The morphology of calaverite (AuTe2) from data of 1931. Solution of an old problem of rational indices

Abstract

The anomalous high-index faces (hkl) of the mineral calaverite (Au~_xAgxTe2) measured goniometrically in the year 1931 by Goldschmidt, Palache & Peacock [Neues Jahrb. Mineral (1931), 63, 1-58] are re-interpreted and related to the wave vector q of the displacive incommensurate modulation which was recently found in the crystal structure. All crystal * Present address: Philips Research Laboratories, PO Box 80.000, NL-5600 JA Eindhoven, The Netherlands.faces (including the high-index ones) can be given four low indices (hklm), using q as a fourth basis vector. From this an almost hundred-year-old anomaly in crystal morphology is in principle solved. I. IntroductionThe present investigation started from a non-applicability of the law of rational indices to crystal growth forms of calaverite Aul_pAgpTe2 (p < 0.15) (Smith, 1902;Goldschmidt, Palache & Peacock, 1931).Classically (Bravais-Friedel) the morphology of crystals is related to the existence of a crystal lattice. This crystal lattice is even implicitly suggested by Haiiy's law of rational indices. The latter can be extended to the use of four or more rational indices in order to describe the crystal forms of modulated crystals (Janner, Rasing, Bennema & van der Linden, 1980;Dam & Janner, 1983, 1986. The fact that the number of indices can be larger than three reflects the fact that the number of fundamental periodicities can also be larger than three, which is a key concept for the understanding of crystal faces. Hence, in the description of crystal morphology one is not necessarily restricted to the three periodicities generating a three-dimensional space lattice. In the onedimensional modulated case the modulation wave vector q = aa* + fib* + yc* has to be added as a fourth basic vector to the three ones of the undistorted reciprocal-lattice unit cell: a*, b*, c*. Any crystal face of a one-dimensionally modulated crystal is then labeled by the four indices (hklm) of the corresponding face normal given by k--ha*+kb*+lc*+ mq.The use of four indices is closely related to the superspace approach as introduced for incommensurately modulated crystals by de Wolff (1977) and Janner & Janssen (1977. Indeed, the presence of fundamental periodicities in a given crystal does not mean that the structure is periodic in space. One can eventually restore the familiar lattice periodicity by embedding the crystal in a larger space, the superspace, with as many dimensions as there are fundamental periodicities. In Dam & Janner (1986) the details of this approach are explained and applied to the morphology of the modulated phases of the crystal tetramethylammonium tetrachlorozincate [(CH3)aN]2ZnCI 4. The most recent experimental result of morphological research on that compound is the observation of a roughening of a satellite orientation (hklm) upon a change of the modulation wave vector q as a function of temperature (Dam, 1985). Note that superspace embedding is necessary for a symmetry characterization in terms of Euclidean space groups, but not if that is not req...

Equilibrium morphology of incommensurately modulated crystals: a superspace description

“…The faces that determine the morphology of incommensurately modulated crystals are perpendicular to Fourier wave vectors of the ~-module. It was shown that the faces of Aul_xAgxTe2 could be labelled, very satisfactorily, with four low integral indices (Dam, Janner & Donnay, 1985;Janner & Dam, 1989). The same behaviour has been found for many other incommensurately modulated crystals (see Bennema et al, 1991).…”

“…In Fig. 1, the result, as was given by Janner & Dam (1989) is reproduced for a typical twin crystal of Aul_xAgxTe 2. A crystal face is called a main face if the fourth index is zero.…”

“…It has been shown experimentally (Janner, Rasing, Bennema & van der Linden, 1980;Dam, Janner, Bennema, van der Linden & Rasing, 1983;Dam et al, 1985;Janner & Dam, 1989;Bennema et al, 1991) that the orientation of faces on incommensurately modulated crystals can be described very accurately with four integral indices hi, h 2, h 3 and h 4. These indices defme a vector H(hlhEhah4) perpendicular to the face.…”

Pressure-Induced Superconductivity in Mineral Calaverite AuTe2

“…In the actual structure, incommensurate modulation (q ¼ À0:4076a à þ 0:4479c à ) induces the breaking up of zigzag chains and the formation of isolated Te 2 dimers with a short interatomic distance of 2.88 # A. 15) Although the charge density wave state or mixed-valence state of Au þ (5d 10 ) and Au 3þ (5d 8 ) is proposed to be the origin of such incommensurate modulation, 16) X-ray photoelectron spectroscopy and first principles calculation indicate the homogeneous monovalent Au þ state of AuTe 2 to be the case. 17,18) In addition, the calculated Fermi surface of the average structure shows that the modulation cannot be understood in terms of Fermi-surface nesting.…”

Qu’est-ce qu’un cristal ?

“…Ce vecteur définit les positions de raies de Bragg, dites raies satellites des raies de Bragg de la structure moyenne. Ainsi une base de rang 4 décrit toute l'image de diffraction, avec introduction simultané-ment de quatre indices de Miller h, k, l, m : G hklm = h a* + k b* + l c* + m q. Cette interprétation vaut pour la calavérite, mentionnée précédemment, l'ensemble des faces de ce cristal se décrivant aussi avec quatre indices [6].…”