# 11 – Crystallography

**KEY CONCEPTS**

- All crystals are made of basic building blocks called unit cells.
- Unit cells may have any of 7 fundamental shapes.
- Unit cells fit together in one of 14 ways to make crystals.
- We make inferences about unit cell shape and lattice type based on crystal habit and symmetry.
- The symmetry of an atomic structure depends on unit cell shape, the lattice, and the locations of atoms in the unit cell.
- There are 230 possible atomic-structure symmetries that can only be distinguished using X-ray techniques.
- Crystals contain one or several different kinds of crystal faces.
- We differentiate crystal faces based on their orientations with respect to a coordinate system based on unit cell edges.

## 11.1 Observations in the Seventeenth through Nineteenth Centuries

In 1669 Nicolaus Steno studied many quartz crystals and found angles between adjacent prism faces, termed *interfacial angles*, to be 120^{o} no matter how the crystals had formed. For example, Figures 11.2 to 11.5 show four varieties of quartz with nearly identical crystal shapes and angles between faces. Steno could not make precise measurements, and some of his contemporaries argued that he was overlooking subtle differences.

blank

A century after Steno, in 1780, more accurate measurements became possible when Arnould Carangeot invented the *goniometer*, a protractor-like device used to measure interfacial angles on crystals. Carangeot’s measurements confirmed Steno’s earlier observations. Shortly after, Romé de l’Isle (1782) stated the first law of symmetry, a law called the “constancy of interfacial angles,” which we commonly call *Steno’s law*. This law states that:

blank■ *Angles between equivalent faces of crystals of the same mineral are always the same.
*

Steno’s law acknowledges that the size and shape of the crystals may vary.

In 1784 René Haüy studied calcite crystals and found that they had the same shape, no matter what their size. Haüy hypothesized the existence of basic building blocks called *integral molecules* and argued that large crystals formed when many integral molecules bonded together. Haüy erroneously concluded that integral molecules formed basic units that could not be broken down further. At the same time, Jöns Jacob Berzelius and others established that the composition of a mineral does not depend on sample size. And Joseph Proust and John Dalton proved that elements combined in proportions of small rational numbers. Scientists soon combined these crystallographic and chemical observations and came to several conclusions:

blank• Crystals are made of small basic building blocks.

blank• The blocks stack together in a regular way, creating the whole crystal.

blank• Each block contains a small number of atoms.

blank• All building blocks have the same atomic composition.

blank• The building block has shape and symmetry that relate to the shape and symmetry of the entire crystal.

Figure 11.6 is the same figure we saw in Chapter 1. It shows the arrangement of atoms and basic building blocks in a fluorite crystal. Notice that the overall crystal has a sort of cube shape, and the building blocks are also cubes. This relationship between crystal symmetry and building block symmetry is at the heart of crystallography.

Early in the 1800s, several researchers found that crystals of similar, but not identical, chemical composition could have identical shapes. W. H. Wollaston (*c.* 1809) showed that calcite (CaCO_{3}), magnesite (MgCO_{3}), rhodochrosite (MnCO_{3}), and siderite (FeCO_{3}) all commonly formed the same distinctive rhombohedron-shaped crystals (Figures 11.7, 11.8, 11.9, and 11.10).

Those who studied sulfate compounds also found that crystals of different compositions had the same crystal shape. Both the rhombohedral carbonates and the sulfates are examples of* isomorphous series*. Wollaston and others concluded that when crystal shapes in such series are truly identical, the distribution of atoms within the crystals must be identical as well, even if the compositions are not. Minerals with identical atomic distributions are termed* isostructural* even if (unlike carbonates or sulfates) they have significantly different compositions.

Sometimes isostructural minerals form solid solutions because they can mix to form intermediate compositions. Fayalite (Fe_{2}SiO_{4}) and forsterite (Mg_{2}SiO_{4}) are isostructural and form a complete solid solution; olivines can have any composition between the two end members. In contrast, halite (NaCl) and periclase (MgO) are isostructural but do not form solid solutions. Calcite, magnesite, rhodochrosite, and siderite are isostructural but their mutual solubility is limited. They form only limited solid solutions, also called *partial solid solutions*.

In 1821 Eilhard Mitscherlich, a student of Berzelius’s, discovered that the same elements may combine in different atomic structures. For instance, calcite and aragonite both have composition CaCO_{3}, but they form different crystal shapes and have different physical properties. We call such minerals *polymorphs* because, although identical in composition, they have different atomic arrangements and crystal morphologies. Mineralogists have now studied several other CaCO_{3} polymorphs, but none except vaterite occur naturally. In calcite and vaterite the basic building blocks have rhomb-shaped faces, while in aragonite the faces are rectangles.

Figure 11.11 shows calcite and siderite crystals that do not have the same shapes as those above in Figures 11.7 and 11.10. Many minerals, especially carbonate minerals have multiple common habits. Figure 11.12 shows aragonite crystals and Figure 11.13 shows vaterite crystals. Both minerals, like calcite, have composition CaCO_{3}.

By the early/mid 1800s it was clear that there was no direct correlation between the shapes of building blocks and crystal composition, as Haüy had originally thought. Despite flaws in some of his ideas, however, we must credit René Haüy as one of the founders of crystallography. In later years, he pioneered the application of mathematical concepts to crystal properties which established the basis for modern crystallography and crystallographers still use much of his work today. We now accept that:

• All crystals have basic building blocks called *unit cells*.

• The unit cells are arranged in a pattern described by points in a *lattice*.

• The relative proportions of elements in a unit cell are given by the chemical formula of a mineral.

• Crystals belong to one of seven crystal systems. Unit cells of distinct shape and symmetry characterize each crystal system.

• Total crystal symmetry depends on both unit cell symmetry and lattice symmetry.

## 11.2 Translational Symmetry

In the previous chapter we discussed symmetry due to rotation, reflection, and inversion. These are all types of *point symmetry*. The orderly repetition of patterns due to *translation* is another form of symmetry, called *space symmetry*. Space symmetry differs from point symmetry.

Space symmetry repeats something an infinite number of times to fill space, while point symmetry repeats something a discrete number of times and only describes symmetry localized about a central point. Point symmetry operators return a dot or a crystal face to its original position and orientation after 1, 2, 3, 4, or 6 repeats of the operation, but space symmetry operators do not. Translation goes on (almost) forever!

We often envision translational symmetry by thinking about a *lattice*, a set of an infinite number of points related by translations. In two dimensions, a lattice is called a *plane lattice*. In three dimensions, a lattice is a *space lattice*. Whether two or three dimensional, lattices are imaginary – they are not physical entities – they are patterns that we use to describe translational symmetry.

Figure 11.14a shows an example of how we depict a plane lattice. The vectors *t _{1}* and

*t*are translation operators that generate the entire lattice from a single starting point. This drawing only shows part of the lattice; in your mind you should picture it extending infinitely in all directions.

_{2}Translational symmetry, like all symmetry, is a property that objects may have. If we have an object or a drawing, we can describe its translational symmetry. But it does not work the other way around because many different objects or drawings may have identical symmetry.

For example, Figure 11.14*b* (above) shows cats related by the same symmetry depicted in Figure 11.14*a*. Figure 11.14*c* shows parallelograms related by the same symmetry. And Figure 11.14*d* shows colored circles (that might represent atoms) related by the same symmetry.

The four drawings in Figure 11.14 have more than just translational symmetry. They also contain multiple 2-fold rotation axes perpendicular to the plane of the page. And there are inversion centers, too. So we see that translational symmetry can combine with other kinds of symmetry.

Repetitive patterns are composed of motifs, which are designs that repeat, such as a flower pattern that might be on wallpaper. The term *motif* is used in an analogous way in music to refer to a sequence of notes that repeat. Interior decorators or clothing designers use the term to describe a common theme or element in their work.

A motif (whether it comprises two cats, a parallelogram, or colored circles) and empty space around it, make up* unit cells* that repeat to create an infinite pattern. The lattice describes the nature of the repetition, with each lattice point representing one motif. But lattice points do not correspond to any particular point on a motif. We could choose them at the motif’s center or one of its corners and the result would be the same.

Figure 11.15 shows *unit cells* that can be used to generate the patterns seen in Figure 11.14. These unit cells are parallelograms with dimensions and angles that match the translation vectors of the lattice. Notice that each one of these unit cells has a vertical 2-fold axis of symmetry and an inversion center. This symmetry must be present to be consistent with the lattice.

## 11.3 Unit Cells and Lattices in Two Dimension

What possible shapes can unit cells have? Mineralogists begin answering this by considering only two dimensions. This is much like imagining what shapes we can use to tile a floor without leaving gaps between tiles. Figure 11.16 shows some possible tile shapes. In the right-hand column of this figure, a single dot has replaced each tile to show the plane lattice that describes how the tiles repeat. This figure does not show all possible shapes for tiles or unit cells, but as we will see later the number of possibilities is quite small.

What happens when gaps occur between tiles? Figure 11.17 shows the two possibilities: The gaps may be either regularly (11.17*a*) or randomly (11.17*b*) distributed. If regularly distributed, we can define a unit cell that includes the gap, as we have in drawing 11.17*a*. The two vectors, *t _{1}* and

*t*and the lattice they create describe how the unit cell repeats to make the entire tile pattern.

_{2}If gaps between tiles are random, the entire structure is not composed of identical building blocks that fit together in a regular way. It is not repetitive and we cannot describe it with a unit cell and a lattice. It does not, therefore, represent the symmetry of a possible crystal structure and we need not consider it further.

Various complex patterns can appear on tiled floors, but the tiles are usually simple shapes such as squares or rectangles. For example, suppose we wish to tile a floor in the pattern shown in Figure 11.18*a*. We could use L-shaped tiles, shown in red in Figure 11.18*b*. However, parallelogram- or square-shaped tiles would get us the same pattern (Figures 11.18*c* and *d*). Other shapes, too, would work.

We can make any repetitive two-dimensional pattern, no matter how complicated, with tiles of one of four fundamental shapes: parallelogram, rhomb (a parallelogram with sides of equal length), rectangle, and square. These shapes are relatively simple compared to more complex ones we could choose, and they reveal symmetry that is present. So, they are used by mineralogists and preferred by tile makers.

For reasons we will see later, we usually distinguish two types of rhombs (parallelograms with sides of equal length): those with *nonspecial angles* between sides and those with 60^{o} and 120^{o} angles between sides. For the rest of this chapter, we will refer to the general type of rhomb as a *diamond*, and the term *rhomb* will be reserved for shapes with only 60^{o} and 120^{o} angles (and sides of equal length). So, Figure 11.19 shows the five basic shapes, the only ones needed to discuss two-dimensional unit cells, and their symmetries.

We can now explain why we only considered 1-, 2-, 3-, 4-, and 6-fold rotational symmetry in the previous chapter. Two-dimensional shapes with a 5-fold axes of symmetry, for example, cannot be unit cells because they cannot fit together without gaps. We can verify this by drawing equal-sided pentagons on a piece of paper. No matter how we fit them together, space will always be left over. The same holds true for equal-sided polyhedra with seven or more faces. They cannot fit together to tile a floor. And if the shapes cannot fit together in two dimensions, they cannot fit together in three dimensions.

### 11.3.1 Motifs and Unit Cells

We can think of atomic arrangements as starting with one motif. The motif is then reproduced by translating (moving) it a certain distance and reproducing it. The distance of the translation is the distance between lattice points. If a dot replaces each motif, we get the lattice.

Figure 11.20 shows a pattern made of two atoms, symbolized by different sized orange circles. The entire pattern consists of motifs composed of one of each kind of atom. The pattern contains multiple horizontal mirror planes but no other symmetry.

The figure shows several possible choices of unit cells (labeled *a* through *d*). In the individual unit cell drawings, solid black lines show mirror planes of symmetry.

Unit cells *a*, *b*, and *c* contain only one motif in total (one small orange circle and one larger lighter orange circle). We call them *primitive* because they are the smallest unit cell choices possible. All primitive unit cells contain exactly one motif, but atoms within a primitive unit cell may be in parts as in unit cell *c*. Add up the parts and you get one motif. We term unit cell *d* *doubly primitive* because it contains two motifs. Triply primitive unit cells contain three motifs, etc.

We can always choose a primitive unit cell for a repetitive pattern of atoms, but sometimes a primitive cell does not show symmetry clearly. By convention, we usually choose the smallest unit cell that contains the same symmetry as the entire pattern or structure. In Figure 11.20, unit cell *b* would be the choice of most crystallographers because it is simple, primitive, does not contain any partial atoms, and contains a horizontal mirror plane (which is the same symmetry as the entire pattern). Note that unit cell *a*, while also being primitive, does not have a mirror plane of symmetry within it; the other choices do.

The pattern shown in Figure 11.20 has relatively simple symmetry; the patterns in Figures 11.21 and 11.22 have more. Figure 11.21 shows an orthogonal (containing lots of 90̊ angles) arrangement of atoms. The repeating motif contains six atoms – four green and two larger light blue ones. The letters *a*, *b*, *c*, and *d* designate four possible choices for unit cell. Cells *c* and *d* are primitive, containing six atoms total, although some are on the edges in cell d so only half of each green atom is in the unit cell. Cells *a* and *b* are doubly primitive but they are rectangular shapes. The symmetries of two of the unit cells are shown in the bottom of the figure. Solid black lines designate mirrors and lens shapes designate 2-fold axes. Unit cell *a*, although being double primitive, is the one that best shows the symmetry elements and so would be the choice of most crystallographers.

Figure 11.22 shows an overall hexagonal pattern. The motif consists of one large turquoise atom and two smaller green atoms. Four possible choices for unit cells are shown. Drawings in the bottom of the figure show symmetry of two of the potential unit cell with solid black lines as mirrors, lens shapes as 2-fold axes, and hexagons as 6-fold axes. Two of the potential unit cells are rhomb shaped and two are hexagonal. Because the overall pattern has hexagonal symmetry, unit cells *c* and *d* are preferred choices compared with the other two. Unit cell *d* is primitive, but does not include all the symmetry. (It does not have 2-fold rotation axes.). So cell *c* is the best choice. This pattern demonstrates why we treat rhombs with 60̊^{o }and 120̊^{o }angles at their corners as special cases – they sometimes result in hexagonal symmetry while diamonds with general angles at their corners cannot.

Although crystallographers follow standard conventions, they are often forced to make choices. If they choose primitive unit cells, each lattice point corresponds to one unit cell and one motif. If they choose nonprimitive unit cells, lattices represent the way motifs repeat, but not the way unit cells repeat because there is more than one lattice point per unit cell.

### 11.3.2 What are the Possible Plane Lattices?

Above we looked at a number of different examples of lattices and unit cells. To get a complete list of possibilities let’s look at all the possible plane lattices. In a plane lattice, lattice points are related by two vectors describing translation. The vectors may be *orthogonal* (perpendicular to each other) or not. And they may be equal length or not.

**Square Net:** Let’s consider the simplest possibility – that the vectors are orthogonal and of equal length. This produces a *square net *(lattice), as shown on the left in Figure 11.23, below. If we choose a square unit cell and put a motif (five atoms) with square symmetry within it, we get the pattern seen in the drawing on the right. This pattern has the same symmetry as the lattice and of the unit cell. The symmetry includes 4-fold and 2-fold rotation axes, and mirror planes that intersect at 45^{o} and 90^{o}. The small P next to the unit cell reminds us that it is primitive. Because the overall atomic arrangement contains a 4-fold axis and two differently oriented mirrors, we describe its symmetry as *P4mm*.

**Orthonet:** Suppose the two vectors relating lattice points are orthogonal but not of equal length. If so, we get an *orthonet*, equivalent to a *rectangular lattice*, as shown in Figure 11.24, below. We can choose a rectangular unit cell and add a rectangular motif that contains nine atoms. We get the pattern shown on the right in Figure 11.24. Note that the lattice, the primitive (P) unit cell, and the pattern have equivalent symmetry that includes mirror planes and 2-fold rotation axes. The overall atomic arrangement contains a 2-fold axes with horizontal and vertical mirrors; we describe its symmetry as *P2mm*.

**Diamond Net:** A third general possibility is that the vectors relating lattice points are of equal magnitude but intersect at a non-special angle. This combination produce a *diamond net* (lattice) like the one below in Figure 11.25. We can add a motif of six atoms to a diamond-shaped unit cell to get the atomic arrangement shown on the right in Figure 11.25. But, notice that we can also choose a rectangular unit cell that is doubly primitive – it has an extra lattice point at its center and contains two motifs. Either unit cell will generate the same pattern. We use the letter *P* to indicate the primitive unit cell, and the letter* C* to indicate the cell with an extra lattice point in its center. The rectangular unit cell shows the vertical and horizontal perpendicular mirror planes and the 2-fold rotation axes more clearly than the diamond shaped one. So, we describe the symmetry of the pattern as *C2mm*.

**Clinonet:** The fourth possibility is that the two translation vectors are of different lengths and are not orthogonal. This produces a *clinonet* (Figure 11.26) that corresponds to a primitive unit cell with the shape of a parallelogram. Adding a motif containing six atoms gives us the atomic arrangement on the right in Figure 11.26. There are no mirror planes here – the only symmetry is 2-fold rotation perpendicular to the page. The arrangement of atoms has symmetry *P2*.

**Hexanet:** The fifth possible net is the created by the special case when the two translation vectors are of equal magnitude and intersect at 60^{o}. This combination gives us a *hexanet* (hexagonal lattice) like the one shown in Figure 11.27, below. As with the diamond net, we can choose either a primitive rhomb-shaped unit cell or a triply-primitive hexagonal unit cell. The lattice and the overall atomic arrangement have hexagonal symmetry, so the hexagonal unit cell (which also has hexagonal symmetry) is generally chosen. A 6-fold axis and two kinds of mirrors at different orientations characterize this pattern; the symmetry is designated *C6mm*.

### 11.3.3 How May Motif’s and Lattices Combine?

Above, we saw examples where the lattice, the motif, and the overall pattern had the same symmetry. This was true in Figures 11.23, 11.24, 11.25, and 11.26. But, it was not true in Figure 11.27 where the motif and a primitive unit cell consisted of 3 atoms and a mirror plane of symmetry, while the lattice and overall pattern have hexagonal symmetry. Notice that in Figure 11.27, we could choose a triply primitive unit cell that had the same symmetry as the lattice and overall pattern. Because the triply primitive cell shows the 6-fold symmetry, it is the best choice.

This leads to an important second law of crystallography:

blank■ *If a motif has certain symmetry, the lattice must have at least that much symmetry.*

A motif with a 4-fold axis of symmetry requires a square plane lattice because it is the only plane lattice with a 4-fold axis. A motif may have less symmetry than a lattice. If a motif has a 2-fold axis of symmetry, it may be repeated according to any of the five plane lattices because they all have 2-fold axes of symmetry.

Halite (NaCl) is an excellent example of a mineral in which the simplest motif has less symmetry than the lattice. Figure 11.28 shows a two-dimensional model of the atomic arrangement in halite. The lattice is on the left and the motif, consisting of a Cl^{–} anion and an Na^{+} cation is shown in the center of the figure.

In two dimensions the lattice has square symmetry (although it is tilted to 45̊^{o} in this drawing), but the motif, consisting of one Na and one Cl atom, does not. Putting a motif at every lattice point gives the atomic arrangement on the right. A primitive unit cell (P), containing two atoms (outlined in red), is rectangular and does not have square symmetry. Notice, however, that we can choose a nonprimitive unit cell containing four atoms and two motifs (outlined in blue) that has square symmetry. This would not be possible if the lattice did not have “at least as much symmetry as the motif.”

Figure 11.29 shows an impossible combination of a lattice and a motif. The lattice has square symmetry, the motif has 3- fold symmetry. But the combination yields a structure that has no symmetry. Looking at that this figure should give you the sense that something is wrong. And, it is. If the motif really has 3-fold symmetry, it is identical in three directions 120̊^{o}apart. So bonds around it should be the same in three directions 120̊^{o}apart. But, they are not in the structure drawing. A motif with 3-fold symmetry requires a lattice that has at least 3-fold symmetry. There is only one and it is a hexanet. The lattice cannot be a square lattice.

## 11.4 Unit Cells and Lattices in Three Dimensions

In two dimensions, patterns are made of unit cells and lattices describe how unit cells and motifs repeat. These relationships are the same in three dimensions. Figure 11.30, for example, shows a single atom repeated an indefinite number of times and disappearing into the distance. Choose any four nearest neighbor atoms and you can connect them to get a 3D unit cell that is the basis for the entire atomic array.

In two dimensions, two vectors describe the translation that relates lattice points. To move from two dimensions (planes) to three dimensions (space), we need only to define a third vector, *t _{3}*, that translates a plane lattice some distance where it is repeated. We continue this process many times to get a space lattice. Figure 11.31 shows an example.

We can start with any of the two-dimensional plane lattices. The third translation may be orthogonal to one or both of the first two translations, but it need not be. Its magnitude may be the same as one or both of the first two translations, or not. We can envision space lattices as identical points that repeat indefinitely in three-dimensional space.

(Note that, for convenience, in this and subsequent drawings, we are choosing unit cells with lattice points at their corners. But we could choose unit cells with lattice points at their centers or anywhere else in the cell, because lattice points are simply patterns showing how unit cells repeat. It makes no difference if the lattice point corresponds to the corner, center, or any other point in the motif.)

In Figure 11.31, we start with a square lattice, and choose a third translation vector (*t _{3}*) that is equal in magnitude and also perpendicular to the first two. The result is in an overall cubic lattice. Connecting eight nearest-neighbor lattice points gives us a

*cubic unit cell*. The plane lattices had symmetry 4mm; the cubic unit cell has symmetry

^{4}/

*3*

_{m}^{2}/

*(described in the previous chapter).*

_{m}If we stack a square net directly above another square net, as in Figure 11.31, the 4-fold rotation axis and the mirror planes line up and we preserve all symmetry. But, suppose the third translation is not perpendicular to the first two, and each square net is slightly offset from the ones above and below it. If so, rotation axes and mirrors in the different nets will not line up. The resulting space lattice may contain no 4-fold axes and no mirrors.

Figure 11.32 shows an example of losing symmetry when nets are stacked with an offset. In this figure, we started with an orthonet. The third translation (*t _{3}*) is neither perpendicular to, nor of the same magnitude as, either of the other two. Each plane lattice is offset slightly to the right compared with the one below it. The result is a unit cell that has rectangular faces on four sides and parallelograms on two. It is called a

*monoclinic*unit cell (because one angle at each corner is not 90̊

^{o}. An individual orthonet contains mirror planes and 2-fold axes perpendicular to the net. But because, in this example, the third translation was not perpendicular to the first two, those mirrors and rotation axes do not persist in the space lattice, nor in the unit cell. The orthonet has symmetry 2mm; the unit cell has symmetry

^{ 2}/

*. The*

_{m}^{2}/

*axis is perpendicular to the front parallelogram face.*

_{m}If we consider all possible combinations of plane lattices and a third translation, we come up with seven fundamental unit cell shapes (Figure 11.33). Two of the unit cells, the hexagonal prism and the rhombohedron, are commonly grouped because they both derive from stacking hexanets. If we stack hexanets one above another, we preserve the 6-fold rotation axis and get a hexagonal prism. If the nets are offset, we may instead preserve a 3-fold rotation axis and get a rhombohedron.

You can think of these 3D shapes as the shapes that bricks can have if they fit together without spaces between them. We made a similar analogy to tiles when we were looking at 2D symmetry earlier in this chapter. Just like the tile shapes, more complex brick shapes are possible. But it can be shown that they are all equivalent to one of the seven below.

These seven unit cell shapes have unique symmetries: ^{4}/* _{m}*3

^{2}/

*(cube),*

_{m}^{4}/

_{m}^{2}/

_{m}^{2}/

*(tetragonal prism),*

_{m}^{2}/

_{m}^{2}/

_{m}^{2}/

*(orthorhombic prism),*

_{m}^{6}/

_{m}^{2}/

_{m}^{2}/

*(hexagonal prism),*

_{m}^{2}/

*(monoclinic prism), 1 (triclinic prism), and 3*

_{m}^{2}/

*(rhombohedron). Each unit cell corresponds to the one of the seven crystal systems introduced in the previous chapter: cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, and rhombohedral. The symmetry of each unit cell seen here is the same as the symmetry of the general form in each system. A third law of crystallography is that:*

_{m}blank *■The symmetries of the unit cells are the same as the point groups of greatest symmetry in each of the crystal systems.*

All minerals that belong to the cubic system have cubic unit cells with symmetry ^{4}/* _{m}*3

^{2}/

*. Their crystals, however may have less symmetry. Similarly, all minerals that belong to the tetragonal system have tetragonal unit cells. Tetragonal crystals may have*

_{m}^{4}/

_{m}^{2}/

_{m}^{2}/

*symmetry but may have less. This same kind of thinking applies to crystals in the other 5 systems, too.*

_{m}We describe the shapes of unit cells with *unit cell parameters* that include *a*, *b*, and *c*, the length of unit cell edges, and *α*, *β*, and *γ*, the angels between cell edges. By convention, the angle between the *a* and *b* edges is *γ* (*c* in the Greek alphabet), the angle between *a* and *c* is *β* (*b* in the Greek alphabet), and the angle between *b* and *c* is *α* (*a* in the Greek alphabet). As seen in Figure 11.33, in cubic unit cells, *a*,* b*, and *c* are equal, and all angles are 90^{o}. In tetragonal unit cells, *a* and *b* are equal and all angles are 90^{o}. In orthorhombic unit cells, the three cell edges have different lengths, but all angles are 90^{o}. In hexagonal unit cells, *a* and *b* have equal lengths, and the angle between *a* and *b* (*γ*) is 120^{o}. In monoclinic unit cells, *a*, *b* and *c* are all different. *α* and *γ* are 90^{o}, but *β* can have any value. (This is the general convention, but sometimes the non-90^{o} angle is chosen as one of the other angles instead of *β*.) In triclinic unit cells, all cell edges are different lengths and all angles are different and not 90^{o}. And in rhombohedral unit cells, the cell edges are all the same length, and the angles are all the same but are not 90^{o}.

The seven distinct unit cell shapes shown above in Figure 11.33 are all that are possible. The seven are primitive – they contain one lattice point in total. But stacking plane lattices can lead to other, nonprimitive, unit cells that have one of the seven shapes we have seen. Figure 11.34 shows three examples.

In Figure 11.34*a*, square nets have been stacked so that every other net is offset. One square from each of three layers is shown. The offset layer puts an extra lattice point (blue) in the center of a tetragonal prism. This is an example of a *body-centered* unit cell, symbolized by the letter *I*.

In Figure 11.34*b*, we put a hexanet directly above another. The result is a hexagonal prism with extra lattice points (blue) in the centers of its top and bottom. This is an example of an *end-centered* unit cell, symbolized by the letters *A*, *B*, or *C*, depending on which pair of faces contain the extra lattice points.

In Figure 11.34*c*, diamond nets are stacked with the middle layer offset. This produced an orthorhombic prism that contains extra lattice points at the centers of every face. This is an example of a* face-centered* unit cell, symbolized by the letter *F*.

### 11.4.1 Bravais Lattices

When all possibilities have been examined, we can show that only 14 distinctly different space lattices exist. We call them the 14 *Bravais lattices*, named after Auguste Bravais, a French scientist who was the first to show that there were only 14 possibilities. The 14 Bravais lattices correspond to 14 different unit cells that may have any of seven different symmetries, depending on the crystal system.

Figure 11.35 lists the 14 Bravais lattices and shows a unit cell for each. In this figure, red lattice points are at the corners of unit cells and blue spheres are the extra points due to centering. Symbols such as 1P, 2P, 2C, 222P, etc., beneath each figure describe their symmetries and are standard labels we use for each of the 14 lattices.

Seven of the possible unit cells are primitive (P), one in each system (shown in Figure 11.33). In these unit cells, eight cells share each of the eight lattice points at the corners, so the total number of lattice points per cell is one.

The other eight Bravais lattices involve nonprimitive unit cells containing two, three, or four lattice points. Body-centered unit cells (I), for example, contain one extra lattice point at their center (Figure 11.34*a*). End-centered unit cells (C) contain extra lattice points in two opposite faces (Figure 11.34*b*). But the extra points are only half in the cell, so the total number of lattice points is 2. Face-centered unit cells (F) have lattice points in the centers of six faces (see Figure 11.34*c*). Each of the six points is half in the unit cell and half in an adjacent cell. The total number of lattice points is therefore 1 (at the corners) + 3 (in the faces) = 4.

As pointed out in the discussion of two-dimensional lattices, we sometimes make arbitrary decisions when choosing unit cells. The 14 Bravais lattices, in fact, do not represent the only 14 that we could list. For example, in the monoclinic system, the end-centered (*2F*) unit cell is equivalent to a body-centered one (*2I*) with different dimensions. Most crystallographers consider only the end-centered cell because that is the way it has been done since the time of Bravais. The important thing to realize is that no matter what unit cells and lattices we consider, only 14 are distinct. Furthermore, the 14 in Figure 11.35 are the simplest and the ones used by most crystallographers.

For a good discussion of Bravais Lattices, watch this video:

blank▶️ Video 11-1: Bravais Lattices (6 minutes)

### 11.4.2 Unit Cell Symmetry and Crystal Symmetry

Consider fluorite (CaF_{2}), spinel (MgAl_{2}O_{4}), the garnet almandine (Fe_{3}Al_{2}Si_{3}O_{12}), and the rare mineral tetrahedrite (Cu_{12}As_{4}S_{13}). Tetrahedrite has a primitive cubic unit cell; fluorite and spinel have face-centered cubic unit cells; almandine has a body-centered cubic unit cell. In fluorite, Ca^{2+} is found at the corners of the unit cell and at the center of each face, while F^{–}occupies sites completely within the unit cell (see Figure 11.6). Spinel, almandine, and tetrahedrite have more complex structures, in large part because they contain more than one cation.

Many unit cells together make up a crystal. The crystal may or may not have the same symmetry as a single unit cell. In fluorite, spinel, and almandine crystals, unit cells stack together so that all symmetry elements (rotation axes and mirror planes) are preserved. However, fluorite crystals are typically cubes, spinel crystals are typically octahedra, and almandine crystals are typically dodecahedra. Figure 11.36 shows how cubes can combine to make these shapes. Figure 11.36*e* shows an example of a crystal, made with a cubic unit cell, that contains two forms.

Figures 11.37 through 11.40 show mineral examples. We also saw a beautiful dodecahedral garnet crystal in Figure 10.48 (Chapter 10) and well formed fluorite crystals in Figures 4.3 and 4.37 (Chapter 4). The cube, octahedron, and dodecahedron are all special forms that have the same symmetry as the unit cell, ^{4}/* _{m}*3

^{2}/

*. Tetrahedrite crystals (Figures 11.36*

_{m}*d*and 11.40), in contrast with the other three, do not have the same symmetry as their unit cell. Although made of cubic unit cells, the crystals lack 4-fold rotation axes. Instead, when euhedral, tetrahedrite crystals typically form pyramids that have symmetry 4

^{3}/

*.*

_{m}Figure 11.41 shows how tetragonal unit cells can be stacked to produce crystals. The results include a tetragonal prism and a ditetragonal pyramid that have the same symmetry as the unit cell (^{4}/_{m}^{2}/_{m}^{2}/* _{m}*). But tetragonal unit cells can also create a pyramid (with symmetry 4mm) or a tetragonal disphenoid (with symmetry 42m). Other shapes with other symmetries are possible too. Note that the crystal in Figures 11.41

*b*and

*d*contain a single form; the other two crystals contain two forms.

The preceding discussion points out a fourth important law of crystallography is:

blank■ *If a crystal has certain symmetry, the unit cell must have at least as much symmetry.*

As a corollary to this law, because crystals consist of unit cells, the symmetry of a crystal can never be more than that of its unit cell. If a crystal has a 4-fold axis of symmetry, it must have a unit cell that includes a 4-fold axis of symmetry. A mineral that forms cubic crystals must have a cubic unit cell. If a crystal has certain symmetry, it is certain that the unit cell and crystal’s atomic structure have at least that much symmetry. They may have more, as in the case of tetrahedrite.

This fourth law is the basis for the crystal systems we introduced in the previous chapter. The symmetry of a crystal requires certain symmetry in the crystal’s unit cell. And we divide the 32 possible symmetries – the 32 point groups – into systems based on that understanding. The six crystal systems are defined by symmetry and named by their unit cell shape. And crystals in the same crystal system all have the same-shaped unit cells, even if the crystals themselves do not look the same.

If, for example, a crystal has symmetry 4, 4, ^{4}/* _{m}*, 422, 4

*mm*, 42

*m*, or

^{4}/

_{m}^{2}/

_{m}^{2}/

*, it must have a unit cell that is a tetragonal prism. If a crystal has symmetry 222,*

_{m}*mm*2, or

*mmm*, it must have a unit cell that is an orthorhombic prism, and so forth. The table below summarizes these relationships. By examining a crystal’s morphology, we can often determine the point group, system, and unit cell shape. Determining whether a unit cell is primitive, face-centered, body-centered, or end-centered, however, is not possible without additional information, which usually comes from X-ray diffraction studies.

Crystal Systems and Point Groups |
||

crystal system |
unit cell shape |
point groups |

triclinic | triclinic prism | 1, 1 |

monoclinic | monoclinic prims | 2, m, ^{2}/_{m} |

orthorhombic | orthorhombic prism | 222, mm2 , mmm |

tetragonal | tetragonal prism | 4, 4, ^{4}/, 422, 4_{m}mm, 42m, ^{4}/_{m}^{2}/_{m}^{2}/_{m} |

rhombohedral | rhombohedron | 3, 3, 32, 3m, 3 m |

hexagonal | hexagonal prism | 6, 6, ^{6}/, 622, 6_{m}mm, 62m, ^{6}/_{m}^{2}/_{m}^{2}/_{m} |

cubic | cube | 23, ^{2}/3, 432, 43_{m}m, ^{4}/3_{m}^{2}/_{m} |

It is important to remember that the symmetry of a crystal depends not only on the symmetry of the unit cell, but also on how the unit cells combine to make the crystal. Some minerals, such as halite and other salts, tend to develop euhedral crystals with obvious symmetry, making it easy to infer the symmetry of the unit cell. Others develop crystals with faces that are nearly identical, suggesting the presence of symmetry but leaving some uncertainty. While size and shape of faces can vary because of accidents of crystal growth, the angles between faces vary little from the ideal (Steno’s law). Consequently, crystallographers often rely on angles instead of face shapes to infer the symmetry of the unit cell. Even if a crystal is anhedral and poorly formed, its internal structure is orderly and its unit cells all have the same atomic arrangement.

## 11.5 Symmetry of Three Dimensional Atomic Arrangements

In the preceding sections, we discussed the shapes and symmetries of crystals. We now turn our attention briefly to space symmetry, the symmetry of three-dimensional atomic structures. What possible symmetry can be present? An atomic arrangement in a crystal consists of groups of atoms (an atomic motif) that repeat an infinite number of times according to one of 14 space lattices. Thus, the overall symmetry depends on both the arrangement of atoms in a motif and the lattice type.

Atomic motif symmetry may involve rotation axes, rotoinversion axes, mirror planes or inversion centers. So motifs may have symmetries equivalent to any of the 32 point groups. Consider an orthorhombic mineral, for example. Its atomic motif may have symmetry mmm, ^{2}/_{m}^{2}/_{m}^{2}/* _{m}*, or

*mm*2. Its lattice may be 222P, 222C, 222I, or 222F. That gives 12 possible combinations. If we do this calculation for all crystal systems, we come up with 61 possible symmetries for 3D atomic arrangements. But there are many more than that. Because in 3D, we find two additional symmetry operators that we have not considered previously.

### 11.5.1 Space Group Operators

In 3D, we assign atomic arrangements to different *space groups* that have different *space symmetries*. Each space group is characterized by a combination of one of the 14 Bravais lattices with a unit cell that has a particular symmetry. We call the different kinds of possible symmetry operators, collectively, *space group operators*. Space symmetry includes the point group symmetries that we discussed previously. And it also includes *glide planes* and *screw axes*. These two are special kinds of space group operators that involve combinations of point symmetry and translational symmetry, in much the same way that rotoinversion axes involve rotation and inversion applied simultaneously.

##### 11.5.1.1 Glide Planes

Glide planes differ from normal mirror planes because they involve translation before reflection. Figure 11.42 shows some examples. In drawing *a*, a single atom is repeated by a horizontal glide plane (dashed red line). The red arrows show how the atom repeats by translation followed by reflection. In drawing *b*, a three-atom motif repeats according to a horizontal glide plane. And in drawing *c*, we see a 2D pattern with glide plane symmetry. The pattern contains no mirrors, rotation axes, and no inversion center. Yet, it contains significant symmetry.

Space group operators and other symmetry elements combine in many ways. For example, Figure 11.42*d* shows a pattern that contains horizontal glide planes and vertical mirror planes. As we have seen previously, combination of symmetry elements often requires other symmetry to be present. The combination of a glide plane with a mirror means that this pattern contains additional (horizontal) mirror planes and 2-fold rotation axes.

Figure 11.43 shows a glide plane in three dimensions; a motif of 8 atoms repeats by vertical translation and reflection. This figure involves a simple motif with lots of space around it. The arrangements of atoms in most minerals are generally more complicated and difficult to draw in 3D in a way that shows their symmetries clearly. In part, this is because a glide plane affects all atoms in a structure, not just the ones adjacent to the plane as seen in Figure 11.43.

Atomic arrangements may contain any of six symmetry elements involving reflection, singly or in combination. These include proper mirror planes and 5 different kinds of glide planes. Glide planes may have any of the orientations that mirror planes can have. So, they may be parallel to the *a-*, *b-*, or* c-*axis, parallel to a face diagonal, or parallel to a main (body) diagonal. For most glide planes, the magnitude of the translation (*t*) is half the unit cell dimension in the direction of translation. The direction of gliding distinguishes different glides and allows us to classify them. The table below summarizes the possibilities.

Space Symmetry Operators Involving Reflection |
|||

operator |
type of operation |
orientation of translation |
translation* |

mabcnd |
proper mirror axial glide axial glide axial glide diagonal glide diamond glide |
none parallel to a parallel to b parallel to c par. to a face diagonal par. to main diagonal |
none^{1}/_{2} a^{1}/_{2} b^{1}/_{2} c^{1}/_{2} t^{1}/_{4} t |

*t = the unit cell dimension in the direction of translation |

##### 11.5.5.2 Screw Axes

Screw axes result from the simultaneous application of translation and rotation. We combine 2-, 3-, 4-, or 6-fold rotation operators with translation to produce these symmetry elements. Many combinations are possible. Figure 11.44 shows a few examples.

A screw axis has the appearance of a spiral staircase. We rotate a motif, translate it, and get an additional motif. As with proper rotation axes (rotation axes not involving translation), each *n*-fold screw operation involves rotation of 360^{o}/*n*. After *n* repeats, the screw has come full circle. The translation associated with a screw axis must be a rational fraction of the unit cell dimension or the result will be an infinite number of atoms, all in different places in different unit cells.

We label screw axes using conventional symbols. In Figure 11.44, they are 6_{1}, 4_{1}, 3_{1}, and 2_{1}. In the labels, the large 6, 4, 3, or 2 signifies 6-fold, 4-fold, 3-fold, or 2-fold rotation. The subscript tells the translation distance. A 6_{1} screw axis, for example, involves translation that is ^{1}/_{6} of the unit cell dimension in the direction of the screw axis. 4_{1}, 3_{1}, and 2_{1} axes involve translations that are ^{1}/_{4}, ^{1}/_{3}, and ^{1}/_{2} the unit cell dimension. Similarly, 6_{2}, 6_{3}, 6_{4}, and 6_{5} screw axes (not shown) would involve translations of ^{2}/_{6}, ^{3}/_{6}, ^{4}/_{6}, and ^{5}/_{6} of the cell dimension.

Figure 11.45*a* shows application of a 4_{2} operator to a single atom. The translation distance is^{ 1}/_{2} the unit cell height, and we must go through four 90^{o} rotations and two unit cells to get another atom that is directly above the starting atom. All unit cells must be identical, but the 4_{2} operation gives a bottom and top unit cell with atoms in different places. The only way this operator can be made consistent is to add the extra atoms shown in Figure 11.45*b*. In other words, the presence of a 4_{2} axis requires the presence of a 2-fold axis of symmetry.

Figure 11.45*c* shows a 4_{3} axis and 11.44*d* shows a 4_{1} axis. After four applications of either operator the total rotation is 360°, bringing the fourth point directly above the first. For the 4_{1} axis, after four applications the total translation is equivalent to one unit cell length. But for the 4_{3} axis it is three unit cell lengths because the translation is ^{3}/_{4} of the unit cell. The only way the 4_{3} operator can be made consistent is to add the extra atoms shown in Figure 11.45*d*. Note that the 4_{3} and 4_{1} axes produce patterns that are mirror images of each other. The two axes are an *enantiomorphic pair*, sometimes called *right-handed* and* left-handed* screw axes.

When all combinations are considered, we get the 21 possible rotation axes (either proper rotation axes or screw axes) listed in the table below. As with proper rotational axes, some screw axes are restricted to one or a few crystal systems. For example, 3_{1} and 3_{2}, which are an enantiomorphic pair, only exist in the rhombohedral system. Similarly, the 6* _{n}* axes only exist in the hexagonal system.

Space Symmetry Operators Involving Rotation |
|||

operator |
type of operation |
rotation angle |
translation* |

1 1 2 2 _{1}2 3 3 _{1}3 _{2}3 4 4 _{1}4 _{2}4 _{3}4 6 6 _{1}6 _{2}6 _{3}6 _{4}6 _{5}6 |
identity inversion center proper 2-fold 2-fold screw mirror proper 3-fold 3-fold screw 3-fold screw 3-fold rotoinversion proper 4-fold 4-fold screw 4-fold screw 4-fold screw 4-fold rotoinversion proper 6-fold 6-fold screw 6-fold screw 6-fold screw 6-fold screw 6-fold screw 6-fold rotoinversion |
360^{o}360 ^{o}180 ^{o}180 ^{o}180 ^{o}120 ^{o}120 ^{o}120 ^{o}120 ^{o}90 ^{o}90 ^{o}90 ^{o}90 ^{o}90 ^{o}60 ^{o}60 ^{o}60 ^{o}60 ^{o}60 ^{o}60 ^{o}60 ^{o} |
none none none 1/2 tnone none 1/3 t1/3 tnone none 1/4 t2/4 = 1/2 t3/4 tnone none 1/6 t2/6 = 1/3 t3/6 = 1/2 t4/6 = 2/3 t5/6 tnone |

*t = the unit cell dimension in the direction of translation |

## 11.6 Space Groups

When we combine the space group operators in the tables above with the 14 possible space lattices, we get 230 possible space groups. See Box 11-1. They represent all possible symmetries crystal structures can have, although most of them are not represented by any known minerals. Deriving them all is not trivial, and crystallographers debated the exact number until the 1890s when several independent studies concluded that there could only be 230.

Crystallographers use several different notations for space groups; the least complicated is that used in the* International Tables for X-ray Crystallography* (ITX) (Hahn, 1983). ITX space group symbols consist of a letter indicating lattice type (*P, I, F, R, A, B,* or *C*) followed by symmetry notation similar to conventional Hermann-Mauguin symbols. An example is P4_{2}/_{m}2_{1}/_{n}2/_{m}, the space group of rutile. Rutile has a primitive (*P*) tetragonal unit cell, a 4_{2} screw axis perpendicular to a mirror plane, a 2_{1} screw axis perpendicular to an *n* glide plane, and a proper 2-fold axis perpendicular to a mirror plane. Rutile crystals have symmetry ^{4}/_{m}^{2}/_{m}^{2}/_{m}.

Or consider garnet. Garnet crystals have point group symmetry ^{4}/* _{m}*3

^{2}/

*. Garnet’s space group is*

_{m}*I4*32/

_{1}/_{a}_{d}. This describes a body-centered unit cell, a 4

_{1}screw axis perpendicular to an

*a*glide plane, a 3-fold rotoinversion axis, and a proper 2-fold axis perpendicular to a

*d*glide plane. Rather than using an entire symbol, crystallographers often use abbreviations for space groups (and occasionally for point groups). Thus, they would say that garnet belonged to the space group

*Ia*3

*d*.

The translations associated with lattices, glide planes, and screw axes are very small, on the order of tenths of a nanometer, equivalent to a few angstroms. Detecting their presence by visual examination of a crystal is impossible, with or without a microscope. A crystal with symmetry ^{4}⁄_{m}^{2}/_{m}^{2}/* _{m}* could belong to the space group I4

_{1}/

_{a}2/

_{c}2/

_{d}, but there are also 19 other possibilities. We say that the 20 possibilities are

*isogonal*, meaning that when we ignore translation they all have the same symmetry. Without detailed X-ray studies, telling one isogonal space group from another is impossible, and we are left with only the 32 distinct point groups.

For some supplementary information and a different perspective on space groups, check out the video that is linked here:

blank▶️ Video 11-2: Space group operators and space groups (8 minutes)

blank

## ●Box 11-1 Why Are There Only 230 Space Groups?In Chapter 10 we showed that point groups may have one of 32 symmetries; in this chapter we determined that crystal structures must have one of 14 Bravais lattices. When we combine point groups with Bravais lattices, and consider all possible space group operators, we get 230 possible space groups. The 230 space groups are the only 3D symmetries that a crystal structure can have. They were tabulated in the 1890s by a Russian crystallographer, E. S. Federov; a German mathematician, Artur Schoenflies; and a British amateur, William Barlow, all working independently. Why are there only 230 space groups? The answer is that symmetry operators, as we have already seen, can only combine in certain ways. In the discussion of point group symmetry, we concluded that mirrors and rotation axes can only combine in 32 ways. Some combinations required that other symmetry be present. Other combinations were redundant or led to infinite symmetry, which is impossible. The same is true of space group operators and Bravais lattices; only certain combinations are allowed and some combinations require other symmetry to be present. Triclinic lattices (P1 and P1) may not be combined with 2-fold axes of any sort. Similarly, 3, 3 |

## 11.7 Crystal Habit and Crystal Faces

Why do halite and garnet, both cubic minerals, have different crystal habits? It is not fully understood why crystals grow in the ways they do. Cubic unit cells may lead to cube-shaped crystals such as typical halite crystals, octahedral crystals such as spinel, dodecahedral crystals such as garnet, and many other shaped crystals. Why the differences? Crystallographers do not have complete answers, but the most important factor is the location of atoms and lattice points within a unit cell. Crystals of a particular mineral tend to have the same forms, or only a limited number of forms, no matter how they grow. Haüy and Bravais noted this and used it to infer that atomic structure controls crystal forms. In 1860 Bravais observed what we now call the *Law of Bravais*:

blank■ *Faces on crystals tend to be parallel to planes having a high density of lattice points.*

This means that, for example, crystals with hexagonal lattices and unit cells often have faces related by hexagonal symmetry. Crystals with orthogonal unit cells (those in the cubic, orthorhombic, or tetragonal systems) tend to have faces at 90° to each other.

The relationship between lattice/unit cell symmetry and crystal habit can be seen in Figure 11.46. The figure shows two choices for calcite’s unit cell. The rhombohedron is a doubly primitive unit cell and the hexagonal equivalent contains four CaCO_{3} motifs. The drawings below the unit cells show some common shapes for natural calcite crystals. There is noticeable resemblance between unit cell shapes, which represent the lattice symmetry, and some of the crystal shapes. Thus, Bravais’s Law works well. The photos below in Figures 11.47 – 11.52 show natural calcite crystals that match quite well the drawings in Figure 11.46.

Unfortunately, some minerals, including pyrite (FeS_{2}) and quartz (SiO_{2}), appear to violate Bravais’s Law. Bravais’s observations were based on considerations of the 14 Bravais lattices and their symmetries, but in the early twentieth century, P. Niggli, J. D. H. Donnay, and D. Harker realized that space group symmetries needed to be considered as well. By extending Bravais’s ideas to include glide planes and screw axes, Niggli, Donnay, and Harker explained most of the biggest inconsistencies. They concluded that crystal faces form parallel to planes of highest atom density, a slight modification of the Law of Bravais.

As a crystal grows, different faces grow at different rates. Some may dominate in the early stages of crystallization while others will dominate in the later stages. The relationship, however, is the opposite of what we might expect. Faces that grow fastest are the ones that eventually disappear. Figure 11.53 shows why this occurs. If all faces on a crystal grow at the same rate, the crystal will keep the same shape as it grows (the green crystal in Figure 11.53*a*). However, this is not true if some faces grow faster than others.

In Figure 11.53*b*, the diagonal faces (oriented at 45^{o} to horizontal) grew faster than those oriented vertically and horizontally. Eventually, the diagonal faces disappeared; they “grew themselves out.” The final crystal has a different shape, and fewer faces, than when it started growing. We observe this phenomenon in many minerals; small crystals often have more faces than larger ones.

## 11.8 Quantitative Aspects of Unit Cells, Points, Lines, and Planes

### 11.8.1 Unit Cell Parameters and Crystallographic Axes

Earlier in this chapter, we introduced the unit cell parameters *a, b, c, α, β,* and *γ*. *a, b,* and *c* are the lengths of unit cell edges; *α, β*, and *γ* are the angles between the edges. *α* is the angle between *b* and *c, β* is the angle between *a* and *c*, and *γ* the angle between *a* and *b* (Figure 11.54*a*).

Unit cell edges define a coordinate system used by crystallographers when they wish to describe the locations of atoms or other features within a cell. Some crystal systems are orthogonal (cubic, tetragonal, orthorhombic) but others are not. Cubic crystals, and others that belong to an orthogonal system, use a standard Cartesian coordinate system. Figure 11.54*a* shows an example.

Figure 11.54*b* shows axes that are inclined with respect to each other. In this example, they correspond to a triclinic unit cell.* α, β,* and *γ,* the angles between axes, are not equal to 90^{o} and (in contrast with the cube) the edges of the unit cell have different lengths along different axes.

Figure 11.55 shows coordinate axes for each of the crystal systems. We have labeled all angles in the top drawing (a cube); in the other drawings, only the non-90^{o} angles are identified. In the orthogonal systems, *α, β,* and *γ* all equal 90^{o}. In the hexagonal and rhombohedral systems, *β* = 120^{o} and the other angles are 90^{o}. Other constraints are in the right-hand column of this figure. We used calcite as an example of a rhombohedral mineral. In much of the literature, however, calcite’s unit cell is described using a hexagonal prism.

Although it makes no practical difference which edges of a unit cell we call *a, b,* or *c,* or which angles are *α, β*, and *γ*, mineralogists normally follow certain conventions. In triclinic minerals, none of the angles are special and *a, b,* and *c* are all different lengths. Although the literature contains exceptions, by modern convention edges are chosen so that *c* < *a* < *b*. In monoclinic minerals, such as sanidine, only one angle in the unit cell is not 90°. By convention, the non-90° angle is* β*, the angle between *a* and *c*. For historical reasons, this convention is called the *second setting for monoclinic minerals*. In orthorhombic crystals, we choose axes so that *c* > *a* >* b*. In tetragonal and hexagonal crystals, the *c*-axis always corresponds to the 4-fold or 6-fold axis.

In this book we use a, b, and c to designate the three crystallographic axes, but crystallographers sometimes use subscripts to indicate axes, and thus cell edges, that must be identical lengths because of symmetry. Instead of a, b, and c, the three axes of cubic minerals might be designated a_{1}, a_{2}, and a_{3}. The axes of tetragonal crystals can be designated a_{1}, a_{2}, and c.

For hexagonal crystals, crystallographers have historically used four axes: a_{1}, a_{2}, a_{3}, and c (Figure 11.56*a*). The three a-axes, a_{1}, a_{2}, and a_{3}, are parallel to edges of a nonprimitive hexagonal unit cell. Although the third a-axis is redundant for describing symmetry or points in 3D space, it has been included in the past to emphasize that there are three identical a-axes perpendicular to the c-axis. Because only three axes are used in much of the modern literature (Figure 11.56*b*), we will only briefly mention the fourth axis in the rest of this chapter.

Although mineralogists historically used angstroms (1 Å = 10^{-10 }m) to give cell dimensions, much recent literature uses nanometers (1 nm = 10 Å). Typical mineral unit cells have edges of 2 to 20 Å (0.2 to 2 nm). The angles *α, β,* and *γ* are normally given as a decimal number of degrees (for example, 94.62°). These angles between edges vary greatly, although unit cells are often chosen so that angles are close to 90°.

Instead of using angstroms (or nanometers) to give distances, we may also use unit cell dimensions as a scale. For example, we might say that a certain plane intersects the axes at distances of 3*a*, 2*b*, and 2*c* from the origin. It is implicit that 3*a* refers to a distance equal to three unit cell edge lengths along the a-axis, 2*b* a distance equal to two unit cell lengths along the b-axis, and 2*c* a distance equal to two unit cell lengths along the c-axis.

The symmetry of a unit cell always affects the relationships between *a, b,* and *c*. So, in the cubic system, *a* = *b* = *c*, but in the tetragonal and hexagonal systems *a* = *b* ≠ *c*. The relationships implied by crystal systems mean that, for systems other than triclinic, we need not give six values to describe unit cell shape. For example, for orthorhombic, hexagonal, tetragonal, and cubic minerals, we do not specify any angles because they are all defined by the crystal system. The table below gives examples of unit cell parameters for minerals from each of the crystal systems; unnecessary information has been omitted. These same example minerals are the ones listed in Figure 11.55.

Unit Cell Parameters (a, b, c, α, β, and γ), Z (number of formulas per unit cell), and V (unit cell volume) for One Mineral from Each of the Crystal Systems* |
||||||

fluoriteCaF_{2} |
rutileTiO_{2} |
berylBe_{3}Al_{2}Si_{6}O_{18} |
calciteCaCO_{3} |
enstatiteMg_{2}Si_{2}O_{6} |
sanidineKAlSi_{3}O_{8} |
albiteNaAlSi_{3}O_{8} |

cubicZ = 4a = 5.46V = 162.77 |
tetragonalZ = 2a = 4.59c = 2.96V = 62.36 |
hexagonalZ = 2a = 9.23c = 9.19V = 2034.09 |
rhombohedralZ = 2a = 4.98c = 17.06 |
orthorhombicZ = 4a = 18.22b = 8.81c = 5.21V = 836.30 |
monoclinicZ = 4a = 8.56b = 13.03c = 7.17γ = 115.98V = 799.72 |
triclinicZ = 4a = 8.14b = 12.8c = 7.16α = 94.33β = 116.57γ = 87.65V = 746.01 |

*a, b, and c are in angstroms; α, β, and γ are in degrees; and V is in cubic angstroms. |

Consider the triclinic mineral albite, a feldspar with composition NaAlSi_{3}O_{8}. The last column in the table lists albite’s unit cell parameters. Because none of the angles are special and none of the cell edges are equal, we need six parameters to describe the cell shape. In contrast, fluorite (the first column of the table) has a cubic unit cell, so we need to give only one cell dimension, the length of the cell edge. It is implicit that all angles are 90° and all cell edges are the same length.

Physical dimensions do not completely describe a unit cell. We must also specify the nature and number of atoms within the unit cell. To provide some of this information, the table above lists two other things besides dimensions and angles: mineral formulas and *Z*, the number of formulas in each unit cell. For example, fluorite has the formula CaF_{2} and *Z* = 4. This means there are 4 CaF_{2} molecules (4 Ca and 8 F atoms) in each unit cell. Albite has the formula NaAlSi_{3}O_{8} and *Z* = 4. This means that four NaAlSi_{3}O_{8} formulas are in each unit cell. In other words, each unit cell contains 4 Na atoms, 4 Al atoms, 12 Si atoms, and 32 O atoms.

### 11.8.2 Points in Unit Cells

Mineralogists describe the locations of atoms or other points in unit cells by giving their coordinates. As examples, consider the orthorhombic and triclinic crystals shown in Figure 11.57. In the orthorhombic crystal the three axes (shown in red) intersect at 90°, in the triclinic crystal they do not. Assume both crystals have cell dimensions *a* = 5.20 Å, *b* = 18.22 Å, and *c* = 8.80 Å, and the axes’ origins are at the center of the unit cells as shown. The coordinates of point **P**, at the lower right-hand corner in drawings *a* and *b*, are therefore 5.20/2 = 2.60 Å, 18.22/2 = 9.11 Å, and 8.80/2 = -4.40 Å. The negative sign arises because the points are in a negative direction from the origin along the c-axis.

We do not, however, normally report coordinates in angstroms (Å). Instead, we report distances relative to unit cell dimensions. Another way to describe the location of point **P** in Figure 11.57*a* and *b* would be to say it has coordinates 1/2*a*, 1/2*b*, -1/2*c*. Typically, we normalize coordinates to unit cell dimensions by dropping the *a, b,* and *c*. The coordinates then become 1/2 , 1/2 , -1/2. The points at the corners must have coordinates ±1/2, ±1/2, ±1/2, if the origin is at the center of the unit cell (Figure 11.57*c*). Note that the system to which a crystal belongs does not affect coordinate values.

Traditionally, crystallographers have used the variables *u, v,* and *w* to represent the coordinates of a point if the coordinates are rational numbers, and *x, y,* and *z* if irrational. For simplicity, in this text we will use *uvw* in all instances (Figure 11.58*a*). We can say that *uvw* represents a general point anywhere in the cell.

Suppose two coordinates, for example *v* and *w*, are zero. All the points described by the coordinates *u*00 constitute a set of special points, lying along the a-axis (Figure 11.58*b*). Similarly, 0*v*0 and 00*w* refer to sets of special points on the b- and c-axes, respectively. If only one coordinate equals zero, a point lies in a plane including two axes. We might, for example, talk about special points located at *uv*0. These are points on a plane that includes the a- and b-axes, as shown in Figure 11.58*c*. Note that no parentheses or brackets are used when giving coordinates of a point.

Figure 11.59*a* shows a unit cell of sphalerite, ZnS, with *uvw* coordinates for each atom. The axes origin is in the bottom back left corner. Because three-dimensional drawings are sometimes difficult to draw and see, crystallographers often use projections, which contain the same information in a less cluttered manner (Figure 11.59*b*). In the projection, which is a view down the c-axis, we need not specify *u* and *v* values because they can be estimated from the location of the atoms in the drawing. The numbers are *w* values. Sometimes we omit *w* values for atoms at unit cell corners; by convention, this means the atoms are found on both the top and bottom of the cell as it appears in projection.

### 11.8.3 Lines and Directions in Crystals

The absolute location of a line in a crystal is not often significant, but directions (vectors) have significance because they describe the orientations of symmetry axes, zones, and other linear features. Crystallographers designate directions with three indices in square brackets, [*uvw*]. As with point locations, we give numbers describing directions in terms of unit cell dimensions. In Figure 11.60*a*, direction **V **has indices [132] – it goes from the origin to a point at 1*a*, 3*b*, 2*c*.

The drawings in 11.60*b* and 11.60*c* show other vectors pointing the same direction as [132]. We clear fractions and divide by common denominators when giving the indices of a direction. So directions [½ 1½ 1] and [264] would be simplified and described as [132]. Although no commas are included in [132], reading sequentially, as we do for point locations, we articulate it “one-three-two.” By convention, commas separate indices only if they have more than one digit.

Figure 11.61 shows several lines on a two-a dimensional lattice. Parallel lines have identical indices, and choosing an origin is unimportant. The direction [240] is the same as [120], and when we clear the common denominator [240] becomes [120]. For line [120], the bar over the 1 is equivalent to a negative sign, indicating that the line goes in the negative a-direction. We articulate the indices as “bar-one-two-zero.” Because all directions in Figure 11.61 lie in the a-b plane, they have indices [*uv*0]. This notation indicates that the first two indices may vary, but the third is 0.

### 11.8.4 Planes in Crystals

While crystals and crystal faces vary in size and shape, Steno’s law tells us that angles between faces are characteristic for a given mineral. The absolute location and size of the faces are rarely of significance to crystallographers, while the relative orientations of faces are of fundamental importance. We can use face orientations to determine crystal systems and point groups, so having a simple method to describe the orientation of crystal faces is useful.

Figure 11.62*a* shows three axes with different unit cell lengths; small red dots show unit increments along the axes. The axes are orthogonal, so this arrangement corresponds to the orthorhombic system. Consider a plane parallel to a crystal face. Such a plane may be parallel to one axis (Figure 11.62*b*), parallel to two axes (Figure 11.62*c*) or it may intersect all three axes (Figure 11.62*d*).

We can describe the orientation of a plane by listing its intercepts with the axes. For a face running parallel to an axis, the intercept is at ∞ (because the face never intersects the axis). In Figure 11.62*b*, the plane is parallel to the c-axis but intersects the a-axis and b-axis one unit from the origin. So, it has axial intercepts 1*a*, 1*b*, and ∞*c*, which we shorten to 1, 1, ∞. In Figure 11.62*c*, the plane is parallel to two axes and has axial intercepts ∞*a*, ∞*b*, 3*c*, or simply ∞, ∞, 3.

Crystal faces are commonly parallel to axes, but the use of ∞ to describe their orientations can be confusing and awkward. A second problem with using intercepts to describe face orientation is that parallel planes, such as plane D and plane D´ Figure 11.10*d*, have different intercepts. Crystallographers find this inconvenient because crystals may be large or small, but face orientation, not size, is generally the feature of greatest significance. To avoid these and other problems, crystallographers do not report axial intercepts. They use *Miller indices* instead.

## 11.9 Miller Indices

Miller indices were first developed in 1825 by W. Whewell, a professor of mineralogy at Cambridge University. We use them to describe the orientation of crystal faces, and also the orientations of cleavages and other planar properties. They are named after W. H. Miller, a student of Whewell’s, who promoted and popularized their use in 1839. The general symbol for a Miller index is (*hkl*), in which the letters *h, k,* and* l* each stand for an integer. Parentheses enclose the resulting Miller index. As with directions, bars above numbers show negative values and we do not include commas unless numbers have more than one digit.

We calculate Miller indices for a plane from its axial intercepts (Figure 11.62). The procedure is as follows:

·First, we invert axial intercept values. (∞ becomes zero after inversion.)

·Then we clear all fraction by multiplying by a constant.

·And we divide by a constant to eliminate common denominators.

Consider the plane in Figure 11.62*b*. It has intercepts 1, 1, ∞. Inverting these values gives us (110), the Miller index of the plane. And the plane in Figure 11.62*c* has axial intercepts ∞, ∞, 3. Inverting these values gives us 0, 0, ^{1}/_{3}. Clearing the fraction gives us (001), the Miller index. We articulate it as “oh-oh-one.”

Figure 11.62*d* contains two planes. One has intercepts 1, 1, 2. The other has intercepts 3, 3, 6. Inverting intercepts for the first plane gives us 1, 1, ^{1}/_{2}. Clearing fractions yields the Miller index (221). Inverting intercepts for the second plane gives ^{1}/_{3}, ^{1}/_{3}, ^{1}/_{6}. Clearing fractions yields (221). So we see that parallel planes have the same Miller index. If the planes shown in this figure were crystal faces, we would call them the “two two one face” no matter their size or shape. Thus the Miller index describes the orientation of a crystal face with respect to crystallographic axes, but not the absolute size or location of the face. The relationship between planes and directions in a crystal depends on the crystal system. Except in the cubic system, the direction [*uvw*] is neither perpendicular nor parallel to planes with the Miller index (*uvw*).

Because crystal faces are parallel to rows of lattice points, they are parallel to planes that intercept crystal axes at an integral number of unit cells from the origin. Consequently, inversion of intercepts and clearing fractions always yields integers. This observation is known as the *Law of Rational Indices*. Another observational law, called *Haüy’s Law:
*blank■ Miller indices of faces generally contain low numbers.

For example, (111) is a common face in crystals, while (972) is not. Haüy’s law is really a corollary to the *Law of Bravais,* already discussed, which states that faces form parallel to planes of high lattice point density; planes with low values in their Miller index have the greatest lattice point density.

As mentioned previously, crystallographers have in the past used four axes for crystals in the hexagonal system. This yields a Miller index with four numbers (*hkil*). One of the first three values *h, k,* or *i* is redundant because we can always describe the location of a plane in three-dimensional space with three variables. In all cases: *h* + *k* + *i* = 0. Because of the redundancy, and to be consistent with other crystal systems, many crystallographers today use only three indices for hexagonal minerals.

## ●Box 11-2 The Miller Indices of Planes within a Crystal StructureWe use Miller indices to describe the orientation of crystal faces, but we also use them to describe planes within a crystal structure. For example, we may be interested in knowing which planes in a unit cell contain the most atoms. For planes within a cell, we calculate Miller indices as we do for crystal faces, except that we do not clear common denominators after inversion of axial intercepts. For example, if we calculate an index of (633) for a set of planes, we do not divide by 3 to give (211), as we do when calculating a Miller index for a crystal face. We do not do this because, besides orientation, the spacing and location of planes are important when we are talking about atomic arrangements and other aspects of crystal structures, and not talking about crystal faces. Figure 11.63 Drawing Different families of planes, with the same orientation but different spacings, have different indices. The drawings in Figure 11.63 Because the Miller indices (120) and (240) are proportional, the two families of planes are parallel but the (240) planes are spaced apart half as much as the (120) planes. This is always the case – proportional indices mean planes are parallel and larger indices mean planes are closer together. |

## 11.10 Crystal Forms and the Miller Index

The replacement of unknown or variable numbers in a Miller index with *h, k,* or *l* allows us to make generalizations. The index (*hk*0) describes the family of faces with their third index equal to zero. A Miller index including a zero describes a face is parallel to one or more axes (Figure 11.64). The family of faces described by (*hk*0) is parallel to the c-axis (Figure 11.63); faces with the Miller index (00*l*) are parallel to both the a-axis and the b-axis (Figure 11.62*c*).

Figure 11.64*a* shows an orthorhombic prism with six faces. Some faces cannot be seen, but the Miller indices of all six are (100), (010), (001), (100), (010), and (001). Although this crystal contains three forms, indices for all faces contain the same numbers (two zeros and a one) but the order of numbers and the + or – sign changes.

Figure 11.64*b* shows an orthorhombic dipyramid. It contains only one form, and all the faces have the same numbers in their Miller index (332). This is always true for faces that belong to the same form; they always have similar Miller indices. This relationship is especially clear for crystals in the cubic system because high symmetry means that many forms may contain many identical faces.

The six identical faces on the cube in Figure 11.65*a* have indices (001), (010), (100), (001), (010), and (100). We symbolize the entire form {100}, and the { } braces indicate the form contains all faces with the numerals 1, 0, and 0 in their Miller index, no matter the order.

The four faces on the tetrahedron in Figure 11.65*b*, and the eight faces on the octahedron in Figure 11.65*c* are all equilateral triangles. For both, the form is {111}. As Figure 11.65*b* and *c* demonstrate, two crystals of different shapes can have the same form if they belong to different point groups. The tetrahedron in Figure 11.65*b* belongs to point group 43m; the octahedron in Figure 11.65*c* belongs to point group ^{4}/* _{m}*3

^{2}/

*. If we know the point group and the form, we can calculate the orientation of faces. If a crystal contains only one form, we then know the shape of the crystal. Note that the cube, octahedron, and dodecahedron all belong to point group*

_{m}^{4}/

*3*

_{m}^{2}/

*. The cubic form is {100}, the octahedral form is {111}, and the dodecahedral form in Figure 11.65*

_{m}*d*is {110}.

Figure 11.65*e* shows a crystal containing three forms: cube {100}, octahedron {111}, and dodecahedron {110} . Because they all belong to point group ^{4}/* _{m}*3

^{2}/

*, we know the faces are oriented as shown. However, the crystal in Figure 11.64*

_{m}*f*belongs to the same point group and contains the same forms, but the size and shape of corresponding faces are different. We do not know the crystal shape if more than one form is present, unless we know some extra information.

Crystallographers sometimes label faces of the same form with the same letter as we have done in Figure 11.65. For some forms, the letter is just the first letter of the form name. For example, *o* indicates the octahedral form and *d* the dodecahedral form in the cubic system. Usually, however, the symbols are less obvious (we normally designate cube faces, for example, by the letter *a*); labels also vary from one crystal system to another.

### 11.10.1 Zones and Zone Axes

As discussed in Chapter 9, a set of faces parallel to a common direction defines a *zone*. We designate zones by the common direction, which we call the *zone axis*. Figure 11.66 shows examples of tetragonal, hexagonal, and orthorhombic crystals. The green faces in the tetragonal crystal (Figure 11.66*a*) form a zone. The faces are parallel to the a-axis. Three other faces, not visible are also part of the zone. The zone axis is [100], parallel to a. Zones may contain faces from more than one form, as in this example.

Figure 11.66*b* shows a common zone in hexagonal minerals: 6 (blue) faces (three not seen) are parallel to the c-axis. The zone axis is [001]. And in Figure 11.66*c*, the zone (purple faces) comprises three forms and the zone axis is [010]. In all three drawings, additional zones are present besides those just mentioned. In these three examples, the zones axes have simple indices because they are parallel to crystal axes. Zones need not be parallel to axes, and consequently the indices may contain values other than 0 and 1.

## ●Box 11-3 General Forms, Special Forms, and Miller IndicesThe relationship between special forms and general forms is the same as the one between special points and general points discussed in the previous chapter. Faces of a general form are neither parallel nor perpendicular to any symmetry element, while faces of special forms are. Thus, in special forms, symmetry relates fewer equivalent faces. The ^{2}/ point group; its Miller index contains three different values. The simplest index is {123}, indicating that _{m}h ≠ k ≠ l and that none of the indices are zero. The hexoctahedron contains 48 triangular faces. We can think of this form as octahedron with each of its eight faces replaced by six smaller triangular faces; hence, its name.All other forms with the same symmetry are special forms in the hexoctahedral class. Crystals in this class may contain any of 15 forms singly or in combinations. Figure 11.67 shows four examples. See Box 10-3 and the table in Chapter 10 for other examples. Special forms have two or more identical indices in their Miller index, or have an index with value zero; Figure 11.67 contains examples. For example cube faces belong to the { As shown in Figure 11.67, if we combine groups of faces on a hexoctahedron and replace them with a single face, we get a cube, octahedron, dodecahedron, or trapezohedron with no change in symmetry. Faces of the special forms all coincide with symmetry elements – that is what makes them special. Cube faces are perpendicular to 4-fold rotation axes, octahedron faces are perpendicular to 3-fold rotoinversion axes, dodecahedron faces are perpendicular to 2-fold axes, and trapezohedron faces are perpendicular to mirror planes. The examples seen in Figure 11.67 are all from a single point group, ^{2}/. But the principles discussed apply to all 32 point groups. However, symmetry relationships and the nature of special forms are more difficult to see for some point groups, especially those with little symmetry._{m} |

## ●Figure CreditsUncredited graphics/photos came from the authors and other primary contributors to this book. 11.1 Topaz from Pakistan, Didier Descouens, Wikimedia Commons Video 11-1: Bravais lattices, Keith Purtirka, YouTube |