1 Crystals, Chemistry, and Space Groups Sean McAfee University of Utah September 1, 2015
2 Symmetry The (sometimes vague) idea of symmetry permeates all areas of math and science.
3 Symmetry Physics:
4 Symmetry Biology:
8 Symmetry Chemistry:
9 Symmetry Mathematics: i) x 2 + y 2 + z 2 ii) x 3 + 7x 2 y + 7xy 2 + y 3 iii) x 3 + y 3 + z 3 + xyz 37 The study of so-called symmetric polynomials like these led to the creation of the field of group theory.
10 Groups A group is an object used by mathematicians to study symmetry. Roughly speaking, different groups describe different flavors of symmetry.
11 Groups More precisely, a group (usually denoted G) is defined by two things: i) A set of objects, called elements of G, ii) A rule (denoted by and called multiplication) for combining any two elements of G to get another element of G, such that a) there is an identity element 1 G so that 1 g = g for any g G, b) for each g G, there is an inverse element g 1 G such that g g 1 = 1, and c) the group multiplication is associative: g 1 (g 2 g 3 ) = (g 1 g 2 ) g 3.
12 Groups This definition may look somewhat contrived at first; try thinking of a group as a sort of self-contained machine, where any action has an opposite action.
13 Groups Groups pop up everywhere; you are already familiar with several examples, such as...
14 Groups Example 1: The set of integers Z, using normal addition as the rule for combining two integers. i) 0 + n = n, n Z ii) n + ( n) = 0, n Z iii) m + (n + p) = (m + n) + p, m, n, p Z
15 Groups Example 2: A vector space V is (among other things) a group, with vectors as elements and vector addition as the rule for combining elements. i) 0 + v = v, v V ii) v + ( v) = 0, v V iii) u + (v + w) = (u + w) + v, u, v, w V
16 Groups Slightly more abstractly, we can consider a triangle floating in two dimensional space. The ways we can manipulate the triangle while keeping its location in space fixed form a group of six elements, called D 3.
17 Groups Yet another group is the set of translations in the plane along the x-axis in integral increments. Notice that this group can be considered the same as Z.
18 Groups We want to understand how groups are used to study symmetry. In particular, we will be interested in how groups describe symmetries of objects in 1,2, and 3-dimensional space.
19 Crystals One of the more striking examples of symmetry in nature are in the form of crystals. By definition, a crystal is a solid whose atoms have a periodic arrangement. We can think of them as a single three-dimensional pattern repeated forever in three given directions.
23 Crystals We would like a mathematical, methodical way of classifying three dimensional crystals. The trick will be to associated a given crystal with a unique group that describes its symmetry. These are called space groups. We will say that two crystals are the same if they are both described by the same space group. To understand what these groups look like, we need a little more background.
24 Space Groups An isometry of three dimensional space is a motion of the space that preserves distances between points. Some example of isometries are the translations described earlier, rotations around a point, reflections through a fixed plane, and glide reflections (a translation and a reflection at the same time).
25 Space Groups With a little work, it can be shown that all isometries φ of a three-dimensional vector space V act on vectors v V by the formula φ(v) = Av + w, Where A is a 3 3 orthogonal matrix and w is a chosen, fixed vector in V.
26 Space Groups In other words, every isometry acts on a vector by first pointing it in a different direction (while preserving its length) and then translating that vector by some distance in the new direction it is pointing.
27 Space Groups Luckily, for our purposes, the set of isometries of V form a group! You can check the requirements for a group (the identity element is just the isometry that does nothing), but the important thing to notice is that the composition of two isometries is another isometry: A(Bv + w B ) + w A = ABv + (Aw B + w A ).
28 Space Groups The isometries of V = R 3 form a gigantic (uncountably infinite) group. The space groups that will describe symmetries of crystals will be subsets (called subgroups) of this larger group. More precisely, a space group G is a discrete subgroup of Isom(R 3 ) such that the quotient space R 3 /G is compact.
29 Space Groups This definition fits exactly with our problem of classifying types of crystals. If we can determine all possible space groups sitting inside of Isom(R 3 ), we will have classified every type of periodic arrangement of solids. To see how this is done, we will look at the easier cases of one and two-dimensional space.
30 Space Groups In one dimension, there are exactly two types of space groups: those that describe objects with purely translational symmetry and those that describe objects with both translational and reflection symmetry.
31 Space Groups Case 1: The one dimensional, purely translational symmetry can be seen in the following picture: If we think of each circled dot as lying on an integer, the left and right translations can be described exactly by a choice of positive or negative integer. In other words, the space group G that describes this one-dimensional crystal is Z.
32 Space Groups Case 2: The one dimensional symmetry that involves both translations and reflections can be seen in this picture: Here we think of the circled dots and center small dots as lying on integers; our symmetries amount to a choice of distance to translate and a choice of dot to reflect across. In group theory language, this is described by the space group G = Z Z 2. This is known as the infinite dihedral group D.
33 Space Groups The two-dimensional case becomes more complicated. There turn out to be 17 different space groups, known as the wallpaper groups.
34 Space Groups
35 Space Groups Like in the one-dimensional case, these wallpaper groups can be classified by hand: here is an example of a portion of the flow chart used to do this:
36 Space Groups As an example, consider the following wallpaper pattern: This pattern has translational symmetry in both the vertical and horizontal direction, as well as symmetry by reflections through vertical lines.
37 Space Groups The space group that describes this symmetry is known as pm. Convince yourself that the two patterns below are the same in our symmetrical sense:
38 Space Groups
39 Space Groups
40 Space Groups
41 Space Groups Not surprisingly, the three-dimensional case is much more complicated. We shall have mercifully little to say about the 230 space groups... -N.W. Ashcroft and N.D. Mermin, Solid State Physics
42 Space Groups The 230 space groups in three dimensional space were classified by hand (with a couple mistakes), in 1891 by Evgraf Federov.
43 Space Groups The method he used was similar to our flow chart from two dimensions; it involved first classifying the solids involved (called fundamental regions) into 7 classes, such as the triclinic solid shown here:
44 Space Groups An example of a crystal from the triclinic categroy is microcline:
45 Space Groups In fact... Another category of crystal is the monoclic. The fundamental region in this category has translation symmetry in three directions and reflection symmetry through parallel vertical planes. An example of this is the crystal gypsum:
46 Space Groups A box full of (three dimensional) Billy Crystals has the same space group as gypsum!
47 Space Groups So, there are 2 space groups of one dimension, 17 of two dimensions, and 230 of of three dimensions. This raises the natural question: what about space groups of n dimensions? Are there always finitely many for a given dimension? This question was formally stated by David Hilbert in 1900 at the International Congress of Mathematicians:
48 Space Groups Is there in n-dimensional Euclidean space...only a finite number of essentially different kinds of groups of motions with a compact fundamental domain?
49 Space Groups This question, which is part of a larger question known as Hilbert s 18th Problem, was answered affirmatively by Bierbach in In four dimensions, there are 4894 types of space group. In five dimensions, there are 222,097. In six dimensions, there are 28,927,915. There is no known formula for the number of space groups in n dimensions.
50 Space Groups Modern mathematicians use much more sophisticated tools than flow charts to describe space groups. For a given space group G, we can consider the subgroup M (the lattice) which is responsible for translations, and the subgroup H = G/M (the point group), which is responsible for the symmetries of the fundamental region.
51 Space Groups Unfortunately, the datum from M and H is not enough to define G. We look at an example:
52 Space Groups
53 Space Groups
54 Space Groups Both tilings have translational symmetry in the direction of e 1 and e 2, and both have a fundamental region whose symmetries are given by D 4. However, these tilings do not have the same symmetry since in the first case we can rotate around the intersection of two lines of symmetry, and in the second case we cannot.
55 Space Groups To find the missing information needed to describe G in each case, we need the powerful tool of group cohomology. First we state two facts without proof: Fact 1: If G is a space group, it fits into an exact sequence M G H. Fact 2: In such a case, H acts (faithfully) on M; i.e., M is an H-module.
56 Space Groups Group Cohomology in 1 Minute
57 Space Groups Let C n (H, M) be the set of all functions from H n M. This is an abelian group. There exist homomorphisms d n : C n (H, M) C n+1 (H, M) such that d n d n 1 = 0 n. Let Z n (H, M) = Ker(d n ) and B n (H, M) = Im(d n 1 ). Let H n (H, M) = Z n (H, M)/B n (H, M). This is called the n th cohomology group of H with coefficients in M.
58 Space Groups It turns out that the second cohomology, H 2 (H, M) can be identified with the set of space groups G which fill the exact sequence from before! In the case of our example diagrams, we have that H 2 (D 4, Z Z) = Z 2.
59 Space Groups There is much more to be said about space groups and their applications to crystallography. A good start is to read Crystallography and Cohomology of Groups, a short but very well written paper by Howard Hiller.
60 Representations of Groups The classification of space groups is really only half the story; now that we know what they look like, how do we study them in a meaningfull way? One particularly useful tool for this is a group representation.
61 Representations of Groups The motivation for representations comes from the fact that linear algebra is perhaps the most useful and well understood area of math.
62 Representations of Groups Math Problem Is it linear? yes Use linear algebra. no Linearize it! Use linear algebra.
63 Representations of Groups A representation ρ of a group G is a homomorphism from the group to the linear transformations of a chosen n-dimensional vector space V, which can be viewed as elements of GL n (C): ρ : G GL n (C).
64 Representations of Groups In other words, a representation ρ of G assigns an invertible matrix to each element of G so that the multiplication in the group is preserved in the image of G: ρ(g 1 g 2 ) = ρ(g 1 )ρ(g 2 ).
65 Representations of Groups As an example, consider the dihedral group D 3 discussed earlier:
66 Representations of Groups We can concretize D 3 by representing these actions on the triangle by matrices that act on, say, the complex plane: ( ) ( ) 1 0 e 2πi/3 0, e 2πi/3 ( ) ( e 2πi/3, 1 0 e 2πi/e 0 ( e 4πi/3 0, 0 e 4πi/3 ), ), ( ) 0 e 4πi/3 e 4πi/3 0
67 Representations of Groups Once we have represented a group in this way, we can talk about it in linear algebraic terms and consider things such as bases, eigenvalue, eigenvectors, Jordan normal form, etc.
68 Representations of Groups We can also use representations to filter out noise and focus on specific symmetry properties while ignoring others. For example, we can look at our representation of D 3 as acting on four-dimensional complex space by just mapping elements of D 3 to two copies of the original inside a 4 4 block diagonal matrix:
69 Representations of Groups ρ(rotation by 120 degrees) = e 2πi/ e 2πi/ e 2πi/ e 2πi/3.
70 Representations of Groups This representation can be thought of as acting on two triangles simultaneously, one living in the first two complex dimensions, and the other living in the third and fourth. Notice that this action moves vectors living in the first two dimensions inside their own subspace:
71 Representations of Groups e 2πi/ e 2πi/ e 2πi/ e 2πi/3 a b 0 = 0 ae 2πi/e be 2πi/3 0 0.
72 Representations of Groups Such a representations is called reducible. It means that we can peel off unneccessary layers of symmetry and focus on smaller things.
73 Representations of Groups An example from chemistry is the symmetry of the water molecule:
74 Representations of Groups The water molecule has rotational, reflection, and translation symmetry when viewed as living inside a large collection of molecules. There is also more subtle symmetry in the form of molecular vibrations: the periodic motion of the bonds between the atoms in the molecule.
75 Representations of Groups By using representations to ignore the non-vibrational symmetries, we can classify the possible motions between the atoms in water:
76 Representations of Groups This construction is too involved for this talk, but it is handled in a very readable, elementary way in the link below: http: // nancy2010/souvignier_irrep_syllabus.pdf
77 Representations of Groups This is the end of the symmetry story for now, I hope this has encouraged you to look more into the fascinating mathematics behind crystallography, group cohomology, and representation theory!
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