UCSD NANO106 - 04 - Symmetry in Crystallography

Symmetry in Crystallography
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego

Readings
¡Chapter 8 of Structure of Materials
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 4
2

Concept of Symmetry
¡ A geometric figure (object) has symmetry if there is an
isometry that maps the figure onto itself (i.e., the object
has an invariance under the transform).
Can you identify all the symmetry elements in the following pictures?
3

Notation used for symmetry operations
¡Two major schools
¡ International notation or Hermann-Mauguin notation
¡ Schoenflies notation (commonly used in physics and
chemistry)
¡Important to know both notations
4

Overview of the Types of Symmetry
Rotation ReflectionTranslation Inversion
Operations of first kind Operations of second kind
Screw axis
Mirror-rotation
Glide plane
Roto-inversion
5

Operations of first and second kind
¡ Operations of the first kind does not change handedness,
while operations of the second kind does. Best
demonstrated using an object with no intrinsic symmetry.
Rotation (1st kind) Reflection (2nd kind)
6

Rotation
¡ Characterized by
¡ rotation axis, denoted by [uvw]
¡ rotation angle, as a fraction of 2π (radians), e.g., 2π/n where n is an
integer.
7

International notation
¡ Symbol: n (Cn) – where n is order of rotation
¡ When rotation axis is not aligned with c-axis, need to
explicitly provide axis, either as a direction vector [uvw] or
equation of the line.
¡ E.g., 3 (C3) [111] refers to a 3-fold rotation axis aligned
along the [111] direction of the crystal. In the International
Tables of Crystallography, this is represented as 3 (C3) x,
x, x because the line corresponds to the direction where
x=y=z.
8

Rotation matrices
¡ Consider an arbitrary rotation of an arbitrary point P by angle α about
the c-axis (coming out of the page).
Rotation matrix, D
Blackboard
x'
y'
z'
!
"
#
#
#
$
%
&
&
&
=
cosθ −sinθ 0
sinθ cosθ 0
0 0 1
!
"
#
#
#
$
%
&
&
&
x
y
z
!
"
#
#
#
$
%
&
&
&
9

Properties of Rotation Matrices
¡ Orthonormality, i.e., columns and rows are mutually pependicular. This
in turns implies that the transpose of the rotation matrix is its own
inverse, e.g.,
¡ Determinant = +1
cosθ −sinθ 0
sinθ cosθ 0
0 0 1
"
#
$
$
$
%
&
'
'
'
−1
=
cosθ −sinθ 0
sinθ cosθ 0
0 0 1
"
#
$
$
$
%
&
'
'
'
T
=
cosθ sinθ 0
−sinθ cosθ 0
0 0 1
"
#
$
$
$
%
&
'
'
'
=
cos(−θ) −sin(−θ) 0
sin(−θ) cos(−θ) 0
0 0 1
"
#
$
$
$
%
&
'
'
'
Blackboard proof
10

Rotation matrices in the Cartesian
frame
¡ In the last two slides, we have worked in the Cartesian frame (ex and ey in the
picture below, ez is perpendicularly out of the page) of reference. Let us
consider a 6-fold rotation about the c-axis. The angle of rotation is therefore
2π/6 = π/3. In the Cartesian frame, the rotation matrix is therefore:
D(
π
3
) =
cos
π
3
−sin
π
3
0
sin
π
3
cos
π
3
0
0 0 1
"
#
$
$
$
$
$
$
$
%
&
'
'
'
'
'
'
'
=
1
2
−
3
2
0
3
2
1
2
0
0 0 1
"
#
$
$
$
$
$
$
$
$
%
&
'
'
'
'
'
'
'
'
ex
ey
11

Rotation matrices in the crystal frame
¡ Let us now see what happens when we work in the
crystal frame (a1 and a2). After a rotation of π/3, these
basis vectors are transformed to a1‘ and a2’.
¡ By inspection, we observe that
a1
a2 a1
!
a2
!
a1
! = a1 + a2
a2
! = -a1
a3
! = a3
a1
!
a2
!
a3
!
"
#
$
$
$
$
$
%
&
'
'
'
'
'
=
1 1 0
−1 0 0
0 0 1
"
#
$
$
$
%
&
'
'
'
a1
a2
a3
"
#
$
$
$$
%
&
'
'
''
=DT
Dcrys
=
1 −1 0
1 0 0
0 0 1
"
#
$
$
$
%
&
'
'
'
12

Stereographic Projection of 3D objects
13

Stereographic representations of
rotations
14

Crystallographic Restriction Theorem
¡ We have thus far seen 2D nets with 2, 4 and 6-fold
rotational symmetry. How do we know these are all the
possible rotational symmetries?
¡ It can be demonstrated mathematically that only 2, 3, 4
and 6-fold rotational symmetries are compatible with
crystallographic lattices.
15

Proof of the Crystallographic
Restriction Theorem
Blackboard proof
16

Translation
¡Valid symmetry only in infinite solids.
¡In crystals, only translations based on lattice
vectors are allowed.
¡Symbol: t(u,v,w) or t[uvw]
¡Can we represent this symmetry operation as a
matrix multiplication? (Clearly can’t be done with
3x3 matrices)
r'= r + t
17

Homogenous coordinates
¡ Work in 4D coordinates, with the last coordinate set to 1 (known as
homogenous or normal coordinates)
¡ Home exercise: Show that once you discard the 1, the above matrix
operation represents t (u, v, w).
¡ The matrix can be represented by the Seitz symbol, which is effectively a
decomposition of the matrix as follows:
x'
y'
z'
1
!
"
#
#
#
#
$
%
&
&
&
&
=
1 0 0 u
0 1 0 v
0 0 1 w
0 0 0 1
!
"
#
#
#
#
$
%
&
&
&
&
x
y
z
1
!
"
#
#
#
#
$
%
&
&
&
&
D11 D12 D13 u
D21 D22 D23 v
D31 D32 D33 w
0 0 0 1
!
"
#
#
#
#
#
$
%
&
&
&
&
&
D t
written as (D(θ)|t)
Examples:
• (D|0) – pure rotation/other symmetry
operations
• (E|t) – pure translation
18

Reflection or mirror
¡ Symbol:
¡ Sometimes or to distinguish
between horizontal and vertical mirror
planes
¡ Orientation of mirror plane can also
be provided by specifying normal to
plane, e.g., or
¡ Can also be represented as matrices,
e.g., the following represents a mirror
operation in the plane formed by the b
and c axes.
−1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
m (σ )
σh σv
m (σ ) [110] m (σ ) x,−x,0
Represented by thick lines
19

Inversion
¡ Symbol:
¡ Exists only in 3D.
¡ Maps r to –r.
¡ Matrix representation:
1 (i)
−1 0 0
0 −1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
20

Sequence of symmetry operations
¡ Using homogenous coordinates, all symmetry operations
can be represented as matrices
¡ Consecutive application of symmetry operations is simply
a sequence of matrix multiplications:
x'
y'
z'
1
!
"
#
#
#
#
$
%
&
&
&
&
=
1 0 0 1
0 1 0 2
0 0 1 3
0 0 0 1
!
"
#
#
#
#
$
%
&
&
&
&
cosπ 0 −sinπ 0
0 1 0 0
sinπ 0 cosπ 0
0 0 0 1
!
"
#
#
#
#
$
%
&
&
&
&
−1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
!
"
#
#
#
#
$
%
&
&
&
&
x
y
z
1
!
"
#
#
#
#
$
%
&
&
&
&
Represents inversion, followed by a 2-fold rotation about the b-axis,
followed by a translation of [123].
21

Symmetry operations that do not pass
through the origin
¡ Thus far, we have limited our discussion to operations that
pass through the origin.
¡ All symmetry operations that do not pass through the
origin can be decomposed into three consecutive
symmetry operations
Blackboard
22

Symmetry operations and their matrix
representations
¡ As seen earlier, all rotation matrices have det(D) = +1
¡ Inspecting the matrix representations of all the symmetry
operations thus far, we find that:
¡ Symmetry operations of the first kind have det(D) = +1
¡ Symmetry operations of the second kind have det(D) = -1
23

Symmetry Matrices in the Crystal
Frame
¡Always comprised of integers
¡General approach
¡ Determine how the symmetry operation changes the
crystal lattice vectors
¡ Express lattice vectors after symmetry operation as a
linear combination of the lattice vectors before symmetry
operation
¡ Construct transformation matrix mapping original lattice
vectors to lattice vectors after symmetry operation
¡ Take transpose of transformation matrix.
24

Examples
¡Let’s work out the following symmetry
matrices in the crystal frame:
i. Two-fold rotation about b-axis in
monoclinic lattice
ii. Reflection about a-c plane in the
orthorhombic lattice.
iii. 3-fold rotation axis parallel to [111] in
rhombohedral lattice
iv. Reflection plane coinciding with c-axis
and diagonal between a and b lattice
vectors in tetragonal lattice
Blackboard
25

Overview of the Types of Symmetry
Rotation ReflectionTranslation Inversion
Operations of first kind Operations of second kind
Screw axis
Mirror-rotation
Glide plane
Roto-inversion
In the next few slides, we will be talking about combinations
of symmetry operations.
26

Some preliminaries
¡ Not all combinations will result in new symmetry
operations, i.e., some combinations are identical to
existing symmetry operations (this is a prelude to the
symmetry “group” concept in the next lecture).
27

Roto-inversion
¡ Rotation + an inversion center on that axis.
¡ Symbol: (Schonflies notation depends on value of n)
¡ Matrix representation: Multiplication of rotation and inversion matrices
Equivalent to
m (σ )
n
Just an
inversion center
28

Mirror-rotation
¡ Rotation axis + perpendicular mirror plane
¡ Historically, Hermann-Mauguin uses roto-inversions, while Schoenflies uses
mirror-rotation. Using the roto-inversion order (n) as the starting point:
¡ Symbol:
Where is ?!6
n = 4N, n = !n, Symbol: n (Sn )
n odd, n = 2!n, Symbol: n (Cni )
n = 4N + 2, n =
!n
2
=
n
2
m
, Symbol:
n
2
m
(Cn
2
h
)
=
3
m
29

Screw Axis
¡ n-fold rotation + translation parallel to rotation axis by
¡ Symbol: nm
¡ Example: for n = 3, possible values of
τ =
mt
n
τ =
t
3
,
2t
3t
τ =
t
3
τ =
2t
3
31
32
30

Screw Axis, contd.
¡ nm and nn-m are mirror images of each other
(enantiomorphous)
¡ Screw axes where m < n/2 are right-handed, while those
where m > n/2 are left-handed. For m=n/2, the screw axis
is without hand.
D11 D12 D13 u
D21 D22 D23 v
D31 D32 D33 w
0 0 0 1
!
"
#
#
#
#
#
$
%
&
&
&
&
&
D(θ)| τ( )
1 −1 0 0
1 0 0 0
0 0 1
1
6
0 0 0 1
"
#
$
$
$
$
$$
%
&
'
'
'
'
''
Example: 61 using
hexagonal reference frame
6-fold rotation matrix
derived earlier
Translation by 1/6 of
lattice vector
31

Glide planes
¡ Mirror + translation over half a
lattice vector or half a centering
atom
¡ Symbols:
Mirror m None
Axial glide a
b
c
a/2
b/2
c/2
Diagonal
glide
n A, B, C, I
Diamond
glide
d (a + b)/4, (b + c)/4, (c + a)/4
(a + b + c)/4
D(m)| τ( )
32

Properties of Symmetry Matrices
¡ Operations of the first kind (proper motions)
¡ Operations of the second kind (improper motions)
¡ As all symmetry operations are orthogonalmatrices and the product of
orthogonal matrices is also an orthogonalmatrix, we can infer that the
combination of symmetry operations result in another symmetry
operation.
¡ Furthermore, since det(AB) = det(A)det(B), successive combinations of
operations of the first kind leave handedness unchanged.Successive
odd number of combinations of operations of the second kind changes
handedness,and successive even number of operations of the second
kind leave handednessunchanged.
33
det(DI
) = +1
det(DII ) = −1

Combinations of Symmetry Operations
34
Combination of two
intersecting mirror
planes at an angle
αto each other result
in rotation operation
of α
Combination of two parallel
mirror planes result in
translation operation
Mathematical proof for Parallel Mirror Planes
Mathematical proof for intersecting mirror
planes is somewhat more complicated and will
be a problem set exercise
Blackboard

Point symmetry
¡ With the exception of the translation operation (and all other
operations containing a translation), all other operations are point
symmetries. A point symmetry is an operation that leaves a point
unchanged. What point is unchanged for all symmetry operations?
35
D11 D12 D13 0
D21 D22 D23 0
D31 D32 D33 0
0 0 0 1
!
"
#
#
#
#
#
$
%
&
&
&
&
&

Combining rotations
¡ In some of our earlier derivations, we have seen that a
Bravais lattice can have more than one rotation axis. (e.g.,
a cubic lattice has 4, 3 and 2-fold rotation axes)
¡ We have also mathematically shown that for
crystallographic lattices, only rotations of order 2, 3, 4 and
6 are possible.
¡ Before we embark on deriving all the point groups, we
start by asking a fundamental question – what
combinations of rotation axes are possible?
¡ To answer this question, we rely on Euler’s theorem.
36

Euler’s theorem
¡ Consider three rotation axes represented by the
poles A, B and C.
¡ Euler’s theorem states that if the angle between the
great circles AB and AC is α, the angle between the
great circles BA and BC is β, and the angle between
the great circles CA and CB is γ , then a clockwise
rotation about A through the angle 2α, followed by a
clockwise rotation about B through the angle 2β is
equivalent to a counterclockwise rotation about C
through the angle 2γ.
¡ For crystals,
¡ Also, from the cosine rule in spherical geometry, we
know that
37
2α,2β,2γ ∈ 2π,
2π
2
,
2π
3
,
2π
4
,
2π
6
"
#
$
%
&
'

38Possible
rotation
combinations
We can separate out the
combinations into two separate
classes:
- 2-fold rotation axis
perpendicular to a 2, 3, 4 or 6-
fold rotation (known as the
dihedral groups)
- 2-fold rotation that is not
perpendicular to other rotation
axes.

UCSD NANO106 - 04 - Symmetry in Crystallography

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UCSD NANO106 - 04 - Symmetry in Crystallography