Crystallographic excursion in superspace

Abstract

After studying more than 100 different samples of calaverite Au_1-pAg_pTe₂ (p < 0.15) three famous mineralogists declared in a well documented 1931 publication the invalidity of the law of rational indices. What could lead them to draw such an extreme conclusion from the observation law of Hauÿ dating from more than two centuries? Their observations remained unexplained for the next forty years. In the 1970's, the room temperature structure of γ-Na₂CO₃ resisted any attempt for a precise structural analysis. The appearance of satellite reflections was noted on single crystal diffractograms which led to the generalisation of the concept of crystal. This generalisation consisted in using at least four integers to fully characterise each individual diffractions peaks. The theory of periodic crystals in space of higher dimension, i.e. the superspace was then developed to deal with the new experimental observations. Later, a new class of materials called composite crystals and still later, the discovery of quasicrystals only reinforced the validity of the superspace concept to describe any material requiring more than three integers to index their diffraction pattern.

What is the essence of superspace to describe crystalline structures? Any crystal structure requesting more than three integers to index its diffraction pattern can be described as a periodic object in superspace with dimension equal to the number of required integer. The structure observed in our real word is a three dimensional cut of the superspace description. In general this cut is irrational and consequently the crystal is aperiodic. Calaverite and γ-Na₂CO₃ are examples of aperiodic crystals which includes incommensurately modulated crystals, composite and quasi-crystals. Rational cuts are also possible. In this case, the structure is periodic and is usually called a superstructure.