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Symmetries in Crystal Structure

Introduction of Crystal structure

The solid substances existing in nature can be divided into two categories: crystalline and amorphous. The essential difference between crystalline and amorphous structures is how the atoms, ions, molecules, and other particles that make up the structures are arranged. Crystals are regularly arranged (long program), while the particles in amorphous substances are stacked together irregularly (short program). In the field of crystallography, researchers study the symmetry of crystal structures in order to perform data collection and categorize substances. However, an understanding of the group structure needs to be added to the chemistry-only approach to crystallography. Through group theory and rigid motion analysis, mathematics allows us to organize crystal structures.

How Symmetry is used to Categorize Crystals

Crystals are a classification of solids. Any solid with a well-defined atomic pattern visible on its surface is a crystal. Having a well-defined pattern of atoms forces the crystals to have translational symmetry. This means you can take an internal piece of a crystal and translate it, which will perfectly overlap another section of the crystal. The core principle of crystal symmetry is that the pieces must be translationally symmetric.

We discussed group theory with symmetry in class and dealt with many kinds of rigid motions. A rigid motion is any motion that preserves the size and shape of the moving object. There are four rigid motions in two dimensions: translations, rotations, reflections, and glide reflections. An example of a motion that is not rigid would be squeezing a square or pattern or extending it. This would not retain the object’s shape and size, so it is not a rigid motion. The tricky part is figuring out which rigid motions can be applied to a given crystal structure to classify it. However, we can start defining crystals based on their symmetry once we figure this out.

A chemist and a mathematician would approach this problem very differently. A chemist would memorize what a crystal structure would look like and associate that directly with a type of crystal. On the other hand, a mathematician would classify a crystal structure according to group theory. The best way to approach this problem mathematically is first to isolate the smallest part of the pattern that would be able to recreate the whole pattern, given enough rigid motions. When choosing this piece of the pattern, ensure not to include any duplicate sections of the pattern. Next, draw as many axes of reflection as possible (if you are working in three dimensions, these would be planes), and find all roto centers with rotational symmetry. These are the basic motions, but they can be combined to form new motions, such as the glide reflection. Furthermore, if we expand our view to the third dimension, we can see that we get even more complex motions, such as the screw/helical axes. These are combinations of rotation, translation, and sometimes reflections.

The use of Group Theory for Classification

Group theory addresses how a certain collection of mathematical “objects,” in this case, crystals, are related. Each crystal structure has its own group of rigid motions that can be applied while preserving translational symmetry, allowing us to categorize them. For group theory, we also require an operation that can be applied to the elements of a group and elements. An element would be a rigid motion that preserves translational symmetry, and the operation is a composition of two elements. For instance, take the first element, apply it to the structure, then apply the second element. This is the same as applying the composition of the first and second elements of the group to the structure.

There are also classifications of these groups. For instance, if there is a 4-fold rotational axis in the crystal structure, which means that it has an axis with four rotations that preserve translational symmetry, then that means that the crystal is a tetragonal system. Other classifications of crystal symmetry groups are isometric, hexagonal, orthorhombic, monoclinic, and triclinic.

Ethics in Crystallography

Crystallography is often used in novel research, and our understanding of crystals is constantly improving. Also, almost any material can be crystalized, so crystallography applications are widespread. Moreover, the classification of crystals, which is what we are doing by analyzing crystal symmetry, is essential because many properties of crystals are determined by what groups they fall under or what types of symmetry they have. Crystallography has often been used to find important information about the world around us, such as the applications of x-rays, the discovery of penicillin, or the way DNA replicates.

Many people believe that the next steps in scientific discovery lie behind the discovery of new materials with more desirable properties. This is important to crystallography because the symmetry groups of a crystal affect its tensor groups, which describe a material’s physical properties. With that in mind, crystallography will continue to be a critical area of study in the near future. With the many benefits of crystallography research, it is hard to find enough downsides to outweigh the interest in the area. However, a couple of problems arise, such as the legitimacy of synthesized crystals, the environmental impact of crystal mining, and how crystal synthesizing affects job availability in different parts of the world.

While synthesized crystals have been proposed as an alternative to mined crystals for uses such as jewelry, many people oppose the notion of lab-made crystals and gems. Some people refuse to buy synthesized crystals because they mean less or are less legitimate, while others say it is unethical. The reliance on synthesizing crystals can take away jobs and devastates communities that depend on mining as the primary driver of their economies. On top of that, synthesizing crystals produces more CO2 emissions than mining does. So the positive impact on the environment is still under debate. However, the mining industry has recently declined due to publications and movies about the unethical practices of many mining companies, such as child labor, terrible pay, and dangerous working conditions. Thus, whether ethical or not, we are seeing a steadily growing percentage of crystals made in labs rather than mined.