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Chapter 1 Crystal Structure

• 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

Outline

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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 of the formR→

→→→→

++= 332211 anananR

1ar

Where are any three vectors and are not all in the same

plane, and n1, n2, and n3 range through all integral values2

ar

3ar

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A general 2-D Bravais lattice of no particular symmetry

1ar

2ar

'

212122 aaaaPrrrr

+−=+=

O

P

Q

'

2

'

121 2 aaaaQrrrr

+−=+−=

'

2ar

Primitive vectors are not unique

Primitive vectors: 1ar

2ar

or1

ar '

2ar

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

P

Q

R

P and Q are equivalent

P and R are not

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Simple cubic (sc) structure

xaa ˆ1

=r

yaa ˆ2

=r

zaa ˆ3

=r

1ar

3ar

2ar

Atoms per cubic cell: 1

Body centered cubic (bcc)

A

B

A and B are equivalent

It is a Bravais lattice

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x y

z

1ar

2ar

3ar

)ˆˆˆ(2

1

);ˆˆˆ(2

1

);ˆˆˆ(2

1

3

2

1

zyxaa

zyxaa

zyxaa

+−=

++−=

−+=

r

r

r

Primitive vectors for bcc structure

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An alternative set of primitive vectors for bcc structure

x

y

z

1ar

2ar3

ar

)ˆˆˆ(2

ˆ

ˆ

3

2

1

zyxa

a

yaa

xaa

++=

=

=

r

r

r

Less symmetric compared to previous set

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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

)ˆˆ(2

1

)ˆˆ(2

1

)ˆˆ(2

1

3

2

1

xzaa

zyaa

yxaa

+=

+=

+=

r

r

r

1ar

2ar

3ar

x

y

z

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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.

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The obvious primitive cell:

332211axaxaxrrrrr

++=

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

1ar

2ar

3ar

Disadvantage: the primitive cell defined as above does not

reflect full symmetry of the Bravais lattice

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Primitive cell of a bcc Bravais lattice

1ar

2ar3

ar

Primitive cell is a rhombohedron with edge a2

3

Volume of primitive cell is half of the cube

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x

y

z

1ar

2ar

3ar

Primitive cell of a fcc Bravais lattice

Volume of the primitive cell is ¼ of the cube

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x

y

z

Simple hexagonal (sc) Bravais lattice and its primitive cell

1ar

2ar

3ar

zca

ya

xaa

ya

xaa

ˆ

ˆ)2

(ˆ)2

3(

ˆ)2

(ˆ)2

3(

3

2

1

=

+−=

+=

r

r

r

zca

yaa

ya

xaa

ˆ

ˆ

ˆ)2

(ˆ)2

3(

3

2

1

=

=

+=

r

r

r

An alternative set:

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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

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

hexagon

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Wigner-Seitz cell of bcc

Wigner-Seitz cell of fcc

Note: the surrounding

cube is not the cubic cell

Truncated octahedron

14 faces (8 regular hexagons and 6 squares)

12 faces (parallelograms)

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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 non-

primitive 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: 0 )ˆˆˆ(2

zyxa

++

)ˆˆ(2

yxa

+ )ˆˆ(2

zya

+

fcc: Simple cubic unit cell

Basis: 0 )ˆˆ(2

zxa

+

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Diamond structure

• Not a Bravais lattice

• Two interpenetrating fcc Bravais lattice

Bravais lattice : fcc

basis0 )ˆˆˆ(

4

1zyx ++

Coordination number : 4

Four nearest neighbors of each point form the vertices of a

regular tetrahedron

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0 1/2 0

1/2 0 1/2

0 1/2 0

3/4 1/4

1/4 3/4

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

surface

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

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Hexagonal close-packed (hcp) structure

Bravais lattice: simple hexagonal

Basis: 0321

2

1

3

1

3

2aaarrr

++

Two interpenetrating simple hexagonal Bravais lattice

Four neighboring atoms form the vortices of a tetrahedron

1ar

2ar

3ar

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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

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

layer

Try calculating packing density

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Basis consisting of different atoms

NaCl

CsCl

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

1ar

2ar

3ar

x1

x2

x3

• 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 constants1

ar

2ar

3ar

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

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 symmetry operation is known as space group

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

Rigid operation: operations that preserve the distance between all lattice points.

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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

tetragonal

orthorhombic

monoclinic

triclinic

trigonal

hexagonal

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

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Cubic system: 3 Bravais lattices

Simple cubicBody centered cubic Face centered cubic

Tetragonal system: 2 Bravais lattices

Obtained by pulling on two opposite faces of a cube

Simple tetragonalCentered tetragonal

No distinction between face centered and body centered tetragonal

One lattice plane

Next lattice

plane c/2 above

Two ways of viewing the same lattice along c axis

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

stretchstretch

Simple orthorhombicBase centered orthorhombic

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

stretch stretch

Body-centered orthorhombicface-centered orthorhombic

Symmetry reduced, two structures distinguishable

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not a right angle

Monoclinic system: 2 Bravais lattices

Base centered orthorhombic

Reduce orthorhombic symmetry by Distorting the rectangular faces perpendicular to c axis

Simple orthorhombic

Simple monoclinic

Distort rectangular shape into parallelogram

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Distort rectangular shape into parallelogram

Body-centered orthorhombicface-centered orthorhombic

Centered monoclinic

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

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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 latticecubic

tetragonal

orthorhombic

monoclinic

triclinic

hexagonal

trigonal

Arrow: direction of symmetry reduction