CRYSTAL STRUCTURE

Prof. H. K. Khaira

Professor

Deptt. of MSME

M.A.N.I.T., Bhopal

Topics Covered

• Crystal Structure

• Miller Indices

• Slip Systems

Types of Solids

• Solids can be divided into two groups based on the arrangement of the atoms. These are

–Crystalline

–Amorphous

Types of Solids

• In a Crystalline solid, atoms are arranged in an orderly manner. The atoms are having long range order.– Example : Iron, Copper and other metals,

NaCl etc.

• In an Amorphous solid, atoms are not present in an orderly manner. They are haphazardly arranged.– Example : Glass

• Non dense, random packing

• Dense, regular packing

Dense, regular-packed structures tend to have lower energy.

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

ENERGY AND PACKING

5

6

Some Definitions

Crystalline material is a material comprised of one or many crystals. In each crystal, atoms or ions show a long-range periodic arrangement.

Single crystal is a crystalline material that is made of only one crystal (there are no grain boundaries).

Grains are the crystals in a polycrystalline material.

Polycrystalline material is a material comprised of many crystals (as opposed to a single-crystal material that has only one crystal).

Grain boundaries are regions between grains of a polycrystalline material.

• 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" = NoncrystallineAdapted from Fig. 3.18(b), Callister 6e.

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

From Callister 6e resource CD.

Atomic PACKING

7

Space Lattice

A CRYSTAL STRUCTURE is a periodic arrangement of atoms in the crystal that can be described by a LATTICE

LATTICE

A Space LATTICE is an infinite, periodic array of mathematical points, in which each point has identical surroundings to all others.

8

Important Note: • Lattice points are a purely mathematical concept,

whereas atoms are physical objects. • So, don't mix up atoms with lattice points. • Lattice Points do not necessarily lie at the center of

atoms.

  In Figure (a) is the 3-D periodic arrangement of atoms, and Figure (b) is the corresponding space lattice.  In this case, atoms lie at the same point as the space lattice.    

Space Lattice

9

Crystalline Solids: Unit Cells

Fig. 3.1 Atomic configuration in Face-Centered-Cubic

Arrangement

R

R R

R

a

Unit Cell: It is the basic structural unit of a crystal structure. Its geometry and atomic positions define the crystal structure.

A unit cell is the smallest part of the unit cell, which when repeated in all three directions, reproduces the lattice.

10

Unit Cells and Unit Cell Vectors

a

b

c

Lattice parametersaxial lengths: a, b, cinteraxial angles:

unit vectors:

a

b

c

All unit cells may be described via these vectors and angles.

11

Possible Crystal Classes

12

Possible Crystal Classes

13

The 14 Bravais Lattices!

14

Unit Cells Types

Primitive Face-Centered

Body-Centered End-Centered

A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition.

• Primitive (P) unit cells contain only a single lattice point.• Internal (I) unit cell contains an atom in the body center.• Face (F) unit cell contains atoms in the all faces of the planes composing the cell.• Centered (C) unit cell contains atoms centered on the sides of the unit cell.

KNOW THIS!

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Orderly arrangement of atoms in Crystals

Orderly arrangement of atoms in Crystals

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Atom

Translation Vectors

a1

a3

a2a1, a2 ,a3

AMORPHOUS

CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT

CRYSTALLINE

There exists at least one crystalline state of lower energy (G) than

the amorphous state (glass) The crystal exhibits a sharp melting point “Crystal has a higher density”!!

Factors affecting the formation of the amorphous state

When the free energy difference between the crystal and the glass is small Tendency to crystallize would be small

Cooling rate → fast cooling promotes amorphization “fast” depends on the material in consideration Certain alloys have to be cooled at 106 K/s for amorphization Silicates amorphizes during air cooling

Types of Lattices

Types of Lattices

• There are 14 lattice types• Most common types:

– Cubic: Li, Na, Al, K, Cr, Fe, Ir, Pt, Au etc.

– Hexagonal Closed Pack (HCP):Mg, Co, Zn, Y, Zr, etc.

– Diamond: C, Si, Ge, Sn (only four)

Unit Cell

• Unit Cell is the smallest part of the lattice which when repeated in three directions produces the lattice.

• Unit Cell is the smallest part of the lattice which represents the lattice.

Metals

• Most of the metals are crystalline in solid form. They normally have the following crystal structures.

– Body Centered Cubic (BCC)– Face Centered Cubic (FCC)– Hexagonal Close Packed (HCP) or Close

Packed Hexagonal (CPH)

Unit Cell of BCC Lattice

•Fe (Up to 9100 C and from 14010C to Melting Point), W, Cr, V, \

A less close-packed structure is Body-Centered-Cubic (BCC). Besides FCC and HCP, BCC structures are widely adopted by metals.

• Unit cell showing the full cubic symmetry of the BCC arrangement. • BCC: a = b = c = a and angles = = 90°.• 2 atoms in the cubic cell: (0, 0, 0) and (1/2, 1/2, 1/2).

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Unit Cell of FCC lattice

• Al, Cu, Ni, Fe (9100 C-14010 C)

ABCABC.... repeat along <111> direction gives Cubic Close-Packing (CCP)• Face-Centered-Cubic (FCC) is the most efficient packing of hard-spheres of any lattice.• Unit cell showing the full symmetry of the FCC arrangement : a = b =c, angles all 90°• 4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0)

Self-Assessment: Write FCC crystal as BCT unit cell.

29

A

B

C

FCC Stacking

Highlighting the faces

Highlighting the stacking

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FCC Unit Cell

Highlighting the ABC planes and the cube.

Highlighting the hexagonal planes in each ABC layer.

31

CLOSE PACKING

A B C

+ +

FCC

=

Note: Atoms are coloured differently but are the same

FCC

Unit Cell of Hexagonal closed packed (HCP)

• Zn, Mg

ABABAB.... repeat along <111> direction gives Hexagonal Close-Packing (HCP)• Unit cell showing the full symmetry of the HCP arrangement is hexagonal • Hexagonal: a = b, c = 1.633a and angles = = 90°, = 120°• 2 atoms in the smallest cell: (0, 0, 0) and (2/3, 1/3, 1/2).

34

A

B

HCP Stacking

Highlighting the cellFigure 3.3

Highlighting the stacking

A

Layer A

Layer A

Layer B

Self-Assessment: How many atoms/cell?35

FCC HCP

Looking down (111) plane!

Looking down (0001) plane

Comparing the FCC and HCP Planes Stacking

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Important Properties of unit cell

• Number of atoms required to represent a unit cell

• Effective number of atoms per unit cell

• Coordination number

• Atomic packing factor

Number of atoms required to represent a unit cell

• It is the number of atoms required to show a unit cell

Number of atoms required to represent

a unit cell of BCC = 9

Number of atoms required to represent

a unit cell of FCC = 14

Number of atoms required to represent

a unit cell of HCP = 17

Effective Number of Atoms

• It is the total number of atoms belonging to a unit cell

Counting Number of Atoms Per Unit Cell

Counting Atoms in 3D CellsAtoms in different positions are shared by differing numbers of unit cells. • Corner atom shared by 8 cells => 1/8 atom per cell.

• Edge atom shared by 4 cells => 1/4 atom per cell.

• Face atom shared by 2 cells => 1/2 atom per cell.

• Body unique to 1 cell => 1 atom per cell.

Simple Cubic

8 atoms but shared by 8 unit cells. So, 8 atoms/8 cells = 1 atom/unit cell

How many atoms/cell forBody-Centered Cubic?

And, Face-Centered Cubic?

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Effective Number of atoms per unit cell for BCC

It will be 8*1/8 + 1*1 = 2

Effective Number of atoms per unit cell for FCC

It will be 8*1/8 + 6*1/2 = 4

William D. Callister, Jr., Materials Science and Engineering, An Introduction, John Wiley & Sons, Inc. (2003)

Effective Number of atoms per unit cell for Hexagonal closed packed (HCP)

•It will be 6*1/6 + 2*1/2 + 3*1 = 6

William D. Callister, Jr., Materials Science and Engineering, An Introduction, John Wiley & Sons, Inc. (2003)

Coordination Number

• It is the number of nearest equidistant neighbours of an atom in the lattice

Consider a simple cubic structure.

Six yellow atoms are are touching the blue atom.

Hence, the blue atom is having six nearest neighbors.

Therefore, for a Simple cubic lattice: coordination number, CN = 6

Coordination Number of a Given Atom

48

Coordination Number

Atomic Packing Factor

• Atomic Packing Factor is the fraction of volume occupied by atoms in a unit cell.

Atomic Packing Factor (APF)

Face-Centered-CubicArrangement

APF = vol. of atomic spheres in unit cell total unit cell vol.

Depends on: • Crystal structure.• How “close” packed the atoms are.• In simple close-packed structures with hard sphere atoms, independent of atomic radius

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

52

a

a

2R

R22

R22

3

3

44 RVatoms 333

_ 216)22( RRaV cellunit

Basic Geometry for FCC

2a 4R

Geometry:

4 atoms/unit cell

Coordination number = 12

Ra 22

Geometry along close-packed direction give relation between a and R.

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Atomic Packing Factor for FCC

Face-Centered-CubicArrangement

APF = vol. of atomic spheres in unit cell total unit cell vol.

How many spheres (i.e. atoms)? What is volume/atom?

What is cube volume/cell?

How is “R” related to “a”?

4/cell

4R3/3

a3

= 0.74APF =

a3

4

3( 2a/4)34

atoms

unit cell atomvolume

unit cell

volumeIndependent of R!

2a 4R

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

54

Atomic Packing Factor for BCC

Geometry:

2 atoms/unit cell

a

a2

4R a 3

Again, geometry along close-packed direction give relation between a and R.

APF VatomsVcell

243

a 34

3

a3

38

0.68

4R a 3

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Important Properties of unit cell

SC BCC FCC HCP

Relation between atomic radius (r) and lattice parameter (a)

a = 2r

Effective Number of atoms per unit cell

1 2 4 6

Coordination Number (No. of nearest equidistant neighbours)

6 8 12 12

Packing Factor 0.52 0.68 0.74 0.74

ra 43 ra 42 ra 2

Miller Indices

• Miller Indices are used to represent the directions and the planes in a crystal

• Miller Indices is a group of smallest integers which represent a direction or a plane

z y z

c [ 1 1 1 ] [01 2 ] x [111]

[021] y [100]

a [110] b x

Miller Indices of a Direction

• Select any point on the direction line other than the origin

• Find out the coordinates of the point in terms of the unit vectors along different axes

• Specify negative coordinate with a bar on top

• Divide them by the unit vector along the respective axis

• Convert the result into the smallest integers by suitable multiplication or division and express as <uvw>

Miller Indices of directions

z y z

c [ 1 1 1 ] [01 2 ] x [111]

[021] y [100]

a [110] b x

Miller Indices of planes

- Find the intercepts of the plane with the three axes: (pa, qb, rc)

- Take the reciprocals of the numbers (p, q, r)- Reduce to three smallest integers (h, k, l) by suitable

multiplication or division.

- Miller indices of the plane: (hkl) - Negative indices are indicated by a bar on top - Same indices for parallel planes

- A family of crystallographically equivalent planes (not necessarily parallel) is denoted by {hkl}

Index System for Crystal Planes (Miller Indices)

Example: determine Miller indices of a plane z

• Intercepts: (a/2, 2b/3, ∞)

• p= ½, q= 2/3, r= ∞ c • Reciprocals: h = 2, k = 3/2, l = 0

y • Miller indices: (430) a

b x

Crystal Planes

Important planes in cubic crystals

• {100} family: (100) (010) (001)

• {110} family: (110) (101) (011) ( 1 1 0) ( 10 1 ) (01 1 )

• {111} family: (111) ( 1 1 1 ) ( 1 11 ) ( 11 1 )

{100} planes {110} planes

Hexagonal indices

• A special case for hexagonal crystals• Miller-Bravais indices: (hkil)

h= Reciprocal of the intercept with a1-axisk= Reciprocal of the intercept with a2-axisi = -(h + k)

l= Reciprocal of the intercept with c-axis

c c

( 1 1 00) ( 11 2 0)

a2 a2

(0001 ) a1 a1

FCC

Packing Densities in Crystals: Lines and Planes

Linear and Planar Packing Density which are independent of atomic radius!

Also, Theoretical Density

Concepts

66

Linear Density in FCC

LD =Number of atoms centered on a direction vector

Length of the direction vector

Example: Calculate the linear density of an FCC crystal along [1 1 0].

ANSWERa. 2 atoms along [1 1 0]

in the cube.b. Length = 4R

ASKa. How many spheres along blue line? b. What is length of blue line?

LD110 2atoms

4R

12R

XZ = 1i + 1j + 0k = [110]

Self-assessment: Show that LD100 = √2/4R. 67

Planar Density in FCC

Ra 22

R4

PD =Number of atoms centered on a given plane

Area of the plane

Example: Calculate the PD on (1 1 0) plane of an FCC crystal.

• Count atoms within the plane: 2 atoms• Find Area of Plane: 8√2 R2

PD 2

8 2R2

1

4 2R2Hence,

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Planar Packing Density in FCC

Ra 22

R4

PPD =Area of atoms centered on a given plane

Area of the plane

Example: Calculate the PPD on (1 1 0) plane of an FCC crystal.

• Find area filled by atoms in plane: 2R2

• Find Area of Plane: 8√2 R2

PPD 2R2

8 2R2

4 20.555Hence,

Always independent of R!

Self-assessment: Show that PPD100 = /4 = 0.785. 69

Density of solids atomic weight from periodic table (g/mol) • Mass per atom: m =

A 23 6. 02 × 10 (atoms/mol )

4 3 • Volume per atom: V = πR A A 3 • Number of atoms per unit cell (N)

2 for bcc, 4 for fcc, 6 for hcp

Volume occupied by atoms in a unit cell = NVA

NV • Volume of unit cell (VC)

A Packing Factor = V

- Depends on crystal structure C

Nm • Mass density: A ρ = V C

Example: Copper

n AVcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell

(cm3/unit cell)Avogadro's number (6.023 x 1023 atoms/mol)

• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 cm)-7

Compare to actual: Cu = 8.94 g/cm3

Result: theoretical Cu = 8.89 g/cm3

Theoretical Density

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3

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Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen

Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H

At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008

Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------

Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------

Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------

Adapted fromTable, "Charac-teristics ofSelectedElements",inside frontcover,Callister 6e.

Characteristics of Selected Elements at 20 C

72

(g

/cm

3)

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

1

2

20

30Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass,

Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix). 10

3 4 5

0.3 0.4 0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

Graphite Silicon

Glass -soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Metals have... • close-packing (metallic bonds)

• large atomic mass

Ceramics have... • less dense packing (covalent bonds)

• often lighter elements

Polymers have... • poor packing (often amorphous)

• lighter elements (C,H,O)

Composites have... • intermediate values

Data from Table B1, Callister 6e.

DENSITIES OF MATERIAL CLASSESmetals > ceramics > polymers

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Slip Systems

• Slip system is a combination of a slip plane and a slip direction along which the slip occurs at minimum stress

• Slip plane is the weakest plane in the crystal and the slip direction is the weakest direction in that plane.

• The most densely packed plane is the weakest plane and the most densely direction is the weakest direction in the crystal.

Most Closely Packed Plane and directions

CLOSE PACKING

A B C

+ +

FCC

=

Note: Atoms are coloured differently but are the same

FCC

An FCC unit cell and its slip system

Slip occurs along <110> - type directions {111} - type planes (4 planes) have

(A-B,A-C,D-E) within the {111} planes. all atoms closely packed.

FCC has 12 slip systems - 4 {111} planes and each has 3 <110> directions

A B

+

HCP

=

A

+

Note: Atoms are coloured differently but are the same

A plane

B plane

A plane

HCPShown displaced for clarity

* A crystal will be anisotropic (exhibit directional properties) if the slip systems are less than 5. It will be isotropic (have same properties in all the directions) if the slip systems are more than 5.