Crystallography

Crystallography is a branch of geometry that deals with indefinitely repeating patterns. Two-dimensional crystallography can be used, for example, to describe the way tiles cover a floor. Extending the field into three dimensions allows a general description of the way atoms or molecules arrange themselves into crystals. The three-dimensional crystallography was proven to be complete over a century ago. The fact that the mathematics itself cannot be advanced without some change of its axioms has meant that it is studied less often as pure mathematics, than as a means of understanding the details of complex structures in matter. Two fields are particularly reliant on it: materials scientist use it to describe the structure of engineering materials, often with particular attention to crystallographic defects; biochemists use it to describe the structure of biopolymers (see proteomics for an example), which usually must be processed laboriously before they form crystals. In mathematical terms, a crystal is an object with translational symmetry, i.e. it can be moved some distance and remain the same. This type of symmetry is fundamentally different from the more familiar mirror symmetry (the human face) or rotational symmetry (an airplane propeller), in that objects we imagine to represent translational symmetry must be larger than ourselves. We can imagine passing Alice through the looking glass, or spinning a propeller by one blade's fraction of a rotation, while we stand still. For an experience of translational symmetry, however, we must move ourselves, and not notice the difference. This can happen in an ocean, a desert, or a large suburb, if every wave, or dune, or tract home looks exactly like the next. Just as no eye is the exact mirror of its opposite, and no propeller is perfectly balanced, no physical crystal is perfect. There will always be a boundary that gets nearer or farther after a unit of translation. Strictly speaking, any true crystal must fill the entire universe.

Abstraction & categorizationEdit

If we imagine a "perfect" housing development (dystopian though it may be) which covers a two-dimensional plane with an infinitely-repeating pattern of homes, and want to apply crystallography to it, we can save a lot of work by eliminating all the geometric complexity of garages and sprinkler heads and such. To make things as simple as possible, we could abstract every house down to a single point, although we need to keep track of each house's orientation.

The set of all these points is conventionally known as a lattice, and it contains all information about repetition. It is defined as a set of points, each with an identical environment. To keep the description complete, we could also create a blueprint for a generic home in this development. This second set of information is called the motif, and it describes what is repeated. As long as each point on the lattice is taken to be the same point on the blueprint for every house, it doesn't matter where that point is. It could be the top of the gable, the doorknob, the house's cener of gravity, or one particular corner of the lot. It could even be a point several miles from the house, but since all lattice points are the same by definition, it's more convenient to choose a point that's closer. In any case, the combination of lattice and motif will still produce the same result.

The definition of a lattice imposes some restrictions which may not be obvious at first. While there are many types of symmetry, only a few can be applied to a true lattice. For instance, while rotational symmetry exists for any fraction of a revolution, only 2-, 3-, 4-, and 6-fold rotations will allow all the points of a lattice to align with one another.

NotationEdit

Throughout this wikibook, we will use standard crystallographic notation to describe points, directions, and planes according to the crystallographic lattice. All of these geometric concepts are associated with vectors, but the special properties of an infinite lattice mean that standard Cartesian coordinates are not always the most helpful. The origin should be a point on the lattice, and the basis vectors should be chosen to connect lattice points. Furthermore, the space we create should be modular, to reflect the fact that translation to a new lattice point is congruent to no translation at all. See High School Mathematics Extensions/Primes/Modular Arithmetic for a review of the concept of modularity.

Constructing a modular space also frees us to consider only a small area (or volume) the size of the separation between lattice points, rather than an infinite space. Conceptually, this small space has sharp boundaries where passing out one side is the same as passing in from the opposite side, like a game of Asteroids.

The basis vectors are chosen to connect lattice points. The only requirement is that enough vectors are chosen that the set of vectors spans the space; in three dimensions, any three vectors that don't lie in the same plane are enough.

The Cartesian coordinate system is convenient because it is orthornormal; it saves a lot of work if we construct our space using vectors that are orthogonal, or normalized, or both, but the highest priority is that the basis vectors connect lattice points. Fortunately, many common crystals (silicon, iron, copper, diamond, table salt) are cubic, which allows the use of an orthonormal basis set. For non-cubic crystals, we must construct a metric tensor in order to make use of the full range of vector operations. Fortunately, the list of all possible metric tensors for crystal lattices is relatively short and easy to generate.

Having defined the space in which a crystal lattice sits, we can then represent vectors by ordered sets of numbers. These numbers are presented differently depending on what the vector means:

Position                (u,v,w)
Direction               [u v w]
Family of directions    <u v w>
Plane                   (u v w)
Family of planes        {u v w}

In all these cases, a negative value is represented by adding a bar over the term. For instance, [1 2 0] is the direction found by adding \vec{a} -2\vec{b}.

Last modified on 19 December 2011, at 10:37