Solid State Physics (1) Phys3710

Crystal structure 2Lecture 2

Dr Mazen AlshaaerSecond semester 2013/2014

Department of Physics

1Ref.: Prof. Charles W. Myles, Department of Physics, Texas Tech University

2

Crystal structures Crystal: atoms are arranged so that their positions are periodic in allthree dimensionsAtoms are bound to one another →well defined equilibrium separations;many identical atoms →minimum energy requires everyidentical atom to be inidentical environment → 3D periodicity

Ideal crystal: perfect periodicity

Real crystals are never perfect:

•surface•impurities and defects•thermal motion of atoms (lattice vibrations)

3

4

5

Basic definitions

The periodic array of points is called crystal lattice.For every lattice point there is a group of atoms (orsingle atom) called basis of the lattice

Don't confuse with a1, a2, a3 - basis vectorsparallelogram formed by the basis vectors –unit cell

if a unit cell contains only one lattice point, it iscalled a primitive cell (minimum volume)Bravais lattices – all lattice pointsare equivalent

Crystal Structure 6

Crystal Lattice

Bravais Lattice (BL) Non-Bravais Lattice (non-BL)

All atoms are of the same kind All lattice points are equivalent

Atoms can be of different kind Some lattice points are not equivalentA combination of two or more BL

7

Bravais Lattice (1)

A fundamental concept in the description of crystalline solids is that of a“Bravais lattice”. A Bravais lattice is an infinite arrangement of points (oratoms) in space that has the following property:The lattice looks exactly the same when viewed from any lattice point

A 1D Bravais lattice:

A 2D Bravais lattice:

b

b

c

8

Bravais Lattice (2)

b

d

c

9

Bravais Lattice (3)A Bravais lattice has the following property:The position vector of all points (or atoms) in the lattice can be written asfollows (Translational Lattice Vectors) :

Example (1D):

Example (2D):

10

Bravais Lattice (4)

11

Bravais Lattice (5)

12

Non-Bravais Lattice

Not only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice.

The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right.

Red (and blue) sides are equivalent and have the same appearance

Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.

Honeycomb

13

Five Bravais Lattices in 2D

General or Oblique Rectangular Centered rectangular

Hexagonal Square

The Primitive Lattice Vectors a1,a2,a3

aren’t necessarily a mutually orthogonal set!Usually

Usually, they are neither mutually perpendicular nor all the same length!

• For examples, see Fig. 3b (3 dimensions):

14

2-Dimensional Unit Cells

Unit Cell The smallest component of the crystal (group

of atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole

crystal.

S

ab

S

S

S

S

S

S

S

S

S

S

S

S

S

S

15

Unit Cell The smallest component of the crystal (group of

atoms, ions or molecules), which, when stacked together with pure translational repetition, reproduces the whole crystal.

The choice of unit cell is not unique!

S

S

S

16

2-Dimensional Unit CellsArtificial Example: “NaCl”

Lattice points are points with identical environments.

17

2-Dimensional Unit Cells: “NaCl”

The choice of origin is arbitrary - lattice points need not be atoms - but the unit cell size must always be the same.

18

2-Dimensional Unit Cells: “NaCl”These are also unit cells -

it doesn’t matter if the origin is at Na or Cl !

19

2-Dimensional Unit Cells: “NaCl”These are also unit cells -

the origin does not have to be on an atom!

20

2-Dimensional Unit Cells: “NaCl”

These are NOT unit cells - empty space is not allowed!

21

2-Dimensional Unit Cells: “NaCl”

In 2 dimensions, these are unit cells – in 3 dimensions, they would not be.

22

2-Dimensional Unit CellsWhy can't the blue triangle be a unit cell?

23

3-Dimensional

Unit Cells

24

3-Dimensional Unit Cells3 Common Unit Cells with Cubic Symmetry

Simple Cubic Body Centered Cubic Face Centered Cubic (SC) (BCC) (FCC)

Simple Cubic Body Centered Cubic Face Centered Cubic (SC) (BCC) (FCC) 25

Conventional & Primitive Unit Cells

Unıt Cell Types

Primitive

A single lattice point per cellThe smallest area in 2 dimensions, orThe smallest volume in 3 dimensions

Simple Simple CCubicubic (sc)(sc)ConventionalConventional Cell Cell == Primitive cell Primitive cell

More than one lattice point per cell Volume (area) = integer multiple of

that for primitive cell

Conventional (Non-primitive)

Body Centered Cubic (bcc)Body Centered Cubic (bcc)ConventionalConventional Cell Cell ≠≠ Primitive cell Primitive cell

26

27

Bravais Lattices in 3DThere are 14 different Bravais lattices in 3D that are classified into 7 different crystal systems (only the unit cells are shown below)

28

Simple Cubic (SC) Structure

It is very cumbersome to draw entire lattices in 3D so some small portion of the lattice, having full symmetry of the lattice, is usually drawn. This small portion when repeated can generate the whole lattice and is called the “unit cell” and it could be larger than the primitive cell

Unit Cell:

Simple Cubic Lattice:

Face Centered Cubic (FCC) Structure

Conventional & Primitive Unit Cells Face Centered Cubic Lattice

Primitive Lattice Vectors

a1 = (½)a(0,1,0)

a2 = (½)a(1,0,1)

a3 = (½)a(1,1,0)

Primitive Lattice Vectors

a1 = (½)a(0,1,0)

a2 = (½)a(1,0,1)

a3 = (½)a(1,1,0)

Conventional Unit Cell (Full Cube)

Conventional Unit Cell (Full Cube)

Primitive Unit Cell(Shaded)

Primitive Unit Cell(Shaded)

Lattice

Constant

Lattice

Constant

Elements That Form Solids with the FCC Structure

Body Centered Cubic (BCC) Structure

Conventional & Primitive Unit Cells Body Centered Cubic Lattice

Primitive Lattice Vectorsa1 = (½)a(1,1,-1)a2 = (½)a(-1,1,1)a3 = (½)a(1,-1,1)

Note that the ai’s are

NOT mutuallyorthogonal!

Primitive Lattice Vectorsa1 = (½)a(1,1,-1)a2 = (½)a(-1,1,1)a3 = (½)a(1,-1,1)

Note that the ai’s are

NOT mutuallyorthogonal!

Primitive Unit CellPrimitive Unit Cell

Lattice Constant

Lattice Constant

Conventional Unit Cell (Full Cube)

Conventional Unit Cell (Full Cube)

Elements That Form Solids with the BCC Structure

Conventional & Primitive Unit Cells Cubic Lattices

a

b c

a

b cSimple Cubic (SC)

Primitive Cell = Conventional CellFractional coordinates of lattice points: 000, 100, 010, 001, 110,101, 011, 111

Body Centered Cubic (BCC)Primitive Cell Conventional Cell

Fractional coordinates of lattice points in conventional cell: 000,100, 010,

001, 110,101, 011 111, ½ ½ ½

a

bc

Conventional & Primitive Unit Cells Cubic Lattices

Face Centered Cubic (FCC)Primitive Cell Conventional Cell

Fractional coordinates of lattice points in conventional cell: 000,100, 010, 001, 110,101, 011,111, ½

½ 0, ½ 0 ½, 0 ½ ½ ½1 ½ , 1 ½ ½ , ½ ½ 1

a

bc

Simple Hexagonal Bravais Lattice

Conventional & Primitive Unit Cells

Hexagonal Bravais LatticePrimitive Cell = Conventional Cell

Fractional coordinates of lattice points in conventional cell: 100, 010, 110, 101, 011 111, 000, 001

Points of Primitive Cell

a

b

c

120

o

Hexagonal Close Packed (HCP) Structure:(A Simple Hexagonal Bravais Lattice with a 2 Atom Basis)

The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

General Unit Cell Discussion• For any lattice, the unit cell &, thus,

the entire lattice, is UNIQUELY determined by 6 constants (figure):

a, b, c, α, β and γwhich depend on lattice geometry.

• As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure, each unit cell is associated with (8) (1/8) = 1 lattice point.

Primitive Unit Cells & Primitive Lattice Vectors

• In general, a Primitive Unit Cell is determined by the parallelepiped formed by the Primitive Vectors a1 ,a2, & a3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure.

• As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectors

T = n1a1 + n2a2 + n3a3.

• The Primitive Unit Cell volume can be found by vector manipulation:

V = a1(a2 a3)

• For the cubic unit cell in the figure, V = a3

Primitive Unit Cells

• Note that, by definition, the Primitive Unit Cell must contain ONLY ONE lattice point.

• There can be different choices for the Primitive Lattice Vectors, but the Primitive Cell volume must be independent of that choice.

A 2 Dimensional Example!

P = Primitive Unit CellNP = Non-Primitive Unit Cell