Date Published: January 01, 2018
Publisher: International Union of Crystallography
Author(s): Walter Steurer.
The state of the art of quasicrystal research is critically reviewed. Fundamental questions that are still unanswered are discussed and experimental limitations are considered.
Dan Shechtman was the first to identify a rapidly solidified intermetallic phase as a representative of a novel class of long-range-ordered (LRO) phases with icosahedral diffraction symmetry (Shechtman et al., 1984 ▸). The term ‘quasicrystal’ for such intermetallics was coined by Dov Levine and Paul J. Steinhardt in an article published only a few weeks after Shechtman’s seminal paper (Levine & Steinhardt, 1984 ▸). This term refers to the class of ‘quasiperiodic functions’, which were introduced as such by the mathematician Harald Bohr (Bohr, 1924 ▸, 1925 ▸, and references therein). Amazingly, quasicrystals (QCs) were prepared in the course of studies of intermetallic phase diagrams unknowingly, long before Shechtman’s discovery (see, e.g., Hardy & Silcock, 1956 ▸; Palenzona, 1971 ▸). Since, in most cases, only X-ray powder-diffraction methods were routinely used for sample characterization at that time, the fivefold diffraction symmetry characteristic of icosahedral QCs did not immediately catch the eye, unlike Shechtman’s electron diffraction patterns. Furthermore, it seems that nature created QCs aeons earlier. According to findings in meteorites, QCs may have been formed billions of years ago (see Bindi et al., 2009 ▸, 2016 ▸, and references therein).
QCs are binary or ternary intermetallic compounds, in many cases accompanied by low-order ACs with slightly different chemical compositions. So far, there is no known case of a quasicrystal transforming to an approximant crystal with exactly the same chemical composition, either as a function of temperature or with increasing pressure. Up to now, only QCs with icosahedral, pentagonal or decagonal symmetry have been found to be thermodynamically stable [Fig. 2 ▸; compare also Figs. 5.24 and 5.33 in Steurer & Dshemuchadse (2016 ▸)]. One hypothesis explaining the prevalence of fivefold symmetry in intermetallic QCs is based on cluster symmetry and packing possibilities. For instance, structural subunits with eight-, nine- or 12-fold rotational symmetry could be easily accommodated and packed in tetragonal, trigonal and hexagonal crystal structures, respectively. There is no need for potentially more efficient quasiperiodic packings that might be obtained by sacrificing the advantages of periodicity (phonons and Bloch waves are periodic). Indeed, there are only two dodecagonal QCs known so far. One, the telluride (Ta,V)1.6Te, is of rather poor quality and probably metastable (Krumeich et al., 2012 ▸, and references therein). The other is a Mn-rich quaternary alloy, Mn72.0−xCr5.5+xNi5.0Si17.5 with x = 0 or 2.0, which seems to be more stable, but is also of low crystal quality (Iwami & Ishimasa, 2015 ▸). Sevenfold symmetry is very rare in intermetallic compounds but frequently found in some borides, which can be seen as ACs (Steurer, 2007 ▸; Orsini-Rosenberg & Steurer, 2011 ▸). However, no quasicrystal with sevenfold symmetry has been found so far. Potential QCs with 11-, 13- or 15-fold rotational symmetry could be energetically unfavourable for steric reasons. Indeed, structural subunits with such or higher symmetries are not currently known in intermetallics. Furthermore, quasiperiodic structures with these symmetries would have significantly lower degrees of average periodicity. This means that their atomic sites would show large deviations from their (in these cases badly defined) periodic average structures (Deloudi & Steurer, 2012 ▸), which can be of importance for the propagation of phonons and the formation of Bloch waves for electrons.
There has been a long discussion, which is still ongoing, about whether QCs are ‘energy- or entropy-stabilized’. In other words, whether quasiperiodic structural order can be a ground state of condensed matter (thermodynamically stable at 0 K) or has to be stabilized by entropic contributions from phonons, phasons and structural disorder. In the case of entropic stabilization, QCs would be high-temperature (HT) phases, only stable above a specific threshold temperature. In the early days of quasicrystal research, many QCs were discovered based on the working hypothesis of electronic stabilization by the Hume–Rothery mechanism (for a review see Tsai, 2003 ▸). Indeed, in several cases a pseudogap has been identified at the Fermi energy, originating either from Fermi-surface/pseudo-Brillouin zone nesting or from the hybridization between d and p states (see Lin & Corbett, 2007 ▸; Tamura et al., 2004 ▸; Suchodolskis et al., 2003 ▸; Mizutani, 2016 ▸; and references therein). The role of entropic contributions to the stabilization of QCs, in particular that of the phason modes, is still an ongoing matter of debate (de Boissieu, 2006 ▸, and references therein). Unfortunately, the mechanism for the stabilization of mesoscopic and macroscopic quasiperiodic arrangements cannot be directly transferred to intermetallic QCs because of the different interaction mechanisms.
Why are we interested in the analysis of crystal structures at all? At present, the main structural databases contain more than 1 200 000 entries for periodic crystal structures [>188 000 inorganics/intermetallics in the Inorganic Crystal Structure Database (ICSD); >875 000 organics/metalorganics in the Cambridge Structural Database (CSD); >107 000 proteins in the RCSB Protein Data Bank (PDB)] and 145 of incommensurate structures in the Bilbao Incommensurate Structures Database (B-IncStrDB). The number of quantitatively determined QC structures is much smaller (≃20) than the number of known stable QCs (≃50), but this number includes representatives of all QC structure types known to date (Steurer & Deloudi, 2009 ▸, and references therein). For examples of state-of-the-art n-dimensional (nD) structure analyses of IQCs and DQCs see, for instance, Takakura et al. (2007 ▸) and Logvinovich et al. (2014 ▸), respectively; for tiling-based analyses of DQCs at ambient and high temperatures see Kuczera et al. (2012 ▸) and (2014 ▸), respectively.
‘For this reason, I was somewhat doubtful that nature would actually produce such ‘quasi-crystalline’ structures spontaneously’ said Roger Penrose (Thomas, 2011 ▸). ‘I couldn’t see how nature could do it because the assembly requires non-local knowledge’. The growth of quasiperiodic structures is still not fully clarified; however, some realistic models and simulations have been published recently (see Kuczera & Steurer, 2015 ▸, and references therein). It should be borne in mind that no atomistic growth models exist for complex intermetallics either. How does the 1000th atom find its site in a giant unit cell with thousands of atoms? When does a structure grow periodically, when quasiperiodically?
What do we want to know about the structure of a periodic crystal? The answers are: the content of the unit cell, its structure and the kind of chemical bonding based on quantum-mechanical calculations. If the LRO is periodic, then there is no need to study it except in the case of intrinsic disorder. In the case of a quasiperiodic structure, however, both the SRO and the LRO need to be determined. In some cases this is only possible up to some limit, but this may still be sufficient to gain a good understanding. Our absolute limit, at present, is that we are not able to use quantum-mechanical calculations for QCs in the same way as for periodic crystals in order to identify when, for instance, it is advantageous for a structure to become quasiperiodic instead of forming a high-order AC.