Two-point correlation function for the triangular Ising antiferromagnet

The vertebrate muscle superlattice: discovery, consequences, and link to geometric frustration

Early x-ray diffraction studies of muscle revealed spacings larger than the basic thick filament lattice spacing and led to a number of speculations on the mutual rotations of the filaments in the myosin lattice. The nature of the arrangements of the filaments was resolved by John Squire and Pradeep Luther using careful electron microscopy and image analysis. The intriguing disorder in the rotations, that they termed the myosin superlattice, remained a curiosity, until work with Rick Millane and colleagues showed a connection to "geometric frustration," a well-known phenomenon in statistical and condensed matter physics. In this review, we describe how this connection gives a satisfying physical basis for the myosin superlattice, and how recent work has shown relationships to muscle mechanical behaviour.

Geometric frustration in the myosin superlattice of vertebrate muscle

Geometric frustration results from an incompatibility between minimum energy arrangements and the geometry of a system, and gives rise to interesting and novel phenomena. Here, we report geometric frustration in a native biological macromolecular system---vertebrate muscle. We analyse the disorder in the myosin filament rotations in the myofibrils of vertebrate striated (skeletal and cardiac) muscle, as seen in thin-section electron micrographs, and show that the distribution of rotations corresponds to an archetypical geometrically frustrated system---the triangular Ising antiferromagnet. Spatial correlations are evident out to at least six lattice spacings. The results demonstrate that geometric frustration can drive the development of structure in complex biological systems, and may have implications for the nature of the actin--myosin interactions involved in muscle contraction. Identification of the distribution of myosin filament rotations with an Ising model allows the extensive results on the latter to be applied to this system. It shows how local interactions (between adjacent myosin filaments) can determine long-range order and, conversely, how observations of long-range order (such as patterns seen in electron micrographs) can be used to estimate the energetics of these local interactions. Furthermore, since diffraction by a disordered system is a function of the second-order statistics, the derived correlations allow more accurate diffraction calculations, which can aid in interpretation of X-ray diffraction data from muscle specimens for structural analysis.

Probing magnetic interactions in metal-organic frameworks and coordination polymers microscopically

Materials with magnetic interactions between their metal centres play a tremendous role in modern technologies and can exhibit unique physical phenomena. In recent years, magnetic metal-organic frameworks and coordination polymers have attracted significant attention because their unique structural flexibility enables them to exhibit multifunctional magnetic properties or unique magnetic states not found in the conventional magnetic materials, such as metal oxides. Techniques that enable the magnetic interactions in these materials to be probed at the atomic scale, long established to be key for developing other magnetic materials, are not well established for studying metal-organic frameworks and coordination polymers. This review focuses on studies where metal-organic frameworks and coordination polymers have been examined using such microscopic probes, with a particular focus on neutron scattering and density-functional theory, the most-well established experimental and computational techniques for understanding magnetic materials in detail. This paper builds on a brief introduction to these techniques to describe how such probes have been applied to a variety of magnetic materials starting with select historical examples before discussing multifunctional, low dimensional and frustrated magnets. This review highlights the information that can be obtained from such microscopic studies, including the strengths and limitations of these techniques. The article then concludes with a brief perspective on the future of this area.

Diffraction by a frustrated system: the triangular Ising antiferromagnet

Expressions are derived for diffraction by the triangular Ising antiferromagnet, a disordered lattice system consisting of two kinds of scatterer and exhibiting geometric frustration. Analysis of the expressions shows characteristics of the diffraction patterns, including the presence of Bragg and diffuse diffraction, superlattice reflections, and their behavior with temperature. These characteristics are illustrated by numerical simulations. The results have application to diffraction imaging of disordered systems.