STRUCTURE OF SOLIDS

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STRUCTURE OF SOLIDS

• Types of solids based on structure
• Types of solids based on bonding

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UNIVERSE
STRONG WEAK ELECTROMAGNETIC GRAVITY
HYPERBOLIC EUCLIDEAN SPHERICAL
ENERGY
SPACE
nD t
FIELDS
PARTICLES
METAL SEMI-METAL SEMI-CONDUCTOR INSULATOR
NON-ATOMIC
ATOMIC
BAND STRUCTURE
STATE / VISCOSITY
LIQUID CRYSTALS
GAS
SOLID
LIQUID
STRUCTURE
CRYSTALS
RATIONAL APPROXIMANTS
AMORPHOUS
QUASICRYSTALS
SIZE
NANO-QUASICRYSTALS
NANOCRYSTALS
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CLASSIFICATION OF SOLIDS BASED ON ATOMIC
ARRANGEMENT
QUASICRYSTALS
CRYSTALS
AMORPHOUS
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!!

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CLASSIFICATION OF SOLIDS BASED ON ATOMIC
ARRANGEMENT
QUASICRYSTALS
CRYSTALS
AMORPHOUS
ADDITIONAL POSSIBLE STRUCTURES
Incommensurately Modulated structures
Modulated structures
Liquid crystals
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THE ENTITY IN QUESTION
GEOMETRICAL
PHYSICAL
E.g. Atoms, Cluster of AtomsIons, etc.
E.g. Electronic Spin, Nuclear spin
ORDER
POSITIONAL
ORIENTATIONAL
Order-disorder of POSITION, ORIENTATION,
ELECTRONIC NUCLEAR SPIN
ORDER
TRUE
PROBABILISTIC
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ORIENTATIONAL
Perfect
POSITIONAL
A
B
Positionally ordered
PROBABILISTIC OCCUPATION
Probability of occupation A ? 50B ? 50
Probabilistically ordered
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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 Short Range Long Range Long Range
Class/example(s) ? 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
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Crystal Physics, G.S. Zhdanov, Oliver Boyd,
Ediburgh, 1965
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Factors affecting the formation of the amorphous
state

• When primary bonds are 1D or 2D and secondary
  bonds aid in the formation of the crystal
• The crystal structure is very complex

• 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

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CRYSTALS
Non-molecular
Molecular
COVALENT

• Molecule held together by primarycovalent bonds
• Intermolecular bonding is Van der walls

IONIC
METALLIC
CLASSIFICATION OF SOLIDS BASED ON BONDING
IONIC
METALLIC
COVALENT
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Approximate Strengths of Interactions between
atoms
Bond Type kJ/mol
Covalent Bond 250
Electrostatic 5
van der Waals 5
Hydrogen bond 20
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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 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, a-Mn, Pu

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FCC
CLOSE PACKING



A
B
C
FCC
Note Atoms are coloured differently but are the
same
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Shown displaced for clarity
HCP



HCP
B
A
A
Unit cell of HCP (Rhombic prism)
Note Atoms are coloured differently but are the
same
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Atoms (0,0,0), (?, ?,½)
Note diagrams not to scale
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IDEAL c/a
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PACKING FRACTION / Efficiency
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
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/(a2?2) 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) lt (110) lt (100) (110) lt (100) lt (111) (111) lt (100) lt (110)
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(100)
(110)
(111)
SC
FCC
BCC
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ATOMIC DENSITY (area covered by atoms/unit area)
SC SC FCC FCC BCC 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 3?2?/16 0.833
(111) 1/(?3a2) 0.577(?/4) 0.453 4/(?3a2) ?/(2?3) 0.9068 1/(?3a2) ?3?/16 0.34

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VOIDS
FCC
OCTAHEDRAL
TETRAHEDRAL
At body centre ½, ½, ½ face centering
translations
¼ way along body diagonal ¼, ¼, ¼, ¾, ¾, ¾
face centering translations
Note Atoms are coloured differently but are the
same
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FCC- OCTAHEDRAL
Site for octahedral void
½, ½, ½ ½, ½, 0 1, 1, ½ ? 0, 0, ½
Equivalent site for an octahedral void
Face centering translation
Note Atoms are coloured differently but are the
same
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FCC voids Position Voids / cell Voids / atom
Tetrahedral ¼ way from each vertex of the cube along body diagonal lt111gt ? ((¼, ¼, ¼)) 8 2
Octahedral Body centre 1 ? (½, ½, ½) Edge centre (12/4 3) ? (½, 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 the new atom
e
Size of the largest atom which can fit into the
Octahedral void of FCC
2r 2x a
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VOIDS
HCP
OCTAHEDRAL
TETRAHEDRAL
Coordinates (? ?,¼), (?,?,¾)
These voids are identical to the ones found in FCC
Note Atoms are coloured differently but are the
same
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The other orientation of the tetrahedral void
Octahedral voids occur in 1 orientation,
tetrahedral voids occur in 2 orientations
Note Atoms are coloured differently but are the
same
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Note Atoms are coloured differently but are the
same
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Octahedral voids
Tetrahedral void
Note Atoms are coloured differently but are the
same
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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)
Atoms in HCP crystal (0,0,0), (?, ?,½)
Check below
HCP voids Position Voids / cell Voids / atom
Tetrahedral (0,0,3/8), (0,0,5/8), (?, ?,1/8), (?,?,7/8) 4 2
Octahedral (? ?,¼), (?,?,¾) 2 1
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A
B
A
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VOIDS
BCC
Distorted OCTAHEDRAL
Distorted TETRAHEDRAL
a?3/2
a
a?3/2
a
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
rVoid / ratom 0.155
rvoid / ratom 0.29
Actually an atom of correct size touches only
the top and bottom atoms
Note Atoms are coloured differently but are the
same
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0, 0, ½)
BCC voids Position Voids / cell Voids / 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
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a
BCC Distorted Tetrahedral Void
a?3/2
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Distorted Octahedral Void
As the distance OA gt OB the atom in the void
touches only the atom at B (body centre). ? void
is actually a linear void
a?3/2
This implies
a
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FCC
FeFCC
C
Void (Oct)
N
O
Relative sizes of voids w.r.t to atoms
BCC
FeBCC
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Ignoring the atom sitting at B and assuming the
interstitial atom touches the atom at A
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Summary of void sizes
rvoid / ratom rvoid / ratom rvoid / ratom rvoid / ratom
SC BCC FCC DC
Octahedral (CN 6) 0.155(distorted) 0.414 -
Tetrahedral (CN 4) 0.29 (distorted) 0.225 1(½,½,½) (¼, ¼, ¼)
Cubic (CN 8) 0.732


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FCC

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

Note Atoms are coloured differently but are the
same
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ADDITION OF ALLOYING ELEMENTS
Segregation / phase separation
1
Interstitial
Solid solution
Element Added
2
Substitutional
Ordered
3
Compound /Intermediate structure(new crystal
structure)
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Segregation / phase separation
1

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

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Valency compounds (usual) Electrochemical
compounds Zintl Mg2Sn, Mg2Pb, MgS etc.
Interstitial Phases Hagg Determined by Rx / RM
ratio W2C, VC, Fe4N etc.
3
Chemical compounds
Electron compoundsspecific e/a ratio 21/14,
21/13, 21/12CuZn, Fe5Zn21, Au3Sn
Size Factor compoundsLaves phases, Frank-Kasper
Phases
Etc.
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Chemical compounds
Zintl PhasesElectrochemical 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

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Solid solution
2
Interstitial
Substitutional

• 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 ? ?, ?, ?

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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

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Examples of pairs of elements satisfying Hume
Rothery rules and forming complete solid solution
in all proportions
System Crystal structure Radius of atoms (Å) Valency Electronegativity
Ag-Au Ag FCC 1.44 1 1.9
Ag-Au Au FCC 1.44 1 2.4
Cu-Ni Cu FCC 1.28 1 1.9
Cu-Ni Ni FCC 1.25 2 1.8
Ge-Si Ge DC 1.22 4 1.8
Ge-Si Si DC 1.18 4 1.8
A continuous series of solid solutions may not
form even if the above conditions are satisfied
e.g. Cu-? Fe
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Counter example of a pair of elements not forming
solid solution in all proportions
35 Zn in Cu
Zn
Cu
1 Cu in Zn
HCP Valency 2
FCC Valency 1
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In a strict sense this is not a crystal !!
Ordered Solid solution
High T disordered
BCC
470ºC
G H ? TS
Sublattice-1
Sublattice-2
SC
Low T ordered
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• 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
  in magnetic 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 ? 11, 12, 13 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

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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 Å), N (r 0.71 Å), H (r 0.46
  Å)