Symmetry and Properties of Crystals (MSE638) Symmetry ...home.iitk.ac.in/~bsomnath/mse638/WWW/2dsym.pdfSymmetry and Properties of Crystals (MSE638) Symmetry operations in 2D and planar

Symmetry and Properties of Crystals(MSE638)

Symmetry operations in 2D and planar lattices

Somnath Bhowmick

Materials Science and Engineering, IIT Kanpur

January 18, 2019

One step symmetry operations in 2D

Three one step symmetry operations in 2D.

Translation: (x, y)T−→ (x+ma, y + nb), where

T = (1a, 2b).

Reflection: (x, y)m−→ (−x, y).

Rotation: (x, y)A2−−→ (−x,−y).

Translation – a vector (having magnitude anddirection but no unique origin).

Only reflection changes the chirality – lefthanded motif becomes right handed and viceversa.

Question: how many 1 step operations arethere in 1D and 3D?

1D: Translation and reflection

3D: Translation, reflection, rotation, inversion2 / 21

Transformation Matrix

Multiply [1 2]T with any 2× 2 matrix: transform the vectorCase 1: [

1 10 −1

] [12

]=

[3−2

]Case 3: [

0 −11 0

] [12

]=

[−21

]

Case 2:[−4 10 −1

] [12

]=

[−2−2

]Case 4: [

−1 00 1

] [12

]=

[−12

]We are mainly interested in rotation and reflection matrices

3 / 21

Rotation matrix

In 2D, z−axis is the rotation axisRotation does not change lengthx′ = r cos(θ + α) = x cosα− y sinα, y′ = x sinα+ y cosαMatrix equation: [

cosα − sinαsinα cosα

] [xy

]=

[x′

y′

]Rotation matrix in 2D (counter clockwise rotation):

R(α) =

[cosα − sinαsinα cosα

]Note that det[R] = 1 4 / 21

Reflection matrix

Reflection about a line y = px; reflection does not change lengthx′ = r cos(2α− θ) = x cos 2α+ y sin 2α, y′ = x sin 2α− y cos 2αMatrix equation: [

cos 2α sin 2αsin 2α − cos 2α

] [xy

]=

[x′

y′

]Reflection matrix:

m(α) =

[cos 2α sin 2αsin 2α − cos 2α

]α = arctan(p) if p ≥ 0 and α = arctan(|p|) + π2 if p < 0Note that det[m] = −1 5 / 21

Properties of rotation and reflection matrix

Both R and m are orthogonal matrix (rotation/reflection does notchange length)

Matrix A is orthogonal if A preserves the length of vectors:Av ·Av = v · v (check this for R and m)A also preserves angles: Au ·Av = u · vNote that R−1(α) = R(−α) = RT (α)General property of orthogonal matrix: A−1 = AT ⇒ AAT = 1Check this for R and m:

A =

[a bc d

], A−1 =

1

det[A]

[d −b−c a

], AT = aji =

[a cb d

]Reflection matrix is a symmetric matrix also: m = mT

A Reflection is it’s own inverse: m = m−1

6 / 21

Adding two translations

Can the second translation be parallel to T1?If T2 is parallel to T1, it has to be an integer multiple of T1But then, we are just repeating the old set of lattice pointsSo, T1 and T2 are non-collinear and forms the 2D latticeUnit cell: area uniquely associated to one (primitive unit cell) or more(non-primitive unit cell) lattice pointsIs the choice of unit cell unique? How to select?

I pick shortest T1 & T2I pick T1 & T2 such that the unit cell has symmetry of the lattice - do

not pick a parallelogram unit cell for a square lattice 7 / 21

Adding translation and rotation

Translation limits the number of possible rotations in a crystal.

Rotation axis denoted by Aα or n, where α =2πn

Denoted by n fold rotation axis

cosα = 1−m2

m cosα α n Geometrical symbol

0 +12 60 6 71 0 90 4 �2 −12 120 3 43 -1 180 2 ()

-1 +1 0/360 1 .

8 / 21

List of symmetry operations in 2D and notation

Symmetry element Chirality change Analytical symbol Geometrical symbol

Translation No T →Reflection Yes m |

6-fold rotation No 6 74-fold rotation No 4 �3-fold rotation No 3 42-fold rotation No 2 ()

1-fold rotation No 1 .

Next: combine translation (T) to each of the symmetry elements (m, 2, 3,4, 6) and derive the 2D lattices

9 / 21

2D lattices

Lattice Translations a1∠a2 SymmetryOblique a1 6= a2 α 2

Rectangle a1 6= a2 90◦ 2, mDiamond a1 = a2 α 2, m

Square a1 = a2 90◦ 2, m, 4

Hexagonal a1 = a2 60◦, 120◦ 2, m, 3, 6

Hexagonal lattice is also known as a triangular lattice

You may find more names in the literature

Example: centered rectangular lattice– same as diamond lattice

Example: honeycomb lattice (famous for graphene)– this is ahexagonal lattice with two point basis

Example: Kagome lattice– this is a hexagonal lattice with three pointbasis

10 / 21

Honeycomb and Kagome lattice

11 / 21

2D unit cells

Unit cell Translations a1∠a2 SymmetryOblique (P) a1 6= a2 α 2

Rectangle (P) a1 6= a2 90◦ 2, mDiamond (P) a1 = a2 α 2, m

Centered rectangle (NP) a1 6= a2 90◦ 2, mSquare (P) a1 = a2 90

◦ 2, m, 4

Hexagonal (P) a1 = a2 60◦, 120◦ 2, m, 3, 6

P = Primitive (one lattice point per unit cell).

NP = Non-primitive (multiple lattice points per unit cell).

Lattice translation vectors: sensible choice of unit cell for a givenlattice

But there is no unique unit cell for a given lattice - one draw anoblique unit cell in a square lattice.

Symmetry of lattice does not depend on symmetry of unit cell !!

Symmetry of unit cell ≤ symmetry of lattice.12 / 21

Wigner-Seitz cell

Wigner-Seitz cell – a special primitive cell.

Area enclosing the space closer to a particular lattice point than anyother is defined as the WS cell of that particular lattice point.

How to draw – join neighbors of a lattice point by lines and drawperpendicular bisectors of those lines.

13 / 21

Reciprocal lattice

Rectangle

a1

a2

b1

b2

Hexagonal

a1

a2

b1

b2

Corresponding to ~a1 and ~a2 (real space lattice translation vectors),there exist a pair of vectors ~b1 and ~b2 such that ~ai · ~bj = δij .Using ~b1 and ~b2 as two translation vectors, we can construct a lattice,known as the reciprocal space lattice.

Dimension of ~bi is 1/length and it’s magnitude is inverselyproportional to ~ai.

Wigner-Seitz cell in reciprocal space: known as first Brillouin zone.

Useful for studies of diffraction, solid state physics etc.

14 / 21

Real and reciprocal lattice in 2D

b1

b2

a1a2

ObliqueRectangle

a1

a2

b1

b2

a1

a2

b1

b2

Square

Hexagonal

a1

a2

b1

b2

Diamond

b1

b2

a1

a2

Real lattice vectors ~a1 and ~a2Reciprocal lattice vectors ~b1 and ~b2Satisfies: ~ai · ~bj = δijWigner-Seitz cell and first Brillouin zone – rotated by 90◦ w.r.t. eachother

15 / 21

Experimental observation of real lattice

STM image of graphene (Nature Materials volume 10, pages 443449 (2011))

16 / 21

Experimental observation of reiprocal lattice

Electron diffraction pattern of graphene (PNAS)17 / 21

Directions and Intercepts in real space lattice

In crystallography, directions and intercepts are measured with respectto the lattice translation vectors (not necessarily same as Cartesian).However, lattice translation vectors represented in Cartesian system.Direction – represented by vectors in ~a1 and ~a2 basis.Intercepts – represented w.r.t. ~a1 – ~a2 coordinate system.

Example: hexagonal lattice – ~a1 = (a, 0), ~a2 = (−a2 and√3a2 ).

a1a2

[1,2] [2,2]

[2,1]

a1a2 (1,2)

(2,2)

(2,1)

[hk] direction: ~R = h~a1 + k ~a2(hk) intercept: fractional coordinates (a1h ,

a2k )

In general, [hk] is not perpendicular to (hk) interceptSquare lattice: [hk] is perpendicular to (hk) interceptSpacing between (hk) intercept: dhk =

a√h2+k2

(square lattice only)18 / 21

Reciprocal space lattice

(0,0)(1,0)

(2,0)

(2,1)

(2,2)(1,2)

(0,2)(1,1)

(0,1)

b1b2

(2,1)(1,2)

(2,2)

(hk) direction, ~Ghk = h~b1 + k~b2Looks like (hk) direction in reciprocal lattice is perpendicular to the(hk) intercept in the real lattice (prove it)Perpendicular distance of the intercept from the origin: dhk =

1| ~Ghk|

Reciprocal lattice points represent orientation and spacing ofintercepts (2D) or planes (3D) in real latticeReciprocal lattice: alternate geometrical construction of real latticeUsing ~Ghk, one can find interplanar distance (in real space)Using ~Ghk, one can find angle between planes (in real space)

19 / 21

Reciprocal lattice: important applications

~ai · ~bj = δij : easy to solve for the components of ~b in 2DGeneral definition: ~b1 =

~a2× ~a3~a1·( ~a2× ~a3) ,

~b2 =~a3× ~a1

~a1·( ~a2× ~a3) ,~b3 =

~a1× ~a2~a1·( ~a2× ~a3)

In case of 2D, ~a3 = a3ẑ

Reciprocal of (~b1,~b2) is (~a1,~a2)~R = u~a1 + v~a2 can be written as ~R =

∑j(~R ·~bj)~aj

Compare it with Cartesian coordinate system !!

exp[2πι ~G · (~r + ~R)] = exp[2πι ~G · ~r]Any function u(~r), having the periodicity of the lattice can be writtenas: u(~r) =

∑~GC(

~G) exp(2πι ~G · ~r)Solid state physics (Bloch theorem) : eigenfunction of an electron ina crystal (pediodic potential) expressed as ψ(~r) = u(~r) exp(~k · ~r),where u(~r) has the periodicity of the lattice

Observed peaks in a diffraction pattern (structure factor calculation):Fhkl =

∑Nj=1 f~rj exp(2πι

~Ghkl · ~rj)20 / 21

Summary

Translation restricts the possible rotational symmetries in a crystal –only 2, 3, 4 and 6-fold rotation axes possible.

There exist five plane lattices in 2D – oblique, primitive rectangle,diamond or centered rectangle, square and hexagonal.

No unique unit cell for a given lattice - most sensible choice is the cellformed by lattice translation vectors ~a1 and ~a2.

Symmetry of unit cell ≤ symmetry of lattice.In crystallography, directions and intercepts are measured with respectto the lattice translation vectors.

However, lattice translation (both real and reciprocal) vectorsrepresented in Cartesian system.~Ghk is perpendicular to the (h,k) intercept in the real lattice.

Perpendicular distance of the intercept from the origin: dhk =1| ~Ghk|

.

Reciprocal lattice points represent orientation and spacing ofintercepts in real lattice.

Reciprocal lattice: alternate geometrical construction of real lattice.21 / 21