STRUCTURE OF CRYSTALLINE SOLIDS

Prepared by: Drs. Arcan Dericioğlu & Mert Efe & Caner Şimşir @ METU, based on Course Material of Callister, 6th Ed. 2003.

CHAPTER 3:

STRUCTURE OFCRYSTALLINE SOLIDS


What is a Crystal?


What is a Polycrystal?

Most efficient packing


What is a Crystal?


Analogy to Polycrystalline Structure

Intergranular Fracture

Transgranular Fracture

GRAIN

Surface of the Grain:GRAIN

BOUNDARY


• atoms pack in periodic, 3D arrays• long-range order• typical of: -metals

-many ceramics-some polymers

• atoms have no periodic packing• occurs for: -complex structures

-rapid cooling

Si Oxygen

crystalline SiO2

noncrystalline SiO2

"Amorphous" = Noncrystalline

Materials and packing:Types of solids


Si (crystalline)

SiO2 (amorphous)


• Non dense, random packing

• Dense, regular packing

Dense, regular-packed structures tend to havelower energy.

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy and packing


How about METU Forest?


METU Forest


Online modules:

1) http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.php

2) http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php


Atomic hard sphere model is used to describe the arrangement of atoms.

Lattice: A 3D array of points in space coinciding with atom positions (or sphere centers).

Hard sphere model and lattice:


Unit cell:(Lattice cell) Smallest structural unit that decribes the whole crystal structure.

Unit cell:

Reduced sphere model:


As an easy and simple way of representation we use cubes or similar geometrical tools to represent a unit cell.


Different variations for unit cells are possible. Parameters describing the unit cells are 1. Angles 2. Translation factors (Dimensions)

IN THIS COURSE WE WILL BE LIMITED TO SIMPLE SYTEMS where

(a=b=c)(===90o)

CUBIC SYSTEMS (most of the time)

Parameters defining a unit cell:


within the contentof this coursecourse

Total of SEVEN CRYSTAL STRUCTURES

Prepared by: Drs. Arcan Dericioğlu & Mert Efe & Caner Şimşir @ METU, based on Course Material of Callister, 6th Ed. 2003

METALLIC CRYSTALS


• tend to be densely packed.

• have several reasons for dense packing:-Typically, only one element is present, so all atomicradii are the same.

-Metallic bonding is not directional.-Nearest neighbor distances tend to be small inorder to lower bond energy.

• have the simplest crystal structures.

Most common crystal structures in metals: BCC, FCC, HCP

METALLIC CRYSTALS


Rare for metals due to poor packing (only Po has this structure)Close-packed directions are cube edges.

• Coordination # = 6(# nearest neighbors)

Simple Cubic (SC)

Examples: Po


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

close-packed directions

a

R=0.5a

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

ATOMIC PACKING FACTOR


Body-Centered Cubic (BCC)

Examples: Fe(), Cr, Mo


Face-Centered Cubic (FCC)

Examples: Cu, Al, Ag, Au


Hexagonal Close-Packed (HCP)

Examples: Mg, Ti, Zn,


HCPFCC


METU Forest


• ABCABC... Stacking Sequence• 2D Projection

A sites

B sites

C sitesB B

B

BB

B BC C

CA

A

• FCC Unit CellA

BC

FCC Stacking SequenceClose-Packed Crystal Structures

• APF = 0.74

• Coordination # = 12


• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Hexagonal Close-Packed Structure (HCP)


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

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

Compare to actual: Cu = 8.94 g/cm3Result: theoretical Cu = 8.89 g/cm3

DENSITY (Computation


(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

Why?Metals have...• close-packing

(metallic bonding)• large atomic mass

Ceramics have...• less dense packing

(covalent bonding)• often lighter elements

Polymers have...• poor packing

(often amorphous)• lighter elements (C,H,O)

Composites have...• intermediate values

DENSITIES OF MATERIAL CLASSES


PolymorphismSome metals and nonmetals having

more than one Crystal Structure.

When found in elemental solidsthe condition is called Allotropy.

Example:C Graphite is stable at ambient temperature.

Diamond at extremely high pressures.

Fe BCC at Room Temperature.FCC above 912 C.


Crystallographic Direction

How to determine?

A vector between to points in a Crystal Structure.

Pass a vector of convenient length through origin.

Determine vector projection on each axis in terms of a, b, c.

Reduce the numbers by a common factor.

Find [u v w].


Example:

For some crystal structures several nonparallel directions with different indices are actually equivalent meaning that the spacing of atoms along each direction is the same.

Equivalent directions are grouped in a family <u v w>.


Crystallographic Planes

Except for Hexagonal Crystal system 3-axis coordinate system is used.

Crystallographic planes are specified by 3 Miller indices (h k l)


How to determine Miller indices (h k l) ? 1. If the plane passes through origin draw a parallel plane or shift

the coordinate system appropriately.

2. Find the intercept of the plane with each axis..

3. Take the reciprocal of the intercepts (if the plane is parallel to an axis, the intercept is and its reciprocal is 0).

4. If necessary convert the numbers to the smallest integer.

5. Determine (h k l).


Parallel planes are equivalent and have identical indices.

Reversing the directions of all indices specifies another plane parallel to, on the opposite side and equidistant from the origin.

In cubic crystals planes and directions having the same indices are perpendicular to each other.

Crystallographically equivalent planes, having the same atomic packing, is grouped in a family of planes {h k l}.


Atomic Arrangements Atomic spacing along a direction or atomic arrangement of

a crystallographic plane depends on the crystal structure.

[1 1 0] direction in FCC unit cell

(1 1 0) plane in FCC unit cell

(1 1 0) plane in BCC unit cell


Linear and Planar Atomic Densities Linear density (LD) is the number of atom per unit length whose centers lie on a specific direction.

Planar density (PD) is the number of atoms per unit area whose centers lie on a specific plane.

(1 1 0) plane in FCC unit cell[1 1 0] direction in FCC unit cell

NAtoms= 2

LXZ= 4R

LD= 2/4R= 1/(2R)

NAtoms= 2

AACDF= 82 R2

PD= 2/(82 R2)=1/(82 R2

STRUCTURE OF CRYSTALLINE SOLIDS

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