Crystal structure

Learning ObjectiveStudents should be able to:• Differentiate between crystalline, noncrystalline,

single crystal and polycrsytal.

• Derive the relationship between unit cell edge length and atomic radius for face centered and body centered cubic

• Compute density and atomic packing factors for crystal

• Specify the miller indices for crystallography plane and direction

• Relate the crystal structure with material properties

Fundamental Concept

a crystal structure is a unique arrangement of atoms in a crystal.

Composed of a unit cell

• Formation crystal structure depends– Chemistry of fluid– Condition under which is being solidified– Ambient pressure

• Crystallization-Process forming crystalline structure.

Crystal Structure

Crystal structure

Crystalline MaterialNoncrsytalline material

(Amorphous)

Single Crystal polycrystal

Crystalline Material

• Crystalline material- atoms, molecules or ion packed in a regularly ordered, repeating pattern, extending in 3 spatial dimension.

• Single crystal -the periodicity of the pattern extends throughout a certain piece of material.

• Polycrystalline material- comprised of many single crystal or grain

• atoms pack in periodic, 3D arrays• typical of:

Crystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

Si Oxygen

crystalline SiO2

noncrystalline SiO2"Amorphous" = Noncrystalline

Adapted from Fig. 3.18(a), Callister 6e.

MATERIALS AND PACKING

Polycrystalline material

grains

Crystalline Amorphous

19

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 m

SINGLE VS POLYCRYSTALS

• Unit cell - smallest structural unit or building block that can describe the crystal structure. Repetition of the unit cell generates the entire crystal.

• Primitive unit cell- smallest possible unit cell one can construct.

• Lattice parameters- spacing between unit cells in various direction.

Unit cell

Unit Cells?

Concept test

which one is unit cell

Crystal system• Point group of lattice

• 7 unique crystal system– Cubic– Hexagonal– Tetragonal– Rhombohedral– Orthorhombic– Monoclinic– Triclinic

• By adding additional lattice point to 7 basic shapes – form 14 Bravais lattice

Crystal system

Metallic crystal structure

• Most found crystal structure in common metal– Body centered cubic (BCC)– Face centered Cubic (FCC)– Hexagonal close-packed (HCP)

• Simple cubic (SC)– one lattice point at each of the eight  corners

a= lattice parameter

a=2r

n= no. of atom per unit cell

n= 1

coordination no : no of adjacent atom that touch atom at lattice point

= 6

• Body Centered Cubic (BCC)

3

4ra

n = 2

coordination no: 8

• Face Centered Cubic (FCC)

n= 4

coordination no = 12

22ra

• Hexagonal Close Packed Structure

c/a =1.633

Atomic Packing Factor (APF)

• Efficiency of atomic arrangement in a unit cell.

Exercise: calculate APF for SC,BCC and FCC crystal structure

Vc

nVs

APF

cellunit of vol.

sphere x vol.ofatom of no.

Theoretical density

ex; If a hypothetical metal crystalline with BCC crystal structure. Calculate its density. Given A= 26.98 amu/atom, atomic diameter 0.286nm

AcNV

nA

so'no.avogadr x cell)/unit cell(cmunit of vol.

(g/mol) weight atomic x atom of no. 3

Polymorphism allotropic transformation• the ability of a solid material to exist

in more than one form or crystal structure.

• Example: Carbon and iron

Allotropy : iron

Allotropy: Carbon

Graphite Diamond

Crystallography Direction and Plane

(by using miller indices)

Crystallography direction

• Line between two point or a vector

Axis x y z

Head (H)

0 1 1

Tail (T) 0 0 1

Projection (H-T)

0 1 0

Enclosed

[ O 1 O]

Enclosed in square bracket

Lets do another example…

Axis x y z

Head (H)

1 ½ 0

Tail (T) 0 1 1

Projection (H-T)

1 -½ -1

Reduction (x 2)

2 -1 -2

Enclosed

[ 2 ]

1 2

Try it by yourself!!

(0,0,0)

Axis x y z

Head (H)

Tail (T)

Projection (H-T)

Reduction (x )

Enclosed [ ]

You have to come to the class to get more examples….

Crystallography PlaneThe procedure:

1. If the plane passes through the selected origin• Another parallel plane must be constructed OR• Establish new origin

2. Determine either the plane is intersects or parallels with three axes.

3. Get the reciprocals.

4. If necessary, reduce the number.

5. Enclosed with parentheses , ( )

Remember the integer in enclosure is not separated by commas

Example:

Axis x y z

Intercepts

1

reciprocals

0 1 0

Reduction(if necessary)

- - -

Enclosed ( 0 1 0 )

Reciprocal: 1/(value)

Example 2:

Axis x y z

Intercepts

1 1

reciprocals

1 1 0

Reduction(if necessary)

- - -

Enclosed ( 1 1 0 )

Try it by yourself!!

(0,0,0)

You have to come to the class to get more examples….

Axis x y z

Intercepts

reciprocals

Reduction(if necessary)

Enclosed ( )