Determination of near-coincident cells for hexagonal crystals. Related DSC lattices

Abstract

Crystallographic data concerning geometric properties of hexagonal lattices of C,~, Zn, B%, Ti,~, Zr~, Mg and Cd are obtained from two different computation techniques. These properties are related to the relative orientations of identical hexagonal lattices 1 and 2 which superimpose two multiple cells M1 and M2 within a given small deformation. These orientations are listed for ratios 27 = Ivolume of cells M1 (or M2)/volume of the unit cell l varying from 1 to 25. Their number are limited by choosing all the principal strains transforming M1 into M2 less than or equal to 1%. IntroductionThree techniques have been used to determine the relative orientations of identical hexagonal crystals which give rise to a near coincidence of two cells of the two crystal lattices 1 and 2, cells denoted hereinafter M 1 and M2 respectively. [This has been referred to by previous workers as a 'coincidence' or 'near-coincidence site lattice' or 'orientation de macle',* where the number 271 (or 2~2) is defined by the ratio volume of M1 (or M2)/volume of the unit cell.] Two techniques depend either upon searching for vectors of common length arising from rational values of (c/a) 2 (Fortes, 1973;Warrington, 1975) or in searching for coincidences arising from rotation about specific axes, chosen a priori, of (low) crystallographic index (Bruggeman, Bishop & Hartt, 1972). The first method derives from that used by Warrington & Bufalini * We note for the benefit of our readers whose mother tongue is English that the term 'orientation de macle' as defined by Friedel (1964) is more general a concept than the nearest English equivalent of'twin' used in its restrictive sense.0567-7394/81/020184-06501.00 (1971) for cubic crystals; the second from the use of a 'generation function' typified by Ranganathan (1966) and Goux (1961). The third technique (Bonnet & Cousineau, 1977), tested on Zn/Zn and NiaA1 (cubic)/ Ni3Nb(orthorhombic ), depends on a numerical method of calculation capable of treating the case of general lattices and envisaged in part by Santoro & Mighell (1973). It takes into account the experimental [rather than idealised or rational values of (c/a) 2] values of the lattice parameters. The technique determines relative orientations that, with additional imposed constraints (which may be chosen arbitrarily small) on M1, will give full or true coincidence with M2. Determination of relative orientationsIn searching for a 'constrained coincidence' the worker must compromise and set limits on the deviation from exact coincidence that is to be allowed. In the numerical method (Bonnet & Cousineau, 1977) this is represented by a maximum value of S = It, ll + l e21 + lea1 where e 1, e 2, e a are the principal strains of the pure deformation D transforming lattice 1 into lattice 2. D -~ transforms lattice 2 into a fictitious lattice denoted lattice 2', which can be exactly superposed onto lattice 1 (Bonnet & Durand, 1975). The unit cell of this CSL (coincidence site lattice) is defined either by M1 or by the deformed cell M2, d...

In situ TEM study of twin boundary migration in sub-micron Be fibers

“…• perfect twin misorientation [57]. Indeed, the observed twin boundary presents a misorientation ≈ 85.2…”

Magnetoresistance of antidot lattices with grain boundaries

“…1(a), which corresponds to the Σ7 coincidence site lattice within one of the standard classification schemes for GBs 26,27 , meaning that one out of 7 lattice sites coincide in an overlay of the two grains. It belongs to the set of energetically favorably GBs according to various models, 28 and occurs frequently in real systems see, e.g.…”

Fundamentals for the description of hexagonal lattices in general and in coincidence orientation

“…rotations generating CSL's in hexagonal lattices (Warrington, 1975;Fortes & Smith, 1976;Bonnet, Cousineau & Warrington, 1981;Hag~ge, Nouet & Delavignette, 1980;Bleris, Nouet, Hag~ge & Delavignette, 1982). This last paper, which will be referred to as BNHD, uses an axis-angle description in lattice coordinates for the rotations, which turns out to be convenient for deriving the coincidence rotations.…”