Most Important Element in life Most important Element in Engineering Most important element in Nanotechnology C

Graphite Diamond Buckminster Fullerene 1985 Carbon Nanotubes 1991 Graphene 2004 Allotropes of C

Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices Reciprocal lattice

A 3D translationaly periodic arrangement of atoms in space is called a crystal. Crystal, Lattice and Motif Crystal ?

Lattice? A 3D translationally periodic arrangement of points in space is called a lattice. Crystal, Lattice and Motif

A 3D translationally periodic arrangement of atoms Crystal A 3D translationally periodic arrangement of points Lattice Crystal, Lattice and Motif

Motif? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point Crystal, Lattice and Motif

“Nothing that is worth knowing can be taught.” Oscar Wilde

+ Love Pattern (Crystal) Love Lattice+ Heart (Motif) = = Lattice + Motif = Crystal Love Pattern

Air, Water and Earth Maurits Cornelis Escher Dutch Graphic Artist

Every periodic pattern (and hence a crystal) has a unique lattice associated with it

Cu CrystalNaCl Crystal Lattice FCC Motif1 Cu + ion1 Na + ion + 1 Cl - ion Crystal Crystal, Lattice and Motif

Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices

Translational Periodicity One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Unit Cell Unit cell description : 1

The most common shape of a unit cell is a parallelopiped with lattice points at corners. UNIT CELL: Primitive Unit Cell: Lattice Points only at corners Non-Primitive Unit cell: Lattice Point at corners as well as other some points

Lattice Parameters: 1. A corner as origin 2. Three edge vectors {a, b, c} from the origin define a CRSYTALLOGRAPHIC COORDINATE SYSTEM 3. The three lengths a, b, c and the three interaxial angles , ,  are called the LATTICE PARAMETERS    a b c Unit cell description : 4

Wigner-Seitz Unit Cells FCC BCC Rhombic Dodcahedron Tetrakaidecahedron

The six lattice parameters a, b, c, , ,  The cell of the lattice lattice crystal + Motif

Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices

7 crystal Systems Crystal SystemConventional Unit Cell 1. Cubica=b=c,  =  =  =90  2. Tetragonala=b  c,  =  =  =90  3. Orthorhombica  b  c,  =  =  =90  4. Hexagonal a=b  c,  =  = 90 ,  =120  5. Rhombohedral a=b=c,  =  =  90  OR Trigonal 6. Monoclinic a  b  c,  =  =90  7. Triclinic a  b  c,  Unit cell description : 5

Crystal SystemBravais Lattices 1.CubicPIF 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP ? Why so many empty boxes? E.g. Why cubic C is absent? P: Simple;I: body-centred; F: Face-centred;C: End-centred

The three cubic Bravais lattices Crystal systemBravais lattices 1.CubicPIF Simple cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F

Orthorhombic C End-centred orthorhombic Base-centred orthorhombic

Monatomic Body-Centred Cubic (BCC) crystal Lattice: bcc CsCl crystal Lattice: simple cubic BCC Feynman! Corner and body-centres have the same neighbourhood Corner and body-centred atoms do not have the same neighbourhood Motif: 1 atom 000 Motif: two atoms Cl 000; Cs ½ ½ ½ Cs Cl

½ ½ ½ ½ Lattice: Simple hexagonal hcp latticehcp crystal Example: Hexagonal close-packed (HCP) crystal x y z Corner and inside atoms do not have the same neighbourhood Motif: Two atoms: 000; 2/3 1/3 1/2

Crystal SystemBravais Lattices 1.CubicPIF 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP ? Why so many empty boxes? E.g. Why cubic C is absent? P: Simple;I: body-centred; F: Face-centred;C: End-centred

End-centred cubic not in the Bravais list ? End-centred cubic = Simple Tetragonal

14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIFC 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP

Face-centred cubic in the Bravais list ? Cubic F = Tetragonal I ?!!!

14 Bravais lattices divided into seven crystal systems Crystal systemBravais lattices 1.CubicPIFC 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP

Couldn’t find his photo on the net Auguste Bravais 1850: 14 lattices 1835: 15 lattices ML Frankenheim Aug 2009: 13 lattices !!! AML750 IIT-D X 1856: 14 lattices History:

Why can’t the Face- Centred Cubic lattice (Cubic F) be considered as a Body-Centred Tetragonal lattice (Tetragonal I) ?

Primitive cell Non- primitive cell A unit cell of a lattice is NOT unique. UNIT CELLS OF A LATTICE Unit cell shape CANNOT be the basis for classification of Lattices

What is the basis for classification of lattices into 7 crystal systems and 14 Bravais lattices?

Lattices are classified on the basis of their symmetry

What is symmetry?

If an object is brought into self- coincidence after some operation it said to possess symmetry with respect to that operation. Symmetry

NOW NO SWIMS ON MON

Lattices also have translational symmetry Translational symmetry In fact this is the defining symmetry of a lattice

If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n- fold rotation axis where  =180   =90  Rotation Axis n=2 2-fold rotation axis n=44-fold rotation axis

Rotational Symmetries Z 180  120  90  72  60   8 Angles: Fold: Graphic symbols

Crsytallographic Restriction 5-fold symmetry or Pentagonal symmetry is not possible for Periodic Tilings Symmetries higher than 6-fold also not possible Only possible rotational symmetries for periodic tilings …

Proof of The Crystallographic Restriction A rotation can be represented by a matrix If T is a rotational symmetry of a lattice then all its elements must be integers (wrt primitive basis vectors) N0123  180°120°90°60°0°0° n-fold23461

Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4 “Fig. 1-4 is an invented arrangement for ice, and although it contains many of the correct features of the ice, it is not the true arrangement. One of the correct features is that there is a part of the symmetry that is hexagonal. You can see that if we turn the picture around an axis by 120°, the picture returns to itself.” Hexagonal symmetry

Correction: Shift the box Michael Gottlieb’s correction: But gives H:O = 1.5 : 1

QUASICRYSTALS (1984) Icosahedral symmetry (5-fold symmetry) Lack strict translational periodicity -> Quasiperiodic Icosahedron Penrose Tiling Diffraction Pattern External Morphology

Reflection (or mirror symmetry)

Symmetry of lattices Lattices have Rotational symmetry Reflection symmetry Translational symmetry

The group of all symmetry elements of a crystal except translations (e.g. rotation, reflection etc.) is called its POINT GROUP. The complete group of all symmetry elements including translations of a crystal is called its SPACE GROUP Point Group and Space Group

Classification of lattices Based on the space group symmetry, i.e., rotational, reflection and translational symmetry  14 types of lattices  14 Bravais lattices Based on the point group symmetry alone (i.e. excluding translational symmetry  7 types of lattices  7 crystal systems Crystal systems and Bravais Lattices Classification of Lattices

7 crystal Systems SystemRequired symmetry CubicThree 4-fold axis Tetragonalone 4-fold axis Orthorhombicthree 2-fold axis Hexagonalone 6-fold axis Rhombohedralone 3-fold axis Monoclinicone 2-fold axis Triclinicnone

Tetragonal symmetry Cubic symmetry Cubic C = Tetragonal P Cubic F  Tetragonal I

The three Bravais lattices in the cubic crystal system have the same rotational symmetry but different translational symmetry. Simple cubic Primitive cubic Cubic P Body-centred cubic Cubic I Face-centred cubic Cubic F

Neumann’s Principle Symmetry elements of a physical property of a crystal must include all the symmetry elements of its point group (i.e., all its rotational axes and mirror planes). Electrical resistance of a cubic crystal is isotropic (spherical symmetry) All properties that can be represented by tensors of rank up to 2 are isotropic for cubic crystals

Jian-Min Zhanga,Yan Zhanga, Ke-Wei Xub and Vincent Ji J Phys. Chem. Solids, 68 (2007) Elasic modulus (4 th. Rank Tensor) is not isotropic Representation surfaces of Young’s modulus of fcc metals Ag Au Ni Pb Cu Al

QUESTIONS?

Contents Crystal, Lattice and Motif Unit cells, Lattice Parameters and Projections Miller Indices & Miller-Bravais Indices Directions and Planes Classification of Lattices: 7 crystal systems 14 Bravais lattices

1. Choose a point on the direction as the origin. 2. Choose a coordinate system with axes parallel to the unit cell edges. x y3. Find the coordinates of another point on the direction in terms of a, b and c 4. Reduce the coordinates to smallest integers. 5. Put in square brackets Miller Indices of Directions [100] 1a+0b+0c z 1, 0, 0 Miller Indices 2 a b c

y z Miller indices of a direction: only the orientation not its position or sense All parallel directions have the same Miller indices [100] x Miller Indices 3

x y z O A 1/2, 1/2, 1 [1 1 2] OA=1/2 a + 1/2 b + 1 c P Q x y z PQ = -1 a -1 b + 1 c -1, -1, 1 Miller Indices of Directions (contd.) [ ] __ -ve steps are shown as bar over the number Direction OA Direction PQ

Miller indices of a family of symmetry related directions [100] [001] [010] = [uvw] and all other directions related to [uvw] by the symmetry of the crystal = [100], [010], [001] = [100], [010] Cubic Tetragonal [010] [100] Miller Indices 4

Miller indices of slip directions in CCP y z [110] [101] [011] [101] Slip directions = close-packed directions = face diagonals x Six slip directions: [110] [101] [011] [110] [101] All six slip directions in ccp: Miller Indices 5

Vectors vs Directions: Miller Indices of a direction Exception: The Burger’s vector [110] is a direction along the face diagonal of a unit cell. It is not a vector of fixed length FCC BCC A vector equal to half body diagonal A vector equal to half face diagonal

5. Enclose in parenthesis Miller Indices for planes 3. Take reciprocal 2. Find intercepts along axes 1. Select a crystallographic coordinate system with origin not on the plane 4. Convert to smallest integers in the same ratio (111) x y z O

Miller Indices for planes (contd.) origin intercepts reciprocals Miller Indices A B C D O ABCD O 1 ∞ ∞ (1 0 0) OCBE O* 1 -1 ∞ (1 1 0) _ Plane x z y O* x z E Zero represents that the plane is parallel to the corresponding axis Bar represents a negative intercept

Miller indices of a plane specifies only its orientation in space not its position All parallel planes have the same Miller Indices A B C D O x z y E (100) (h k l )  (h k l ) _ _ _ (100)  (100) _

Miller indices of a family of symmetry related planes = (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal {hkl } All the faces of the cube are equivalent to each other by symmetry Front & back faces: (100) Left and right faces: (010) Top and bottom faces: (001) {100} = (100), (010), (001)

Slip planes in a ccp crystal Slip planes in ccp are the close-packed planes AEG (111) A D F E D G x y z B C y ACE (111) A E D x z C ACG (111) A G x y z C A D F E D G x y z B C All four slip planes of a ccp crystal: {111} D E G x y z C CEG (111)

{100} cubic = (100), (010), (001) {100} tetragonal = (100), (010) (001) Cubic Tetragonal Miller indices of a family of symmetry related planes x z y z x y

[100] [010] Symmetry related directions in the hexagonal crystal system = [100], [010], [001] Not permutations Permutations [110] x y z = [100], [010], [110]

(010) (100) (110) x {100} hexagonal = (100), (010), {100} cubic = (100), (010), (001) Not permutations Permutations Symmetry related planes in the hexagonal crystal system (110) y z

Problem: In hexagonal system symmetry related planes and directions do NOT have Miller indices which are permutations Solution: Use the four-index Miller- Bravais Indices instead

x1x1 x2x2 x3x3 (1010) (0110) (1100) (hkl)=>(hkil) with i=-(h+k) Introduce a fourth axis in the basal plane x1x1 x3x3 Miller-Bravais Indices of Planes x2x2 Prismatic planes: {1100} = (1100) (1010) (0110) z

Miller-Bravais Indices of Directions in hexagonal crystals x1x1 x2x2 x3x3 Basal plane =slip plane =(0001) [uvw]=>[UVTW] Require that: Vectorially a1a1 a2a2 a2a2

a1a1 a1a1 -a 2 -a 3 a2a2 a3a3 x1x1 x2x2 x3x3 x1:x1: x2:x2: x3:x3: Slip directions in hcp Miller-Bravais indices of slip directions in hcp crystal:

Some IMPORTANT Results Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl): h u + k v + l w = 0 Weiss zone law True for ALL crystal systems h U + k V + i T +l W = 0

CUBIC CRYSTALS [hkl]  (hkl) Angle between two directions [h 1 k 1 l 1 ] and [h 2 k 2 l 2 ]: C [111] (111)

d hkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell O x (100) B O x z E

Orientation Relationships In solid state phase-transformation, the new crystalline phase has a particular orientation relationship with the parent phase. When proeutectioid ferrite forms from austenite in steels the following orientation relationship, known as Kurdjumov-Sachs relationship is observed:

Orientation relationship between TiC and B2 phase in as-cast and heat-treated NiTi shape memory alloys Z. Zhang et al. Mat. Sci. Engg. A, 2006

Dendrite Growth Directions FCCBCCHCP

[uvw]Miller indices of a direction (i.e. a set of parallel directions) Miller indices of a family of symmetry related directions (hkl)Miller Indices of a plane (i.e. a set of parallel planes) {hkl}Miller indices of a family of symmetry related planes [uvtw]Miller-Bravais indices of a direction, (hkil) plane in a hexagonal system Summary of Notation convention for Indices

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