This review provides a summary of the state-of-the-art in terms of description, interpretation and classification of crystal structures in mineralogy. Among the various methods, the focus is on atomic packing (including both anion and cation arrays), coordination polyhedra (both cation- and anion-centred), the concept of fundamental building blocks and related ideas and the use of networks, graphs and tilings (space partitions). The basic concepts under discussion in modern structural mineralogy include structure hierarchy (specification and compositional hierarchies are considered separately), modularity (representation of crystal structures as constructed from modules extracted from simple archetype structures) and complexity (with emphasis on static or informational and algorithmic complexities). Short historical notes are given for all of the topics considered.

In the last twenty-five years, relevant theoretical, methodological and experimental advances have been made in the development and application of the X-ray powder diffraction (XRPD) method. In particular, attention has been devoted to the interpretation of XRPD data. The XRPD approach is used currently in mineralogical as well as in many other scientific fields (solid-state chemistry, pharmacology, materials science, etc. ) to address a wide range of scientific purposes: qualitative analysis for the identification of the crystalline phases constituting a powder sample; quantitative analysis for estimating the weight fraction of each phase in a mixture; structure solution; microstructural analysis for the inspection of crystalline domain size effects and lattice defects; investigation of highly complex materials: compounds with incommensurate structures, nanoparticles, amorphous materials; studies at non-ambient conditions, in situ , time-resolved and in operando for the description of thermal or compressional behaviour, phase stability and structural evolution. The aim of this chapter is to provide an overview of some basic principles and significant aspects of the XRPD method and examples of its applications to mineralogical problems.

Analysis of the structures of minerals is an important part of mineralogical investigations. Mineral assemblies are often formed by micro- or even nanocrystals, and it is most interesting to be able to shed light on the crystallography of individual grains, their structure, crystallinity and other properties. This can be done efficiently by the techniques of transmission electron microscopy. Transmission electron microscopy offers not only an ultimate spatial resolution in the imaging mode, but also the possibility of performing diffraction experiments and structure analysis of very small crystals, as well as a range of spectroscopic techniques revealing chemical composition and other properties of the crystals. This chapter reviews the basic techniques of transimssion electron microscopy and their application in mineralogical crystallography. A special emphasis is put on the methods of structure analysis of nanocrystals. This field has seen a rapid evolution in recent years, and has transformed from a being niche technique to a widely accepted and commonly used method of structure analysis. Its applications in mineralogy are especially rich and attractive.

Total scattering experiments using high-energy synchrotron X-rays and spallation neutrons are providing new insights into the structures of nanoscale and poorly crystalline materials of environmental and mineralogical relevance. The pair distribution function (PDF) derived from these total scattering data is a real-space depiction of the atomic arrangements over short (<3–5 Å), intermediate (up to ~20 Å), and even longer length scales. Structural information can be extracted both directly from the PDF and through modelling. PDF analysis approaches are described using selected examples of natural and synthetic nanoparticles as well as a sample that is a mixture of amorphous and crystalline structural phases. Several applications include combined analysis of the real- and reciprocal-space forms of the scattering data. Greater application of the total scattering and PDF methods to environmental minerals that are nanoscale and poorly crystallized will provide new insight to structure, including structural disorder at different length scales, and help to develop further structure-property relationships.

The three-dimensional periodic nature of crystalline structures was so strongly anchored in the minds of scientists that the numerous indications that seemed to question this model struggled to acquire the status of validity. The discovery of aperiodic crystals, a generic term including modulated, composite and quasicrystal structures, started in the 1970s with the discovery of incommensurately modulated structures and the presence of satellite reflections surrounding the main reflections in the diffraction patterns. The need to use additional integers to index such diffractograms was soon adopted and theoretical considerations showed that any crystal structure requiring more than three integers to index its diffraction pattern could be described as a periodic object in a higher dimensional space, i.e. superspace, with dimension equal to the number of required integers. The structure observed in physical space is thus a three-dimensional intersection of the structure described as periodic in superspace. Once the symmetry properties of aperiodic crystals were established, the superspace theory was soon adopted in order to describe numerous examples of incommensurate crystal structures from natural and synthetic organic and inorganic compounds even to proteins. Aperiodic crystals thus exhibit perfect atomic structures with long-range order, but without any three-dimensional translational symmetry. The discovery of modulated structures was soon followed by the discovery of composite structures consisting of structural entities with partly independent translations and finally by the discovery of quasicrystals. In recent years, the use of CCD and imaging plate systems improved considerably the sensitivity of data collection for aperiodic structures and in particular modulated structures and, therefore, there was a need for further improvement of the methods. Today, several computer programs are able to solve and refine incommensurately modulated structures using the superspace approach. In nature, it is uncommon to find minerals which have strong and sharp incommensurate satellites that could be used for a higher dimensional refinement. Here we describe several cases of aperiodic minerals (natrite, calaverite, melilite, fresnoite, pearceite–polybasite, cylindrite) including the first examples of natural and stable quasicrystalline structures (icosahedrite and decagonite) which settle beyond doubt any questions which remain about the long-term stability of quasicrystals.

At the dawn of structural crystallography, Walther Friedrich, Paul Knipping and Max von Laue carried out the first experiments and developed the theory of X-ray diffraction. From the early days, when even the simpler inorganic structures filled an entire PhD study, structural crystallography evolved at its own pace and found new partners in chemistry, physics, materials science, biology and other fields of physical sciences. Both morphological and structural crystallography, however, have remained as important instruments in the mineralogist’s toolbox until today. Efforts to enhance the existing instrumentation, to improve our understanding of the theory of diffraction, to study nanoparticulate or poorly ordered materials, and to master large, complex structures continue in all fields of physical sciences. Mineralogy can thus use the fruits of this labour and include them in its toolbox.