Crystallography and Computational Quantum Mechanics Part III

Conventional and Primitive Lattices

In the previous section we saw that three-dimensional lattices can be separated into seven crystal systems, defined by their holohedry, or rotational symmetry. Somewhat confusingly we found that we only need six lattices to describe the seven systems.

When you look at any text on solid state physics or crystallography, such as Ashcroft and Mermin or any version of Kittel, you'll see that there are fourteen three-dimensional lattices. Where did those other eight lattices come from?

It turns out that those lattices have additional translational symmetry which does not affect their overall rotational symmetry. That means that each of these new lattices can be put into one of the seven crystal systems.

In this section we'll scroll through all seven crystal systems and find which of these Bravais lattices fit into each system.

A more formal version of this listing can be found in Mehl (2017), including different views of the standard and Wigner-Seitz unit cells.

The Seven Crystal Systems and the Fourteen Bravais Lattices

System I: The Triclinic Crystal System

As we found last time, the only rotational symmetry in the triclinic system is the complete 360° rotation. A triclinic lattice can be described by the primitive vectors

$\begin{array}{ccc} {\bf A}_{1} & = & a \, \hat{x} \\ {\bf A}_{2} & = & b \, \cos\gamma \, \hat{x} + b \, \sin\gamma \, \hat{y} \\ {\bf A}_{3} & = & c_{x} \, \hat{x} + c_{y} \, \hat{y} + c_{z} \, \hat{z} \end{array}$   ,   (1) where $\begin{array}{ccc} c_{x} & = & c \, \cos\beta \\ c_{y} & = & c \, (\cos\alpha - \cos\beta \cos\gamma) / \sin\gamma \\ c_{z} & = & \sqrt{c^2 - c_{x}^2 - c_{y}^2} \end{array}$   ,   (2) and with unit cell volume $V = a \, b \, c_{z} \, \sin\gamma$   .   (3) The lattice constants a, b, c, α, β, and γ are defined in the Part I of this tutorial. The can have any values that do not lead to a vanishing unit cell or which give the lattice additional rotational symmetry — see Table I of Part II for the list of lattice constants which generate more rotations.

Lattice 1: The Triclinic Bravais Lattice

There is only one lattice in the triclinic crystal system, the triclinic lattice. Its lattice vectors are identical to (1):

$\begin{array}{ccc} {\bf a}_{1} & = & a \, \hat{x} \\ {\bf a}_{2} & = & b \, \cos\gamma \, \hat{x} + b \, \sin\gamma \, \hat{y} \\ {\bf a}_{3} & = & c_{x} \, \hat{x} + c_{y} \, \hat{y} + c_{z} \, \hat{z} \end{array}$   ,   (4) with c_x, c_y, and c_z, defined by (2) and unit cell volume (3).

The only difference between (1) and (4) is that the former lattice vectors are defined with capital letters A_n and the later with lower case letters, a_n. This is a deliberate choice on our part with will be examined more closely in the next section. Suffice it to say that any lattice in a given crystal system can be defined by the conventional cell for that system, and we will use the A_n vectors to describe that cell. The individual Bravais lattices in that system are defined by their primitive cells, using a_n to define that lattice. Every crystal system has one Bravais lattice identical to the conventional cell, as here, but most systems have other Bravais lattices as well.

The primitive vectors, unit cell, and Wigner-Seitz cell for the conventional triclinic cell and a representative triclinic lattice are shown in Fig. 1. Since the lattices described by (1) and (4) are identical the two figures are identical.

System II: The Monoclinic Crystal System

The monoclinic crystal system has one 2-fold (180°) rotation axis. In the unique axis b ^† setting of the conventional lattice can be generated from (1) by setting α = γ = 90°, giving the primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & b \, \hat{y} \\ {\bf A}_3 & = & c \, \cos\beta \, \hat{x} + c \, \sin\beta \, \hat{z} \end{array}$   .   (5) with unit cell volume $V = a \, b \, c\, \sin\beta$   .  (6)

There are two lattices in the monoclinic system, simple and base-centered, as shown in Fig. 2.

Lattice 2: The Simple Monoclinic Bravais Lattice

The simple monoclinic lattice is identical to the conventional monoclinic lattice (5):

$\begin{array}{ccc} {\bf a}_1 & = & a \, \hat{x} \\ {\bf a}_2 & = & b \, \hat{y} \\ {\bf a}_3 & = & c \, \cos\beta \, \hat{x} + c \, \sin\beta \, \hat{z} \end{array}$   .   (7) with unit cell volume $V = a \, b \, c\, \sin\beta$   .  (8) The “simple” label merely indicates that it is identical to the conventional lattice.

The top drawings in Fig. 2 show the lattice vectors and unit cells for for the conventional and simple (or primitive) monoclinic lattice. Since the simple monoclinic lattice is the same as the conventional lattice, the cells are identical.

Lattice 3: The Base-Centered Monoclinic Bravais Lattice

Let's look at a set of primitive vectors we apparently pulled out of a hat:

$\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, a \, \hat{x} - \frac12 \, b \hat{y} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac12 \, b \hat{y} \\ {\bf a}_3 & = & c \, \cos\beta \, \hat{x} + c \, \sin\beta \, \hat{z} \end{array}$   .   (9) The unit cell of this lattice has the volume $V = \frac12 \, a \, b \, c\, \sin\beta$   .   (10)

The interesting thing about (9) is that we can write these vectors in terms of the conventional monoclinic lattice (5):

$\begin{array}{ccc} {\bf a}_{1} = \frac12 {\bf A}_{1} - \frac12 {\bf A}_{2} \\ {\bf a}_{2} = \frac12 {\bf A}_{1} + \frac12 {\bf A}_{2} \\ {\bf a}_{3} = {\bf A}_{3} \end{array}$   ,   (11) or visa versa $\begin{array}{ccc} {\bf A}_{1} = {\bf a}_{1} - {\bf a}_{2} \\ {\bf A}_{2} = {\bf a}_{1} + {\bf a}_{2} \\ {\bf A}_{3} = {\bf a}_{3} \end{array}$   ,   (12) This relationship shows that we can construct the lattice (9) from the lattice (5) or (7). It follows that the lattice described by (9) has the same holohedry as (5) and so belongs to the monoclinic crystal system.

This might just seem like a lot of words, so let's look at a picture. The bottom of Fig. 2 shows the lattice with the vectors given by (5). Its unit cell takes up half of the volume of a simple monoclinic cell, and we can easily see the relationships shown in (12). We can the think of equation (9) as describing a monoclinic lattice with an additional translational symmetry. This new lattice is called the base-centered monoclinic lattice for reasons that are obvious from the picture. This lattice has the same holohedry as the simple/conventional monoclinic lattice – the only rotational symmetry is a 2-fold axis – and so it belongs to the monoclinic system.

System III: The Orthorhombic Crystal System

The orthorhombic crystal system has three perpendicular 2-fold rotation axes, usually aligned along the Cartesian directions. The conventional cell is described by the vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & b \, \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   ,   (13) and the unit cell volume is $V = a \, b \, c$   .   (14) This structure can be generated from (5) by setting β = 90°.

There are four lattices in the orthorhombic system, the most of any crystal system: simple, base-centered, body-centered, and face-centered. These are shown in Fig. 3.

Lattice 4: The Simple Orthorhombic Bravais Lattice

Just as with the previous crystal systems, the simple orthorhombic lattice is identical to the conventional lattice, with primitive vectors

$\begin{array}{ccc} {\bf a}_1 & = & a \, \hat{x} \\ {\bf a}_2 & = & b \, \hat{y} \\ {\bf a}_3 & = & c \, \hat{z} \end{array}$   ,   (15) and unit cell volume $V = a \, b \, c$   .   (16) A sample set of primitive vectors and their unit cell can be seen on the top right of Fig. 3.

Lattice 5: The Base-Centered Orthorhombic Bravais Lattice

Like the base-centered monoclinic lattice, the base-centered monoclinic lattice is generated by taking

$\begin{array}{ccc} {\bf a}_{1} = \frac12 {\bf A}_{1} - \frac12 {\bf A}_{2} \\ {\bf a}_{2} = \frac12 {\bf A}_{1} + \frac12 {\bf A}_{2} \\ {\bf a}_{3} = {\bf A}_{3} \end{array}$   ,   (19) starting with the conventional cell (13). This gives the primitive vectors $\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, a \, \hat{x} - \frac12 \, b \hat{y} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac12 \, b \hat{y} \\ {\bf a}_3 & = & c \, \hat{z} \end{array}$   .   (20) which are just (9) with β = 90°. The volume of the unit cell is $V = \frac12 \, a \, b \, c$   .   (21) A sample set of primitive vectors and their unit cell can be seen in the center right drawing in Fig. 3.

Lattice 6: The Body-Centered Orthorhombic Bravais Lattice

In the base-centered orthorhombic lattice (20), often abbreviated as bco, there is a primitive vector pointing toward the center of the base of the conventional orthorhombic cell. In the body-centered cell the primitive vector points toward the center of the conventional orthorhombic cell, as shown at the bottom of Fig. 3.

There are many possible choices for the primitive vectors in this case. For example we could chose simply chose to keep the first two vectors the same as in the simple orthorhombic case and point the third vector toward the center of the conventional cell, giving the primitive vectors

$\begin{array}{ccc} {\bf a}_1 & = & a \, \hat{x} \\ {\bf a}_2 & = & b \, \hat{y} \\ {\bf a}_3 & = & + \frac12 \, a \, \hat{x} + \frac12 \, b \, \hat{y} + \frac12 \, c \, \hat{z} \end{array}$   ,   (22) The Encyclopedia we have chooses a more symmetric form, $\begin{array}{ccc} {\bf a}_1 & = & - \frac12 \, a \, \hat{x} + \frac12 \, b \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_2 & = & + \frac12 \, a \, \hat{x} - \frac12 \, b \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_3 & = & + \frac12 \, a \, \hat{x} + \frac12 \, b \, \hat{y} - \frac12 \, c \, \hat{z} \end{array}$   ,   (23) as shown in Fig. 3. Of course both of these cells have $V = \frac12 \, a \, b \, c$   .   (24)

Lattice 7: The Face-Centered Orthorhombic Bravais Lattice

The final orthorhombic lattice has the primitive vectors pointing toward three faces of the conventional orthorhombic unit cell rather than the base or the center. Not surprisingly it is called the face-centered orthorhombic lattice, and can be abbreviated as fco. Again we can conceive of many different sets of primitive vectors, but the Encyclopedia standard is given by

$\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, b \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac12 \, c \, \hat{z} \\ {\bf a}_3 & = & \frac12 \, a \, \hat{x} + \frac12 \, b \, \hat{y} \end{array}$   .   (25) with unit cell volume $V = \frac14 \, a \, b \, c$   .   (26) The center-left sketch in Fig. 3 shows this lattice and the corresponding unit cell.

System IV: The Tetragonal Crystal System

The tetragonal crystal system replaces one of the 2-fold rotation axes of the orthorhombic system with a 4-fold rotation axis. We can generate the lattice vectors by taking an orthorhombic lattice and choosing two of the vectors to have equal length, usually b = a. That leaves us with conventional cell unit with primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & a \, \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   ,   (27) and unit cell volume is $V = a^2 \, c$   .   (28) This makes one side of the conventional unit cell a square, system a 4-fold (90°) rotation axis.

There are only two lattices in the tetragonal system, simple tetragonal and body-centered tetragonal, as shown in Fig. 4. What happened to the other base-centered and face-centered structures? We'll explain as we go along.

Lattice 8: The Simple Tetragonal Bravais Lattice

The simple tetragonal lattice is, well, simple. By now you realize that its primitive vectors are just

$\begin{array}{ccc} {\bf a}_1 & = & a \, \hat{x} \\ {\bf a}_2 & = & a \, \hat{y} \\ {\bf a}_3 & = & c \, \hat{z} \end{array}$   ,   (29) and unit cell volume is $V = a^2 \, c$   .   (30)

Now we can see what happened to the base-centered tetragonal lattice: if we set b = a in (20), the vectors a₁ and a₂ and have equal length and are perpendicular to one another, giving us a lattice (24) with a $\rightarrow$ a/$\sqrt{}$2, or a standard tetragonal lattice.

Lattice 9: The Body-Centered Tetragonal Bravais Lattice

The body-centered tetragonal lattice (abbreviated bct) can be generated from the bond-centered orthorhombic lattice by setting b = a. If we use the standard bco vectors (23) as a starting point we find

$\begin{array}{ccc} {\bf a}_1 & = & - \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_2 & = & + \frac12 \, a \, \hat{x} - \frac12 \, a \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_3 & = & + \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} - \frac12 \, c \, \hat{z} \end{array}$   ,   (31) with the vectors and unit cell shown in at the bottom of Fig. 4. Of course both of these cells have $V = \frac12 \, a^{2} \, c$   .   (32)

Now we can answer the question of what happened to the face-centered tetragonal vectors. If we perform the b $\rightarrow$ a transformation to the face-centered orthorhombic cell in (25) we find the vectors

$\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, a \, \hat{y} + \frac12 \, c \, \hat{z} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac12 \, c \, \hat{z} \\ {\bf a}_3 & = & \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} \end{array}$   .   (33) Rotate this system by 45° about the z-axis. This takes $\begin{array}{ccc} \hat{x} & \rightarrow & \frac1{\sqrt2} \, \hat{x}' - \frac1{\sqrt2} \, \hat{y}' \\ \hat{y} & \rightarrow & \frac1{\sqrt2} \, \hat{x}' + \frac1{\sqrt2} \, \hat{y}' \end{array}$   .   (34) If we rotate this the coordinate system by 45° about the z-axis, placing the a'₁ vector along the new x-axis, and do a little manipulation of the vectors, we find $\begin{array}{ccc} {\bf a}'_1 & = & - \frac{1}{\sqrt2} \, a \hat{x}' + \frac{1}{\sqrt2} \, a \hat{y}' + \frac12 \, c \hat{z} \\ {\bf a}'_2 & = & + \frac{1}{\sqrt2} \, a \hat{x}' - \frac{1}{\sqrt2} \, a \hat{y}' + \frac12 \, c \hat{z} \\ {\bf a}'_3 & = & + \frac{1}{\sqrt2} \, a \hat{x}' + \frac{1}{\sqrt2} \, a \hat{y}' - \frac12 \, c \hat{z} \end{array}$   .   (35) This is exactly the same set of primitive vectors as (31), with a $\rightarrow$ a/$\sqrt{}$2. What we've shown is that a face-centered tetragonal lattice is just a rotated body-centered tetragonal lattice. By convention, we denote all of them as body-centered. In either case (31) and (32) represent the same lattice.

System V: The Trigonal Crystal System

The trigonal crystal system has a 3-fold rotation axis. We can generate a conventional trigonal lattice from the monoclinic lattice (5) by setting β = 60° or 120° and taking a = c. This would make the b axis the 3-fold axis. However, convention, driven by the way we usually start with an x-y plane and add a z-axis, says that we should put the 3-fold axis along c. Convention^‡ also tells us to set γ = 120°, and to make the a₁ and a₂ vectors look symmetric. All of this makes puts the primitive vectors of the conventional trigonal (and hexagonal) lattice into the form

$\begin{array}{ccc} {\bf A}_1 & = & \frac12 \, a \, \hat{x} - \frac{\sqrt{3}}2 \, a \, \hat{y} \\ {\bf A}_2 & = & \frac12 \, a \, \hat{x} + \frac{\sqrt{3}}2 \, a \, \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   .   (36) with unit cell volume $V = \frac{\sqrt{3}}2 \, a^{2} c$   .  (37)

Somewhat perversely the lattice described by (36) is called the hexagonal lattice. The reason for this can be seen at the bottom of Fig. 6, which shows the Wigner-Seitz cell for this lattice and the corresponding hexagonal lattice. As we can see, the Wigner-Seitz cell forms a hexagonal prism, giving rise to its name.

There are two lattices in the trigonal system, shown in Fig. 5. Some texts, including both Ashcroft and Mermin and Kittel, only list the rhombohedral lattice here, placing simple trigonal crystals in the hexagonal system. This is incorrect, as the crystal system is determined by the holohedry of the lattice. We will discuss this more fully in the hexagonal crystal system section.

Lattice 10: The Simple Trigonal Bravais Lattice

It should come as no surprise that the simple or “primitive” trigonal lattice is just the same as the conventional lattice (36) except in lower case:

$\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, a \, \hat{x} - \frac{\sqrt{3}}2 \, a \, \hat{y} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac{\sqrt{3}}2 \, a \, \hat{y} \\ {\bf a}_3 & = & c \, \hat{z} \end{array}$   .   (38) with unit cell volume $V = \frac{\sqrt{3}}2 \, a^{2} c$   .  (39) A sketch of the lattice and its unit cell is shown on the top right in Fig. 5.

Lattice 11: The Rhombohedral Lattice

The rhombohedral lattice has three primitive vectors of equal length, with the angle between any two of the vectors the same as the angle between any two others. We can generate it from the triclinic lattice (4) by taking b = c = a and β = γ = α, however the vectors are usually oriented to have the 3-fold rotation axis in the c direction. The Encyclopedia standard for the primitive vectors is

$\begin{array}{ccc} {\bf a}_{1} & = & \frac12 \, a \, \hat{x} - \frac1{2\sqrt{3}} \, a \, \hat{y} + \frac13 \, c \, \hat{z} \\ {\bf a}_{2} & = & \frac1{\sqrt{3}} \, a \, \hat{y} + \frac13 \, c \, \hat{z} \\ {\bf a}_{3} & = & - \frac12 \, a \, \hat{x} - \frac1{2\sqrt{3}} \, a \, \hat{y} + \frac13 \, c \, \hat{z} \end{array}$   .   (40) where a and c are the lattice constants associated with the conventional trigonal cell (36). The volume of the cell is one-third of that of the conventional cell, $V = \frac1{2 \sqrt{3}} \, a^{2} c$   ,  (41) and a sketch of the cell is shown at the bottom of Fig. 5. In terms of the conventional lattice (36) we can express the vectors (40) as $\begin{array}{ccc} {\bf a}_{1} & = & \frac23 \, {\bf A}_{1} + \frac13 \, {\bf A}_{2} + \frac13 \, {\bf A}_{3} \\ {\bf a}_{2} & = & - \frac13 \, {\bf A}_{1} + \frac13 \, {\bf A}_{2} + \frac13 \, {\bf A}_{3} \\ {\bf a}_{3} & = & - \frac13 \, {\bf A}_{1} - \frac23 \, {\bf A}_{2} + \frac13 \, {\bf A}_{3} \end{array}$   .   (42)

There is a duality in the expression of the rhombohedral lattice vectors. All of the primitive vectors (40) have length

$a = b = c = \sqrt{a^2/3 + c^2/9}$   ,   (43) where the first set of lengths (a,b,c) is not the same as the lengths (a,c) in (40). The angle between the vectors is $ \alpha = \beta = \gamma = \cos^{-1}[(2 c^2 - 3 a^2)/( 6 a^2 + 2 c^2) ]$   ,   (44) In terms of the length a from (43) and the angle α from (44) we can write the primitive vectors as $\begin{array}{ccc} {\bf a}_{1} & = & \frac1{\sqrt{3}} \, a \, \left[ \, \sqrt{3 (1 - \cos\alpha)} \, \hat{x} - \sqrt{1 - \cos\alpha} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha)} \, \hat{z} \right] \\ {\bf a}_{2} & = & a \, \left[ \sqrt{2 (1 - \cos\alpha)} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha)} \, \hat{z} \right] \\ {\bf a}_{3} & = & \frac1{\sqrt{3}} \, a \, \left[ \, \sqrt{3 (1 - \cos\alpha)} \, \hat{x} + \sqrt{1 - \cos\alpha} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha)} \, \hat{z} \right] \end{array}$   ,   (45) and the volume of the cell is $V = a^3 (1 - \cos\alpha) \sqrt{1 + 2 \cos\alpha}$   .   (46)

The literature is maddeningly inconsistent in the use (a,c) from (40) or (a,α) from (45) to describe a rhombohedral lattice. Some authors use the former while others use the later. The important thing to remember is that the a in (40) is not the same a used in (45). The conversion between the two sets of coordinates is given by (43) and (44). The Encyclopedia always uses the (a,c) notation, converting from authors' values of (a,α) as necessary.

System VI: Hexagonal Crystal System

The only difference between the trigonal and hexagonal crystal systems is the holohedry of the lattice: the trigonal system has a 3-fold rotation axis, and the hexagonal system a 6-fold rotation axis. Thus the primitive vectors of the conventional hexagonal lattice are given by (36), with unit cell volume (37). The top left part of Fig. 5 shows a sketch of the conventional hexagonal lattice, while Fig. 6 shows the standard unit cell and the Wigner-Seitz cell. There is only one lattice in the hexagonal system:

Lattice 10 (redux): The Simple Hexagonal Bravais Lattice

Like its conventional counterpart, the simple hexagonal lattice is identical to the simple trigonal lattice. We don't even give it a new number. Top right sketch in Fig. 5 and the standard/Wigner-Seitz cells in Fig. 6 describe it perfectly.

So how do we know when the lattice is trigonal or hexagonal? We don't, at least not until we determine the holohedry of the system and at the moment we do not have enough information to determine the holohedry. Fig. 7 shows the problem. This shows some of the Wigner-Seitz cells looking down the z-axis of the trigonal/hexagonal lattice (38). There is obviously a 6-fold axis about the origin, but there is also a 3-fold axis, and even a couple of 2-fold axes.

The only way we can distinguish between a simple trigonal and a simple tetragonal lattice is to look at the basis. Fig. 8 lets us do just that, as we add a basis to Fig. 7. On the left we have a system with a 3-fold rotation axis along the z-axis, so it is trigonal. On the right there is a 6-fold rotation axis, so the system is hexagonal.

     

Is this the way it should be? That could be debated, but it is the way that is has been since Paul Niggli published the first modern set of space group tables in 1919, and it persists through the current International Tables.

All of that said, from this point on we will refer to the lattice specified by (38) as the hexagonal lattice, regardless of the crystal system. This is the standard practice, and we regret the confusion.

System VII: Cubic Crystal System

This is the last crystal system. It has three 4-fold rotation axes through the origin of the cell, so it is a super tetragonal lattice. It also has four 3-fold rotation, one along each cube diagonal, so it is also a super rhombohedral lattice. We can construct the conventional lattice from the the conventional tetragonal cell (27) by setting c = a, or it can be generated from the rhombohedral lattice (45) by setting α = 90°. This is equivalent to using the hexagonal-like representation (40) by setting c = $\sqrt{3/2}$ a. In the standard orientation, the lattice vectors are

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & a \, \hat{y} \\ {\bf A}_3 & = & a \, \hat{z} \end{array}$   ,   (47) and unit cell volume is $V = a^3$   .   (48) There are three Bravais lattices in the crystal system. They are shown in Fig. 9 and described below.

The cubic lattice (48) can be regarded as a special case of the tetragonal lattice (27) with c = a. It can also be regarded as a special case of the rhombohedral lattice (45) with α = 90°. Not surprisingly all of the primitive lattices in the cubic crystal system will have similar relationships.

Lattice 12: The Simple Cubic Bravais Lattice

Of course the simple cubic lattice is identical to the conventional lattice, where we just change the A vectors to a:

$\begin{array}{ccc} {\bf a}_1 & = & a \, \hat{x} \\ {\bf a}_2 & = & a \, \hat{y} \\ {\bf a}_3 & = & a \, \hat{z} \end{array}$   ,   (49) and unit cell volume is $V = a^3$   .   (50) This cell is often abbreviated as the sc (simple cubic) lattice.

There is no base-centered cubic lattice for the same reason that there is no base-centered tetragonal lattice: if we try to construct one, we get a simple tetragonal lattice.

Lattice 13: The Body-Centered Cubic Lattice

This lattice, abbreviated bcc, can be generated from the body-centered tetragonal lattice (31) by stetting c = a, or from the rhombohedral lattice by setting α = cos^-1(-1/3) = 109.471° in (45) or c = $\sqrt{3/8}$ a in (40). The standard form is taken from (31) with c = a:

$\begin{array}{ccc} {\bf a}_1 & = & - \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} + \frac12 \, a \, \hat{z} \\ {\bf a}_2 & = & + \frac12 \, a \, \hat{x} - \frac12 \, a \, \hat{y} + \frac12 \, a \, \hat{z} \\ {\bf a}_3 & = & + \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} - \frac12 \, a \, \hat{z} \end{array}$   ,   (51) with the vectors and unit cell shown in at the bottom of Fig. 8. Of course both of these cells have $V = \frac12 \, a^{3}$   .   (52)

Lattice 14: The Face-Centered Cubic Lattice

The final^††3-dimensional Bravais lattice is the face-centered cubic lattice (fcc). It is generated from the face-centered orthorhombic lattice (25) by setting b = c = a:

$\begin{array}{ccc} {\bf a}_1 & = & \frac12 \, a \, \hat{y} + \frac12 \, a \, \hat{z} \\ {\bf a}_2 & = & \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{z} \\ {\bf a}_3 & = & \frac12 \, a \, \hat{x} + \frac12 \, a \, \hat{y} \end{array}$   .   (53) with unit cell volume $V = \frac14 \, a^{3}$   .   (54) It can also be generated from the rhombohedral lattice (45) by setting α = 60°. Alternatively we can generate it from (40) by taking c = $\sqrt$6 a. The cell is shown in the center of Fig. 8.

You might (should?) be surprised to find a face-centered lattice here, since there isn't a separate face-centered tetragonal lattice. However if we take the special value c = a in (33) we find that the cubic symmetry has returned.

The Close of the Bravais Lattices

That's it! We've gone thorough all fourteen Bravais lattices that can exist in three dimensions. Except for distinguishing between trigonal and hexagonal systems, though, we've neglected the atoms that appear in a crystal. When we do that, we'll find many more symmetries than the translational and rotational ones we find here. Stay tuned.

Acknowledgments

We are very grateful to David Hicks for providing the originals of the figures showing the lattices and their unit cells. These originally appeared in (Mehl, 2017).

Resources

AFLOW

AFLOW (Automatic FLOW) is an open-source package which can be used to generate and run first-principles electronic structure calculations for a variety of codes. It can also be used to analyze and compare crystal structures, including the production of Crystallographic Information Files (CIFs). This code is the primary resource used to generate the structures in the Encyclopedia of Crystallographic Prototypes.

gnuplot

gnuplot is a freely-distributable code for plotting graphs. We use it extensively in these tutorials and in other sections of the Encyclopedia.

Jmol

Jmol is an open-source Java viewer which can be used to visualize crystal structures as well as molecules. Many of the figures shown here were drawn with Jmol.

Glossary

Here is a brief definition of some of the terms used in this article:

Bravais Lattice:

In three dimensions, one of the fourteen allowed lattices. Each Bravais lattice belongs to one of the crystal systems.

Conventional Cell:

The unit cell describing all of crystals in a given crystal class. A lattice in this system may have additional translational symmetry, which leads to a different primitive lattice. We will discuss this in the next section, Conventional and Primitive Lattices.

Crystal:

A periodically repeated collection of objects in n-dimensions.

Crystallographic Information File (CIF):

The Crystallographic Information File (CIF) is a standard format for presenting the structure of a crystal, including information on the stoichiometry, lattice, basis, thermal displacement of the atoms, and other experimental information. All the structures found in the Encyclopedia of Crystallographic Prototypes are generated using CIF files.

Crystal System:

The collection of all lattices with the same holohedry.

Holohedry:

The point group of a lattice which describes its rotational symmetry, without translations, mirrors, glides, or inversion. In two dimensions the only possibilities are 1-, 2-, 3-, 4-, and 6-fold rotations (rotations by 360°, 180°, 120°, 90°, and 60°, respectively) about the origin.

Lattice:

A periodically repeated collection of points in n-dimensions.

Pearson Symbol

A method of specifying the crystal class (first letter), lattice type (second letter), and the number of atoms in the cell. (Pearson_1967)

Primitive Cell:

The lattice vectors describing a given crystal system. The primitive lattice may be the conventional lattice for the crystal system, or it may contain additional translational symmetry. This will be covered in the next section, Conventional and Primitive Lattices.

Primitive Vectors:

A set of vectors that defines the allowed shifts in the origin of the lattice that do not violate translational symmetry.

Rotational Symmetry:

A rotation of the crystal about an axis which produces a structure indistinguishable from the original.

Translational Symmetry:

A shift of the origin of a crystal that produces a structure indistinguishable from the original.

Unit Cell:

The (non-unique) smallest area (smallest volume in three dimensions) of space that reproduces all of the information about the crystal structure, and which can be periodically tiled to create the entire structure.

Wigner-Seitz Cell:

A uniquely defined unit cell consisting of all spatial points closer to a given lattice point than to any other lattice point. Ordinarily the point chosen is designated the origin, but it could be anywhere in the system.

Footnotes

^† We'll end up with a system having monoclinic symmetry, but if we start with a monoclinic lattice we have to decide between the “unique axis b” and “unique axis c” settings, and everything from here on out is “unique axis c”, so we punted. We can make the final cell orthorhombic by adding more symmetry operations, which for now are left as an exercise for the reader.

^‡ “Convention” and “conventional” do a lot of heavy lifting here.

^†† It's OK to celebrate, it's been a long road.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, Orlando, 1976), A downloadable copy is available through the Internet Archive.
• D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comput. Mater. Sci. 161, S1–S1011 (2019), doi:10.1016/j.commatsci.2018.10.043. (arXiv link)
• Charles Kittel, Introduction to Solid State Physics, 8^th edition (John Wiley & Sons, 2005). This is one of the premiere texts of condensed matter (aka solid state) physics, along with Ashcroft and Mermin. Unlike the later, this book has gone through numerous editions. We chose the to highlight the eighth edition because it is freely available online or as an ebook through the Internet Archive. The basic introduction to solid state physics remains the same in each edition, but newer editions add different topics.
• M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comput. Mater. Sci. 136, S1–S828 (2017), doi:10.1016/j.commatsci.2017.01.017. (arXiv link)
• International Tables for Crystallography (2016). Volume A, Space-group symmetry. Online access (Paywall). For freely available tables of space groups, see the Hypertext Book of Crystallographic Space Group Diagrams and Tables. For space group tables and much more, see the Bilbao Crystallographic Server.
• Paul Niggli, Geometrische Kristallographie des Diskontinuums, (Verlag vo Gebrüder Borntraeger, Leipzig, 1919). Available through the Internet Archive.
• W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, Volume 2, N.R.C. No. 8752 in International Series of Monographs on Metal Physics and Physical Metallurgy (Pergamon Press, Oxford, London, Edinburgh, New York, Paris, Frankfort, 1967)