structure of solids types of solids based on structure types of solids based on bonding
TRANSCRIPT
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
+
Note: Atoms are coloured differently but are the same
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
Note: Atoms are coloured differently but are the same
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
Note: Atoms are coloured differently but are the same
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
Note: Atoms are coloured differently but are the same
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
Note: Atoms are coloured differently but are the same
Note: Atoms are coloured differently but are the same
Note: Atoms are coloured differently but are the same
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
Note: Atoms are coloured differently but are the same
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)