A non-linear theory of strong interactions

Abstract

High-resolution transmission electron microscopy and selected-area electron diffraction show that all phases of the general formula [Ba$_x$Cs$_y$] [(Al, Ti)$^{3+}_{2x+y}$ Ti$^{4+}_{8-2x-y}$]O$_{16}$, 1.08 $\leqslant$ x+y $\leqslant$ 1.51 have the hollandite-type substructure. These hollandites display commensurate and incommensurate superlattices owing to the ordered insertion of large cations (Ba$^{2+}$, Cs$^+$) into the (2, 2) tunnel interstices of the octahedral (Al, Ti) O$_6$ framework. Multiplicity (m) of a supercell is defined as d$_{\mathrm{supercell}}$ divided by d$_{002}$ for the subcell. Ordering may be one-dimensional, in which case the cation sequences between (2, 2) channels are independent, three-dimensional with lateral correlation between tunnels, or a combination of both. One-dimensional superstructures yield commensurate multiplicities of 4 in all phases except an aluminous caesium hollandite where m = 6. Three-dimensional superstructures are both incommensurate and commensurate, with 4.5$_0$ $\leqslant m \leqslant$ 6.5$_9$. Multiplicities correlate directly with caesium content per formula unit, establishing a maximum in caesium-rich hollandites. Among barium ($y$ = 0) and caesium endmembers, ($x$ = 0) multiplicities increase modestly with increasing Al$^{3+}$: (Al + Ti)$^{3+}$ content. Superstructure dimensionality is largely determined by the nature and proportions of the trivalent species, rather than the tunnel cations; one-dimensional order is commonplace in hollandites rich in trivalent titanium but rare in aluminous hollandites. High-resolution electron microscopy supports the interpretation of incommensurate superstructures as fine-scale intergrowths of commensurate microdomains with $m$ = 4, 5, 6 or 7. For aluminous hollandites, rare examples of structural modifications involving tunnels of different cross-sectional dimensions may be found, i.e. T(2, $n$), 1 $\leqslant n \leqslant$ 3 intergrowths. As all specimens are sensitive to the electron beam, prolonged irradiation at high electron fluxes can initiate the transformation of single-crystal hollandite to single-crystal rutile. A mechanism for this transformation is proposed, whereby the hollandite crystals initially adjust their multiplicity to six. Growth fronts on {101}$_{\mathrm{holl}}$ subsequently propagate through the crystals consuming hollandite and leaving rutile: the structure of the interface between the phases is believed to contain components of rutile possessing antiphase boundaries. In this reconstructive transformation, [100] of the newly formed rutile invariably lies almost parallel to [110] of the original hollandite. Less severe electron irradiation or argon ion beam milling causes crystals to twin polysynthetically. The superlattice properties of [Ba$_x$ Cs$_y$] [(Ti, Al)$^{3+}_{2x+y}$ Ti$^{4+}_{8-2x-y}$] O$_{16}$ hollandites are integrated with those of other hollandites to identify and evaluate the factors responsible for the stoichiometries and preferred superstructures of hollandites in general. These factors include electrostatic repulsion between large cations in the same tunnel, interaction between cations in neighbouring tunnels, the shielding capacity of the octahedral framework, and kinetic effects.