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STRUCTURE OF SOLIDS Types of solids based on structure
Types of solids based on bonding
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RATIONAL APPROXIMANTS
UNIVERSE STRONG WEAK ELECTROMAGNETIC GRAVITY HYPERBOLIC EUCLIDEAN SPHERICAL ENERGY SPACE nD + t PARTICLES FIELDS METAL SEMI-METAL SEMI-CONDUCTOR INSULATOR NON-ATOMIC ATOMIC BAND STRUCTURE STATE / VISCOSITY LIQUID CRYSTALS GAS SOLID LIQUID STRUCTURE CRYSTALS AMORPHOUS QUASICRYSTALS RATIONAL APPROXIMANTS SIZE NANO-QUASICRYSTALS NANOCRYSTALS
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CLASSIFICATION OF SOLIDS BASED ON ATOMIC ARRANGEMENT
QUASICRYSTALS CRYSTALS AMORPHOUS 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”!!
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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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ORDER ORDER THE ENTITY IN QUESTION GEOMETRICAL PHYSICAL
E.g. Atoms, Cluster of Atoms Ions, etc. E.g. Electronic Spin, Nuclear spin ORDER ORIENTATIONAL POSITIONAL Order-disorder of: POSITION, ORIENTATION, ELECTRONIC & NUCLEAR SPIN ORDER TRUE PROBABILISTIC
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Perfect ORIENTATIONAL Average Perfect POSITIONAL Average A B Positionally ordered PROBABILISTIC OCCUPATION Probability of occupation: A 50% B 50% Probabilistically ordered
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Order Spatial Temporal
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Crystals*/ Quasicrystals
Range of Spatial Order Short Range (SRO) Long Range Order (LRO) Class/ example(s) Short Range Long Range 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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CLASSIFICATION OF SOLIDS BASED ON BONDING
CRYSTALS Molecular Non-molecular COVALENT Molecule held together by primary covalent bonds Intermolecular bonding is Van der walls IONIC METALLIC CLASSIFICATION OF SOLIDS BASED ON BONDING COVALENT IONIC METALLIC
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Approximate Strengths of Interactions between atoms
Bond Type kJ/mol Covalent Bond 250 Electrostatic 5 van der Waals 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 ( ºC), 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
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FCC FCC + + = CLOSE PACKING A B C
Note: Atoms are coloured differently but are the same
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HCP HCP + + = Shown displaced for clarity 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), (⅔, ⅓,½) Note: diagrams not to scale
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h 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 Atoms / cell 1 2 4 8 Lattice points / cell No. of nearest neighbours 6 12 Packing fraction = 0.52 = 0.68 = 0.74 = 0.34 * Crystal formed by monoatomic decoration of the lattice
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SC FCC BCC (100) 1/a2 = 1/a2 2/a2 = 2/a2 (110) 1/(a22) = 0.707/a2
ATOMIC DENSITY (atoms/unit area) SC FCC BCC (100) 1/a2 = 1/a2 2/a2 = 2/a2 (110) 1/(a22) = 0.707/a2 2/a2 = 1.414/a2 (111) 1/(3a2) = 0.577/a2 4/(3a2) = 2.309/a2 Order (111) < (110) < (100) (110) < (100) < (111) (111) < (100) < (110)
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(100) (110) (111) SC FCC BCC
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ATOMIC DENSITY (area covered by atoms/unit area)
SC FCC BCC Atoms / Area Area / Area (100) 1/a2 /4 = 0.785 2/a2 3/16 = 0.589 (110) 2/(2a2) 0.707(/4) = 0.555 2/a2 2/8 = 0.555 32/16 = 0.833 (111) 1/(3a2) 0.577(/4) = 0.453 4/(3a2) /(23) =0.9068 3/16 = 0.34
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FCC VOIDS OCTAHEDRAL TETRAHEDRAL At body centre
¼ way along body diagonal {¼, ¼, ¼}, {¾, ¾, ¾} + face centering translations At body centre {½, ½, ½} + face centering translations Note: Atoms are coloured differently but are the same
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Site for octahedral void
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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¼ way from each vertex of the cube along body diagonal <111>
FCC voids Position Voids / cell Voids / atom Tetrahedral ¼ 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
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Size of the largest atom which can fit into the Octahedral void of FCC
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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These voids are identical to the ones found in FCC
HCP OCTAHEDRAL TETRAHEDRAL Coordinates: (⅓ ⅔,¼), (⅓,⅔,¾) 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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Central atom 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), (⅔, ⅓,½) Check below HCP voids Position Voids / cell Voids / 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) 1
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A B A
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BCC VOIDS a3/2 a a3/2 a rVoid / ratom = 0.155 rvoid / ratom = 0.29
Distorted OCTAHEDRAL** Distorted TETRAHEDRAL a3/2 a a3/2 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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Distorted Tetrahedral
{0, 0, ½}) BCC voids Position Voids / cell Voids / atom Distorted Tetrahedral 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) 3
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BCC: Distorted Tetrahedral Void
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Distorted Octahedral Void
As the distance OA > OB the atom in the void touches only the atom at B (body centre). void is actually a ‘linear’ void a3/2 This implies: a
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FCC BCC Void (Oct) FeFCC C N O FeBCC
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 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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Segregation / phase separation
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 Segregation / phase separation 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 compounds specific e/a ratio [21/14, 21/13, 21/12] CuZn, Fe5Zn21, Au3Sn Size Factor compounds Laves phases, Frank-Kasper Phases Etc.
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Zintl Phases: Electrochemical 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 Substitutional Interstitial
2 Solid solution Substitutional Interstitial 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 Au 2.4 Cu-Ni Cu 1.28 Ni 1.25 2 1.8 Ge-Si Ge DC 1.22 4 Si 1.18 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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Ordered Solid solution
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 → 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]
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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)
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