Space groups are important in materials science because they capture all of the essential symmetry in a crystal structure.
Space groups are mathematical constructs that capture every way an object can be repeated through space, through translation, rotation, reflection, screws, and gliding. In 3 dimensions, there are 230 space groups.
Outline
How Many Space Groups are There?
There are 230 space groups in 3-dimensions. In 2-dimensions, there are 17 plane groups (also called “wallpaper groups”). You can think of space groups as the combination of Bravais lattices and point groups.
You may wonder then, why are there only 230 space groups? Although 14 Bravais lattices x 32 point groups = 448, several of those resulting 448 space groups are duplicates of each other. After removing duplicates, we end up with 230 space groups.
Symmetry Operations
To understand space groups, you first need to know what symmetry operations are:
Reflection is a symmetry operation which causes a set of points to be mirrored across a plane. We call this plane the “mirror plane.” Any point (X,Y,Z) becomes (-X,Y,Z) if there is a mirror axis perpendicular to the X direction.
Rotation is a symmetry operation which causes a set of points to be rotated around a point. We call this point an “axis of rotation.” In polar coordinates, any point (R, θ, φ) becomes (R, θ + 360º/n, φ) for an n- fold rotation axis perpendicular to θ.
Inversion is a symmetry operation which pulls every point through an “inversion center” to the other side. Any point (X, Y, Z) becomes (-X, -Y, -Z) if there is an inversion center at the origin. You can combine rotation with inversion to produce the rotoinversion symmetry operation.
Translation is a symmetry operation that moves a set of points through space. Point groups DO NOT involve translation. Any point (X,Y,Z) becomes (X+a, Y, Z) for translation of length a in the X-direction.
Glide is a symmetry operation which combines translation + reflection. Objects are translated in the direction, and rotated across, the glide plane.
Screw is a symmetry operation which combines translation + rotation. Objects are translated down, and rotated around, the screw axis.
Space Group Notation
The international standard notation for space groups is shortened Hermann–Mauguin (H-M) notation. There is also a full H-M notation system which identifies all symmetry operations in each space group, but the shortened HM notation eliminates symmetry operations which are not not necessary to uniquely identify a space group.
Space groups are also ordered by number.
In HM notation, the first letter is capitalized, and explains what kind of centering is present in the Bravais lattice. “P” stands for primitive, “C” stands for base-centered, “F” stands for face-centered, and “I” stands for body-centered.
After the centering letter, there are symmetry operations which operate on 3 different axes.
Positive numbers n means that there is n-fold rotation. Negative number n means there is n-fold rotoinversion. Numbers with a subscript XY indicated screw axes. The letter “m” stands for mirror planes. Any other lowercase letters refer to glide planes, such as “a,” “b,” “c,” “e” (glide along one face),”n” (glide along half diagonal of a face), and “d” (glide along ¼ diagonal of a face).
For example, space group 8 is C 1 m 1. The first letter is C, for base-centered. There is no symmetry in the first axis, a mirror plane in the 2nd axis, and nothing in the 3rd axis.
Another example, space group 47 is P 2/m 2/m 2/m. The P means there is no centering, and all 3 axes have both a 2-fold rotation and a mirror plane.
The same effect can be produced by combining multiple symmetry operations. This leads to abbreviated H-M notation, which omits some of these symmetry operations to the minimum which can uniquely identify a space group.
For example, going back to space group 8 and 47. C 1 m 1 just becomes Cm, because the 1s are not telling you anything special. P 2/m 2/m 2/m becomes Pmmm, because 3 mirror planes must be perpendicular to each other, which automatically creates a 2-fold rotation in each axis.
The abbreviated Hermann–Mauguin notation is the international standard, and is the most common notation system you will see.
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