Crystallography and mathematics
Crystallography is the scientific study of the orderly arrangement of particles within solid materials. This field has its roots in the early 20th century, primarily through the development of X-ray diffraction techniques by Max von Laue. Crystalline solids are characterized by having distinct melting points, a feature that sets them apart from amorphous materials like glass. While most materials tend to adopt a crystal form due to its energetically favorable state, imperfections are common in real crystals.
Mathematics plays a crucial role in crystallography, as it is used to describe and classify various crystal structures and tackle complex problems related to packing and arrangement. The foundational concept in crystallography is the lattice, specifically the Bravais lattice, which is essential for understanding the periodic nature of crystal arrangements. There are 14 types of lattices, each defined by specific translation directions that allow the crystal structure to repeat itself in three-dimensional space.
Moreover, the symmetries inherent in crystal structures are mathematically analyzed, leading to the identification of 230 distinct symmetry groups in three-dimensional space. Notably, certain substances like snowflakes exhibit a fascinating variety of forms while maintaining underlying symmetry, demonstrating the complexity and diversity within the realm of crystallography. New discoveries, such as quasicrystals, have expanded the understanding of crystalline structures by revealing materials that defy conventional translational symmetry.
Subject Terms
Crystallography and mathematics
Summary: Various mathematical principles are inherent in the structure of crystals and are used to study and classify them.
Crystallography is the study of the periodic structural arrangements of particles in solids. The first discoveries of the crystallographic structure of materials were made in the early twentieth century with the X-ray diffraction technique pioneered by Max van Laue. Solids that have crystal structures have a sharp melting point, which distinguishes them from amorphous substances, such as glass, which has neither a sharp melting point nor a crystal structure.
All matter tends to crystallize, since a crystal form is the lowest energy state. In reality, most physical crystals will have flaws rather than a perfect geometric structure. The chemical composition of a substance does not determine its crystal form. Calcareous spar, for example, has at least three distinct crystal types. Although crystals exist in three dimensions, some substances, such as graphite, form strong bonds between molecules in a plane, and only weak bonds between parallel planes. Mathematics is inherently connected to crystallography, as mathematicians describe and classify crystal structures and also use crystallographic methods to solve mathematical questions, such a packing problems. Despite almost a century of the existence of the modern science of crystallography, scientists do not have a good understanding of how local ordering principles produce large-scale order.
Lattices
The first consideration in crystal structure is the lattice, also known as the Bravais lattice, after August Bravais. There are 14 types of lattices. In a crystal structure, a translation is a motion in space in a certain direction through some distance. The arrangement of atoms, ions, and molecules must be periodic, and there must be three nonunique axes of translation. An axis of translation specifies a direction in which the structure repeats. If the whole structure is moved the proper distance in the direction of an axis, it will exactly cover itself. The lattice can be considered to be all the points to which any given particle can be translated by a translation, which also moves the entire crystal structure onto itself. Thus, the lattice consists of all the points that a given point or particle is moved to by a translation. From every point in the lattice, the view of the rest of the crystal is exactly the same. The portion of the crystal obtained by starting with a particle and moving it the smallest possible distance in each of the three translation directions is known as the unit cell.
Symmetries in Crystals
The geometry of a crystal structure is characterized by its symmetries. Besides translations, other symmetries include reflections in a plane, rotations through an angle about an axis, glide reflections (translation combined with a reflection), and screw translations (translation with a rotation). A crystal structure can only have rotations that are one-half, one-third, one-fourth, or one-sixth of a complete revolution. Mathematically speaking, two crystallographic structures are the same if their symmetries are the same. A collection of symmetries for an object is called a “symmetry group.” Yevgraf Federov and Arthur Schoenflies, in the late 1800s, independently discovered that there are 230 distinct crystallographic symmetry groups in three-dimensional space.
Other Crystals
Wilson Bentley provided a wealth of insight into the structure of snow crystals using a photographic microscope, taking thousands of photographs of individual snowflakes over the course of 50 years. His photographs show that although snowflakes always have a basic hexagonal symmetry, they exhibit an endless variety of detail and seem to have a limitless number of forms. The simpler snowflakes grow slowly at high altitudes in low temperatures, and the more complex ones form at higher temperatures at greater humidity. Besides direct examination, information about the structure of snowflakes has been deduced by the forms of halos that they cause around the sun and moon.
In recent years, substances such as various aluminum alloys have been discovered to have regularity of structure but no translational symmetry. These substances are called “quasicrystals,” and unlike true crystals, they can have 5-fold, 8-fold, 10-fold, or 12-fold rotational symmetry.
Bibliography
Bentley, W. A., and W. J. Humphries. Snow Crystals. New York: Dover Publications, 1962.
Burke, John G. Origins of the Science of Crystals. Berkeley: University of California Press, 1966.
Engels, Peter. Geometric Crystallography. Dordrecht, Holland: D. Reidel, 1986.
Kock, Elke, and Werner Fischer. “Mathematical Crystallography.” http://www.staff.uni-marburg.de/~fischerw/mathcryst.htm.
Lord, Eric A., Alan L. Mackay, and S. Ranganathan. New Geometries for New Materials. Cambridge, England: Cambridge University Press, 2006.