UCSD NANO106 - 04 - Symmetry in Crystallography

Symmetry in CrystallographyShyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Readings¡Chapter 8 of Structure of Materials

NANO 106 - Crystallography of Materials 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-rotationGlide 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 0sinθ cosθ 00 0 1

!

"

###

$

%

&&&

xyz

!

"

###

$

%

&&&

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 0sinθ cosθ 00 0 1

"

#

$$$

%

&

'''

−1

=cosθ −sinθ 0sinθ cosθ 00 0 1

"

#

$$$

%

&

'''

T

=cosθ sinθ 0−sinθ cosθ 00 0 1

"

#

$$$

%

&

'''=

cos(−θ ) −sin(−θ ) 0sin(−θ ) cos(−θ ) 00 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(π3) =

cosπ3

−sin π3

0

sin π3

cosπ3

0

0 0 1

"

#

$$$$$$$

%

&

'''''''

=

12

−32

0

32

12

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 + a2a2! = -a1a3! = a3

a1!

a2!

a3!

"

#

$$$$$

%

&

'''''

=1 1 0−1 0 00 0 1

"

#

$$$

%

&

'''

a1a2a3

"

#

$$$$

%

&

''''

=DT

Dcrys =1 −1 01 0 00 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 u0 1 0 v0 0 1 w0 0 0 1

!

"

####

$

%

&&&&

xyz1

!

"

####

$

%

&&&&

D11 D12 D13 uD21 D22 D23 vD31 D32 D33 w0 0 0 1

!

"

#####

$

%

&&&&&

D twritten as (D(θ)|t)Examples: • (D|0) – pure rotation/other symmetry

operations• (E|t) – pure translation

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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 00 1 00 0 1

"

#

$$$

%

&

'''

m (σ )

σ h σ v

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 00 −1 00 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 10 1 0 20 0 1 30 0 0 1

!

"

####

$

%

&&&&

cosπ 0 −sinπ 00 1 0 0

sinπ 0 cosπ 00 0 0 1

!

"

####

$

%

&&&&

−1 0 0 00 −1 0 00 0 −1 00 0 0 1

!

"

####

$

%

&&&&

xyz1

!

"

####

$

%

&&&&

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 latticeii. Reflection about a-c plane in the

orthorhombic lattice.iii. 3-fold rotation axis parallel to [111] in

rhombohedral latticeiv. 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-rotationGlide 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 = !n2=

n2m

, Symbol:

n2m

(Cn2h)

=3m

29

Screw Axis¡ n-fold rotation + translation parallel to rotation axis by

¡ Symbol: nm

¡ Example: for n = 3, possible values of


τ =mtn

τ =t3, 2t3t

τ =t3

τ =2t3

31

32

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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 uD21 D22 D23 vD31 D32 D33 w0 0 0 1

!

"

#####

$

%

&&&&&

D(θ ) | τ( )

1 −1 0 01 0 0 0

0 0 1 16

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 abc

a/2b/2c/2

Diagonal glide

n A, B, C, I

Diamondglide

d (a + b)/4, (b + c)/4, (c + a)/4(a + b + c)/4

D(m) | τ( )

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Properties of Symmetry Matrices¡ Operations of the first kind (proper motions)

¡ Operations of the second kind (improper motions)

¡ As all symmetry operations are orthogonal matrices and the product of orthogonal matrices is also an orthogonal matrix, 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 handedness unchanged.


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det(DI ) = +1

det(DII ) = −1

Combinations of Symmetry Operations


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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 0D21 D22 D23 0D31 D32 D33 00 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, 2π2, 2π3, 2π4, 2π6

"#$

%&'


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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