STRUCTURE OF SOLIDS Types of solids based on structure Types of solids based on bonding.

STRUCTURE OF SOLIDSSTRUCTURE OF SOLIDS

Types of solids based on structure

Types of solids based on bonding

UNIVERSE

PARTICLES

ENERGYSPACE

FIELDS

STRONG WEAKELECTROMAGNETICGRAVITY

METALSEMI-METAL

SEMI-CONDUCTORINSULATOR

nD + t

HYPERBOLICEUCLIDEANSPHERICAL

GAS

BAND STRUCTURE

AMORPHOUS

ATOMIC NON-ATOMIC

STATE / VISCOSITY

SOLID LIQUIDLIQUID

CRYSTALS

QUASICRYSTALS CRYSTALSRATIONAL APPROXIMANTS

STRUCTURE

NANO-QUASICRYSTALS NANOCRYSTALS

SIZE

AMORPHOUS

CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT

QUASICRYSTALS CRYSTALS

Ordered+

Periodic

Ordered+

Periodic

Ordered+

Periodic

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”!!

AMORPHOUS

CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT

QUASICRYSTALS CRYSTALS

ADDITIONAL POSSIBLE STRUCTURES

Modulated structuresIncommensurately

Modulated structures

Liquid crystals

THE ENTITY IN QUESTION

GEOMETRICAL PHYSICAL

E.g. Atoms, Cluster of AtomsIons, etc.

E.g. Electronic Spin, Nuclear spin

ORDER

ORIENTATIONAL POSITIONAL

ORDER

TRUE PROBABILISTIC

Order-disorder of: POSITION, ORIENTATION, ELECTRONIC & NUCLEAR SPIN

ORIENTATIONAL

POSITIONAL

PROBABILISTIC

OCCUPATION

Perfect

Average

Perfect

Average

Positionally ordered

Probabilistically ordered

A B

Probability of occupation:A 50%B 50%

Order

Spatial Temporal

Range of Spatial Order

Short Range (SRO) Long Range Order (LRO)

Class/example(s)

Short Range Long Range

Ordered Disordered Ordered Disordered

Crystals*/Quasicrystals

Glasses# Crystallized

virus$

Gases Notes:* In practical terms crystals are disordered both in the short range (thermal vibrations) and

in the long range (as they are finite)# ~ Amorphous solids$ Other examples could be: colloidal crystals, artificially created macroscopic crystals Liquids have short range spatial order but NO temporal order

Crystal Physics, G.S. Zhdanov, Oliver & Boyd, Ediburgh, 1965

When primary bonds are 1D or 2D and secondary bonds aid in the formation of the crystal

The crystal structure is very complex

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

COVALENT

CLASSIFICATION OF SOLIDS BASED ON BONDING

IONIC METALLIC

Molecular

CRYSTALS

Non-molecular

COVALENT

IONIC

METALLIC

Molecule held together by primarycovalent bonds

Intermolecular bonding is Van der walls

Bond Type kJ/mol

Covalent Bond 250

Electrostatic 5

van der Waals 5

Hydrogen bond 20

Approximate Strengths of Interactions between atoms

METALLIC

Positive ions in a free electron cloud Metallic bonds are non-directional Each atoms tends to surround itself with as many neighbours as possible! Usually high temperature (wrt to MP) → BCC (Open structure) The partial covalent character of transition metals is a possible reason

for many of them having the BCC structure at low temperatures

FCC → Al, Fe (910 - 1410ºC), Cu, Ag, Au, Ni, Pd, Pt BCC → Li, Na, K , Ti, Zr, Hf, Nb, Ta, Cr, Mo, W, Fe (below 910ºC), HCP → Be, Mg, Ti, Zr, Hf, Zn, Cd Others → La, Sm Po, α-Mn, Pu

CLOSE PACKING

A B C

+ +

FCC

=

Note: Atoms are coloured differently but are the same

FCC

A B

+

HCP

=

A

+


HCPShown displaced for clarity

Unit cell of HCP (Rhombic prism)

Note: diagrams not to scale

Atoms: (0,0,0), (⅔, ⅓,½)

aCV4

6

aCV12

6

aVCCFh3

2

632.13

22

2

a

h

a

cIDEAL c/a

h

PACKING FRACTION / Efficiency

Cell of Volume

atomsby occupied VolumeFraction Packing

SC* BCC* CCP DC HCP

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

a = 2r a = 2r

Atoms / cell 1 2 4 8 2

Lattice points / cell 1 2 4 4 1

No. of nearest neighbours 6 8 12 4 12

Packing fraction

= 0.52 = 0.68 = 0.74 = 0.34 = 0.74

ra 43 ra 42 ra 24

3

6

8

36

216

3

3

24rc

6

2

* Crystal formed by monoatomic decoration of the lattice

ATOMIC DENSITY (atoms/unit area)

SC FCC BCC

(100) 1/a2 = 1/a2 2/a2 = 2/a2 1/a2 = 1/a2

(110) 1/(a22) = 0.707/a2 2/a2 = 1.414/a2 2/a2 = 1.414/a2

(111) 1/(3a2) = 0.577/a2 4/(3a2) = 2.309/a2 1/(3a2) = 0.577/a2

Order (111) < (110) < (100) (110) < (100) < (111) (111) < (100) < (110)

FCC

BCC

(100) (110) (111)

SC

a

a2

a2a2

a2

a

aa

a2 a2

ATOMIC DENSITY (area covered by atoms/unit area)

SC FCC BCC

Atoms / Area

Area / Area Atoms / Area

Area / Area Atoms / Area

Area / Area

(100) 1/a2 /4 = 0.785 2/a2 /4 = 0.785 1/a2 3/16 = 0.589

(110) 2/(2a2) 0.707(/4) = 0.555

2/a2 2/8 = 0.555 2/a2 32/16 = 0.833

(111) 1/(3a2) 0.577(/4) = 0.453

4/(3a2) /(23) =0.9068 1/(3a2) 3/16 = 0.34

VOIDS

TETRAHEDRAL OCTAHEDRAL

FCC


cellntetrahedro VV24

1

celloctahedron VV6

1

¼ way along body diagonal{¼, ¼, ¼}, {¾, ¾, ¾}

+ face centering translations

At body centre{½, ½, ½}

+ face centering translations

FCC- OCTAHEDRAL

{½, ½, ½} + {½, ½, 0} = {1, 1, ½} {0, 0, ½}

Face centering translation


Equivalent site for an octahedral void

Site for octahedral void

FCC voids Position Voids / cell Voids / atom

Tetrahedral¼ way from each vertex of the cube

along body diagonal <111>

((¼, ¼, ¼))8 2

Octahedral• Body centre: 1 (½, ½, ½)

• Edge centre: (12/4 = 3) (½, 0, 0)4 1

Size of the largest atom which can fit into the tetrahedral void of FCC

CV = r + x Radius of the new atom

exre

4

6

225.0~12

3 2

r

xre

Size of the largest atom which can fit into the Octahedral void of FCC

2r + 2x = a ra 42

414.0~12 r

x

VOIDS

TETRAHEDRAL OCTAHEDRAL

HCP

These voids are identical to the ones found in FCC


Coordinates: (⅓ ⅔,¼), (⅓,⅔,¾)),,(),,,(),,0,0(),,0,0(: 87

31

32

81

31

32

85

83sCoordinate

Octahedral voids occur in 1 orientation, tetrahedral voids occur in 2 orientations

The other orientation of the tetrahedral void



Octahedral voids

Tetrahedral void

HCP voids PositionVoids /

cellVoids / atom

Tetrahedral(0,0,3/8), (0,0,5/8), (⅔, ⅓,1/8), (⅔,⅓,7/8)

4 2

Octahedral • (⅓ ⅔,¼), (⅓,⅔,¾) 2 1

Voids/atom: FCC HCP

as we can go from FCC to HCP (and vice-

versa) by a twist of 60 around a central atom of

two void layers (with axis to figure) Central atom

Check below

Atoms in HCP crystal: (0,0,0), (⅔, ⅓,½)

A

A

B

VOIDS

Distorted TETRAHEDRAL Distorted OCTAHEDRAL**

BCC

a

a3/2

a a3/2

rvoid / ratom = 0.29 rVoid / ratom = 0.155

Note: Atoms are coloured differently but are the same ** Actually an atom of correct size touches only the top and bottom atoms

Coordinates of the void:{½, 0, ¼} (four on each face) Coordinates of the void:

{½, ½, 0} (+ BCC translations: {0, 0, ½})Illustration on one face only

BCC voids PositionVoids /

cellVoids / atom

DistortedTetrahedral

• Four on each face: [(4/2) 6 = 12] (0, ½, ¼) 12 6

Distorted Octahedral

• Face centre: (6/2 = 3) (½, ½, 0)

• Edge centre: (12/4 = 3) (½, 0, 0)6 3

{0, 0, ½})

From the right angled triange OCM: 416

22 aaOC

5

4a r x

For a BCC structure: 3 4a r (3

4ra )

xrr

3

4

4

5 29.01

3

5

r

x

a

a3/2

BCC: Distorted Tetrahedral Void

2

axrOB

32

4rxr raBCC 43:

1547.013

32

r

x

Distorted Octahedral Void

a3/2

a

aa

OB 5.02

aa

OA 707.2

2

As the distance OA > OB the atom in the void touches only the atom at B (body centre). void is actually a ‘linear’ void

This implies:

A 292.1FeFCCr

A534.0)( octxFeFCC

A 77.0Cr

C

N

Void (Oct)

FeFCC

O

A 258.1FeBCCr

A364.0).( tetdxFeBCC

A195.0).( octdxFeBCC

FCC

BCC

FeBCC

Relative sizes of voids w.r.t to atoms

( . )0.155

FeBCC

FeBCC

x d oct

r

( . )0.29

FeBCC

FeBCC

x d tet

r

A 258.1FeBCCr

2

2A

aOA r x

2 6

3A

rr x

raBCC 43:

2 61 0.6329

3Ax

r

Ignoring the atom sitting at B and assuming the interstitial atom touches the atom at A

0.796AAOX x

0.195ABOY x

A364.0).( tetdxFeBCC

rvoid / ratom

SC BCC FCC DC

Octahedral (CN = 6)

0.155(distorted)

0.414 -

Tetrahedral (CN = 4)

0.29 (distorted)

0.2251

(½,½,½) & (¼, ¼, ¼)

Cubic (CN = 8)

0.732

Summary of void sizes

• The primitive UC for the FCC lattice is a Rhombohedron• Primitive unit cell made of 2T + 1O• Occupies ¼ the volume of the cell

FCC


ADDITION OF ALLOYING ELEMENTSADDITION OF ALLOYING ELEMENTS

Element Added

Segregation / phase separation

Solid solution

Compound /Intermediate structure(new crystal structure)

Interstitial

Substitutional Ordered

1

2

3

Segregation / phase separation

The added element does not dissolve in the parent/matrix phase →in a polycrystal may go to the grain boundary

1

Chemical compounds

Valency compounds (usual) Electrochemical compounds : Zintl

Mg2Sn, Mg2Pb, MgS etc.

Interstitial Phases: HaggDetermined by Rx / RM ratio

W2C, VC, Fe4N etc.

Electron compoundsspecific e/a ratio [21/14, 21/13, 21/12]

CuZn, Fe5Zn21, Au3Sn

Etc.

3

Size Factor compoundsLaves phases, Frank-Kasper Phases

Chemical compounds

Different crystal lattice as compared to the components Each component has a specific location in the lattice

AnBm

Different properties than components Constant melting point and dissociation temperature Accompanied by substantial thermal effect

Zintl Phases:Electrochemical compounds

Solid solution

InterstitialSubstitutional

The mixing is at the atomic scale and is analogous to a liquid solution

NOTE Pure components → A, B, C … Solid solutions → , , … Ordered Solid solutions → ’, ’, ’ …

2

Substitutional Solid Solution

HUME ROTHERY RULES

Empirical rules for the formation of substitutional solid solution The solute and solvent atoms do not differ by more than 15% in diameter The electronegativity difference between the elements is small The valency and crystal structure of the elements is same

Additional rule

Element with higher valency is dissolved more in an element of lower valency rather than vice-versa

SystemCrystal

structureRadius of atoms (Å)

Valency Electronegativity

Ag-AuAg FCC 1.44 1 1.9

Au FCC 1.44 1 2.4

Cu-NiCu FCC 1.28 1 1.9

Ni FCC 1.25 2 1.8

Ge-SiGe DC 1.22 4 1.8

Si DC 1.18 4 1.8

Examples of pairs of elements satisfying Hume Rothery rules and forming complete solid solution in all proportions

A continuous series of solid solutions may not form even if the above conditions are satisfied e.g. Cu- Fe

Counter example of a pair of elements not forming solid solution in all proportions

Cu Zn

FCCValency 1

HCPValency 2

35% Zn in Cu

1% Cu in Zn

Ordered Solid solution

G = H TS

High T disordered

Low T ordered

470ºC

Sublattice-1

Sublattice-2

BCC

SC

In a strict sense this is not a crystal !!

ORDERING A-B bonds are preferred to AA or BB bonds

e.g. Cu-Zn bonds are preferred compared to Cu-Cu or Zn-Zn bonds The ordered alloy in the Cu-Zn alloys is an example of an

INTERMEDIATE STRUCTURE that forms in the system with limited solid solubility

The structure of the ordered alloy is different from that of both the component elements (Cu-FCC, Zn-HCP)

The formation of the ordered structure is accompanied by change in properties. E.g. in Permalloy ordering leads to → reduction inmagnetic permeability, increase in hardness etc. [~Compound]

Complete solid solutions are formed when the ratios of the components of the alloy (atomic) are whole no.s → 1:1, 1:2, 1:3 etc. [CuAu, Cu3Au..]

Ordered solid solutions are in-between solid solutions and chemical compounds

Degree of order decreases on heating and vanishes on reaching disordering temperature [ compound]

Interstitial Solid Solution

The second species added goes into the voids of the parent lattice Octahedral and tetrahedral voids E.g. C (r = 0.77 Å), N (r = 0.71 Å), H (r = 0.46 Å)

STRUCTURE OF SOLIDS Types of solids based on structure Types of solids based on bonding.

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Transcript of STRUCTURE OF SOLIDS Types of solids based on structure Types of solids based on bonding.