What is Crystallography?

Crystallography Defined

What is crystallography?

Crystallography is concerned with the laws governing the crystalline state of solid materials, with the arrangement of atoms in crystals, and with their physical and chemical properties, their synthesis and their growth. (Borchardt-Ott, 2011)

Mineralogists study crystallography because they study minerals, which are, by definition, naturally occurring crystalline materials. Many gemstones are also minerals, so gemologists have an interest in crystallography, too.

The Development of Crystallography

People have been interested in crystals for millennia, but here we will discuss the scientific study of crystalline materials and their properties.

Morphology

Crystallography began as the study and description of the external appearances of crystals. This is known as morphology. Researchers measured the angles between the faces of crystals and created organized principles about crystal shapes based on their findings.

A six-cornered snowflake inspired the German astronomer and mathematician Johannes Kepler (1571-1630) to consider the material's inner structure, which always resulted in six-rayed snowflakes. He compared the arrangement of particles within matter to how vendors stack fruits and vegetables in boxes to display them in markets.

The Danish scientist Nicolas Steno (1638-1686) formulated a fundamental crystallographic principle: the law of the constancy of interfacial angles. It states that the angles between the crystal faces of a particular mineral species are constant, regardless of the crystal's shape, size, or origin. For example, the interfacial angles of morganites, heliodors, and aquamarines are the same, even though they have different colors, because they are all varieties of the beryl mineral species.

Scientists began explaining how the outward appearance of a crystal strongly depends on its chemical composition and arrangement of atoms in crystal lattices. This led to the development of crystallography as a mathematically based scientific field.

Symmetry

You can find symmetry in many objects, from flowers and butterflies to art and architectural designs. Crystals are highly symmetrical. Scientific attempts at explaining symmetry mathematically, combined with studying the atomic structure of matter, gave rise to crystallography.

Symmetry operations and elements are essential in crystallography. While the human body, for instance, displays bilateral symmetry with a mirror plane as its symmetry element, crystals exhibit higher symmetries with multiple symmetry elements, including mirror planes, symmetry centers, and rotation axes. (We will discuss these elements later).

A New Era in Crystallography

The discovery of X-rays by the German physicist W. C. Röntgen in the late 19th century marked a turning point in the field of crystallography. This breakthrough enabled scientists to observe the internal patterns of crystals and better understand the bonds between atoms. This knowledge has made it possible to predict matter's physical and optical properties. Crystallography was no longer merely descriptive.

Modern Crystallography

Nowadays, many disciplines besides mineralogy and gemology also study crystalline materials and their properties, including medicine, microbiology, physical chemistry, materials science, and more. Understanding how the arrangement of atoms in crystals can impact a material's chemical and physical properties is crucial to medicine and materials science. This knowledge enables scientists to create materials with desirable traits.

Modern crystallographic research concentrates on examining internal crystal structures. Advanced spectroscopic and X-ray techniques used in materials science employ the fundamentals of crystallography. Searching for "crystallography" resources online will find research papers on DNA and polymers, not just mineralogy.

How Mineralogists and Gemologists Use Crystallography

Understanding crystallography as a science and crystallographic elements like crystal faces and interfacial angles can help mineralogists and gemologists identify mineral species.

Crystal Systems

Scientists divide crystals into seven systems based on their number of axes, length, and angles.

We can describe each system with a formula. The axes of a crystal are represented by the symbols a, b, and c. The angles at which they meet are represented by the symbols α, β, and γ.

Axes with the same length are equal (=). If they have different lengths, they are not equal (≠).

Angles with the same measurements are equal (=). If they are not the same, they are not equal (≠). We can also specify if certain angles have specific measurements, such as 90º or 120º.

Thus,

• Triclinic system: a ≠ b ≠ с, α ≠ β ≠ γ
• Monoclinic system: a ≠ b ≠ с, α = γ = 90º ≠ β
• Orthorhombic system: a ≠ b ≠ с, α = β = γ = 90º
• Trigonal system: a = b = c, α = β = γ ≠ 90°
• Tetragonal system: a =b ≠ с, α = β = γ = 90º
• Hexagonal system: a = b ≠ с, α = β = 90º, γ = 120º
• Isometric or cubic system: a = b = с, α = β = γ = 90º

For more information on crystal systems, consult What are Crystal Systems and Mineral Habits.

Understanding the properties of crystal systems can help you identify minerals. For instance, if you come across a highly symmetrical crystal with identical axes, lengths, and angles, it likely belongs to the isometric crystal system.

Isotropy and Anisotropy

The properties of minerals stem from their chemical composition and atomic arrangement. One principal difference between minerals is whether their various properties (optical, electrical, magnetic, physical) are the same in all crystal directions. Crystals with the same properties in all directions are called isotropic. Only isometric crystals are isotropic because all their axes and angles are equal. Crystals whose properties differ along different crystal directions are called anisotropic. Crystals from all systems except isometric are anisotropic. All non-isometric crystal systems have some differences in their axes lengths or angles.

Mineralogists and gemologists can use this information when identifying minerals. Properties such as pleochroism (the display of two or three different colors depending on the viewing angle) and birefringence (double refraction) can only occur in anisotropic minerals. Mineral specimens that show the same color regardless of the viewing angle and have only a single refractive index must be isotropic. For example, a transparent red stone that shows no other colors when viewed from different angles might be a garnet or a spinel but can't be corundum (ruby). Corundum forms in the trigonal crystal system, which makes it anisotropic. Of course, more tests are needed for a definitive identification, but such an observation is an excellent first step.

Understanding isotropy and anisotropy can also benefit gem cutters. When choosing an orientation for an anisotropic gemstone, they must consider the crystallographic axes before cutting. This will determine which pleochroic color will be emphasized and how harmoniously the other colors will appear. This is extremely important when cutting and faceting tanzanites, rubies, emeralds, kyanites, and tourmalines.

This gem structure chart summarizes the isotropic or anisotropic properties of the different crystal systems and gives some well-known mineral/gemstone examples.

Crystal Faces

The orientation of striations or grooves on the crystal faces of some minerals can help identify them. Some minerals, such as olivine, scapolite, topaz, tourmaline, and zircon, have crystal faces that show striations running "vertically" or parallel to the direction of crystal elongation. Other minerals, such as corundum, rutile, and quartz, have crystal faces that show striations running "horizontally" or perpendicular to the direction of crystal elongation. All of these examples are anisotropic. Pyrite crystal faces can also show striations, but they have no direction of crystal elongation since they are isotropic.

Petrology, the branch of geology that studies rocks and their formation, also utilizes crystallography. For example, petrologists can estimate the pressure and temperature at which zircon crystallized and, thus, the depth of its rock body formation and other genetic conditions by examining the arrangement of prismatic and pyramidal faces on zircon crystals. (Pupin, 1980)

Basic Crystallography Terms for Mineralogists and Gemologists

Understanding the following basic crystallographic terms is important for novice mineralogists and gemologists.

Unit Cells and Crystal Lattices

A mineral crystal is comprised of atoms, molecules, and ions that fit together in repeating patterns. The unit cell is the smallest possible unit that reflects a mineral's composition and symmetry.

Symmetrical, three-dimensional structural arrangements of atoms, ions, or molecules create a mineral's crystal lattice.

Symmetry Elements and Operations

Mathematical concepts of symmetry can describe how unit cells can be arranged in space. There are symmetry elements and symmetry operations that arrange atoms in various ways.

Symmetry elements are:

• Symmetry center
• Mirror plane
• Rotation axes

Symmetry operations are:

• Rotation (1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotation axes)
• Reflection
• Inversion

Point Groups, Bravais Lattices, and Space Groups

In crystallography, scientists use the term point group to refer to the number of symmetry elements characteristic of a crystal. Only 32 point groups are allowed in crystalline matter based on symmetry elements and symmetry operation combinations. These 32 point groups are also known as the 32 crystal classes in crystallography. 

At the same time, only 14 different types of unit cells or Bravais lattices with specific atomic arrangements can exist. 

Combining 32 point groups with 14 Bravais lattices and considering all possible space group operators, we get 230 possible space groups.

Crystal Habits

Crystal habit refers to a crystal's characteristic macroscopic shape. There are many different descriptive terms for the appearance of crystals. However, here are three fundamental types of habits: equant, planar, and prismatic.

Equant

Crystals with an equant habit have approximately the same height, width, and length. Examples of minerals with equant crystal habits include fluorite, garnet, and pyrite.

Planar

Crystals with a planar or tabular habit are flat, with height and length greater than width. Micas (muscovite and lepidolite) are good examples of minerals with planar crystal habits.

Prismatic

Crystals with a prismatic habit have an elongated, prism-like shape. Their height is greater than their length or width. Examples of minerals with prismatic crystal habits include tourmaline and beryl.

Crystal Forms

The microscopic order of unit cells, symmetry elements, and operations produce the crystal shapes we can observe macroscopically. There are 47 fundamental crystal forms.

The more symmetry elements a crystal has, the more complex its form. Some crystals can occur in simple forms like octahedrons. However, crystals frequently occur in combinations or "mutual germinations" of two forms. For example, quartz's crystal morphology combines two forms: a hexagonal prism and two hexagonal pyramids at the prism's terminations (or ends). Some minerals can also grow into crystals with different forms. However, all the various forms of a mineral species will still have the properties of its crystal system.

The overall morphology of a crystal is the total collection of forms that characterize the crystal.

The following are the crystal forms that make up macroscopic crystal shapes.

Pedion

Also called a monohedron, a crystal form with only a single face.

Dihedron

A type of polyhedron made of two polygonal faces. It looks like a simple house roof. It may be a sphenoid or a dome.

Note: pedions and dihedrons are purely mathematical constructs. There are no naturally occuring pedions or dihedrons.

Pinacoid

Also called a parallelohedron, a crystal form consisting of two parallel and opposite faces.

Rhombic Disphenoid

A type of tetrahedron with four identical scalene triangular faces.

Tetragonal Disphenoid

A type of tetrahedron with four similar isosceles triangular faces.

Pyramid

A pyramid has a polygonal base and flat triangular faces that join at a common point called the apex. There are seven types of pyramids: rhombic pyramid, tetragonal pyramid, ditetragonal pyramid, trigonal pyramid, ditrigonal pyramid, hexagonal pyramid, and dihexagonal pyramid.

Two pyramids joined by their bases create a dipyramid or bipyramid. There are also seven types of dipyramids: rhombic dipyramid, tetragonal dipyramid, ditetragonal dipyramid, trigonal dipyramid, ditrigonal dipyramid, hexagonal dipyramid, and dihexagonal dipyramid.

Prism

A form with two identical polygonal bases on opposite ends, with all the sides that join the bases forming parallelograms. There are also seven types of prisms: rhombic prism, tetragonal prism, ditetragonal prism, trigonal prism, ditrigonal prism, hexagonal prism, and dihexagonal prism.

Trapezohedron

A form with trapezoidal faces. There are three types of trapezohedrons: trigonal trapezohedron, tetragonal trapezohedron, and hexagonal trapezohedron.

Rhombohedron

A form with six rhombic faces.

Scalenohedron

A hemihedral form bounded ideally by scalene triangles. There are tetragonal and ditrigonal scalenohedrons.

Cubic System Crystal Forms

The crystal forms of the cubic system are complex. The simplest forms are the hexahedron (cube), tetrahedron, and octahedron.

Their modifications are the tetartoid, pyritohedron, deltoid-dodecahedron, tristetrahedron, rhomb-dodecahedron, trisoctahedron, trapezohedron, gyroid, hexatetrahedron, tetrahexahedron, and hexoctahedron.

How to Identify Crystal Symmetry

Identifying the symmetry of rough crystals can prove challenging. As you've seen in some of the previous photos, some crystal faces and junctions may be damaged. This can make identifying some symmetry elements difficult. Ideally, you should observe and record all symmetry elements in a crystal (mirror planes and rotation axes) to determine one of the 32 possible point groups. However, this process can be too complicated for beginners and impractical for analyzing actual crystals that don't form in perfect conditions.

The following method focuses on rotation axes, which can speed up the identification process.

Rotation Axes Method

Hold your crystal specimen between your index finger and thumb. If the crystal has a prismatic habit, hold it so you can view the prism from the side, not from its terminations.

1. Rotate the crystal and note if it has a 6-fold axіs. In other words, how many times do you see the same prism face shape? If you see the same shape six times during one complete rotation, about every time you turn the crystal 60º, it has a 6-fold axis. A 6-fold axіs and elongated prismatic habit are signs of the hexagonal crystal system.
2. If the crystal doesn't have a 6-fold axis, look for a 3-fold axіs. This means you can see the same arrangement of faces three times during one complete rotation, about every time you turn the crystal 120º. If this occurs, see if the crystal has only one 3-fold axіs. One 3-fold axis indicates the trigonal crystal system. If you find more than one 3-fold axis and the crystal looks equant, it belongs to the cubic crystal system.
3. If you haven't found 6 or 3-fold axes, look for a 4-fold axis. This means the crystal shows the same arrangement of faces four times during one complete rotation, about every time you turn the crystal 90º of rotation. This indicates the crystal belongs to the tetragonal crystal system.
4. If you still haven't found rotation axes, look for a 2-fold axis. The crystal face arrangement repeats itself twice during one complete rotation, about every time you turn the crystal 180º. This may indicate the orthorhombic crystal system. Next, check that the crystal has two mirror planes — the imaginary plane that cuts the crystal in half — and then confirm if the right and left sides are equal.

What if Your Crystal has no Rotation Axes?

If the crystal has no rotation axes, it may be monoclinic or triclinic. Monoclinic crystals have a mirror plane; triclinic crystals have none.

References for Crystallography

1. Borchardt-Ott, Walter. Crystallography: an Introduction. Third Edition. Springer Science & Business Media, Heidelberg, Dordrecht, London, New York, 2011.
2. Pupin, J. P. "Zircon and granite petrology." Contributions to Mineralogy and Petrology, 73.3 (1980): 207-220. (Accessed 11/3/24 )

Olena Rybnikova, PhD

Olena Rybnikova is a gemologist and mineralogist. She has a PhD in mineralogy and petrology specializing in beryllium minerals and is a certified Applied Jewelry Professional accredited by the Gemological Institute of America. Her passion is actively promoting knowledge and appreciation of nature, geology, and gemstones.

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