The ash heap of crystallography: restoring forgotten basic knowledge

Abstract

A critical analysis of some basic notions often overlooked in crystallographic education is presented to correct some common oversights occurring both in the literature and in textbooks. The crystal forms (face forms), defined in terms of their geometric eigensymmetry, are 47 in number, not 48 as often found in the literature. The split of the dihedron into dome and sphenoid calls for the consideration of the physical properties of the faces building a form; in that case, however, the same criterion should be used for all forms. By taking the handedness of the faces as representative of the physical properties of the faces, the occurrence of 130 crystallographic face forms (97 affine face forms and 33 enantiomorphic pairs) is demonstrated. Next, the correct use of non-coprime Miller indices when a centred unit cell is adopted is shown, and the inconsistent multiplication of Miller indices in the Bravais–Friedel–Donnay–Harker law is pointed out. A geometric derivation of the reflection conditions is reviewed. Finally, the inconsistent presentation of metric restrictions imposed by the structural symmetry is pointed out and corrected.

Direction indices for crystal lattices

“…In the case of centred cells, however, such a simplification is incorrect, inconsistent with the definition of these indices and a potential source of error. After addressing specifically the case of Miller indices in our previous article (Nespolo, 2015a), we have shown that the same type of problem concerns also crystallographic direction indices: these are actually restricted to rational values. We hope that our analysis will help avoid mistakes and oversights by crystallographic software developers and the end users of such software and draw the attention of crystallography lecturers to the necessity to make students well aware of the consequences of adopting centred cells, which may seem trivial but are actually too often overlooked.…”

“…As we have pointed out (Nespolo, 2015a), this is true only when a primitive unit cell is adopted, whereas in the case of multiple unit cells this condition no longer applies and is replaced by characteristic relations between Miller indices that correspond to the integral reflection conditions.…”

Basic Elements of Crystallography. 2nd edition. By Nevill Gonzalez Szwacki and Teresa Szwacka. Pan Stanford Publishing, 2016. Pp. x + 324. Price USD 79.95 (paperback). ISBN 9789814613576.

“…The first edition of this book, published in 2010, was reviewed by Nespolo (2012). I describe here only the changes with respect to the previous edition.…”

“…Bragg's law, structure factors. The presentation is affected by the common mistake of considering Miller indices to be restricted to relatively prime indices independently from the basis vectors chosen (see a discussion in Nespolo, 2015), which leads to the selfcontradictory use of both (100) and (200) notation to indicate different planes of the same family. From a single example (bcc structure) the integral reflection conditions are introduced without further explanation; zonal and serial conditions are not mentioned.…”

Geometry of Crystals, Polycrystals, and Phase Transformations

“…In fact, the projection of a b-unique mB unit cell (equivalent to mP) is incorrectly labelled mC (the in-plane angle should have been right to get an mC unit cell) and the 'trigonal' (which does not exist: read 'rhombohedral' instead) unit cell is shown without any centring nodes, which means that the hP unit cell is shown twice. Miller indices (the correct ones: indices of lattice planes) are given as h 1 ; h 2 ; h 3 instead of h; k; l and are incorrectly defined as never containing common factors, which is true only for primitive (Nespolo, 2015). The screw or glide component of the corresponding screw rotations and glide reflections are said to be 'rational fractions' of a period, which is true for the defining operation of a screw axis or glide plane, but certainly not in general (hint: how would you describe a rotation followed by an integer translation parallel to the axis and relating atoms in different unit cells?).…”