Crystals, Chemistry, and Space
Groups
Sean McAfee
University of Utah
September 1, 2015
Symmetry
The (sometimes vague) idea of symmetry permeates
all areas of math and science.
Symmetry
Physics:
Symmetry
Biology:
Symmetry
Symmetry
Symmetry
Symmetry
Chemistry:
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.
Groups
A group is an object used by mathematicians to
study symmetry. Roughly speaking, different groups
describe different ”flavors” of symmetry.
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:
g1 · (g2 · g3 ) = (g1 · g2 ) · g3 .
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.
Groups
Groups pop up everywhere; you are already familiar
with several examples, such as...
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
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
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 D3 .
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.
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.
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.
Crystals
Crystals
Crystals
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.
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).
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 .
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.
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 + wB ) + wA = ABv + (AwB + wA ).
Space Groups
The isometries of V = R3 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(R3 ) such that the quotient space R3 /G is
compact.
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(R3 ), 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.
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.
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.
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 o Z2 . This is
known as the infinite dihedral group D∞ .
Space Groups
The two-dimensional case becomes more
complicated. There turn out to be 17 different
space groups, known as the wallpaper groups.
Space Groups
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:
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.
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:
Space Groups
Space Groups
Space Groups
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
Space Groups
The 230 space groups in three dimensional space
were classified by hand (with a couple mistakes), in
1891 by Evgraf Federov.
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:
Space Groups
An example of a crystal from the triclinic categroy is
microcline:
Space Groups
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:
In fact...
Space Groups
A box full of (three dimensional) Billy Crystals has
the same space group as gypsum!
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:
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?”
Space Groups
This question, which is part of a larger question
known as Hilbert’s 18th Problem, was answered
affirmatively by Bierbach in 1911. 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.
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.
Space Groups
Unfortunately, the datum from M and H is not
enough to define G . We look at an example:
Space Groups
Space Groups
Space Groups
Both tilings have translational symmetry in the
direction of e1 and e2 , and both have a fundamental
region whose symmetries are given by D4 . 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.
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.
Space Groups
Group Cohomology in 1 Minute
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 nth cohomology group of H with
coefficients in M.
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 (D4 , Z × Z) ∼
= Z2 .
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.
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.
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.
Representations of Groups
Math Problem
Is it linear?
yes Use linear algebra.
no
Linearize it!
Use linear algebra.
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 GLn (C):
ρ : G → GLn (C).
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(g1 · g2 ) = ρ(g1 )ρ(g2 ).
Representations of Groups
As an example, consider the dihedral group D3
discussed earlier:
Representations of Groups
We can concretize D3 by representing these actions
on the triangle by matrices that act on, say, the
complex plane:
1
0
0
1
−2πi/3
−4πi/3
0
e
0
e
0
,
,
,
1
0
e 2πi/3
0
e 4πi/3
1
0
e −2πi/3
0
e −4πi/3
, 2πi/e
, 4πi/3
0
e
0
e
0
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.
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 D3 as acting on four-dimensional
complex space by just mapping elements of D3 to
two copies of the original inside a 4 × 4 block
diagonal matrix:
Representations of Groups
ρ(rotation by 120 degrees)

e −2πi/3
0
0
0
 0
e 2πi/3
0
0 

.
=
0
0
e −2πi/3
0 
0
0
0
e 2πi/3

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:
Representations of Groups
    −2πi/e 
e −2πi/3
0
0
0
a
ae
2πi/3
 0
   2πi/3 
e
0
0 

·b  =  be
.
 0
0
e −2πi/3
0  0  0 
0
0
0
0
0
e 2πi/3

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.
Representations of Groups
An example from chemistry is the symmetry of the
water molecule:
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.
Representations of Groups
By using representations to ignore the
non-vibrational symmetries, we can classify the
possible motions between the atoms in water:
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:
//www.crystallography.fr/mathcryst/pdf/
nancy2010/Souvignier_irrep_syllabus.pdf
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!
Thanks!