crystal structure - uprmacademic.uprm.edu/pcaceres/courses/mateng3045/eme2-1.pdf · radius, nm...

Crystal StructureTypes of Solids

� Amorphous: Solids with considerable disorder in their structures

(e.g., glass). Amorphous: lacks a systematic atomic arrangement.

� Crystalline: Solids with rigid a highly regular arrangement of

their atoms. (That is, its atoms or ions, self-organized in a 3D

periodic array). These can be monocrystals and polycrystals.

Amorphous crystalline polycrystalline

To discuss crystalline structures it is useful to consider atoms as being

hard spheres with well-defined radii. In this hard-sphere model, the

shortest distance between two like atoms is one diameter.

Lattice: A 3-dimensional system of points that designate the positions

of the components (atoms, ions, or molecules) that make up the

substance.

Unit Cell: The smallest repeating unit of the lattice.

The lattice is generated by repeating the unit cell in all three

dimensions

Crystal SystemsCrystallographers have shown that only

seven different types of unit cells are

necessary to create all point lattice

Cubic a= b = c ; α = β = γ = 90

Tetragonal a= b ≠ c ; α = β = γ = 90

Rhombohedral a= b = c ; α = β = γ ≠ 90

Hexagonal a= b ≠ c ; α = β = 90, γ =120

Orthorhombica≠ b ≠ c ; α = β = γ = 90

Monoclinic a≠ b ≠ c ; α = γ = 90 ≠ β

Triclinic a≠ b ≠ c ; α ≠ γ ≠ β ≠ 90

Bravais Lattices

Many of the seven crystal systems have variations of the basic unit

cell. August Bravais (1811-1863) showed that 14 standards unit

cells could describe all possible lattice networks.

Principal Metallic Structures

Most elemental metals (about

90%) crystallize upon

solidification into three densely

packed crystal structures: body-

centered cubic (BCC), face-

centered cubic (FCC) and

hexagonal close-packed (HCP)

Structures of Metallic Elements

Ru

H

Li

Na

K

Rb

Cs

Fr

Be

Mg

Ca

Sr

Ba

Ra

Sc

Y

La

Ac

Ti

Zr

H f

V

Nb

Ta

Cr

Mo

W

Mn

Tc

Re

Fe

Os

Co

Ir

Rh

Ni

Pd

Pt

Cu

Ag

Au

Zn

Cd

Hg

B

Al

Ga

In

Tl

C

Si

Ge

Sn

Pb

N

P

As

Sb

Bi

O

S

Se

Te

Po

F

Cl

Br

I

At

Ne

Ar

Kr

Xe

Rn

He

Primitive Cubic

Body Centered Cubic

Cubic close packing(Face centered cubic)

Hexagonal close packing

0.1470.35840.2950Titanium

0.1250.40690.2507Cobalt

0.1600.52090.3209Magnesium

0.1330.56180.2665Zinc

HCP

0.1440.409Silver

0.1250.352Nickel

0.1440.408Gold

0.1280.361Copper

FCC

BCC

Structure

0.1370.316Tungsten

0.1860.429Sodium

0.2310.533Potassium

0.1360.315Molybdenum

0.1240.287Iron

0.1250.289Chromium

0.1430.405Aluminum

c, nma, nm

Atomic

Radius, nm

Lattice ConstantMetal

• Rare due to poor packing (only Po has this structure)

• Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)

SIMPLE CUBIC STRUCTURE (SC)

• Number of atoms per unit cell= 1 atom

6

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

3π (0.5a)31

atoms

unit cellatom

volume

unit cell

volumeclose-packed directions

a

R=0.5a

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

Atomic Packing Factor (APF)

• Coordination # = 8

• Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of

viewing.

Body Centered Cubic (BCC)

aR

Close-packed directions: length = 4R

= 3 a

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

APF =

a3

4

3π ( 3a/4)32

atoms

unit cell atom

volume

unit cell

volume• APF for a BCC = 0.68

• Coordination # = 12

• Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for

ease of viewing.

Face Centered Cubic (FCC)

a

APF =

a3

4

3π ( 2a/4)34

atoms

unit cell atom

volume

unit cell

volume

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

Close-packed directions: length = 4R

= 2 a

• APF for a FCC = 0.74

Hexagonal Close-Packed (HCP)The APF and coordination number of the HCP structure is the same

as the FCC structure, that is, 0.74 and 12 respectively.

An isolated HCP unit cell has a total of 6 atoms per unit cell.

12 atoms shared by six cells = 2 atoms per cell2 atoms shared by two cells = 1

atom per cell

3 atoms

Close-Packed StructuresBoth the HCP and FCC crystal structures are close-packed structure.

Consider the atoms as spheres:

�Place one layer of atoms (2 Dimensional solid). Layer A

�Place the next layer on top of the first. Layer B. Note that there are

two possible positions for a proper stacking of layer B.

A B A : hexagonal close packed A B C : cubic close packed

The third layer (Layer C) can be placed in also teo different

positions to obtain a proper stack.

(1)exactly above of atoms of Layer A (HCP) or

(2)displaced

A B C : cubic close pack

A

B C

A

120°

90°A

A

B

A B A : hexagonal close pack

Interstitial sitesLocations between the ‘‘normal’’ atoms or ions in a crystal into

which another - usually different - atom or ion is placed.

o Cubic site - An interstitial position that has a coordination number

of eight. An atom or ion in the cubic site touches eight other atoms

or ions.

o Octahedral site - An interstitial position that has a coordination number of six. An atom or ion in the octahedral site touches six

other atoms or ions.

o Tetrahedral site - An interstitial position that has a coordination number of four. An atom or ion in the tetrahedral site touches four

other atoms or ions.

Crystals having filled Interstitial Sites

FCC Lattice has:

3 [=12(¼)] Oh sites at edge centers

+ 1 Oh site at body center

Octahedral, Oh, Sites

FCC Lattice has:

8 Th sites at ¼, ¼, ¼ positions

Tetrahedral, Th, Sites

Interstitial sites are important because we can derive more structures from these basic

FCC, BCC, HCP structures by partially or completely different sets of these sites

Density CalculationsSince the entire crystal can be generated by the repetition of the

unit cell, the density of a crystalline material, ρ = the density of the

unit cell = (atoms in the unit cell, n ) × (mass of an atom, M) / (the

volume of the cell, Vc)

Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)

Mass of an atom, M = Atomic weight, A, in amu (or g/mol) is

given in the periodic table. To translate mass from amu to grams

we have to divide the atomic weight in amu by the Avogadro

number NA = 6.023 × 1023 atoms/molThe volume of the cell, Vc = a3 (FCC and BCC)

a = 2R√2 (FCC); a = 4R/√3 (BCC)where R is the atomic radius.

Density Calculation

ACNV

nA=ρ

n: number of atoms/unit cell

A: atomic weight

VC: volume of the unit cell

NA: Avogadro’s number

(6.023x1023 atoms/mole)

Example Calculate the density of copper.

RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole

n = 4 atoms/cell, 333 216)22( RRaVC ===

3

2338/89.8

]10023.6)1028.1(216[

)5.63)(4(cmg=

×××=ρ

8.94 g/cm3 in the literature

Example

Rhodium has an atomic radius of 0.1345nm (1.345A) and a density of 12.41g.cm-3.

Determine whether it has a BCC or FCC crystal structure. Rh (A = 102.91g/mol)

Solution

ACNV

nA=ρ

n: number of atoms/unit cell A: atomic weight

VC: volume of the unit cell NA: Avogadro’s number

(6.023x1023 atoms/mole)

structure FCC a has Rhodium

01376.04

)1345.0(627.22

627.22)2

4( and 4 n then FCC is rhodium If

0149.02

)1345.0(316.12

316.12)3

4( and 2 n then BCC is rhodium If

01376.0103768.1.10023.6.41.12

.91.102

333

333

333

333

3323

1233

13

nmnmx

n

a

rra

nmnmx

n

a

rra

nmcmxmoleatomsxcmg

molg

N

A

n

a

n

V

A

c

==

===

==

===

===== −

−−

−

ρ

Linear And Planar Atomic Densities

Linear atomic density:

= 2R/Ll

Planar atomic density:

Ll

Crystallographic direction

= 2π R2/(Area A’D’E’B’)

3

4RaLl == = 0.866

A’ E’