Order-disorder transition in the Cd-Ca cubic approximant
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Order-disorder transition in the Cd-Ca cubic approximant M. Widom and M. Mihalkoviˇc1 Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 1 also at: Institute of Physics, Slovak Academy of Sciences, 84228 Bratislava, Slovakia ABSTRACT Recent experiments discovered an order-disorder transition occuring at low temperatures in large unit 1/1 cell cubic approximants of the stable Cd-based binary alloy quasicrystals. The transition is related to correlations among orientational degrees of freedom whose separations are around 12 ˚ A. We analyze the interactions between the degrees of freedom using ab-initio calculations for Cd-Ca alloys and derive an equivalent antiferromagnetic Ising model which shows a similar phase transition. However, the calculated transition temperature is higher than observed experimentally, indicating that the actual structure and its order-disorder transition are more complex than originally proposed. A side-benefit of our study is the discovery of a canonical-cell decoration model for the Cd-Ca icosahedral phase. INTRODUCTION Stable icosahedral quasicrystals occur in the compounds Cd5.7 Yb and Cd5.7 Ca [1, 2, 3]. In each case the phase diagram of the binary alloy contains a 1/1 cubic approximant at a close-by composition [4, 5]. The Cd6 Yb structure [6] can be represented in a conventional simple cubic unit cell with a lattice parameter of a = 15.7 ˚ A. The Pearson symbol for Cd6 Yb is cI176 indicating that it is cubic, body-centered, with 176 atomic positions per simple cubic unit cell. Since 176 is not a multiple of the basic 7-atom stoichiometric unit (6 Cd and 1 Yb), there must be partial occupancy. In fact, the unit cell contains 144 Cd atoms, 24 Yb atoms and 8 vacancies. Sets of 4 vacancies alternate with sets of 4 Cd atoms among vertices of a cube to form tetrahedra located at the center of a dodecahedral cluster (see Fig. 1). There are two such clusters per simple cubic cell, for a total of 8 vacancies.
Figure 1: (Left) Innermost Cd20 dodecahedral shell centered by Cd4 tetrahedron, viewed along cubic [100] axis. (Right) Two alternate tetrahedral orientations (dark=Cd, light=vacancy) according to cI176 structure [6].
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The proper structure type of Cd6 Ca has recently been debated. Initial reports [7] assigned it to prototype Cd6 Y with Pearson symbol cI184. This structure is equivalent to the cI176 structure of Cd6 Yb except for the central Cd4 tetrahedra. In cI184 these Cd atoms occupy 4 out of 12 vertices of a cuboctahedron (resulting in 6 possible orientatons) instead of 4 out of 8 vertices of a cube (resulting in 2 possible orientations). After the discovery of stable Cd-Ca quasicrystals some researchers [8, 9, 10] suggested the structure type of Cd6 Ca might actually be Cd6 Yb.cI176. The full story has yet to be resolved. Indeed a recent study of M Cd6 for a variety of metal atoms M finds evidence for yet additional site types [11]. In order for the overall structure to be truly body-centered, it is necessary that either the tetrahedra at each
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