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At fixed $+2$ oxidation state, increasing effective nuclear charge across the first transition series generally contracts the ionic radius. The first expectation is therefore shorter average metal–nitrogen and metal–sulfur distances toward copper. This is a trend in comparable coordination environments, rather than a requirement that every individual contact shorten monotonically. The occupation of metal–coordination ligand antibonding orbitals, donor identity and lattice distortion all affect the observed distances.

The high-spin $d^5$ and $d^6$ ions both have two electrons in the strongly sigma-antibonding $e_g$ set, so adding the sixth electron to $t_{2g}$ does not itself introduce another sigma-antibonding electron. Copper(II), however, has $e_g^3$ occupation and is subject to the Jahn-Teller effect. A tetragonal Jahn-Teller distortion commonly gives four shorter and two longer contacts, so an elongated copper contact can oppose the overall size trend. Here the nitrogen and sulfur donors are already chemically inequivalent; the supplied schematic alone does not determine which trans pair becomes longest. One should compare actual M–N and M–S sets separately, rather than infer a unique axis or a numerical distance from the magnetic moments. The experimentally determined structures illustrate why the donor geometry and individual contacts matter.^[ref]

The actual ionic radii need not equal the limiting ratio, and electronic charge distributions are not rigid sphere boundaries. In the same unlike-contact hard-sphere model, put $q=r_a/r_c$ and use $a=4(r_a+r_c)/\sqrt{3}$ to getA cation somewhat larger relative to the anion than the limiting size opens gaps between the anions. Starting from $q=1.366$ and reducing $q$ modestly reduces this packing fraction; thus a lower value is expected for that departure from the ideal contact construction. The formula is not monotonic for all imaginable ratios, so no universal inequality should be inferred without specifying radii. Coordination-dependent ionic radii, polarizability, nonionic contributions to bonding, thermal expansion and point defects also limit the model. No numerical real-crystal packing fraction is determined by the supplied data alone.

The host is zirconolite, with ideal end-member composition $\mathrm{CaZrT}{\mathrm{i}}_2{\mathrm{O}}_7$. In the proposed formula, the substitution is formallyIt is an isovalent substitution: the mean charge of the mixed zirconium/plutonium site remains $+4$, soNo compensating oxygen vacancy or aliovalent co-substituent is required by this particular charge account. Suitable oxidation state is essential; the same formula interpreted with only trivalent plutonium would not be charge balanced.

In zirconolite-2M the zirconium site has an oxygen coordination number of seven. The ionic radius of plutonium(IV) is larger than that of zirconium(IV), so increasing its occupancy changes local strain and the preferred arrangement of the cation layers. The observed progression in the investigated substitution series was zirconolite-2M → zirconolite-4M → pyrochlore structure, rather than unlimited substitution in one unchanged monoclinic lattice. The primary study found a 2M solid-solution limit near $x=0.15$, a narrow 4M range near $x=0.41$, and a pyrochlore composition at $x=0.60$ in ${\mathrm{CaZr}}_{1-x}{\mathrm{Pu}}_x\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_7$. These are experimental observations for that series, not universal boundaries independent of preparation and oxidation state.^[ref]

Consequently the nominal $x=0.20$ composition identifies a zirconolite-based waste-form design, but does not by itself prove a pure 2M product. Diffraction and phase-composition measurements are needed. At larger substitution, the fluorite-related pyrochlore arrangement accommodates a different cation distribution and coordination environment.