Introduction to Space Groups

In the previous chapter we have seen that a combination of symmetry operations on the structure and subsequent stacking into a lattice will build up a crystal. Let us investigate this a little further.

Point group operations

The point group symmetry operations applied to a motif can be deduced from the external symmetry of a (perfect) crystal. Normals to the crystal faces intersect in a single point in the centre of the crystal. Examples for point group symmetry operations are Rotation Axes, Mirror Planes, and Inversion Axes. Important restrictions apply to the point group symmetry operations :

  1. The application of a symmetry operation to a motif cannot generate any changes within the motif, i.e. must generate an identical copy of the motif.

  2. The allowable symmetry operations must be compatible with the translational requirements for the specific lattice.

Rotation Axes : Condition (1) limits n-fold rotation axes to n = 1, 2, 3, 4, 6. You cannot create a space filling pattern with 5- fold or 8-fold rotations, but a single molecule may well have 5-fold symmetry (check).
: If the above is true, how comes that one of the platonic bodies, the pentagon dodecahedron, exhibits pentagonal faces and/but has cubic symmetry?

Inversion Axes : Same as above, except that after the 360/n rotation the object is projected through a inversion center on the axis. The axes are denoted with a bar above, I use minus signs here : -1, -2, etc....
Here is a fun exercise that will help you understand the second limitation : draw a simple chiral molecule, and then apply a -2 operation to it. What happened?

Mirror Plane : pretty obvious operation, denoted m. After the exercise above, you should already have found that neither inversion axes nor a mirror plane are acceptable symmetry operations in case of a protein molecule.

A combination of the point group symmetry operations leads to 32 point groups. As they can be deduced from the macroscopic crystal symmetry, they are also referred to as the 32 Crystal Classes. For the same reason, the symmetry elements that give rise to the Crystal Classes are sometimes referred to as external symmetry elements.

Crystal Systems and Bravais Lattices

Let us now consider in which way we can translate our cell contents in 3 dimensions to obtain a crystal. The translations which are allowed create 14 Bravais lattices which belong to 7 crystal systems. Each system has a primitive cell, and some allow face or body centering, as well as the rhombohedral centering in the trigonal system (the rhombohedral lattice can be derived from a cube by pulling along it's space diagonal).

The translational elements also allow for 2 new types of symmetry operation :

Screw Axes : There are n-1 n-fold screw axes Nz , z =1..n-1, n=2,3,4,6. They are combined symmetry elements, resulting from a rotation around 360/n degrees, followed by a translation by z translations of 1/n * x, where x is a lattice vector parallel to the screw axis. For a 2-fold screw axis (21), there is only one translation by 1/2*x, for 4 folds there are 3 possibilities 41, 42, 43 , with translation of 1/4, 2/4, and 3/4 along the parallel lattice vector. Screw axes are very common in protein structures.

Glide Planes : For the same reasons mentioned above, glide planes, a combination of a mirror plane and a translation operation parallel to it, cannot appear in protein structures and are sometimes ignored in macromolecular crystallography textbooks.They are very common in inorganic structures (diamond for example, has, you guessed it, a diamond glide d). Generally, a glide is constructed by a mirror operation followed by a translation along a/2, b/2, c/2 (a,b,c), a face diagonal (a+b)/2, (a+c)/2, (b+c)/2 (in all cases denoted n) or, in case of d along (a+b)/4, (a+c)/4 or (b+c)/4.

Note : Glides and screws are not point group operations, because they involve translations. They only work with (and are compatible with) additional translations in a lattice. As an example, a 43 axis yields an identical orientation of the molecule only after 4 repeated applications - 3 unit cells away (4*3/4 = 3). Clearly, this is not a point group operation. The same holds for glides, and we'll see that all operations involving translations (Bravais centerings, glides and screws) yield observable extinctions (systematically absent reflections) in the diffraction pattern. We will discuss this shortly. One also cannot distinguish a rotation axis from a corresponding screw axis or a glide plane from a mirror plane by just looking at the crystal faces. Thus these symmetry elements involving translations are also referred to as internal symmetry elements, in contrast to the externally observable point group operations (point groups or Crystal Classes). The only difference between 2 and 21 is that every other molecule would stick out of a lattice plane (crystal face) perpendicular to the axis by 1/2 of the unit cell in case of the screw axis. Currently none of the AF microscopes are good enough to see this.

Finally, Space Groups

The combination of all available symmetry operations (point groups plus glides and screws) with the Bravais translations leads to exactly 230 combinations, the 230 Space Groups. The International Tables list those by symbol and number, together with symmetry operators, origins, reflection conditions, and space group projection diagrams. In the next chapter we will go into more detail of space groups and use an interactive program to decode some of the space group symbols. You can learn more about space group symmetry in Chapter 5 - here is another sample page - of my book Biomolecular Crystallography or buy the book from Amazon

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Last revised Dezember 27, 2009 01:40