Symmetry Breaking and Order in the Age of Quasicrystals

The year 2012 marks the centennial of the discovery of the diffraction of X-rays by crystals. This discovery was a great triumph not only for the idea that matter is composed of atoms, but also for the underlying paradigm of modern crystallography, stating that all structurally ordered matter is composed of periodic arrangements of these atoms. Since its early days, modern crystallography treated order and periodicity synonymously, both serving equally to define the notion of a crystal. With that came the so-called “crystallographic restriction,” stating that crystals cannot have certain forbidden symmetries, such as 5-fold rotations. The periodicity of crystals became the underlying paradigm not only for crystallography itself, but also more generally for materials science and solidstate physics or chemistry, whose most basic tools relied on periodicity. The year 2012 also marks the 30th anniversary of the discovery of quasicrystals, which completely shattered this paradigm, leading to a rebirth of crystallography. Cahn described the discovery of quasicrystals as a Kuhnian scientific revolution, and I believe that we are now in the midst of the most exciting stage of this revolution. The old established paradigms, most importantly that of the periodicity of crystals, have been overthrown. The initial skepticism of the scientific community—embodied most vividly in the writings of Pauling—is now gone, replaced by full recognition with the award of the 2011 Nobel Prize in Chemistry to Shechtman, which we are celebrating with this festive issue of the Israel Journal of Chemistry. New notions and paradigms are being tested and carefully adopted. New theories of quasicrystals, aperiodic tilings, and symmetry are being developed. Experimental techniques are undergoing fundamental modifications to encompass aperiodic crystals. All this intense activity is being pursued by hundreds of scientists worldwide, ranging from pure mathematicians and crystallographers to physicists, chemists, materials scientists, and even a few architects. I would like to share some of the exciting surprises encountered in the current paradigm-building phase we are in, by reviewing the successful adaptation of a number of fundamental notions—related to symmetry breaking and order—to the age of quasicrystals. Order, or more specifically long-range order, is a well-established notion, used extensively in theories of phase transitions. The emergence of order is associated with a spontaneous breaking of symmetry, where the less-ordered phase is characterized by a higher degree of symmetry than the ordered one. In fact, it is the change in symmetry that distinguishes between the two phases. In the case of crystals we are talking about the positional order of the atoms. The isotropic pairwise interactions between atoms give rise to a free energy that is isotropic and translationally invariant—one can shift or rotate space by any amount and the free energy remains unchanged. The disordered liquid phase has all these symmetries of the free energy, but the