crystal structures & packing fraction

Presentation on Crystal

structures

“A crystal is a solid in which atoms are arranged in some regular repetition pattern in all directions.”

“Aggregation of molecules with a definite internal structure and the external form of a solid enclosed by symmetrically arranged plane faces.”

“Structure of anything is defined as the framework of its body.”

CRYSTAL

STRUCTURES

Lattice The underlying periodicity of the crystalBasis Entity associated with each lattice points

Lattice how to repeat

Motif what to repeat

Crystal = Lattice+BaseMotif or basis: Typically an atom or a group of atoms associated with each lattice point.

Translationally periodic arrangement of motifs.

Translationally periodic arrangement of points.

Lattice

Crystal

Crystal = Lattice (Where to repeat) +

Motif (What to repeat)

=

+

a

a

2

a

Lattice

Motif

Note: all parts of the motif do not sit on the lattice

point

Crystal

Let us construct the crystal considered before starting with an infinite array of points spaced a/2 apart

Put arrow marks pointing up and down alternately on the points:

What we get is a crystal of lattice parameter ‘a’ and not ‘a/2’!

And the

motif is: +

A strict 1D crystal = 1D lattice + 1D motif The only kind of 1D motif is a line segment.

Lattice

Motif

Crystal

=

+

An unit cell is a representative unit of the structure (finite part of a infinite structure) . Which when repeated gives the whole structure.

1-D Crystal

2D crystal = 2D lattice + 2D motif

Lattice

a

b

+Motif

2-D Crystal

Crystal

=

3D crystal = 3D lattice + 3D motifs

3-D Crystal

CRYSTAL OR SPACE LATTICE

It is defined as an array of points in 3 dimensions in which every point has surroundings identical to every other point in array.

According to BRAVAIS there are 14 possible types of space lattice in 7 basic crystal system

THE 7 CRYSTAL SYSTEM

a = b= c = = = 90º

• Simple Cubic (P) - SC• Body Centred Cubic (I) – BCC• Face Centred Cubic (F) - FCC

Elements with Cubic structure → SC: F, O

BCC: Cr, Fe, Nb, K, W FCC: Al, Ar, Pb, Ni, Ge

Cubic Crystal

• Cubic unit cell is 3D repeat unit • Rare (only Po has this structure)

• Coordination No. = 6 (# nearest neighbors)

SIMPLE CUBIC STRUCTURE

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a)31

atoms

unit cellatom

volume

unit cellvolume

ATOMIC PACKING FACTOR

contains 8 x 1/8 = 1 atom/unit cell

Adapted from Fig. 3.19, Callister 6e.

Lattice constant

close-packed directions

a

R=0.5a

BODY CENTERED CUBIC STRUCTURE

• Coordination No. = 8 (# nearest neighbors)

aR

• APF for a body-centered cubic structure = 3/8 = 0.68

Close-packed directions: length = 4R

= 3 a

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCC

APF = a3

4

3( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

Atoms are arranged at the corners and center of each cube face of the cell.◦ Atoms are assumed to touch along face diagonals

FACE CENTERED CUBIC STRUCTURE

• Coordination No. = 12 (# nearest neighbors)

APF = a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

• APF for a body-centered cubic structure = /(32) = 0.74

Close-packed directions: length = 4R

= 2 a

ATOMIC PACKING FACTOR: FCC

• ABCABC... Stacking Sequence

• FCC Unit CellA

BC

FCC STACKING SEQUENCE

A sites

B sites

C sitesB B

B

BB

B BC C

CA

A

• 2D Projection

HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)

Ideally, c/a = 1.633 for close packingHowever, in most metals, c/a ratio deviates from this value

• Coordination NO.= 12

• ABAB... Stacking Sequence

• APF = 0.74, for ideal c/a ratio of 1.633

• 3D Projection • 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Close Packed Structures

Close packed crystals

A plane

B plane

C plane

A plane

…ABCABCABC… packing[Face Centered Cubic (FCC)]

…ABABAB… packing[Hexagonal Close Packing (HCP)]

Examples of elements with Cubic Crystal Structure

Po

n = 1n = 2 n = 4

Fe Cu

BCC FCC/CCPSC

C (diamond)

n = 8 DC

a = b c = = = 90º

Simple Tetragonal Body Centred Tetragonal -BCT

Elements with Tetragonal structure → In, Sn

Tetragonal Crystal

Example of an element with Body Centred Tetragonal Crystal Structure

BCT

a b c = = = 90º

Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic

Elements with Orthorhombic structure → Br, Cl, Ga

Orthorhombic Crystal

Element with Orthorhombic Crystal Structure

a = b c = = 90º =120º

Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn

Hexagonal Crystal

Simple Hexagonal

Element with Hexagonal Crystal Structure

a = b = c = = 90º

Elements with Trigonal structure → As, B, Bi, Hg

Trigonal/Rhombohedral Crystal

Rhombohedral (simple)

Element with Simple Trigonal Crystal Structure

a b c = = 90º

Elements with Monoclinic structure → P, Pu, Po

Monoclinic Crystal

Simple Monoclinic End Centred (base centered)

Monoclinic

a b c

• Simple Triclinic

Triclinic Crystal

Crystal System Shape of UC Bravais Lattices

P I F C

1 Cubic Cube

2 Tetragonal Square Prism (general height)

3 Orthorhombic Rectangular Prism (general height)

4 Hexagonal 120 Rhombic Prism

5 Trigonal Parallopiped (Equilateral, Equiangular)

6 Monoclinic Parallogramic Prism

7 Triclinic Parallopiped (general)

14 Bravais Lattices divided into 7 Crystal Systems

P Primitive

I Body Centred

F Face Centred

C A/B/C- Centred

A Symmetry based concept ‘Translation’ based concept

+

Face Centred Cubic (FCC) Lattice Two Carbon atom Motif(0,0,0) & (¼, ¼, ¼)

=

Diamond Cubic Crystal