Crystallography and Computational Quantum Mechanics Part VII:

3₁ and 3₂ Screw Axes in Rhombohedral and Cubic Crystals

In the last part of this ePICS tale of crystallographic concepts we introduced the screw axis, a combination of translational and rotational operations that leaves atoms winding along an axis like, well, the thread of a screw. We showed how you can generate 2-fold, 3-fold, 4-fold, and 6-fold screw axes by starting with an appropriate lattice and applying the translation/rotation operations as you go along a lattice vector.

One thing we did not mention in the previous discourse was that we only showed screw axes that were along one of the primitive vectors of the unit cell. In addition we only discussed screw axes in primitive lattices which matched the conventional lattice for a given crystal system. We didn't mention screw axes in base-, body-, or face-centered orthorhombic lattices, for example. This class also includes the body-centered tetragonal (bct) lattice. It turns out that every crystal with a bct lattice has four 2₁ screw axes along the lines (±¼,±¼,z) in lattice coordinates, and those without an inversion site have a 4₁ screw axis along the line (½ ½, z). Neither of these axes are along the standard primitive vectors of the lattice.

The best examples of these neglected screws are the ones we're going to discuss here, the 3₁/3₂ screws that are found in rhombohedral and cubic systems. In rhombohedral systems they are along the lines (⅔ 0 z) and (⅓ 0 z), respectively, in the lattice coordinates of the hexagonal conventional cell. Since cubic lattices are just special cases of the rhombohedral lattice, all cubic systems have these screw axes, but along each of the four <111> diagonals, making a much richer system. This discussion can also serve as a starting point for finding the other off-lattice-vector screws that we mentioned above.

3₁/3₂ Rhombohedral Screw Axes

If you look at a PICStorial diagram of the symmetries of a rhombohedral lattice, such as the one for space group R3 from the Hypertext Book of Crystallographic Space Group Diagrams and Tables, or the Space Group Diagrams in The Fascination of Crystals and Symmetry, you will see six symbols that look like triangles with arms coming out of each vertex. This is the symbol for a 3₁ screw axis, if the arms are pointing counterclockwise, and a 3₂ axis if they are going clockwise. We'd like to examine these axes and find out why they are here.

The Rhombohedral Lattice

We'll begin by reminding ourselves of how to describe a rhombohedral lattice, which belongs to the trigonal crystal system. We can describe it using the hexagonal conventional lattice with lattice constants a_h and c_h,

$\begin{array}{ccc} {\bf A}_1 & = & \frac12 \, a_{h}\, \hat{x} - \frac{\sqrt{3}}2 \, a_{h}\, \hat{y} \\ {\bf A}_2 & = & \frac12 \, a_{h}\, \hat{x} + \frac{\sqrt{3}}2 \, a_{h}\, \hat{y} \\ {\bf A}_3 & = & c_{h}\, \hat{z} \end{array}$   .   (1)

Alternatively, we can describe the system using the the rhombohedral primitive cell. If we use the hexagonal lattice constants a_h and c_h the primitive vectors can be written as

$\begin{array}{ccc} {\bf a}_{1} & = & \frac12 \, a_{h}\, \hat{x} - \frac1{2\sqrt{3}} \, a_{h}\, \hat{y} + \frac13 \, c_{h} \, \hat{z} \\ {\bf a}_{2} & = & \frac1{\sqrt{3}} \, a_{h}\, \hat{y} + \frac13 \, c_{h} \, \hat{z} \\ {\bf a}_{3} & = & - \frac12 \, a_{h}\, \hat{x} - \frac1{2\sqrt{3}} \, a_{h}\, \hat{y} + \frac13 \, c_{h} \, \hat{z} \end{array}$   .   (2) All of these vectors have the same length, which we'll call a_r, and the angle α_r between any pair of vectors is the same as the angle between any other pair. From (2) we find $a_{r} = |{\bf a}_{i}| = \sqrt{a_{h}^3/3 + c_{h}^2/9} ~ , ~ i = 1, 2, 3$   ,     (3) and $\alpha_{r} = {\bf a}_{i} \cdot {\bf a}_{j} / [|{\bf a}_{i}| |{\bf a}_{j}|] = \cos^{-1} [(2 c_{h}^2 - 3 a_{h}^2)/(6 a_{h}^2 + 2 c_{h}^2)] ~ , ~ i \ne j$   .   (4) This lets us rewrite (2) in the form $\begin{array}{ccc} {\bf a}_{1} & = & \frac1{\sqrt{3}} \, a_{r} \, \left[ \, \sqrt{3 (1 - \cos\alpha_{r})} \, \hat{x} - \sqrt{1 - \cos\alpha_{r}} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \\ {\bf a}_{2} & = & a_{r} \, \left[ \sqrt{2 (1 - \cos\alpha_{r})} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \\ {\bf a}_{3} & = & \frac1{\sqrt{3}} \, a_{r} \, \left[ \, \sqrt{3 (1 - \cos\alpha_{r})} \, \hat{x} + \sqrt{1 - \cos\alpha_{r}} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \end{array}$   ,   (5)

Using either (2) or (5) the relationship between the conventional and primitive vectors is

$\begin{array}{ccc} {\bf a}_{1} & = & \frac23 {\bf A}_{1} + \frac13 {\bf A}_{2} + \frac13 {\bf A}_{3} \\ {\bf a}_{2} & = & - \frac13 {\bf A}_{1} + \frac13 {\bf A}_{2} + \frac13 {\bf A}_{3} \\ {\bf a}_{3} & = & - \frac13 {\bf A}_{1} - \frac23 {\bf A}_{2} + \frac13 {\bf A}_{3} \end{array}$   .   (6) Since we're here, we might as well do it the other way, too: $\begin{array}{ccc} {\bf A}_{1} & = & {\bf a}_{1} - {\bf a}_{2} \\ {\bf A}_{2} & = & {\bf a}_{2} - {\bf a}_{3} \\ {\bf A}_{3} & = & {\bf a}_{1} + {\bf a}_{2} + {\bf a}_{3} \end{array}$   .   (7)

The first thing we want to get out of all those equations is that it takes three rhombohedral unit cells to fill the hexagonal conventional cell. That means that if we place an atom at the point

$x \, {\bf A}_{1} + y \, {\bf A}_{2} + z \, {\bf A}_{3}$     (8) in the hexagonal cell then the rhombohedral translational symmetry demands that we also have identical atoms at $(x + \frac23) \, {\bf A}_{1} + (y + \frac13) \, {\bf A}_{2} + (z + \frac13) \, {\bf A}_{3}$     (9) and $(x + \frac13) \, {\bf A}_{1} + (y + \frac23) \, {\bf A}_{2} + (z + \frac23) \, {\bf A}_{3}$     .   (10) Each of these points is in a different rhombohedral cell. As we'll see this is the key to the development of the 3₁ and 3₂ screw axes.

Of course if it takes three rhombohedral cells to make a hexagonal cell, the rhombohedral cell volume must be.

$V_{r} = | {\bf a}_{1} \cdot ( {\bf a}_{2} \times {\bf a}_{3} ) | = \frac{1}{2\sqrt{3}} \, a_{h}^2 c_{h}$   ,   (11) which is ⅓ of the conventional hexagonal cell volume.

Figure 1 shows the conventional and primitive unit cells for the rhombohedral lattice described above. We show the Wigner-Seitz conventional hexagonal cell, but for the primitive cell it is simplest to just use the cell bounded by the primitive vectors (2). Using (11) we know that it takes three rhombohedral cells to fill up the hexagonal cell. It doesn't really look like this in the figure, but if we slice and dice the rhombohedral cells we can in fact fit three cells into the hexagonal cell.

Rhombohedral Symmetry: Rotational and Translational

A crystal with primitive vectors (2) or (5) looks like a rhombohedral lattice, but it doesn't describe a rhombohedral crystal unless that crystal has a 3-fold rotation axis around the A₃ conventional lattice primitive vector. That means that if we have an atom at the point^‡

${\bf R}_{1} = x_{r} \, {\bf a}_{1} + y_{r} \, {\bf a}_{2} + z_{r} \, {\bf a}_{3}$       (12) there is an identical atom ${\bf R}_{2} = y_{r} \, {\bf a}_{1} + z_{r} \, {\bf a}_{2} + x_{r} \, {\bf a}_{3}$       (13) and a third at ${\bf R}_{3} = z_{r} \, {\bf a}_{1} + x_{r} \, {\bf a}_{2} + y_{r} \, {\bf a}_{3}$   .   (14) If we combine these positions equations with (8) through (10) we see that every atom in a primitive rhombohedral cell is connected to eight other atoms in the conventional hexagonal cell.

This (x_r,y_r,z_r)/(y_r,x_r,z_r)/(z_r,x_r,y_r) symmetry is the defining characteristic of a rhombohedral lattice, and one characteristic of a cubic lattice. If it is not present, say if atoms (12) and (13) are silver and (14) is gold, then this is not a rhombohedral or even a trigonal lattice. It is at best orthorhombic or has even lower symmetry.

Often we'll find it easiest to express the atomic positions in terms of the hexagonal primitive vectors (1). In that case we would write

${\bf R}_{1} = x_h {\bf A}_{1} + y_h {\bf A}_{2} + z_h {\bf A}_{3}$   .   (15) This only represents the same point if $\begin{array}{ccc} x_{r} & = & x_{h} + z_{h} \\ y_{r} & = & -x_{h} + y_{h} + z_{h} \\ z_{r} & = & -y_{h} + z_{h} \end{array}$   .   (16) In that case we can write the atomic positions (12)-(14) in terms of there hexagonal lattice constants, $\begin{array}{ccc} {\bf R}_{1} & = & x_h {\bf A}_{1} + y_h {\bf A}_{2} + z_h {\bf A}_{3} \\ {\bf R}_{2} & = & (y_h - x_h) {\bf A}_{1} - x_h {\bf A}_{2} + z_h {\bf A}_{3} \\ {\bf R}_{3} & = & -y_h {\bf A}_{1} + (x_h - y_h) {\bf A}_{2} + z_h {\bf A}_{3} \end{array}$   .   (17) Again if we are using the conventional hexagonal cell we must add the duplicates (8) through (10) to each of the points in (17) to fully describe the system.

We will use both representations, (12)-(14) and (17) depending on which is most convenient. Usually this will be the later.

The bottom line here is that if we place one atom in a rhombohedral lattice at (13) we are actually determining the positions of three identical atoms in the primitive rhombohedral cell and nine identical atoms in the conventional hexagonal cell. This multiplicity of atoms is generates the 3₁ and 3₂ screw axes in a rhombohedral lattice.

Development of the 3₁ and 3₂ Screw Axes

Let's put all this together by actually looking the steps we need to put together a 3₁ or a 3₂ screw axis. The steps are described below and shown as an animation in Figure 2 and described below.^††

  

Figure 2 shows a visualization of a 3₁ screw axis in a rhombohedral lattice. We'll go through it step by step:

1. Start with a hexagonal lattice described by the vectors (1). We have drawn the Wigner-Seitz cell for that lattice.
2. Add the corresponding rhombohedral lattice (2). We draw a corresponding rhombohedral unit cell (briefly highlighted in yellow), but this is not the Wigner-Seitz cell of the lattice. Instead it is the unit cell bounded by the primitive vectors. Trust us, it's a lot easier to visualize what's going on here if we do it this way.
3. The first screw axis we study is along the line ${\bf r}(z) = -\frac13 \, {\bf A}_{2} + z_{h} \, {\bf a}_{3}$   .   (18) Put an atom somewhere in the rhombohedral unit cell near that line. We're putting it close to one of the screw axes so that we can see the screw when we view it from the top, but any coordinates that don't place the atoms directly along the screw axis will do. We will color the atoms by their height above the z = 0 plane. These first three atoms have a red color (⬤). The color is only used to indicate the height of the atom above in the hexagonal unit cell. In reality all of the atoms in this screw are identical.
4. Because of the rotational symmetry (17) there are two identical axes at hexagonal coordinates (-⅓,0,z_h) and (⅓,⅓,z_h). Rotational symmetry also implies that the atom in (c) above is part of a triplet set. If the first atom was in at the point (x_r,y_r,z_r) in rhombohedral coordinates then the second will be at be at (y_r,z_r,x_r).
5. And the third is at (z_r,x_r,y_r).
6. Now duplicate the unit cell and move it in the a₁ direction. If you like, we're treating the rhombohedral unit cells as building blocks and stacking them on top of one another. This particular stacking transports the atoms from (c), (d), and (e) to new positions. Two of these atoms will be outside the conventional hexagonal cell. Since we don't care about them right now they'll vanish when they cross out of the hexagonal cell (more or less). We can always get them back using translational symmetry. This new atom is in a plane ⅓ c above the atoms in (c)-(e). We will color atoms in this plane green (⬤).
7. Do the same thing, but now move the cell to a₂. Note that we could also have done this by using the rotational operation (13), using the point we found in (f) as our starting point.
8. And add another cell a a₃, or use (14). In either case, we now have three atoms above our original three atoms, slightly offset from the originals.
   So far we've only accounted for six of the nine atoms we claimed were in the unit cell.
9. To start getting the other three atoms, shift our original unit cell out to -a₁. This will give use one atom in the hexagonal unit cell a distance ⅓ c below the atoms in (c)-(e). We will color atoms in this plane blue (⬤).
10. Repeat with -a₂, or apply the rotation operations.
11. Finally add another unit cell -a₃ away from the original.

That's it. We now have a hexagonal unit cell with nine atoms. Figure 3 freezes the top view from Figure 2 at its final from. There are three equivalent screw axes, and winding counterclockwise (blue, red, green) if we view the cell from the top. We've constructed a set of 3₁ screws.

The final hexagonal unit cell in Figure 3 looks kind of empty. There are large open spaces. We can think of the hexagonal cell as being made up of six triangular prisms, but only three of them have atoms in them. What happens if we put atoms in the currently unoccupied prisms?

That result is shown in Figure 4 and Figure 5. We created these PICStures using the identical starting coordinates as in Figure 2 and Figure 3, but we reversed the first two coordinates. We then generate a screw by following the same steps as in Figure 2. Since the screw we generate here is independent of the previous step we use a different color scheme.

  

In this new case the screw rotates clockwise as we come up the z-axis, so it is a 3₂ screw. Here we should again emphasize that this is a different screw then the one shown in Figure 2 and Figure 3. The atoms in the 3₂ screw can be a different species then the atoms in the 3₁ screw. This will not be the case in the cubic system, it's unique to the rhombohedral lattice.

Figure 6 shows both screws, and highlights the six prisms in this system. The white regions have 3₁ screws, while the gray regions have 3₂ screws. It's fair to ask if this is a general rule. If we plop an atom down in the white region and use the rotational and translational symmetry of the rhombohedral lattice, do we always get a 3₁ screw? What about an atom in the 3₂ region? Have we discovered a general rule?

  

The left-hand part of Figure 7 sketches a general proof to this rule. We start with a red atom in one of the pink prisms. The rotational symmetry operations in a rhombohedral crystal, ((12) - (14)) places the other two red atoms. We then take the top red atom and apply the translation -a₂. This places a blue atom in the lower white prism. This atom is below the red atom in the z direction out of the page. Similarly we start from the rightmost red atom and translate it by a₃. This results in the green atom, which is above the red atom.

What's the rotation of these atoms about the screw axis? We can see this in the right-hand part of Figure 7. Starting with the blue atom, the lowest atom in the cell, we move to the next highest atom, the red atom. We go from there to the green atom, which is above it in the z direction. Finally we go from the green atom to another blue atom. This arrow is not the the blue atom in the previous paragraph, but its image

a₁ + a₂ + a₃ = A₃ = c $\hat{z}$ above it. The arrows then trace out a screw which rotates counterclockwise as it comes out of the page, so it is a 3₁ screw. It's obvious that we can place the red atom anywhere in a white area and get a 3₁ screw. If however, we started in the region a similar process would lead to a clockwise turning 3₂ screw.

What about the gray areas? If we do a mirror reflection of the atoms in Figure 7 with the reflection plane containing the a₂ primitive, then all of the atoms in the white areas will be transferred to the yellow areas. Since this is a reflection, the arrangement of the atoms in the prisms will be reflected, as well as the path shown in the right-hand side of the figure. The reflected path will turn clockwise, indicating a 3₂ screw.

All of this proves, or at least sketches the proof of, the theorem that atoms in the white areas form 3₁ screws and atoms in the yellow areas from 3₂ screws.

3₁/3₂ Cubic Screw Axes

Since we've learned that all cubic lattices can be expressed as special cases of the rhombohedral lattice, can we assume that any cubic lattice will also have pairs of 3₁ and 3₂ screw axes? The answer is yes, but things are a bit more complicated than that.

Let's start with the obvious. The simple, body-centered, and face-centered cubic lattices are all special cases of the rhombohedral lattice. We can relate these lattices to the rhombohedral/hexagonal lattice. Remember that the hexagonal lattice has primitive vectors in the form

$\begin{array}{ccc} {\bf a}_{1} & = & \frac12 a_{h} \, \hat{x} - \frac{1}{2\sqrt{3}} a_{h} \, \hat{y} + \frac13 c_{h} \, \hat{z} \\ {\bf a}_{2} & = & \frac{1}{\sqrt{3}} a_{h} \, \hat{y} + \frac13 c_{h} \, \hat{z} \\ {\bf a}_{3} & = & - \frac12 a_{h} \, \hat{x} - \frac{1}{2\sqrt{3}} a_{h} \, \hat{y} + \frac13 c_{h} \, \hat{z} \end{array}$   .   (20) and the rhombohedral lattice looks like this: $\begin{array}{ccc} {\bf a}_{1} & = & \frac1{\sqrt{3}} \, a_{r} \, \left[ \, \sqrt{3 (1 - \cos\alpha_{r})} \, \hat{x} - \sqrt{1 - \cos\alpha_{r}} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \\ {\bf a}_{2} & = & a_{r} \, \left[ \sqrt{2 (1 - \cos\alpha_{r})} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \\ {\bf a}_{3} & = & \frac1{\sqrt{3}} \, a_{r} \, \left[ \, \sqrt{3 (1 - \cos\alpha_{r})} \, \hat{x} + \sqrt{1 - \cos\alpha_{r}} \, \hat{y} + \sqrt{2 (1 + 2 \cos\alpha_{r})} \, \hat{z} \right] \end{array}$   ,   (21) where a_h and c_h are the hexagonal lattice constants, while a_r and α_r describe the rhombohedral lattice. The relationships between the these lattice constants and the corresponding cubic lattice constant a_c are shown in Table 1.

Table 1 The relationships between the cubic lattice constant a_c, the primitive lattice volume V_p, the hexagonal lattice constants (a_h,c_h) and the rhombohedral lattice constants (a_r,α_r) describing the cubic lattices in the hexagonal setting (20) and the rhombohedral setting (21) respectively. The cubic lattice constant a_c describes the conventional cubic lattice.

Lattice	$V_{p}$	$a_{r}$	$\alpha_{r}$	$\cos\alpha_{r}$	$a_{h}$	$c_{h}$
face-centered cubic	$\frac14 a_{c}^3$	$\frac{1}{\sqrt{2}} \, a_{c}$	$60^{\circ}$	$\frac12$	$a_{c}$	$\sqrt{3} \, a_{c}$
simple cubic	$a_{c}^3$	$a_{c}$	$90^{\circ}$	$0$	$\sqrt{2} \, a_{c}$	$\frac{1}{\sqrt{2}} \, a_{c}$
body-centered cubic	$\frac12 a_{c}^3$	$\frac{\sqrt3}2 \, a_{c}$	$109.47122^{\circ}$	$-\frac13$	$\sqrt{2} \, a_{c}$	$\frac{\sqrt{3}}2 \, a_{c}$

Since a cubic system is just a special case of a rhombohedral system it's not surprising that a cubic system must have a 3-fold rotation axis as well. That must be along the A₃ conventional (hexagonal) lattice vector. In terms of the primitive cell that's

${\bf a}_{1} + {\bf a}_{2} + {\bf a}_{3}$   . In the standard representation for a simple cubic case (not the one in (21)) this becomes $a \, \hat{x} + a \, \hat{y} + a \, \hat{z}$   , which is the [111] direction in using the Miller indices notation.

The 3-fold rotation axis only guarantees a rhombohedral crystal. To lock in a cubic system we must also have two 2-fold rotation axis or 2₁ screw axis along each of the lattice vectors of the conventional cubic lattice. If we chose the z-axis, then if we place an atom at the point

$x \, a \, \hat{x} + y \, a \, \hat{y} + z \, a \, \hat{z}$   (23) and assume a 2-fold rotation axis, there must also be an identical atom at $-x \, a \, \hat{x} -y \, a \, \hat{y} + z \, a \, \hat{z}$   .   (24) This is equivalent to a z = 0 mirror plane. In addition there must be x = 0 and y = 0 mirror planes.

This everything everywhere symmetry of any cubic lattice, simple, body-centered or face-centered, means that we could align any of the four <111> directions along the A₃ axis of the hexagonal unit cell shown in Figure 1, which means that each of these diagonals is a 3-fold rotation axis, as shown in Figure 8. In fact (Ubic, 2024) defines a cubic crystal as a system that has four 3-fold rotation axes.

At a minimum, the combination of 2-fold and 3-fold rotation axes means that unless we place an atom on a high symmetry point we are are actually placing at least twelve, and up to forty-eight, atoms in the crystal. We'll provide an example of what happens by placing a phosphorous atom at lattice coordinates (0.64,0.97,0.35) in a simple cubic cell. The system can be described as being in space group P23 #195 with atoms on the (12j) Wyckoff positions. Figure 9 shows how we place the atoms: first around the [111] axis, then around the [-111] axis, followed by [1-11] and [11-1]. The result is a crystal with twelve atoms — higher symmetry cubic lattices could have as many as forty-eight atoms.

Figure 10 shows the final crystal constructed in Figure 9 looking down the [111] axis. In this view we have colored all the atoms identically to emphasize that they are connected by symmetry. This view is reminiscent of that in Figure 6, and indeed this can be seen as a simple cubic lattice embedded in a hexagonal cell. Unlike Figure 6, here there are identical atoms in each of the six prisms. This tells us that the simple cubic crystal will automatically contain both 3₁ and 3₂ screws, and that they will be composed of identical atoms connected by symmetry.

The vast number of atoms in the cubic cell means that it won't be easy to show the construction of the screw axes as we did in the rhombohedral system – we simply do not have enough easily distinguishable colors. Instead we will stop using gnuplot to view the cell and instead use Jmol. Figure 11 shows the same cell as found in Figure 9 and Figure 10, but now we have replaced the colored circles by balls representing phosphorous atoms. If you want to play with this yourself, you can download the Crystallographic Information File (CIF) for this very hypothetical structure here.^‡‡

We obviously can not see the screw axes from that view. The last section showed that we needed to properly stack at least seven unit cells to see the screw. Since the cubic system is richer than the rhombohedral system we massively overkill the problem by using Jmol to produce a 4x4x4 block of the cubic unit cell shown in Figure 11. We then rotate the system so that so we look down the [111] axis, and zoom in so that we can see the conventional hexagonal cell. The result is shown in Figure 12, and again you can play with it yourself using this Jmol state file.

We can see that the phosphorous atoms form both 3₁ and 3₂ screws. Unlike the 3₁ and 3₂ screws in the rhombohedral system these two screws cannot be composed of different atoms. They are connected by the rotational symmetries of the cubic system, and so all the atoms in both sets of screws must be identical. In addition, the cubic symmetries tell us that if we look down a [-111], [1-11], or [11-1] axis, which you can do by manipulating the Jmol state file, we will see the same image. The bottom line is that a cubic system, and we mean any cubic system, has multiple and connected 3₁/3₂ axes along every <111> direction.

Just because the 3₁ and 3₂ are generated from the symmetry-connected phosphorous atoms doesn't mean that the screws look identical. We can't see the difference very well in Figure 10, so we blow up part of the image to produce Figure 13, which shows an enlargement of the upper left side of Figure 12. The left-hand screw turns counter-clockwise, and so is a 3₁ screw. The right-hand screw turns counterclockwise and so is a 3₂ screw. The two screws are obviously different, even the the atoms are all connected by symmetry, and every atom in the 3₁ screw is in a 3₂ screw oriented along another <111> axis, and visa versa. You can see all of this by downloading the the Jmol state file and twirling it around.

The bottom line is that every cubic system has a set of 3₁ and 3₂ screw axes along the <111> directions. Each screw axes contains the same species of atom, but the spacing of atoms in the 3₁ and 3₂ screws can be different.

Readers will note that we haven't discussed the screws in the face-centered and body-centered cubic lattices. They are there, and can be constructed using the same principles described above, but the resulting cells contain a very large number of atoms and don't really show anything new. You are welcome to generate your own fcc and bcc screws using the procedures described above.

Wrapping Up

That's it! We've shown that the 3₁ and 3₂ screw axes in a rhombohedral or cubic system are a natural result of the fact that three rhombohedral or cubic cells fit into one hexagonal cell, and the resulting stacking of primitive cells generates the screws. What's more, we've shown that all cubic lattices have related 3₁ and 3₂ screw axes, with identical looking screws along all four of the <111&rt; directions.

As we've seen, screw axes are a combination of rotations and translations. Next time we'll look at another symmetry operation, the mirror plane, and what happens when you add mirrors and translations (it's called a glide plane).

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.

Cambridge Crystallographic Data Centre (CCDC)

The Crystallographic Data Centre (CCDC) hosts both the organic Cambridge Structural Database and the Inorganic Crystal Structure Database, with a search engine which allows free, albeit somewhat restricted, access to structures in both the CSD and the CCDC.

Cambridge Structural Database (CSD)

Cambridge Structural Database (CSD) contains three-dimensional structural data for organic and metal organic systems. As of 1 January 2025 it contained 1,359,039 structures. There is a paywall, which can be worked around using the CCDC search engine described above.

gnuplot

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

Hypertext Book of Crystallographic Space Group Diagrams and Tables

The Hypertext Book of Crystallographic Space Group Diagrams and Tables has tables and figures listing all of the symmetry operations for each of the 230 three dimensional space groups. Links to a space group in the text will lead to the appropriate page on this site. The Reader's Guide has brief descriptions of all the symmetry operations that can occur in the 230 space groups.

Inorganic Crystal Structure Database (ICSD)

The Inorganic Crystal Structure Database (ICSD) contains structural data for inorganic crystals, though the occasional organic crystal slips in. In early 2025 the ICSD had information for 318901 structures, though many are duplicates. Like the CSD this is paywalled, but you can get any structure from the CCDC search engine if you are patient.

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.

Space Group Diagrams

Frank Hoffmann's extremely useful Fascination of Crystals and Symmetry includes a post devoted to Space Group Diagrams. It is slightly different from the Hypertext Book in that it show rotation axes and atomic positions in two different views, and includes both rhombohedral and hexagonal settings, similar to the paywalled International Tables. At the moment this page incomplete, but it does have the R3 space group diagrams we mentioned above.

Glossary

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

Basis:

The collection of items (atoms, pixels, paint drops) that decorate a lattice to produce a crystal or a wallpaper. Every object in a crystal structure is part of the basis.

Basis Vectors:

The vectors pointing from the origin of the lattice to the individual members of the basis.

Cartesian (Basis) Coordinates:

The positions of the basis vectors relative to the origin given on a standard Cartesian grid.

Crystal:

A periodically repeated collection of objects in n-dimensions.

Lattice:

A periodically repeated collection of points in n-dimensions.

Lattice Coordinates:

The positions of the basis vectors expressed relative to the chosen primitive vectors of the system.

Miller Index

The Miller Index is a system for denoting directions and plane orientations in a crystal lattice. For our purposes we're interested in directions [lmn], where l, m, and n are integers and the type of brackets used are very important. This [lmn] notation indicates a vector parallel to $\ell {\bf a}_{1} + m {\bf a}_{2} + n {\bf a}_{3}$ where the a_i are the primitive vectors of the lattice. The notation <lmn> indicates all directions equivalent to [lmn] by symmetry. In a cubic crystal <100> refers to the [100], [010], and [001] directions, while <111> refers to [111], [-111], [1-11], and [11-1].

Mirror Plane

If a crystal has a mirror plane, the atoms on one side of the plane are a reflection of the atoms on the other side of the plane. To simplify notation the origin of the lattice is often taken to be on the mirror plane, though this does not need to be the case.^†.

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.

Screw Axis

A combination of translational and rotational symmetry: a translation along an axis of some amount combines with a rotation around that axis, leading to a structure which is identical the first structure.

Translational Symmetry:

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

Unit Cell:

The (non-unique) smallest volume (smallest area in two 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.

Wyckoff Positions:

A subgroup of a space group that is itself a group, or irreducible representation. If an atom in a crystal is known to be at a given Wyckoff position, there is an identical atom at all the points in subgroup.

Footnotes

^† As we will see in later discussions this “simplify the symmetry operations” approach does not necessarily define a unique origin. Many space groups have two possible origins which lead to different, but equivalent, “simple” expressions of the symmetries of the group.

^‡ In the space group tables arbitrary coordinates are always given as (x,y,z), even if primitive vectors are not along the Cartesian directions. In all cases except the rhombohedral lattice they are given in terms of the conventional unit cell. In the rhombohedral case they can be given with respect to the hexagonal lattice vectors (1) or the rhombohedral lattice vectors (2). To avoid(?) confusion we will use (x_h,y_h,z_h) as the hexagonal coordinates and (x_r,y_r,z_r) as the rhombohedral coordinates, with the two sets related by (16).

^†† It is doubtful that these steps actually occur in Nature. If a screw axis is formed it occurs because it minimizes the free energy of the system, and the atoms presumably move into position simultaneously.

The structure can also be generated using the AFLOW command
$ aflow A_cP12_195_j:P --params=5,0.64,0.03,0.35 --cif

References

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, Orlando, 1976), chap. 4, pp. 73–75. A downloadable copy is available through the Internet Archive.
2. T. Hahn, ed., International Tables of Crystallography. Volume A: Spacegroup symmetry (Kluwer Academic publishers, International Union of Crystallography, Chester, England, 2002).
   For free versions of most of this information see the Bilbao Crystallographic Server and the Hypertext Book of Crystallographic Space Group Diagrams and Tables.
3. 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)
4. 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)
5. Frank Hoffmann, Space Group Diagrams in The Fascination of Crystals and Symmetry
6. 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)
7. R. Ubic, Crystallography and Crystal Chemistry (Springer, Cham, Switzerland, 2024), doi:10.1007/978-3-031-49752-0"