Crystal

بلور (Ar). Cristal (Fr). Kristall (Ge). Cristallo (It). 結晶 (Ja). Кристалл (Ru). Cristal (Sp).

Definition

(The following applies to solids that, when illuminated by radiation having a wavelength on the order of Å, generate a three-dimensional diffraction pattern primarily characterized by discrete peaks. The entry for subperiodic crystal discusses crystals for which the diffraction pattern is instead primarily characterized by intensity maxima that are layers or rods.)

Historically, crystals were first defined in terms of their external morphologies, with the angular and symmetry relationships between their faces leading, in ca 1800, to the Law of Rational Indices. Over a century later the information from X-ray diffraction resulted in a definition based on the three-dimensional translational periodicity inferred from the periodicity of the array of Bragg peaks in the diffraction patterns of most crystals. More recently crystals have been discovered for which the array of Bragg peaks is not periodic in three dimensions so that the repeating atomic arrangement lacks translational periodicity in at least one direction. Currently there are two alternative (but theoretically equivalent) definitions of a crystal; both are based on the central idea of spatial order. One focusses on direct (or real) space, the other on reciprocal (or diffraction) space. The latter is more compact and elegant; the former can be easier to visualize.

Direct-space definition

A solid is a crystal if its atoms, ions and/or molecules form, on average, a long-range ordered arrangement.

In most crystals the arrangement is a periodic array that is governed by the rules of translational symmetry. In aperiodic crystals (incommensurate and quasicrystals) the arrangement is not periodic in three dimensions but is nevertheless still fully ordered, where the ordering follows particular mathematical rules.

Reciprocal-space definition

A material is a crystal if it has essentially a sharp diffraction pattern.

A solid is a crystal if it has essentially a sharp diffraction pattern. The word essentially means that most of the intensity of the diffraction is concentrated in relatively sharp Bragg peaks, besides the always present diffuse scattering. In all cases, the positions of the diffraction peaks can be expressed by

[math]\textbf{H}=\sum_{i=1}^nh_{i}\textbf{a}_{i}^{*}~~(n\ge 3)[/math]

Here [math]\textbf{a}_{i}^{*}[/math] and [math]h_{i}[/math] are the basis vectors of the reciprocal lattice and integer coefficients respectively and the number n is the minimum for which the positions of the peaks can be described with integer coefficient [math]h_{i}[/math].

The conventional crystals are a special class, though very large, for which n = 3.

[The adjective "known" reflects the fact that it is possible to envisage mathematically-based models that lie outside these definitions. See e.g. Grimm (2015).]

See also

• Acta Cryst. (1992), A48, 928 where the definition of a crystal appears in the Terms of reference of the IUCr Commission on Aperiodic Crystals.

Reference

Grimm, U. (2015). Acta Cryst. B71, 258–274. Aperiodic crystals and beyond