Chapter 1 Crystal Structure Outline • Definition of Crystal and Bravais lattice • Examples of Bravais lattice and crystal structures • Primitive unit cell • Wigner – Seitz unit cell • Miller Indices • Classification of Braivais lattices

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• An ideal crystal is constructed by the infinite repetition in space of identical structural unit. • The structure of all crystals is described in terms of a lattice with a group of atoms attached to each lattice point. Bravais lattice

basis

A Bravais lattice is an infinite array of discrete points and appear exactly the same, from whichever of the points the array is viewed. A (three dimensional) Bravais lattice consists of all points with positions vectors R of the form →

→

→

→

→

R = n1 a1 + n2 a2 + n3 a3 r r r Where are any three vectors a1 a2and a3 are not all in the same plane, and n1, n2, and n3 range through all integral values 2

A general 2-D Bravais lattice of no particular symmetry P r r r r P = a1 + 2a2 = −a1 + 2a2'

r a2'

Q r a2 O

r r r r Q = − a1 + a2 = −2a1' + a2'

r a1

Primitive vectors: or

r a1 r a1

r a2 r' a2

Primitive vectors are not unique

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Vortices of a 2-D honeycomb do not form a Bravais lattice

P and Q are equivalent P and R are not

Q P

R

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Simple cubic (sc) structure

Atoms per cubic cell: 1

r a1 = axˆ

r a3 r a2

r a1

r a2 = ayˆ

r a3 = azˆ

Body centered cubic (bcc)

B A A and B are equivalent It is a Bravais lattice

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Primitive vectors for bcc structure z

r a2

r a3

r a1

x

y

r 1 a1 = a ( xˆ + yˆ − zˆ ); 2 r 1 a2 = a( − xˆ + yˆ + zˆ ); 2 r 1 a3 = a( xˆ − yˆ + zˆ ) 2

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An alternative set of primitive vectors for bcc structure z

r a3

r a1

r a2

y

x r a1 = axˆ r a2 = ayˆ r a a3 = ( xˆ + yˆ + zˆ ) 2

Less symmetric compared to previous set 7

Face centered cubic (fcc) structure

Each point can be either a corner point or a face-centering point It is a Bravais lattice

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Primitive vectors for fcc structure z

r a3

r a2

r a1

y

x

r 1 a1 = a( xˆ + yˆ ) 2 r 1 a2 = a( yˆ + zˆ ) 2 r 1 a3 = a( zˆ + xˆ ) 2 9

Primitive unit cell Primitive unit cell is a volume of space that, when translated through all the vectors in a Bravais lattice, just fills all the space without either overlapping itself or leaving voids.

Two ways of defining primitive cell • Primitive cell is not unique • A primitive cell must contain exactly one lattice point. 10

The obvious primitive cell:

r r r r r = x1a1 + x2 a2 + x3 a3 Is the set of all points of the above form for all xi ranging continuously between 0 and 1. It is the parallelipiped spanned by the primitive vectors

r a3

r a2 r a1

Disadvantage: the primitive cell defined as above does not reflect full symmetry of the Bravais lattice 11

Primitive cell of a bcc Bravais lattice

r a3

r a2

r a1

Primitive cell is a rhombohedron with edge

3 a 2

Volume of primitive cell is half of the cube 12

Primitive cell of a fcc Bravais lattice z

r a2

r a3

r a1

y

x Volume of the primitive cell is ¼ of the cube 13

Simple hexagonal (sc) Bravais lattice and its primitive cell

z

r 3 a a1 = ( a ) xˆ + ( ) yˆ 2 2 r 3 a a2 = −( a ) xˆ + ( ) yˆ 2 2 r a3 = czˆ

r a3

An alternative set:

r 3 a a1 = ( a) xˆ + ( ) yˆ 2 2 r a2 = ayˆ r a3 = czˆ

r a2

y r a1

x 14

Wigner – Seitz primitive cell • a primitive cell with the full symmetry of the Bravais lattice • a W-S cell about a lattice point is the region of space that is closer to that point than to any other lattice point

hexagon

Wigner – Seitz unit cell about a lattice point can be constructed by drawing lines connecting the point to all others in the lattice, bisecting each line with a plane, and taking the smallest polyhedron containing the point bounded by these plane 15

Wigner-Seitz cell of fcc Wigner-Seitz cell of bcc

Truncated octahedron 14 faces (8 regular hexagons and 6 squares)

12 faces (parallelograms)

Note: the surrounding cube is not the cubic cell

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Crystal structure: lattice with a basis A crystal structure consists of identical copies of the same physical unit, called the basis, located at all the points of a Bravias lattice (or, equivalently, translated through all the vectors of a Bravais lattice) Honeycomb net

Basis Lattice: 2-D triangular lattice

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Describe a Bravais lattice as a lattice with a basis by choosing a nonprimitive cell (a unit cell) A unit cell is a region that just fills space without any over-lapping when translated through some subset of the vectors of a Bravais lattice. It is usually larger than the primitive cell (by an integer factor)

bcc:

Simple cubic unit cell Basis:

fcc:

0

a ( xˆ + yˆ + zˆ ) 2

Simple cubic unit cell Basis:

0

a ( xˆ + yˆ ) 2

a ( yˆ + zˆ ) 2

a ( xˆ + zˆ ) 2

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Diamond structure • Not a Bravais lattice • Two interpenetrating fcc Bravais lattice Bravais lattice : fcc basis

0

1 ( xˆ + yˆ + zˆ) 4

Coordination number : 4 Four nearest neighbors of each point form the vertices of a regular tetrahedron

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Atomic positions in the cubic cell of diamond structure projected on (100) surface

0

1/2 1/4

3/4 1/2

1/2

0 1/4

0

0

3/4 1/2

0

Fractions in circles denote height above the base in units of a cube edge.

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Hexagonal close-packed (hcp) structure Two interpenetrating simple hexagonal Bravais lattice Bravais lattice: simple hexagonal Basis:

0

2r 1r 1r a1 + a2 + a3 3 3 2

r a3

r a2

r a1

Four neighboring atoms form the vortices of a tetrahedron

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Both hcp and fcc can be viewed as close-packed hard spheres

hcp fcc Coordination number: 12 for both fcc and hcp

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Fcc is close packed structure

Try calculating packing density

The (111) plane equivalent to the triangular close packed hard sphere layer 23

Basis consisting of different atoms

CsCl NaCl Bravais lattice: fcc Basis: Cl- at (0,0,0) and Na+ at (1/2,1/2,1/2)

Bravais lattice: sc Basis: Cs+ at (0,0,0) and Cl- at (1/2,1/2,1/2)

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Miller indices to index crystal planes

x3

r a3

r a1

x1

r a2

x2

• find the intercepts x1, x2, and x3 • h: k: l = 1/x1 : 1/ x2: 1/x3 Find the intercepts on the axes in terms of lattice constants

r a1

r a2

r a3

Take the reciprocals of these numbers and then reduce to the smallest three integers h, k, and l. The results, enclosed in parentheses (hkl), are known as Miller indices

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Classification of Bravais lattices and crystal structures Symmetry operations: all rigid operations that take the lattice into itself Rigid operation: operations that preserve the distance between all lattice points.

The set of symmetry operation is known as space group All symmetry operations of a Bravais lattice contains only operations of the following form: 1. Translations TR through lattice vectors 2. operations that leave a particular point of lattice fixed (point operation) 3. successive operations of 1 and 2

The set of point operation is known as point group, a subset of space group

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(a) A rotation operation through an axis that contains no lattice points (b) An equivalent compound operation involving a translation and a point operation

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Seven crystal systems (point groups) and fourteen Bravais lattices (space groups)

cubic

monoclinic trigonal

tetragonal

orthorhombic

triclinic

hexagonal

There are only seven distinct point groups that a Bravais lattice can have

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Cubic system: 3 Bravais lattices Simple cubic Body centered cubic Face centered cubic

Tetragonal system: 2 Bravais lattices Obtained by pulling on two opposite faces of a cube Simple tetragonal Centered tetragonal No distinction between face centered and body centered tetragonal Two ways of viewing the same lattice along c axis One lattice plane Next lattice plane c/2 above

Viewed as if it’s “body centered”

Viewed as if it’s “face centered”

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Orthorhombic system: 4 Bravais lattice By stretching tetragonal along one of the a axis Two ways of stretching the same simple tetragonal lattice viewed along c axis

stretch

Simple orthorhombic

stretch

30 Base centered orthorhombic

Two ways of stretching the same centered tetragonal lattice viewed along c axis

stretch

Body-centered orthorhombic

stretch

face-centered orthorhombic

Symmetry reduced, two structures distinguishable

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not a right angle

Monoclinic system: 2 Bravais lattices Reduce orthorhombic symmetry by Distorting the rectangular faces perpendicular to c axis

Base centered orthorhombic

Simple orthorhombic

Distort rectangular shape into parallelogram

Simple monoclinic

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Body-centered orthorhombic

face-centered orthorhombic Distort rectangular shape into parallelogram

No distinction between “face-centered” and “body-centered”

Centered monoclinic

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Triclinic system: 1 Bravais lattice Tilt the c axis of a monoclinic lattice

• no restrictions except that pairs of opposite faces are parallel • a Bravais lattice generated by three primitive vectors without special relationship to one another • the Bravais lattice with the minimum symmetry Think: why there are no face-centered or body-centered triclinic?

Trigonal system: 1 Bravais lattice Stretch a cube along a body diagonal

Hexagonal system: 1 Bravais lattice

cubic

hexagonal trigonal

tetragonal orthorhombic monoclinic

triclinic Arrow: direction of symmetry reduction

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