-barrier tunnelling is possible, but it does not give the large thermal-neutron fission probability of the first channel.

The barrier labels in the original PDF need care: it assigns them to ${}^{235}\mathrm{U}$ and ${}^{238}\mathrm{U}$, whereas the states formed after capture are ${}^{236}{\mathrm{U}}^{\ast }$ and ${}^{239}{\mathrm{U}}^{\ast }$. The comparison above uses the intended capture-channel interpretation. If the quoted numbers are literally barriers of the uncaptured target nuclei, the compound-nucleus barriers are missing and this numerical comparison does not follow from those data alone.

In a thermal reactor, neutron moderation slows fission neutrons so the large low-energy fission cross section of uranium-235 can support a nuclear chain reaction. Fuel composition, moderator choice and geometry must balance production against capture and leakage; capture resonances of uranium-238 during slowing are important losses from that neutron population. The reaction rate per target nucleus depends on the spectrum through $\int \varphi (E){\sigma }_f(E)dE$, where $\varphi (E)$ is differential neutron flux per unit energy. A fast spectrum can access the uranium-238 fission channel and changes this balance. Also, uranium-238 is a fertile material: capture followed by two beta-minus decays can produce plutonium-239, a fissile material, so capture can contribute to breeding even when it causes no immediate fission.

The reduction of reactor reactivity caused by neutron absorption by xenon-135. In a simplified model its abundance is fed by radioactive iodine decay, while its removal is by radioactive decay and neutron capture. Changes in these pathways can delay its response to a change of neutron flux.