structure of solids- metallic

8/10/2019 Structure of Solids- Metallic

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STRUCTURE OF SOLIDS

Types of solids based on structure

Types of solids based on bonding


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UNIVERSE

PARTICLES

ENERGYSPACE

FIELDS

STRONGWEAKELECTROMAGNETICGRAVITY

METALSEMI-METAL

SEMI-CONDUCTOR

INSULATOR

nD + t

HYPERBOLICEUCLIDEAN

SPHERICAL

GAS

BAND STRUCTURE

AMORPHOUS

ATOMIC NON-ATOMIC

STATE / VISCOSITY

SOLID LIQUIDLIQUID

CRYSTALS

QUASICRYSTALS CRYSTALSRATIONALAPPROXIMANTS

STRUCTURE

NANO-QUASICRYSTALS NANOCRYSTALS

SIZE


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AMORPHOUS

CLASSIFICATION OF SOLIDS BASED ON ATOMICARRANGEMENT

QUASICRYSTALS CRYSTALS

Ordered+

Periodic

Ordered+

Periodic

Ordered

+Periodic

There exists at least one crystalline state of lower energy (G) thanthe amorphous state (glass)

The crystal exhibits a sharp melting point

Crystal has a higher density!!


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AMORPHOUS

CLASSIFICATION OF SOLIDS BASED ON ATOMICARRANGEMENT

QUASICRYSTALS CRYSTALS

ADDITIONAL POSSIBLE STRUCTURES

Modulated structuresIncommensurately

Modulated structures

Liquid crystals


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THE ENTITY IN QUESTION

GEOMETRICAL PHYSICALE.g. Atoms, Cluster of Atoms

Ions, etc.E.g. Electronic Spin, Nuclear spin

ORDER

ORIENTATIONAL POSITIONAL

ORDER

TRUE PROBABILISTIC

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


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ORIENTATIONAL

POSITIONAL

PROBABILISTIC

OCCUPATION

Perfect

Average

Perfect

Average

Positionally ordered

Probabilistically ordered

A B

Probability of

occupation:

A 50%

B 50%


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Order

Spatial Temporal


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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 crystalsLiquids have short range spatial order butNOtemporal order


9/52Crystal Physics, G.S. Zhdanov, Oliver & Boyd, Ediburgh, 1965


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When primary bonds are 1D or 2D and secondary bonds aid in theformation 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 issmall Tendency to crystallize would be small

Cooling rate fast cooling promotes amorphizationfast depends on the material in considerationCertain alloys have to be cooled at 106K/s for amorphization

Silicates amorphizes during air cooling


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COVALENT

CLASSIFICATION OF SOLIDS BASED ON BONDING

IONIC METALLIC

Molecular

CRYSTALS

Non-molecular

COVALENT

IONIC

METALLIC

Molecule held together by primary

covalent bonds

Intermolecular bonding is Van der walls


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Bond Type kJ/mol

Covalent Bond 250

Electrostatic 5

van der Waals 5

Hydrogen bond 20

Approximate Strengths of Interactionsbetween atoms


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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 reasonfor many of them having the BCC structure at low temperatures

FCC Al, Fe (910 - 1410C), Cu, Ag, Au, Ni, Pd, Pt

BCC Li, Na, K , Ti, Zr, Hf, Nb, Ta, Cr, Mo, W, Fe (below 910C),

HCP Be, Mg, Ti, Zr, Hf, Zn, Cd

Others La, Sm Po, -Mn, Pu


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CLOSE PACKING

A B C

+ +

FCC

=

Note: Atoms are coloured differently but are the same

FCC


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AB

+

HCP

=

A

+


HCPShown displaced for clarity

Unit cell of HCP (Rhombic prism)


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Note: diagrams not to scale

Atoms: (0,0,0), (, ,)


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aCV4

6

aCV12

6

aVCCFh 3

2

632.13

22

2

a

h

a

cIDEAL c/a

h


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PACKING FRACTION / Efficiency

CellofVolume

atomsbyoccupiedVolumeFractionPacking

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

3

6

2

16

3

3

24rc

6

2

* Crystal formed by monoatomic decoration of the lattice


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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)


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FCC

BCC

(100) (110) (111)

SC

a

a2

a2a2

a2

a

aa

a2 a2

ATOMIC DENSITY ( d b / )


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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


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VOIDS

TETRAHEDRAL OCTAHEDRAL

FCC


cellntetrahedro VV24

1

celloctahedron VV

6

1

way along body diagonal{, , }, {, , }

+ face centering translations

At body centre{, , }

+ face centering translations


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FCC- OCTAHEDRAL

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

Face centering translation


Equivalent site for anoctahedral void

Site for octahedral void


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FCC voids Position Voids / cell Voids / atom

Tetrahedral

way from each vertex of the cube

along body diagonal ((, , ))

8 2

OctahedralBody centre: 1 (, , )

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


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

CV = r + x Radius of thenew atom

e

xre 46

225.0~1

2

32

r

xre

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

2r + 2x = a ra 42

414.0~12r

x


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VOIDS

TETRAHEDRAL OCTAHEDRAL

HCP

These voids are identical to the ones found in FCC


Coordinates: ( ,), (,,)),,(),,,(),,0,0(),,0,0(: 8731328131328583sCoordinate


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

The other orientation of the tetrahedral void



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Octahedral voids

Tetrahedral void


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HCP voids PositionVoids /

cellVoids / atom

Tetrahedral (0,0,3/8), (0,0,5/8), (, ,1/8),(,,7/8) 4 2

Octahedral ( ,), (,,) 2 1

Voids/atom: FCC HCP

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

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

two void layers (with axis to figure) Central atom

Check below

Atoms in HCP crystal: (0,0,0), (, ,)


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A

A

B


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VOIDS


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VOIDS

Distorted TETRAHEDRAL Distorted OCTAHEDRAL**

BCC

a

a3/2

a a3/2

rvoid/ ratom= 0.29rVoid/ ratom= 0.155

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

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

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


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BCC voids PositionVoids /

cellVoids /atom

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

DistortedOctahedral

Face centre: (6/2 = 3) (, , 0)

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

{0, 0, })


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From the right angled triange OCM:416

22aa

OC 5

4a r x

For a BCC structure: 3 4a r (3

4r

a )

xrr

3

4

4

5 29.01

3

5

r

x

a

a3/2

BCC: Distorted Tetrahedral Void


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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 voidtouches only the atom at B (body centre).void is actually a linear void

This implies:

FCC


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A292.1FeFCC

r

A534.0)( octxFeFCC

A77.0Cr

C

N

Void (Oct)

FeFCC

O

A258.1FeBCC

r

A364.0).( tetdxFeBCC

A195.0).( octdxFeBCC

FCC

BCC

FeBCC

Relative sizes of voids w.r.t to atoms

( . )0.155

Fe

BCC

Fe

BCC

x d oct

r

( . )0.29

FeBCC

Fe

BCC

x d tet

r


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A258.1FeBCC

r2

2A

aOA r x

2 6

3A

rr x

raBCC 43:

2 61 0.6329

3

Ax

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).( tetdxFe

BCC


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rvoid / ratom

SC BCC FCC DC

Octahedral(CN = 6)

0.155(distorted)

0.414 -

Tetrahedral(CN = 4) 0.29(distorted) 0.225 1(,,) & (, , )

Cubic(CN = 8)

0.732

Summary of void sizes

FCC


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The primitive UC for the FCC lattice is a RhombohedronPrimitive unit cell made of 2T + 1OOccupies the volume of the cell

FCC



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Segregation / phase separation

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

1

V l d ( l)


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Chemicalcompounds

Valency compounds(usual)Electrochemical compounds: Zintl

Mg2Sn, Mg2Pb, MgS etc.

Interstitial Phases: HaggDetermined by Rx/ RMratio

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


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Chemical compounds

Different crystal latticeas 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

S lid l i2


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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


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Substitutional Solid Solution

HUME ROTHERY RULES

Empirical rules for the formation of substitutional solid solutionThe 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


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SystemCrystal

structureRadius ofatoms () 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-Si

Ge 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

l f f l f l d l ll


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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

In a strict sense this is not a crystal !!


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Ordered Solid solution

G = H TS

High T disordered

Low T ordered

470C

Sublattice-1

Sublattice-2

BCC

SC

In a strict sense this is not a crystal !!


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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 ), N (r = 0.71 ), H (r = 0.46 )

structure of solids- metallic

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