3.1 The lattice

Moving from interactions of quanta and matter, let us consider how atoms are arranged in matter. Atoms exist in a quantum-level universe as well, and repel each other if they are placed too close, but attract each other at longer distances. They can also create lower-energy structures by sharing electrons (chemical bonding) to form molecules or for unbonded charged objects, a pack method that interleaves species with opposite charges in three dimensions will reduce the energy. (This is considered via the Madelung potential). In general, enthalpy is lowered when atoms and molecules self-arrange in dense and regular three-dimensional patterns. Opposing this is entropy, which is increased by randomness. In the macroscopic world that we live in, entropy and enthalpy balance at equilibrium, and all materials, even those well away from equilibrium, exist with both order and disorder present. Nonetheless, it is worthwhile considering the properties and symmetry of a perfectly organized set of atoms or molecules, which we will call an ideal crystal, even though it is an imaginary concept; no perfectly ordered materials exist.

In an ideal crystal, atoms occur only at a set of fixed distances from each other, and the structure is built up of repeating units. For the purposes of this introduction, I will not consider two important forms of crystalline matter, quasicrystals and modulated structures, as these require descriptions in more than three dimensions. Quasicrystals are composed of atoms that organize in long-range patterns, but these patterns do not repeat in regular units. In 1982 Dan Shechtman found clear evidence for such a material, and despite rejection from a significant and highly vocal segment of the scientific community (typified by a comment attributed to Linus Pauling that “there are no quasicrystals, only quasi-scientists”) continued to speak to what he had clearly observed in an electron microscope. The existence of quasicrystals is now fully accepted, and Shechtman was awarded a solo Nobel prize in 2011. Similarly, in modulated structures, incommensurate ordering is superimposed on regular three-dimensional patterns of repetition. While initially, the definition of what we would call crystalline matter was tied to the existence of long-range repetition units of atoms, now the definition has been expanded to any matter that shows long-range atomic organization through the appearance of a sharp diffraction pattern. Since my goal here is to explore the properties from long-range repetition units of atoms and show how sharp diffraction patterns arise from this, I will only consider classical three-dimensional ideal crystals and will use them to introduce two initial concepts, the lattice and the reciprocal lattice.

Our imaginary idealized crystal is built of repeated three-dimensional identical boxes of atoms. At this stage, we do not need to concern ourselves with how the atoms are arranged within each box, but we will stipulate that identical means this arrangement is repeated exactly in every box. The crystal is built up by these boxes stacked in every direction. It has been known since the 1800’s that there are seven shapes of boxes that can be used to build a larger object without any gaps. In the simplest example this box would be a cube – having equal length sides and all angles between sides as multiples of 90 degrees. However, these requirements on dimensions and angles can be relaxed, giving rise to a total of seven options, as listed in the Table 3.1;¹ this gives rise to what is known as the seven crystal systems. These boxes are all parallelepipeds, meaning the boxes are six-sided objects where there are three pairs of parallel sides. Note that while one can tile space with hexagons (or in three dimensions, hexagonal prisms), we consider the hexagonal crystal systems to be constructed from parallelepipeds with two equal sides that have a 120° angle between them and where the third side is perpendicular to the other two. Three of these parallelepipeds can construct hexagonal prisms. Thus, this parallelepiped description is equivalent to one built from hexagonal prisms.

Armed with the concept of crystal systems, we can define an important term in crystallography, that of a lattice. Let us label one arbitrary point in the box. The point we select might be a corner but could also be some location inside the box. For that selected point, we can envision that there will be a fixed arrangement of atoms around that point. Should we move to the same point in any other box, the arrangement of atoms will be identical since the boxes are all identical. If we then imagine selecting that same point in every box, we construct an infinite network of indistinguishable points. This infinite network of points is called the lattice. For a given arrangement of atoms, there are any number of ways to construct the lattice, since we can choose any point in the box as our reference point. We could also choose a more sparse network of points, such as selecting every other point in some direction and thus enlarge our box, but still we can decorate our infinite crystal from that lattice. Thus, there are an infinite number of possible lattices associated with a structure. We will see later that only some will better describe the symmetry of the structure. The term lattice is commonly misused to describe the structure or chemical makeup of the crystal, with poorly considered terms such as “lattice compound.” In truth, a lattice is just a mathematical concept, simply a collection of points.