Solid State

Solid State Gases: no definite shape and volume Solids: definite shape, volume and order. Order: definite pattern of arrangement of atoms or molecules or ions. Liquids: no definite shape but definite volume Solids: definite shape and volume

Intensive properties: do not depend on the amount. Unit Cells Smallest Repeating Unit Unit Cells must link-up  cannot have gaps between them All unit cells must be identical

Choice of the origin is arbitrary

This cannot be a unit cell Unit cells are not identical

This also cannot be a unit cell Space between unit cells not allowed

Unit cells exist in only seven shapes Cubic Orthorhombic Rhombohedral Tetragonal Triclinic Hexagonal Monoclinic

Crystal intercepts & Angles a b c   

Crystal Systems Lattice Parameters Crystal Intercepts Crystal Angles Cubic a = b = c  =  =  = 90 o Orthorhombic a  b  c  =  =  = 90 o Rhombohedral a = b = c  =  =   90 o Tetragonal a = b  c  =  =  = 90 o Triclinic a  b  c       90 o Hexagonal a = b  c =  = 90 o,  = 120 o Monoclinic a  b  c =  = 90 o,   90 o

There are not more than 4 ways of arranging spheres in any shape of unit cell These are Primit ive , Body Centered, Face Centered & End Centered

a = 2r 1 2 4 3 5 6 7 8 Primitive Cubic Simple Cubic Unit Cell shape view Unit Cell arrangement view

Volume occupied by a sphere in the unit cell Total volume occupied by all the spheres in the unit cell Primitive Cubic

Packing Fraction Fraction of the Unit cell’s volume occupied by the spheres Primitive Cubic

Coordination number 6 Primitive Cubic

Body Centered Cubic Unit Cell shape view Unit Cell arrangement view

Body Centered Cubic Packing Fraction Volume occupied by a corner sphere in the unit cell Volume occupied by the central sphere in the unit cell Total Volume occupied by the spheres in the unit cell Packing Fraction

Coordination number 8 Body Centered Cubic

Face Centered Cubic Unit Cell shape view Unit Cell arrangement view

Face Centered Cubic Packing Fraction Volume occupied by a corner sphere in the unit cell Volume occupied by a face centered sphere in the unit cell Total Volume occupied by the spheres in the unit cell Packing Fraction Highest Packing Fraction of all shapes and of all arrangements

Face Centered Cubic Coordination number x y z y-z plane x-z plane x-y plane

Face Centered Cubic Coordination number a/2 a/2

Out of all the twenty eight possible unit cells only 14 exist ! Those arrangements in a given shape that violate even one symmetry element of that shape do not exist in that shape 90 o axis of symmetry

If we do the same with BCC & FCC we will get the same result. Lets try with End Centered

Like this 13 other arrangements in various shapes were rejected. We are left with only 14 unit cells

Crystal Systems Cubic Orthorhombic Rhombohedral Tetragonal Triclinic Hexagonal Monoclinic Bravais Lattices Primitive, FCC, BCC Primitive, FC, BC, EC Primitive Primitive, BC Primitive Primitive Primitive, EC

Layer A Layer arrangement view

Layer C Layer A Layer B Layer C Cubic Close Packing (CCP)

Layer A Layer A Layer B Layer A Hexagonal Close Packing

Hexagonal Primitive Unit Cell shape view Unit Cell arrangement view

Hexagonal Primitive a c a = 2r 2r 2r 2r r O A B 30 o OA = r  AOB = 30 o O B c/2 D D E 2r

Contribution of corner atom Contribution of Face atom Contribution of second layer atoms Total atoms per unit cell Hexagonal Primitive

Hexagonal Primitive Packing Fraction

Packing Fraction depends on: 1. Layout of each layer 2. Placement of one layer over the other

Hexagonal Primitive r 30 o O A B AB = r OA = r tan30 o < r Packing Fraction  same Rank of unit cell  2 Volume of unit cell  1/3 of previous mass of unit cell  1/3 of previous density  same

Voids Two types of voids: Octahedral Tetrahedral Found only in FCC & Hexagonal primitive unit cells Octahedral void in FCC

Voids Each octahedral void located at the edge center is shared by 4 unit cells Total contribution of edge centre voids = Contribution of central void Total contribution of all octahedral voids per unit cell of FCC = 4 No. of Octahedral voids per unit cell = Rank of unit cell

Voids Tetrahedral void in FCC (0,0,0) x-axis y-axis z-axis (a/2, a/2,0) (a/2, 0,a/2) (0, a/2,a/2) (a/4, a/4,a/4)

Voids (0,0,0) (a/2, a/2,0) (a/4, a/4,a/4)

Voids With each corner as origin there are 8 tetrahedral voids in FCC unit cell  No. of tetrahedral voids = 2  no. of Octahedral voids

Voids Voids in Hexagonal Primitive Let us assume that this is the unit cell then according to what we have done in FCC no. of Octahedral voids = 6 & no. of tetrahedral voids = 12 Octahedral voids Octahedral void

Voids Voids in Hexagonal Primitive Let us assume that this is the unit cell then according to what we have done in FCC no. of Octahedral voids = 6 & no. of tetrahedral voids = 12 Tetrahedral voids Contribution of tetrahedral voids formed inside the unit cell is 1 each. The ones formed on the corners of the hexagon have a contribution of 1/3. Total contribution In 3 layers

Minimum r c /r a for various coordination numbers Radius Ratios 2r a B O A 30 o Coordination number - 3

Radius Ratios Coordination number - 4 z-axis A B (0,0,0) (a/4, a/4,a/4)

Radius Ratios Coordination number - 4 (square planar) or 6 (octahedron) B A

Radius Ratios Coordination number - 8 (cube)

Radius Ratios Final Radius Ratios Radius Ratio, r c /r a Co-ordination No. <0.155 2 [0.155, 0.225) 2 or 3 [0.225, 0.414) 2 or 3 or 4 T d [0.414, 0.732) 2 or 3 or 4 T d , 4 sq. pl or 6 O h [0.732, 0.99) 2 or 3 or 4 T d , 4 sq. pl or 6 O h or 8

Structures of Ionic Compounds For ionic compounds of the general formula A x B y the ratio of the coordination number of A to that of B will be the ratio of y:x. 1. Rock Salt Structure (NaCl)  Cl -  Na + Cl - is FCC Na + occupies Octahedral voids No. of Cl - per unit cell = 4 No. of Na + per unit cell = 4  formula is NaCl Coordination no. of Na + = 6 Coordination no. of Cl - = 6

Structures of Ionic Compounds Other compounds which have this structure are: all halides of alkali metals except cesium halide, all oxides of alkaline earth metals except beryllium oxide, AgCl, AgBr & AgI.

Structures of Ionic Compounds Consider the unit cell with Cl - as FCC. Consider the unit cell with Na + as FCC. Similarly, r any alkali metal = r any halide r any akaline earth metal = r oxide Comparing

Structures of Ionic Compounds 2. Zinc Blende (ZnS)  S 2-  Zn 2+ S 2- is FCC Zn 2+ occupies alternate tetrahedral voids No. of S 2- per unit cell = 4 No. of Zn 2+ per unit cell = 4  formula is ZnS Coordination no. of Zn 2+ = 4 Coordination no. of S 2- = 4 Other compound which have this structure is: BeO

Structures of Ionic Compounds 3. Fluorite (CaF 2 )  F -  Ca 2+ Ca 2+ is FCC F - occupies all tetrahedral voids No. of Ca 2+ per unit cell = 4 No. of F - per unit cell = 8  formula is CaF 2 Coordination no. of F - = 4 Coordination no. of Ca 2+ = 8 Other compounds which have this structure are: UO 2 , ThO 2 , PbO 2 , HgF 2 etc.

Structures of Ionic Compounds 4. Anti-Fluorite (Li 2 O)  O 2-  Li + O 2- is FCC Li + occupies all tetrahedral voids No. of O 2- per unit cell = 4 No. of Li + per unit cell = 8  formula is Li 2 O Coordination no. of Li + = 4 Coordination no. of O 2- = 8 Other compounds which have this structure are: Na 2 O, K 2 O, Rb 2 O

Structures of Ionic Compounds 5. Cesium Halide  Cl -  Cs + Cl - is Primitive cubic Cs + occupies the centre of the unit cell No. of Cl - per unit cell = 1 No. of Cs + per unit cell = 1  formula is CsCl Coordination no. of Cs + = 8 Coordination no. of Cl - = 8 Other compounds which have this structure are: all halides of Cesium and ammonium

Structures of Ionic Compounds 6. Corundum (Al 2 O 3 ) Oxide ions form hexagonal primitive unit cell and trivalent ions (Al 3+ ) are present in 2/3 of octahedral voids. No. of O 2- per unit cell = 2 No. of Al 3+ per unit cell = 4/3 Coordination no. of Al 3+ = 6 Coordination no. of O 2- = 4 Other compounds which have this structure are: Fe 2 O 3 , Cr 2 O 3 , Mn 2 O 3 etc.

Structures of Ionic Compounds 7. Rutile (TiO 2 ) Oxide ions form hexagonal primitive unit cell and tetravalent ions (Ti 4+ ) are present in 1/2 of octahedral voids. No. of O 2- per unit cell = 2 No. of Ti 4+ per unit cell = 1 Coordination no. of Ti 4+ = 6 Coordination no. of O 2- = 3 Other compounds which have this structure are: MnO 2 , SnO 2 , MgF 2 , NiF 2  formula is TiO 2

Structures of Ionic Compounds 8. Pervoskite (CaTiO 3 )  O 2-  Ca 2+ (divalent ion) Ca 2+ is Primitive cubic Ti 4+ occupies the centre of the unit cell No. of O 2- per unit cell = 3 No. of Ca 2+ per unit cell = 1  formula is CaTiO 3 Coordination no. of O 2- = 6 Coordination no. of Ti 4+ Other compounds which have this structure are: BaTiO 3 , SrTiO 3  Ti 4+ (tetravalent ion) O 2- occupies face centres No. of Ti 4+ per unit cell = 1 = 6 Coordination no. of Ca 2+ = 12

Structures of Ionic Compounds 9. Spinel & Inverse Spinel (MgAl 2 O 4 ) O 2- ion is FCC Mg 2+ (divalent ion) 1/8 th of tetrahedral voids Al 3+ (trivalent ion) 1/2 of octahedral voids O 2- per unit cell = 4 Mg 2+ per unit cell = 1 Al 3+ per unit cell = 1  formula is MgAl 2 O 4 Spinel Inverse Spinel O 2- ion is FCC divalent ion 1/8 th of tetrahedral voids trivalent ion 1/4 th of octahedral voids & 1/8 th of tetrahedral voids O 2- per unit cell = 4 Divalent per unit cell = 1 Trivalent per unit cell = 1

(i) Lattice of atoms Crystal Defects (a) Vacancy  an atom is missing from its position  density decreases  percentage occupancy decreases (b) Self interstitial  an atom leaves its lattice site & occupies interstitial space  density & percentage occupancy remains same (c) Substitutional impurity  foreign atom substitutes a host atom & occupies its lattice  density & percentage occupancy may change (c) Interstitial impurity  foreign atom occupies occupies the interstitial space  density & percentage occupancy increases

(i) Ionic structures Crystal Defects (a) Schottky Defect  Cation – anion pair are missing  electro neutrality is maintained  density decreases (b) Frenkel Defect  ion leaves lattice position & occupies interstitial space  electro neutrality is maintained  density maintained (c) Substitutional Impurity Defect  Ba 2+ is replaced by Sr 2+  electro neutrality is maintained  density changes (d) Interstitial Impurity Defect  H 2 is trapped in TiC  electro neutrality is maintained  density increases (a) F-Centre  electron replaces anion  electro neutrality is maintained  density decreases  colour is imparted

1. Assuming diamond to be FCC of carbon atoms and that each carbon atom is sp 3 hybridized then which of the following statements is correct. (a) all voids are empty (b) 100% octahedral voids are filled (c) 50% octahedral voids are filled (d) 100% tetrahedral voids are filled (e) 50% tetrahedral voids are filled Sol: If no void is filled then each carbon would be in contact with 12 carbon atoms. This is not possible as each carbon is sp 3 hybridized. If octahedral voids are filled then those carbons in the voids would be in contact with 6 carbon atoms. This also is not possible. If 100% tetrahedral voids are filled then the FCC carbons would be in contact with 8 carbon atoms as they are shared in 8 unit cells and would be in contact with 8 tetrahedral voids. Not possible.  (e)

2. In NaCl calculate: The distance between the first 9 nearest neighbors in a unit cell & their total number in all unit cells

neighbor no. distance no. of neighbors 1 6 2 12 3 8 4 6 5 24 6 24 7 12 8 24 9 8

3. Iron crystallizes in FCC lattice. The figures given below shows the iron atoms in four crystallographic planes. Draw the unit cell for the corresponding structure and identify these planes in the diagram. Also report the distance between two such crystallographic planes in each terms of the edge length ‘a’ of the unit cell.

distance between two such planes is a distance between two such planes is a

distance between two such planes is A B C A

3. Marbles of diameter 10 mm are to be placed on a flat surface bounded by lines of length 40 mm such that each marble has its centre within the bound surface. Find the maximum number of marbles in the bound surface and sketch the diagram. Derive an expression for the number of marbles per unit area. Interpretation: 1. count marbles as 1 each even if some portion goes outside the bound surface 2. count marbles based on the portion that is inside the bound surface 25

To calculate no. of marbles per unit area we need to select the smallest repeating unit. 18 d A B C

Solid State

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Solid State