SYMMETRY

Finite geometric objects (“molecules) are symmetric with respect to

a) rotations b) reflections (symmetry planes, symmetry points) c) combinatons of translations and rotations (screw axes)

2π If an object is symmetric with respect to rotation by angle φ = , it is said to have has an “n-fold rotational axis” n

n=1 is a trivial case: every object is symmetric about a full revolution. Non-trivial axes have n>1.

1 Symmetries of an equilateral triangle

2 2 2 180-rotations (flips)

3’ 1’ about 22’

3 1 3 1

2’   about 33’ about 11’ 1 3 

1 2 2 3

3 2 2π 1 120 degree= rotations 3

3 2 1 3 

3 3

4

2 2 1 1 An equilateral triangle is not symmetric with respect to inversion about its center. But a square is.

5 Gliding (screw) axis

Rotate by 180 degrees about the axis

Slide by a half-period

6 2-fold rotations can be viewed as reflections in a plane which contains a 2-fold rotation axis

7

Crystal structure can be obtained by attaching atoms or groups of atoms --basis-- to sites.

Crystal Structure = Crystal Lattice + Basis

Crystal Structure 88 Partially from Prof. C. W. Myles (Texas Tech) course presentation Building blocks of lattices are geometric figures with certain symmetries: rotational, reflectional, etc.

Lattice must obey all these symmetries but, in addition, it also must obey

TRANSLATIONAL symmetry

Which geometric figures can be used to built a lattice?

9 10 Crystallographic Restriction Theorem:

Crystal lattices have only 2-,3, 4-, and 6-fold rotational symmetries

11 B’ ma A’ n-fold axis  φ-π/2 φ=2π/n a A B 1. Suppose that an n-fold rotational axis is going through A. another n-fold axis is going through B. 2. Rotate the lattice about A by φ=2π/n: BB’. 3. Rotate the lattice about B by φ=2π/n: AA’. 4. Distance A’B’ must be an integer multiple of a.

12 B’ ma A’ n-fold axis  φ-π/2 φ=2π/n a A B

ma = a + 2asin(ϕ − π / 2) = a − 2acosϕ 1− m −1 ≤ cosϕ = ≤ 1; m ≥ 0 ⇒ 0 ≤ m ≤ 3 2

m = 0 ⇒ cosϕ=1/2 ⇒ϕ = π / 3 ⇒ n = 6 m = 1⇒ cosϕ=0 ⇒ϕ = π / 2 ⇒ n = 4 m = 2 ⇒ cosϕ=-1/2 ⇒ϕ = 2π / 3 ⇒ n = 3 m = 3 ⇒ cosϕ=-1⇒ϕ = π ⇒ n = 2 13 Icosahedron Penrose tiling

14 Quasicrystals • Discovered in 1984 in Al-Mn submicron granules • Have long-range five-fold orientational but no translational long-range order • Exhibit sharp diffraction patterns • Hard metals with low heat conductivity • A commercially produced Al–Cu–Fe–Cr thermal spray cookware coating • Hydrogen storage?

15 Dan Shechtman Nobel Prize in Chemistry 2011

2 mkm grains Al-Mn, Fe, Cr (10-14%) 16 Fig. 4 The fivefold (A), threefold (B), and twofold (C) diffraction patterns obtained from a region (red dashed circle) of the granule in Fig.

CuAl, CuAl2 CuAl_2,CuAl

L Bindi et al. Science 2009;324:1306-1309

17

Published by AAAS Simplest type of lattices: Bravais lattices

Two equivalent definitions:

A. An infinite array of discrete points with an arrangement and orientation that appear exactly the same, from whichever of the points the array is viewed. B. A lattice consisting of all points with positions vectors R of the form

R = n1a1 + n2a2 + n3a3

Collorary to B: every point of a can be reached from any other point by a finite number of translations. Non-Bravais lattice contains points which cannot be reached by translations only. Rotations and reflections must be used in addition to translation.

Bravais lattices: basis consists of one element (atom) Non-Bravais lattices: can be represented as Bravais lattices with a basis consisting of more than one element. 18 Bravais lattices in 2D (all sites are equivalent)

Oblique Rhombohedral Orthorhombic (Rectangular) 2π   2π 180 = ⇒ 2 − fold axis 180  2π = 180 = 2 2 2

2π 60 = ⇒ 6-fold axis 6 2π 120 = ⇒ 3-fold axis 3

2π 90 = ⇒ 4-fold axis Tetragonal (Square) Hexagonal (Triangular) 19 4 Is this a Bravais lattice?

Yes, it is orthorhombic (rectangular) Bravais lattice 20 and the answer is ….

NO!

21 “body-centered tetragonal”= tetragonal

22 Bravais lattices in 3D: 14 types, 7 classes

Classes: Ag,Au,Al,Cu,Fe,Cr,Ni,Mb… 1. Cubic ✖3 2. Tetragonal✖2 3. Hexagonal✖1 Ba,Cs,Fe,Cr,Li,Na,K,U,V… α − Po 4. Orthorhombic✖4 5. Rhombohedral✖1 6. Monoclinic✖2 7. Triclinic✖1 He,Sc,Zn,Se,Cd…

S,Cl,Br

F Sb,Bi,Hg 23 Is there a two-dimensional body-centered tetragonal lattice?

simple tetragonal body-centered tetragonal?

A. Yes B. No 24 An example of a non-Bravais lattice

A B

Graphene: Honeycomb

• Even though graphene is a monoatomic compound, the basis consist of two atoms! • Red(A) and blue (B) sites are not equivalent.

2525 Nobel Prize 2010

Konstantin Novoselov Andre Geim 26 Honeycomb lattice: oblique lattice of diatomic molecules

27 From Ashcroft and Mermin Unit cell: an element of lattice that fills the space under translations.

28 Primitive Cell:

The smallest component of the crystal (group of atoms, ions or molecules) that, when stacked together with pure translational repetition, reproduces the whole crystal.

S S

S S S

b S S S S S a S S S S S Primitive unit cell

Crystal Structure 29 From Prof. C. W. Myles (Texas Tech) course presentation Blue square=primitive cell (area=1/2) Red parallelogram=unit cell (area=3/4) Green square=unit cell (area=1) Is the blue triangle a unit cell?

Crystal Structure 30 From Prof. C. W. Myles (Texas Tech) course presentation NO. Space must be filled by unit cells using only translations.

31 The choice even of a primitive cell is not unique

S S’’

b S S’

a Areas: S=S’=S’’

Crystal Structure 32 From Prof. C. W. Myles (Texas Tech) course presentation 2D “NaCl”

Crystal Structure 33 From Prof. C. W. Myles (Texas Tech) course presentation Choice of origin is arbitrary - lattice points need not be atoms - but the primitive cell area should always be the same.

Crystal Structure 34 From Prof. C. W. Myles (Texas Tech) course presentation - or if you don’t start from an atom

Crystal Structure 35 From Prof. C. W. Myles (Texas Tech) course presentation This is NOT a primitive cell - empty space is not allowed!

Crystal Structure 36 From Prof. C. W. Myles (Texas Tech) course presentation Graphene lattice as a Rhombohedral Bravais lattice with a 2-atom basis

✖

✖ ✖

✖ ✖

✖ ✖

✖ ✖

✖ ✖

✖

37 Graphene

a2

a1

One possible choice of unit lattice vectors

The electronic properties of graphene A. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim Another possible choice of unit lattice vectors Rev. Mod. Phys. 81, 109 (2009) shown along with the unit cell

38 Primitive and conventional cells Primitive cell: • A smallest volume (area) of space that, when translated through all the vectors of a Bravai lattice, fills all space without either overlapping or leaving voids. • Symmetry of the primitive cell does not necessarily coincide with that of a Bravais lattice. • The choice of a primitive cell is not unique • All primitive cells have the same volume (area)

Different choices of a primitive cell for an oblique 2D Bravais lattice (from Ashcroft and Mermin) 39 Primitive cells of Bravais lattices contain one atom. If this atom has an odd number of electrons, the element is a metal

40 Basis vectors for a bcc lattice

one possible choice of basis a more symmetric choice vectors for a bcc lattice a a a1 = (yˆ + zˆ − xˆ ), a2 = (zˆ + xˆ − yˆ ), a 2 2 a1 = axˆ, a2 = ayˆ, a3 = (xˆ + yˆ + zˆ) a 2 a3 = (xˆ + yˆ − zˆ) 2

41 bcc lattice as a superposition of two simple cubic lattices

42 Basis vectors of an fcc lattice

a a a a1 = (yˆ + zˆ), a2 = (zˆ + xˆ ), a3 = (xˆ + yˆ ) 2 2 2

43 Volume of a primitive cell: bcc lattice

V = a i a × a 1 ( 2 3 )

a a a a1 = (yˆ + zˆ − xˆ ), a2 = (zˆ + xˆ − yˆ ), a3 = (xˆ + yˆ − zˆ) 2 2 2

2 ⎡ xˆ yˆ zˆ ⎤ 2 ⎛ a⎞ ⎢ ⎥ ⎛ a⎞ V = a1 i(a2 × a3 ) = a1 i 1 −1 1 = a1 i 2(yˆ + zˆ) ⎝⎜ 2⎠⎟ ⎢ ⎥ ⎝⎜ 2⎠⎟ ⎣⎢ 1 1 −1 ⎦⎥ 3 ⎛ a⎞ a3 = 2(yˆ + zˆ − xˆ )i(yˆ + zˆ) = ⎝⎜ ⎠⎟ 2 2

44 (conventional) unit cell:

• A volume (area) that fills all space, when translated by some lattice vectors. • Larger than a primitive cell • Allows to “see” the lattice better 3D face-centered cubic (fcc) lattice

60 degree rombohedron

45 From Prof. C. W. Myles (Texas Tech) course presentation The most important primitive cell: the Wigner-Seitz cell

WS cell about a lattice point: a region of space that is closer to a given lattice point than to any other point.

WS cell has the same symmetry as the lattice itself.

Simple construction method: connect the lattice points by line, choose the middle points, and draw lines normal to the connecting lines. The enclosed volume (area) is the WS cell.

The method of construction is the same as for the in the wavenumber (reciprocal) space.

46 WS cell for a tetragonal (square) Bravais lattice

47 Construction of a WS cell for a rectangular lattice

n.n.

n.n.n. n.n.n.

n.n.

What about next-to-next-to-nearest neighbors? Adding n.n.n.n. does not change the WS cell for a n.n.=nearest neighbor rectangular lattice n.n.n.=next-to-nearest neighbor

48 WS cell for a hexagonal 2D lattice

NB: In 2D, WS cells are either hexagons or rectangulars 49 WS cell for an oblique lattice

n.n.

A

B n.n.

Depending on the shape of the parallelogram, either A or B can be n.n.n.

b If α + β < π / 2, a > b Rectangular: α β If α + β>π /2, a < b α=π /2;β=π /4;α+β>π / 2 ⇒ a < b a 50 Wigner-Seitz Cell - 3D

Crystal Structure 5151 Some common non-Bravais lattices

52 Number of atoms 4 atom inside Each of 8 corner atoms is shared among 8 cell: count as one atom. Each of 6 face atoms is shared among 2 cells: count as 3 atoms. 4+1+3=8 S. Sque (Exter) [111] C (diamond), Si, Ge 3a

3a / 4

[111] 3a

3a / 4

Zincblende (ZnS): GaAs, GaP, InAs, InSb, ZnSe, CdTe … 53 Hexagonal Close-Packed (HCP)

Be, Cd, Ce, Gd, Ho, Mg, Ni, Zr…

6 atoms per unit cell

54 “NaCl”: AgBr, AgCll, MnO, LiF…

55 Lattice Sites in Cubic Unit Cell

Crystal Structure 56 to bcc? a a a a1 = (yˆ + zˆ − xˆ ), a2 = (zˆ + xˆ − yˆ ), a3 = (xˆ + yˆ − zˆ) 2 2 2 2π 2π 2π b = a × a , b = a × a , b = a × a 1 V 2 3 2 V 3 1 3 V 1 2

2 ⎡ xˆ yˆ zˆ ⎤ 2 ⎛ a⎞ ⎢ ⎥ ⎛ a⎞ V = a1i(a2 × a3 ) = a1i 1 −1 1 = a1i2(yˆ + zˆ) ⎝⎜ 2⎠⎟ ⎢ ⎥ ⎝⎜ 2⎠⎟ ⎣⎢ 1 1 −1 ⎦⎥ 3 ⎛ a⎞ a3 = ⎜ ⎟ 2(yˆ + zˆ − xˆ )i(yˆ + zˆ) = ⎝ 2⎠ 2 Direct fcc lattice: 2π 4π 2π a b1 = a2 × a3 = (yˆ + zˆ) = (yˆ + zˆ) a1 = (yˆ + zˆ) V V a 2 2π a b2 = ... = (xˆ + zˆ) a2 = (zˆ + xˆ ) a 2 2π a b2 = ... = (xˆ + yˆ ) a3 = (xˆ + yˆ ) a bccfcc 2

57 Crystal Directions

• Choose one lattice point on the line as the origin: point O. Choice of origin is completely arbitrary, since every lattice point is identical. • Choose the lattice vector joining O to any point on the line: point T. This vector can be written as;

R = n1 a + n2 b + n3c

• To distinguish a lattice direction from a lattice point, the triad is enclosed in square brackets [ ...]: [n1n2n3] [111] direction • [n1n2n3] is the smallest integer of the same quotient.

Crystal Structure 58 Examples

210

X = 1 , Y = ½ , Z = 0 X = ½ , Y = ½ , Z = 1 [1 ½ 0] [2 1 0] [½ ½ 1] [1 1 2]

Crystal Structure 59 Negative directions Negative directions are [n n n ] written as 1 2 3 Z direction

- X direction (origin) O

- Y direction Y direction

[n1n2n3 ] X direction - Z direction

Crystal Structure 60 From Prof. C. W. Myles (Texas Tech) course presentation Examples of crystal directions

X = 1 , Y = 0 , Z = 0 [1 0 0] X = -1 , Y = -1 , Z = 0 [110]

Crystal Structure 6161 From Prof. C. W. Myles (Texas Tech) course presentation Miller indices (hkl)

1 / h,1 / k,1 / l = coordinates of intersection points between the (hkl) plane and coordinate axes

62 (100) (110) x = 1; y = ∞;z = ∞ ⇒ 1 / x = 1 /1 = 1;1 / y = 0;1 / z = 0

63 (111) (210) Brillouin zones for a square lattice 1st BZ=Wigner-Seitz cell in the reciprocal space

64 1st BZ for bcc lattice 1st BZ for fcc lattice

65