Crystallography and Computational Quantum Mechanics Part VI:

Screw Axes

In previous articles, we found that all periodic crystals have translational symmetry, while many crystals have rotational symmetry, at least in the ideal case.

What if we combined the two symmetries: a rotation followed by a translation along the rotation axes (or visa versa)? Figure 1 shows a sample unit cell.

  +     =  

• The left hand side of Figure 1 shows an example of translational symmetry. All the atoms in the unit cell are moved upward by the lattice vector a₃. This forces some atoms to leave the unit cell though the top, but their duplicates simultaneously come into the cell from the bottom. At the end of the movement the cell looks identical to the original cell.
• The crystal in the middle has a 4-fold rotation axis along a₃. We rotate through an angle of 360°, but stop every 90°. After each 90° step the cell looks identical to the original.
• On the right we show a crystal with a 4₁ screw axis along a₃. We simultaneously translate the axis through a complete lattice vector while rotating by 360°. We pause every time we rotate by 90° with a translation of ¼ a₃. At each step the crystal looks identical to the starting configuration, demonstrating a combination of translational and rotational symmetry.

We think of the path on the right as a spiral through the crystal. As we go up the spiral we simultaneously rotate around a₃ and translate along it. The crystal shown is the result of stopping every time we rotate by 90° and translate by ¼ a₃ and place an atom at that point. Viewed over many unit cells we would see a spiral (screw) around the axis.

This is not restricted to 90° rotations. Figure 2 shows γ-selenium, which an be thought of as dropping an atom every time we rotate by 120° while translating by ⅓ a₃. This particular view is an example of a 3₁ screw axis.

We originally defined screw axes when we were talking about chiral space groups, where they figure prominently, but it's worthwhile to explore these symmetries in more detail. Here we'll look a wide variety of screw axes^† in three dimensions. As with rotations, it turns out that we can have 2-, 3-, 4- and 6-fold screw axes, with 5-fold and 7+-fold rotations forbidden by the Crystallographic Restriction Theorem. Unlike rotations, we can multiple screw axes with the same degree of rotation. So which screw axes are allowed? Let's find out.

Preliminaries

Before we get started with a discussion of the allowed screw axes, let's get a some things out of the way:

Notation

A screw axis is described by a label

n_m   , where n is the number of rotations it takes to get get back to the starting point and m represents how many unit cells we go through as we make a complete circle of n rotations?

So what does that mean? Well, we start by placing an atom somewhere in a unit cell. We then do a rotation by an angle of 360°/n around the rotation axes.^‡ We then make a translation along the axis. How big a translation? After we do n translations we want to have gone through m unit cells. If “c” is the length of the primitive vector along the rotation axis, then the translation distance is m c/n along the axis. We place another atom here, and put its duplicates in other unit cells as required by translational invariance. After we've done that n times we're back where we started.

As a to do list it looks like this

1. Place an atom at some point R on the lattice. Add in all its translationally invariant duplicates at R + n₁a₁ + n₂a₂ + n₃a₃
2. Starting at that point rotate around the axis by 360°/n while traversing a distance m c/n along the axis.
3. Suppose this takes you to a point R' which is outside the original unit cell. In that case, translational invariance requires there to be an equivalent atom inside the original unit cell. If a₃ is the rotation axis, this point is at R' - a₃   .
4. Repeat until you've completed n rotations. If you've been keeping track of all of the atoms related by translational symmetry you'll find that your last atom is identical to the first atom. The original unit cell will have n atoms spiraling around the rotation axis.

Confused? Don't worry, we'll step through the construction of all these screws.

For a given value of n, we can have m = 1, 2, 3, … n - 1. m = 0 or m = n would correspond to a rotation with no translation or with a translation of c, so this would be a regular n-fold rotation. As far as we know no one has ever described a regular n-fold rotation axis as a n₀ or n_n screw, but it would be consistent.

(Lack of) Origin

In the examples below, we will always start with an atom at z = 0. This is done entirely for convenience, and is not essential. In general no atom in a screw axis need be at the origin. In some space groups it is required to fulfill other symmetry restrictions, such as an inversion, but that is not something we will discuss here.

Colors

The figures below all show atoms of different colors in a unit cell, with a given color corresponding to the “height” of the atom above the starting z = 0 plane. This is fictional, done only for clarity. In reality each figure only shows one species of atom. In any real crystal, all atoms that are part of a particular screw are identical. If there are multiple screws in the system the atomic species in each screw may be different. We won't look at that here, but if you want to look ahead you can see what happens in rhombohedral and cubic crystals.

With that out of the way, lets look the allowed screw axes in three dimensions. We'll start with the simplest, 2₁.

2-fold Screw Axes

A 2₁ screw can occur in any crystal system with symmetry higher than triclinic. The “2” indicates that we'll be doing two rotations before we get back to the original position, which means that each rotation is 180°. The “1” indicates that before (or after, or during) each rotation we'll do a translation by 1/2 of the lattice vector pointing along the axis. This is best shown by an example, shown in Figure 3.

       

Figure 3 shows the development of a 2₁ screw axis. This type of screw can be found in any crystal system above triclinic. Here for simplicity we use a simple orthorhombic unit cell.

1. Start with simple orthorhombic lattice.^††; a₁ = a $\hat{x}$       
   a₂ = b $\hat{y}$   (1)
   a₃ = c $\hat{z}$        Add an atom at a point (x,y,z) in lattice coordinates, or, since this is an orthorhombic cell, at $a \, x \, \hat{x} + b \, y \, \hat{y} + c \, z \, \hat{z}$   (2) in Cartesian coordinates. Since this is a periodic system, there will be an identical atom at (x,y,1+z), i.e., $a \, x \, \hat{x} + b \, y \, \hat{y} + c \, (1+z) \, \hat{z}$ In the figure we take z = 0 so that we can see both the initial atom and its final image, but this is not a requirement.
2. Now find a point ½ a₃ above the original atom. In lattice coordinates this is (x,y,½). Rotate by 180° about the a₃ axis, keeping track of where that point ends up. Place an atom there. In lattice coordinates it is at (-x,,-y;½). That's the first of our two translation plus rotation operations. We colored this one blue to indicate its height above the plane, but the “atoms” in this picture are actually identical.
3. Translate upward from that atom by ½ a₁, ending a (-x,-y,1). Follow that by another 180° rotation about a₂. That's the second translation plus rotation, and we're now at (x,y,1). We place an atom there. It's red, because it is one lattice translation away from the atom we put down in (a), an so would be there by translation symmetry alone.
4. And here's the final unit cell.

We then rotate the cell around the a₃ axis so that you can see the screw. In the central figure we show the path as a spiral, again showing the screw, and in the right-hand figure we look at the whole process from the top.

Suppose you had three dimensional model of the crystal that looked like the one in Figure 2. If you performed one rotation plus translation as described above you'd get a crystal that looks exactly the same, so this crystal is invariant under the application of a 2₁ screw.

Though we started with an orthorhombic unit cell, if the atoms shown in Figure 2 are the only ones in the unit cell the final crystal symmetry is monoclinic. AFLOW tells us that it is in space group P2₁/m. The 2₁ part of the label is an explicit acknowledgment of the contribution of the screw axis to this space group. To keep the orthorhombic symmetry of the lattice we would have to add additional atoms. We will discuss that when we eventually get tutorial on Wyckoff positions.

When we think about it, we can realize there can only be one type of screw axis with a 180° rotation like this.^††† if we rotate twice and translate twice, we must get back to the original positions. It doesn't matter if we rotate clockwise or counterclockwise (as seen looking down the axis). Higher-order (smaller angle) rotations will have more possibilities, as we shall see.

Finally, Table 1 gives the lattice coordinates of the atoms in a 2₁ screw. This is a form similar to the Wyckoff positions in space groups, and screw axes do form parts of many Wyckoff positions in a multitude of space groups.

Table 1 The lattice coordinates of the two atoms comprising a 2₁ screw axis using an arbitrary value for y. We use the unique axis b representation (1). If this is the only symmetry operation allowed for the crystal it is in monoclinic space group P2₁ #4, and the atoms are at the (2a) Wyckoff positions. Obviously there are many more space groups which have 2₁ screw axes.

Atom	21
Space Group	P21
Number	4
Wyckoff Letter	(2a)
1	(x,y,z)
2	(-x,-y,z+½)

3-fold Screw Axes

The 3-fold screw axes feature three 120° rotations, with either ⅓ c or ⅔ c translations along the rotation axis. These are called 3₁ and 3₂ screw axes, respectively. Three-fold screw axes can be found in trigonal crystals, including both simple trigonal and rhombohedral lattices. Every cubic crystal also has three-fold screw axes. The rhombohedral/cubic case can't be described using one or two primitive unit cells, so we will save that for later.

Screw axes in trigonal (or even cubic) systems start with the hexagonal lattice:

a₁ = $\frac12$ a $\hat{x}$ - $\frac{\sqrt{3}}2$ a $\hat{y}$                 
a₂ = $\frac12$ a $\hat{x}$ + $\frac{\sqrt{3}}2$ a $\hat{y}$     .     (3)
a₃ = c $\hat{z}$           The a₃ lattice vector is the screw axis.

The 3₁ screw evolves much like the 2₁ screw: after three steps, instead of two, we end up at the other end of the unit cell from where we started. The 3₂ screw is a little more difficult to visualize, since three translations of length 2c/3 will takes us all the way across two unit cells. Because of that, and because this problem is going to reoccur with a vengeance for the 4_m and 6_m screws, let's look at the 3₂ screw in some detail.

Figure 4 shows the development of a 3₂ screw. Unlike the 2₁ screw we must transverse two unit cells, so we show two hexagonal cells in the figure. Let's go through this step by step:

1. First we draw two unit cells of the hexagonal lattice, stacked one on top of the other in the a₃ direction. Because of the periodicity of the lattice if we place an atom at any location in the bottom unit cell, there is an identical atom at the same location in the top cell.
2. We put the first atom in the z = 0 plane. As we mentioned before this is purely for our convenience, if we put it anywhere else we would have to draw three unit cells to show all the translations. We'll color this atom red, and for reference we'll call it the first atom.
3. Now put a second atom on top of the first (colored purple for reference), and translate upward by 2/3 a₃, a distance of 2c/3.
4. Rotate that atom by 120° counterclockwise, as seen from above, around the a₃ (z) axis. That's the second atom in the screw.
5. Put another atom on top of the second atom. (We'll color it blue). Translate it upward by 2c/3, to z = 4c/3.
6. Do another 120° counterclockwise rotation. This fixes the location of the third atom in the screw.
7. One last atom: place it on top if the “second” atom, color it red, and translate it upward by 2c/3 so that its z value is 2c.
8. Rotate that atom by 120° around the z-axis. That's the final location of the third atom in our screw. We can see that is exactly 2a₃ away from the first atom we put down in (b), so those two atoms are identical — this is why we started with an atom at z = 0, so that we could see the beginning and the end of the screw.
9. Here's a look at our result.
10. Here we've traced out the path of the screw: it rotates by 360° as we translate a distance of 2c up the z-axis, and we drop off three atoms equally spaced along the z-axis.
    We'll rotate the cell around so that we can get a good look at the screw.
11. But somethings missing — we're supposed to have a lattice with a period of c in the z-direction, and we obviously don't.
12. To fix this we have to invoke the translational symmetry of the lattice described by (3). There is a red atom at z = 0 so there must be another one a₃ away from it, at z = c. Put one down there. Note that if we put another atom a₃ away from that one, we end up at the position of or third atom in (h), so our periodicity will extend beyond the two unit cells we've drawn.
13. There's a purple atom at z = 2c/3, so there must also be a purple atom at z = 5c/3.
14. And finally there's a blue atom at z = 4c/3. By the above arguments that means there is a blue atom a z = 7c/3. That atom is out of the range of our figure. However, periodicity goes down in z as well as up in z, so there must be an atom -a₃ away from the blue atom. That puts our final atom at z = c/3.
15. Our job is done. The bottom unit cell looks exactly like the top unit cell, so the periodicity of (3) is upheld. The three atoms we just added are a part of the screw as well. We'll draw a spiral through those atoms. We'll rotate the lattice around so that we can look at it from all angles.
16. One final view of the structure.

That's a lot of translations and duplications. Worse, we have to continually think about which unit cell we're in. Let's try to simplify this bit.

Every translation/rotation operation in Figure 4 involves at translation by ⅔a₃. But the translational symmetry (3) implies that there is also going to be an equivalent operation if we translate by

⅔a₃ - a₃ = - ⅓a₃   . If we do three of these translations we end up only one unit cell away from our original position rather than two cells away as we did in Figure 4. That's got to be similar, so let's look at that.

Let's step through Figure 5 as we did with Figure 4:

1. Again draw the double unit cell.
2. Once more, put an atom at z = 0.
3. Use periodicity to place another atom at z = c.
4. And at z = 2 c.
5. Instead of starting with the atom in (a), let's start with its identical sibling in (d). We translate that downward by -⅓a₃, so that it is at z = 5c/3.
6. Again do a 120° counterclockwise rotation and place the atom there. We'll color it purple, and it is at exactly the same location as the atom in (g) in the previous figure.
7. Since this is a periodic crystal, there must be a duplicate atom -a₃ away, at z = 2c/3. Of course it is purple, and it's identical to the atom we place in step (f) in Figure 4.
8. Continuing with the atom in (g), translate downward by -⅓a₃ to z = 4c/3. Unlike the previous case we're in the same (top) unit cell.
9. Give this atom a 120° counterclockwise rotation around the z-axis to its final position. We color it blue, and it's at the identical location to the atom in step (i) in Figure 4.
10. This atom has a duplicate -a₃ away, at z = c/3. It's in exactly the same position as the atom in step (j) in the previous plot.
11. Start with the atom in (i) and translate downward by -&frac3;a₃ to z = c. Now we're in exactly the same position as step (k) in Figure 4.
12. Now we can just repeat step (l) from Figure 4, and we're back to one of the original red atoms.

When we look at the final plots in Figure 4 and Figure 5 we see that all of the atoms are in the same positions in both figures. In other words, translating atoms upward by 2c/3 gives exactly the same answer as translating them downward by c/3. Since the later procedure lets us stick to one unit cell, making for more compact drawings, we'll use it whenever making multiple translations takes us out of one unit cell into another.

Now we're ready to look at the 3₁ screw. For convenience, we'll put the Figure 5 description of the 3₂ screw below it, so we can compare the results.

       

       

Figure 6 shows the development of the 3₁ (top) and 3₂ (bottom) screw axes. As with the 2₁ system we show three views: one with the translation followed by the rotation, one with a simultaneous translation and rotation, and the last a view looking down the a₃ lattice vector/rotation axis. Let's go through for both figures, step by step. Note that we'll be combining the translation and rotation into one step.

1. Start with a hexagonal lattice, with primitive vectors (3).
2. Add an atom z = 0 for 3₁ and at z = c for 3₂. Of course translational invariance demands that both positions be occupied in the final crystal. In lattice coordinates the atoms are at (x,y,0) and (x,y,1) respectively. (Remember that the starting points 0 and c are arbitrary, chosen so that we can make more compact drawings.)
3. In the 3₁ screw, move up the axis by ⅓ c   , while in the 3₂ screw move down by - ⅓ c   . After this rotate either cell by 120° counterclockwise (when looking down the rotation axis, as seen on the figures at the right. Place an atom at that position. In lattice coordinates this is (-y,x-y,⅓) for the 3₁ screw and (-y,x-y,⅔) for the 3₂ screw.
4. Repeat the operations starting from the last atomic positions. This will result in a atom at (y-x,-x,⅔) (3₁) or (y-x,-x,⅓) (3₂).
5. Do all of this one more time, putting atoms at (x,y,c) or (x,y,0). Because of the translational symmetry of the lattice this means we're back where we started.

We then again rotate the cells so you can see the screws. Table 2 summarizes the atomic positions.

Table 2 The lattice coordinates of the three atoms comprising a 3₁ and 3₂ screw axis, starting at arbitrary z and using the hexagonal lattice (3). The space group listed is the defining space group for the given screw axis, i.e. this is the first trigonal space group in the International Tables which has the given screw axes. The Wyckoff letter is the one associated with the screw axis. The coordinates given are the coordinates for that Wyckoff letter.

Atom	31	32
Space Group	P31	P32
Number	145	146
Wyckoff Letter	(3a)	(3a)
1	(x,y,z)	(x,y,z)
2	(-y,x-y,z+⅓)	(-y,x-y,z+⅔)
3	(y-x,-x,z+⅔)	(y-x,-x,z+⅓)

The simple structures displayed in Figure 6 — one atom type, three atoms in the unit cell, all lined up along a screw axis – are perfectly good crystal structures, though they do not seem to appear in the Inorganic Crystal Structure Database (ICSD). AFLOW tells us the 3₁ structure is in space group P3₁ #144, and the 3₂ structure is in P3₂ #145. If you've read our Chiral Space Groups tutorial, you'll recall that these are Sohncke Class II space groups, and so are enantiomorphic. In that case, if we find a crystal structure in space group P3₁, we know that a mirror image crystal structure with identical properties can also exist in space group P3₂ – which of the pair actually forms depends on the environment where it exists. We can see this in Figure 6: if we place a mirror between the structures, the reflection of the 3₁ screw will look exactly like the 3₂ screw, with the exception of the fictitious colors on the atoms.

But wait, there's more.

We've left out an important class of 3₁ and 3₂ screw axes. These appear in rhombohedral and cubic crystals. These aren't generated in quite as straightforward a way as the axes we describe on this page, so we will discus them in Part VII of this series.

4-fold Screw Axes

All 4-fold screw axes contain four translation/rotation pairs, where the rotation angle is 360°/4 = 90°. There are three types of screw axes: 4₁, 4₂, and 4₃. 4₁ and 4₃ will be familiar from the our above work, while 4₂ is somewhat different.

As with 4-fold rotations can find 4-fold screw axes in tetragonal and cubic lattices. For our example we will use the simple tetragonal lattice, with primitive vectors

a₁ = a $\hat{x}$            
a₂ = a $\hat{z}$   ,   (4)
a₃ = c $\hat{z}$              but you can follow this work to find the 4-fold screw axes in any tetragonal or cubic system.

4₁ and 4₃ Screw Axes

     

     

The construction of the 4₁ and 4₃ axes are similar, so we'll do them altogether. Figure 7 shows both lattices. The procedure is the same as for the 3₁ and 3₂ axes, so we will not write it out in detail. The only difference is that now each translation is ±¼ rather than ⅓, the rotations are by 90° rather than 120° and we need four operations to get back to the starting position.

As with the 3₁/3₂ screws, 4₁ and 4₃ screws are mirror images of each other with the mirror in the z = 0 plane. For the cases we've shown here, the 4₁ example is in space group P4₁ #76, and 4₃ is P4₃ #78. These Sohncke Class II form an enantiomorphic pair. The tetragonal groups P4₁22/P4₃22 and P4₁2₁2/P4₃2₁2 as well as the cubic groups P4₁32/P4₃32 form similar enantiomorphic pairs. Note, however, that just because a space group has a 4₁ in its name does not mean that there is a similar enantiomorphic group with 4₃. Centered space groups (whose names start with I or F) may have a 4₁ screw axis, but the corresponding 4₃ screw is in the same lattice, so the space group may be chiral (Sohncke Class III), but it does not have an enantiomorphic twin.

The 4₂ Screw Axis

       

The 4₂ screw axis is somewhat different than the ones we encountered before, so we'll go through it in detail in Figure 8:

1. Start with our familiar tetragonal lattice (4).
2. Place an atom at lattice coordinates (x,y,0). Remember that setting z = 0 is done only for convenience.
3. Go up from this position by ½ a₃ = ½ c and then rotate by 90° counterclockwise (as seen from above) leaving us at (-y,x+y,½). Place and atom there.
4. Repeat this operation we're now at (-x,-y,1). Place an atom there.
5. This leaves us at the top of the unit cell. However, translational invariance says that if there is an atom at (-x,-y,1) (in lattice coordinates) there is an identical atom at (-x,-y,0), so translate down by -a₃ and place an atom there. There is no rotation because we're still working with the same atom as in (d).
6. Starting from (-x,-y,0) do the ½c translation/90° rotation again. This takes us to (y,-x,½). Place an atom there.
7. One last translation/rotation takes us to (x,y,1), or (x,y,0) if we invoke translational symmetry. This is where we're started, so we're done.

Take one more look at the structure as we rotate it.

Unlike all the previous screws, we could have rotated clockwise as we went up the lattice rather than counterclockwise, or rotated down rather than going up . As a result, there is only one screw of this kind. If you like, 4₂ is its own mirror image. Because of this, no space group with a 4₂ screw axis is Sohncke Class II, but several are chiral, belonging to Sohncke Class III: tetragonal groups P4₂, P42₁, P4₂22 P4₂2₁2, and cubic group P4₂32.

Table 4 summarizes the operations needed to construct an arbitrary 4_n screw, starting from lattice coordinate (x,y,z) in a tetragonal or cubic crystal.

Table 4 The lattice coordinates of the four atoms comprising the 4_n screw axes, starting at arbitrary z. These can be in a tetragonal system, or in a cubic system if we take c = a in (4). The space group listed is the defining space group for the given screw axis, i.e. this is the first tetragonal space group in the International Tables which has the given screw axes. The Wyckoff letter is the one associated with the screw axis. The coordinates given are the coordinates for that Wyckoff letter.

Atom	41	42	43
Space Group	P41	P42	P43
Number	76	77	78
Wyckoff Letter	(4a)	(4d)	(4a)
1	(x,y,z)	(x,y,z)	(x,y,z)
2	(-y,x,z+¼)	(-y,x,z+½)	(-y,x,z+¾)
3	(-x,-y,z+½)	(-x,-y,z)	(-x,-y,z+½)
4	(y,-x,z+¾)	(y,-x,½)	(y,-x,z+¼)

5-fold Screw Axes

As with regular rotations, 5-fold screw axes in periodic crystals are forbidden by the Crystallographic Restriction Theorem. It seems likely that quasicrystals, considered as a higher-dimensional crystal projected onto three-dimensional space, might have 5-fold screw axes, but quasicrystals are not periodic in three dimensions.

6-fold Screw Axes

In what should be no surprise at this point, the 6_n screw axes consist of six 60° rotations, each with a translation of (n/6) c along the rotation axis, where c is the period length of the crystal along that axis, and n is an integer between one and 5.

Every hexagonal crystal has a hexagonal primitive lattice, described by (3), with the same unit cell used in the 3₁ and 3₂ screws shown in Figure 6. What makes it different from those screw axis crystals is that it takes six translations/rotations get back to the starting point.

The five 6_m screws naturally fall into three categories. The first is 6₁/6₅, shown in Figure 9. This is exactly the same type of procedure we did with the 2₁, 3₁/3₂, and 4₁/4₃ screws: go up (or down) by c/6 and rotate by 60°.

6₁ and 6₅ Screw Axes

       

       

The two screws are mirror images of one another, and any space group with one of these screws also has a Sohncke Class twin. There are two such pairs, P6₁/P6₅ and P6₁22/P6₅._*.

6₂ and 6₄ Screw Axes

The 6₂ and 6₄ screws are shown in Figure 10. Their construction is much the same as previous ones, with a translation of ±c/3 and a rotation of 60° at each step. A minor annoyance is that after three of these operations we have run out of unit cell, so we have to invoke translational invariance yet again to keep things in the same cell. That procedure is the same as it was in the 4₂ screw, and is describe in step (e) in the figure.

       

       

As we've come to expect, the two screws are mirror images of one another, and are associated with pairs of enantiomorphic space groups, in this case P6₂/P6₄ and P6₂22/P6₄22.

Examining these pictures we see that there is a relationship between the 3₁ and 6₂ screws: If we start a 3₁ screw at lattice coordinates (x,y,z), and another one with identical atoms at (-x,-y,z), we will have a structure equivalent to the 6₂ screw. A similar relationship holds between 3₂ and 6₄. This is a reminder that the trigonal and hexagonal crystal systems are not that far apart.

6₃ Screw Axes

The 6₃ screw is akin to 4₂, and is shown in Figure 11. This time we run out of the unit cell twice, so we have to do a -a₃ translation at steps (d) and (g).

       

Like 4₂, the 6₃ screw is its own mirror image, so it may be part of a Sohncke Class III group, but it does not have to be, and does not have an enantiomorphic twin.

We close this section with Table 5, which shows all the operations needed to construct any 6_n screw in the hexagonal lattice (3), and the defining space group and Wyckoff position for each of the screw axes.

Table 5 The lattice coordinates of the six atoms comprising the 6_n screw axes, starting at arbitrary z and using the hexagonal lattice (3). The space group listed is the defining space group for the given screw axis, i.e. this is the first hexagonal space group in the International Tables which has the given screw axes. The Wyckoff letter is the one associated with the screw axis. The coordinates given are the coordinates for that Wyckoff letter.

Atom	61	62	63	64	65
Space Group	P61	P62	P63	P64	P65
Number	168	171	173	172	17
Wyckoff Letter	(6a)	(6c)	(6c)	(6c)	(6a)
1	(x,y,z)	(x,y,z)	(x,y,z)	(x,y,z)	(x,y,z)
2	(x-y,x,z+⅙)	(x-y,x,z+⅓)	(x-y,x,z+½)	(x-y,x,z+⅔)	(x-y,x,z+⅚)
3	(-y,x-y,z+⅓)	(-y,x-y,z+⅔)	(-y,x-y,z)	(-y,x-y,z+⅓)	(-y,x-y,z+⅔)
4	(-x,-y,z+½)	(-x,-y,z)	(-x,-y,z+½)	(-x,-y,z)	(-x,-y,z+½)
5	(y-x,-x,z+⅔)	(y-x,-x,z+⅓)	(y-x,-x,z)	(y-x,-x,z+⅔)	(y-x,-x,z+⅓)
6	(y,y-x,z+⅚)	(y,y-x,z+⅔)	(y,y-x,z+½)	(y,y-x,z+⅓)	(y,y-x,z+⅙)

Finishing Up

Figure 12 shows all of the screw axes derived in this tutorial. We should note a few things:

• All of the screw axes shown have a primitive lattice which is identical to the conventional lattice of the crystal system.
• The screw axis is along one of the natural primitive vectors of the lattice.

That does not have to be the case. For example, all rhombohedral lattices, which are not the same as the conventional trigonal lattice, have a set of 3₁ and 3₂ screw axes which are not along the 3-fold rotation axis. Since all cubic lattices can be considered as special cases of the rhombohedral lattice, they have similar 3₁ and 3₂ screws, parallel to the four body diagonals of the cube. Similar behavior occurs in body-centered tetragonal lattices.

There are still other screw axes that occur due to subtleties of the particular space group under consideration. Prof. Harold Stokes at Brigham Young University enumerated all the screw axes allowed in the 230 three-dimensional space groups. He found that there are 1,028 screw axes in 187 space groups, with 43 space groups containing no screw axis.

We'll explore some of these these screw axes in Part VII of this tutorial series.

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

This 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.

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.

Crystallographic Restriction Theorem:

In two or three dimensions symmetry is only preserved for rotations (and screw rotations) with angles of 30°, 60°, 90°, 120° and 180°.

Enantiomorphic Space Groups

Two space space groups that are mirror images of one another. If a structure exists in one of a pair of enantiomorphic space groups, then its mirror image is in the other one. Computations will show that both structures have exactly the same energies, elastic constants, electronic density of states, and phonon spectra. Which image exists in a given sample depends on the conditions under which it was formed. There are eleven pairs of enantiomorphic space groups in Sohncke Class II.

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.

Primitive Vectors:

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

Quasicrystal:

A non-periodic structure that nevertheless contains axes with 5-fold (or 7+-fold, but usually 5) rotational symmetry in two or three dimensions.

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.

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 at given Wyckoff position, there is an identical atom at all the points in subgroup.

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.

Footnotes

^† From now on when we say “screw axis” or “screw axes” we're implying that the crystal we're talking about is symmetric with respect to that axis: it is unchanged if we perform the stated translation plus rotation.

^‡ The rotation is taken in a counter-clockwise direction as seen looking down the axis. This can be seen by looking at the right-most image in any of the figures shown below, which shows a top view.

^†† We could use any lattice with a higher symmetry, as all non-triclinic crystals can have 2₁ screw axes.

^††† This doesn't mean that there aren't other 2₁ axes in the crystal. For example, the lattice shown in Figure 3 has a screw axis along the a₂ direction starting at corner of each unit cell, and a the midpoints of the cell boundaries as drawn in 2(i).

^‡‡ This statement only applies to trigonal space groups with an explicit screw axis. There are similar conditions for some higher symmetry space groups, but not the statement that every space group with an explicit n_m screw axis is part of an enantiomorphic pair is incorrect.

^* Examination of the space group diagrams for shows that they also have 2-fold and 3-fold screws, but we'll let you look those up.

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. 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)
5. 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)