Localized Basis for Effective Lattice Hamiltonians: Lattice Wannier Functions

A systematic method is presented for constructing effective Hamiltonians for general phonon-related structural transitions. The key feature is the application of group theoretical methods to identify the subspace in which the effective Hamiltonian acts and construct for it localized basis vectors, which are the analogue of electronic Wannier functions. The results of the symmetry analysis for the perovskite, rocksalt, fluorite and A15 structures and the forms of effective Hamiltonians for the ferroelectric transition in $PbTiO_3$ and $BaTiO_3$, the oxygen-octahedron rotation transition in $SrTiO_3$, the Jahn-Teller instability in $La_{1-x}(Ca,Sr,Ba)_xMnO_3$ and the antiferroelectric transition in $PbZrO_3$ are discussed. For the oxygen- octahedron rotation transition in $SrTiO_3$, this method provides an alternative to the rotational variable approach which is well behaved throughout the Brillouin zone. The parameters appearing in the Wannier basis vectors and in the effective Hamiltonian, given by the corresponding invariant energy expansion, can be obtained for individual materials using first- principles density-functional-theory total energy and linear response techniques, or any technique that can reliably calculate force constants and distortion energies. A practical approach to the determination of these parameters is presented and the application to ferroelectric $PbTiO_3$ discussed.Comment: extensive revisions in presentation, 32 pages, Revtex, 7 Postscript figure

O okresowości pewnych ciągów liczb naturalnych

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On the equation x02+x12+⋯+xn2=xn+1kx_0^2 + x_1^2 + \dots + x_n^2 = x_{n+1}^k

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Below the Hall–Petch Limit in Nanocrystalline Ceramics

Reducing the grain size of metals and ceramics can significantly increase strength and hardness, a phenomenon described by the Hall–Petch relationship. The many studies on the Hall–Petch relationship in metals reveal that when the grain size is reduced to tens of nanometers, this relationship breaks down. However, experimental data for nanocrystalline ceramics are scarce, and the existence of a breakdown is controversial. Here we show the Hall–Petch breakdown in nanocrystalline ceramics by performing indentation studies on fully dense nanocrystalline ceramics fabricated with grain sizes ranging from 3.6 to 37.5 nm. A maximum hardness occurs at a grain size of 18.4 nm, and a negative (or inverse) Hall–Petch relationship reduces the hardness as the grain size is decreased to around 5 nm. At the smallest grain sizes, the hardness plateaus and becomes insensitive to grain size change. Strain rate studies show that the primary mechanism behind the breakdown, negative, and plateau behavior is not diffusion-based. We find that a decrease in density and an increase in dissipative energy below the breakdown correlate with increasing grain boundary volume fraction as the grain size is reduced. The behavior below the breakdown is consistent with structural changes, such as increasing triple-junction volume fraction. Grain- and indent-size-dependent fracture behavior further supports local structural changes that corroborate current theories of nanocrack formation at triple junctions. The synergistic grain size dependencies of hardness, elasticity, energy dissipation, and nanostructure of nanocrystalline ceramics point to an opportunity to use the grain size to tune the strength and dissipative properties