= Solution
\b[<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>.
\b[<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.
\b[<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.<Immobilisation of Higher Activity Wastes from Nuclear Reactor Production of 99Mo by M. W. A. Stewart and others>[Section 2, plutonium-immobilisation ceramic development, uranium-235 ingrowth and neutron-absorber retention]{ref}
These are complementary physical purposes, \b[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.
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