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uranium with less uranium-235 than natural uranium. Addition to an immobilised plutonium inventory can dilute uranium-235 subsequently formed by plutonium-239 radioactive decay; chemical coexistence is not irreversible protection against deliberate separation.

Material convertible into a fissile material by neutron capture followed by nuclear transformations. A central example is uranium-238 capturing a neutron and undergoing two beta-minus decays to plutonium-239. Fertile does not mean that thermal-neutron capture immediately causes fission.

Material capable of sustaining a nuclear chain reaction with slow neutrons in a suitable configuration. Fissility involves capture and fission probabilities, neutron production and losses, rather than positive fission Q value alone. Uranium-235 and plutonium-239 are common examples; fissile is more specific than fissionable by sufficiently energetic neutrons.

Neptunium is the chemical element with atomic number 93. Its isotope neptunium-239 occurs between uranium-239 and plutonium-239 in a neutron-capture and beta decay production chain.

gadolinium is primarily a strong neutron absorber. The supplied thermal cross section for gadolinium-157 is very large, so capture removes neutrons that could otherwise sustain nuclear fission. This is a criticality-control role rather than a requirement to complete the host formula. The isotope-specific number must not be assigned unchanged to natural gadolinium.

hafnium supplies an additional neutron poison with favourable retention chemistry. Its supplied natural-abundance thermal cross section is much smaller than that of gadolinium-157 but still appreciable. Chemically, tetravalent hafnium resembles zirconium and can remain associated with a durable titanate host. Keeping the absorber with the fissile inventory during alteration is as important as its initial neutron capture strength: a highly absorbing constituent offers poor long-term protection if it is preferentially lost. These considerations explain using complementary absorbers rather than ranking them solely by their initial thermal cross sections.

uranium addition chiefly addresses isotopic dilution, including the long-term daughter inventory. plutonium-239 alpha decay produces uranium-235. Added uranium-238 lowers the uranium-235 fraction of that future uranium inventory; depleted uranium adds less uranium-235 initially than natural uranium. uranium-238 also absorbs some neutrons, but its small listed cross section does not make it a substitute for the strong absorbers. The long-term daughter issue and the retention of gadolinium and hafnium are explicit concerns in ceramic waste-form development.^[ref]

These are complementary physical purposes, not proof of criticality control in radioactive waste from three cross-section values. neutron energy, isotopic composition, geometry, moderation, absorber distribution and their coupled evolution matter. Incorporation in the same durable waste form helps maintain association, but neither uranium addition nor chemical immobilisation makes deliberate element separation physically impossible.

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.

A fertile material whose neutron capture produces uranium-239. It can undergo neutron-induced fission at sufficiently high incident neutron energies, while slow-neutron capture can instead initiate breeding of plutonium-239.

An unstable uranium isotope produced by neutron capture on uranium-238. Two successive beta-minus decays, through neptunium-239, lead to plutonium-239.