Some Groups of Mathematical
Crystallography
Part Deux
Quick Review
• Crystals are regular arrangements of
atoms/molecules in solids
• Model symmetry using group theory
• We are considering rotations and
reflections-we left off with the discussion of
the dihedral group: groups of 90° rotations
and reflections across diagonals and axes
Overview
•
•
•
•
Chemistry and Physics Viewpoint
Lattices as crystal models
The groups E(2) and O(2)
Crystallographic space groups and their
point groups
• Concept of Equivalence
• Point Group Classification
Chemistry Viewpoint
• “Most solid substances are crystalline in
nature”
• “Every crystal consists of atoms arranged in
a three-dimension pattern that repeats itself
regularly.”
• “It is the regularity of arrangement of the
atoms in a crystal that gives to the crystal its
characteristic properties,…”
Chemistry Viewpoint cont’d
• “The principal classification of crystals is
on the basis of their symmetry.”
• “Chemists often make use of the observed
shapes of crystals to help them in the
identification of substances.”
• -Linus Pauling, General Chemistry
Physics Viewpoint
• In solids, the atoms will arrange themselves
in a configuration that has the lowest energy
possible. This arrangement is infinitely
repetitive in three dimensions.
• “The arrangement of the atoms in a crystalthe crystal lattice-can take on many
geometric forms.”
Physics Viewpoint cont’d
• “…[I]ron has a body-centered cubic lattice
at low temperatures, but a face-centered
cubic lattice at higher temperatures. The
physical properties are quite different in the
two crystalline forms.”
• Richard Feynman, Lectures on Physics
Examples
Face-Centered
Cubic
Body-Centered
Cubic
Source: http://cst-www.nrl.navy.mil/lattice/index.html
Techniques to Study Molecular
Structure
• X-ray diffraction
• Neutron diffraction
• Electron diffraction
Space Groups and
their corresponding
Lattices
Space Groups
• Wishing to examine symmetry groups of
crystals – namely, those symmetries which
map a crystal to itself, we look to Space
Groups.
Space Groups
• Since crystals are repetitive formations of atoms, it
can be said that there is some lattice T with basis
t1, t2 such that any translation is of the form Ta,
where
a = m t1 + n t2
(m,n ε Z)
Note: Any crystal which arises from these
translations, is a map onto itself from the lattice T.
Space Groups
• Def: A crystallographic space group is a subgroup
of E(2) whose translations are a set of the form
{(I , t) | t ε T} where T is a lattice.
Remark: The set of translations {(I,t) | t εT} forms
an abelian subgroup of G (the translation group of
G). Clearly there is a 1-1 correspondence
(I, t)
t between G and the elements of T.
Space Groups
* All translations come in some sense from a fixed
lattice *
Ex. Let T be the lattice with basis (1,0) and (0,1).
The matrices and vectors below are written with
respect to this lattice basis.
1) G = T is a space group
2) Let G be the set consisting of the translation
subgroup T along with all elements of the form
(A,t), t εT where A =   1 0 
 0
 1
Space Groups and their Lattices
1)
2)
Note that the lattice itself does not identify the crystal by
symmetry type – again think of the crystal as having
identical patterns of atoms at each lattice point – the type
of atom pattern determines the full symmetry group.
Lattices as Crystal Models
• Given that the crystal lattice is the
arrangement of atoms in a crystal, we can
model crystals using lattices.
• We’ll do this by defining space groups,
point groups, and their relationships.
The Groups E(2) and O(2)
• O(2) – the orthogonal group in the plane R2
• E(2) – the Euclidean isometry group on R2;
the group (under function composition) of
all symmetries of R2
Crystallographic Space Groups
and Their Point Groups
• The symmetries of a crystal are modeled by a
group called the crystallographic space group
(G  E(2) ).
– The translations for this group can be identified
with a lattice T  G.
– G0 = { A | (A, a)  G }; (A, a) represents Ax + a
where A is an orthogonal matrix
– We can associate a point group (G0) with a space
group: G/T  O(2) where G/T is isomorphic to
G0
Concept of Equivalence
• Two point groups G0 and G0` are equivalent
if they are conjugate as subgroups of all 2 x
2 unimodular matrices.
– A unimodular matrix is one with determinant
±1 with integer entries.
• Two space groups are equivalent if they are
isomorphic and their lattice structure is
preserved.
Point Group Classification
• Finiteness of point groups
• Crystallographic restriction
• The 10 Crystal Classes
Finiteness of Point Groups
The Point Group G0
• THM: The point Group G0 of a space
group MUST be a finite group.
• Proof: First consider a circle about the
origin containing a lattice basis {t1, t2} of T.
• N: # of lattice points in the circle
G0 Proof (Cont.)
G0 Proof (Cont.)
• There are only finitely many lattice points
inside this circle, say n (Note: n  4)
• f = mt1 + nt2 (m,n  )
• || t ||  m 2  n 2  Radius R
 Thus finitely many
G0 Proof (Cont.)
• Matrix A  G0 is distance preserving if a
lattice is moved (A maps lattice points to
lattice points in the circle)
• A permutes the N lattice points in the circle
– N! permutations of N lattice points
 N! A matrices
• Thus, G0 is must be finite
G0 Proof (Cont.)
• Observations: (If A  G0)
A(T) = T
tT
At  T
Finite Subgroups of O(2)
• Finite subgroups of O(2) are either cyclic
or dihedral
• Proof : (next slides)
• Note: R in the next slides is the
rotation in the plane around the origin
Finite Subgroups of O(2)
Proof: Cyclic
• G is a subgroup of O(2), G - finite
G  O(2)
G  {R  , R  ,..., R  }
1
2
n
Finite Subgroups of O(2)
Proof: Cyclic (cont)
•Set

* = least non-zero
R G  n  1
n    (n  1)
*
•Why is this true???
*
Finite Subgroups of O(2)
Proof: Cyclic (cont)
•Union the set when
n 1
[ nΘ , (n  1)Θ
*
n 1
*
)  [Θ , )
*
Finite Subgroups of O(2)
Proof: Cyclic (cont)
n
*
Θ
R ΘR  G
R ΘR
n
*
Θ
 ?  R -n*
Finite Subgroups of O(2)
Proof: Cyclic (cont)
0  Θ  nΘ  Θ
*
Θ  nΘ  0
*
  n
*
*
Finite Subgroups of O(2)
Proof: Cyclic Conclusion
R   R *
R * is a generator of G
 G is cyclic
Finite Subgroups of O(2)
Proof: Dihedral
• F is a subgroup of G - finite
φ  det : G  G  {1,1}
/
/
the identity is 1 in G
determinan ts have this property :
φ(AB)  φ(A) φ(B)
Finite Subgroups of O(2)
Proof: Dihedral (cont)
H  ker φ  rotations in G
φ  G  G (homomorph ism)
/
G HG
|G H| 2
/
Finite Subgroups of O(2)
Proof: Dihedral (cont)
• H is the cyclic group of rotation
matrices
• HF is the reflection coset
H, HF
Finite Subgroups of O(2)
Proof: Dihedral (cont)
H  {I, R, R , R ,..., R }
2
3
n -1
HF  {F, RF, R F,..., R F}
2
n -1
Every group can be written as the
union of distinct cosets
G  H  HF
Finite Subgroups of O(2)
Proof: Dihedral Conclusion
F I
2
 G is dihedral
Crystallographic Restriction
Theorem
(CRT)
Crystallographic Restriction
Theorem (CRT)
• Definition:
– Let R be a rotation in a point group through an
angle 2/n. Then n is 1, 2, 3, 4, or 6.
CRT Proof
• Let R be an element of O(2) with the
matrix:
cosq -sinq
sinq cosq
• The trace of the matrix is 2cosq
CRT Proof cont’d
• The matrix R with respect to a lattice basis
has Z entries, because the matrix is
unimodular. Thus it has an Z trace.
– Note: matrices with the same linear
transformations with respect to different basis
have the same trace.
• Since cosq  0, ±1, or ±1/2, the
corresponding n values are: 1, 2, 3, 4, 6
CRT Proof (Continued)
• Cn and Dn both contain rotations through
2/n, the implications assert that:
– Any point group must be associated with the 10
crystal classes within Cn and Dn, where Cn and
Dn are the cyclic group of order n and dihedral
group of order 2n respectively.
The 10 Crystal Classes
•
•
•
•
•
•
•
•
•
•
C1
C2
C3
C4
C6
D1
D2
D3
D4
D6
Examples of Patterns Formed
Example of orthogonal group of D6
Examples cont’d
• Example of C6 orthogonal group.
References
• “Modern Geometries, 5th Ed.” by James R. Smart,
Brooks/Cole Publishing Company 1998
• “Symmetry Groups and their Applications” by
Willard Miller Jr., Academic Press 1972
• “General Chemistry” by Linus Pauling, Dover
1970
• “The Feynman Lectures on Physics”, Feynman, et
al, Addison-Wesley 1963
• “Applications of Abstract Algebra”, by George
Mackiw, Wiley 1985
Questions?