THE STRUCTURE OF CRYSTALLINE SOLIDS. IE-114 Materials Science and General Chemistry Lecture-3

Transcription

1 THE STRUCTURE OF CRYSTALLINE SOLIDS IE-114 Materials Science and General Chemistry Lecture-3

2 Outline Crystalline and Noncrystalline Materials 1) Single, Polycrystalline, Non-crystalline solids 2) Polycrystalline Materials Crystal Structures 1)Unit cells 2) Metallic crystal structures 4) Crystal Systems (Directions and Planes) 5) Linear and Planar Atomic Densities

3 Crystal Structures Material classification can be made based on the regularity or irregularity of atom or ion arrangement with respect to each other. 1) Crystalline Material Single crystalline Polycrystalline Atoms are situated in a repeating or periodic array over large atomic distances (Longe range order) e.g.all metals, some ceramics and polymers 2) Noncrystalline (amorphous) Material Long range atomic arrangement lacks in this type of materials.

4 Two Dimensional View of Atomic Arrangements Single crystal Polycrystal Amorphous Single crystal diamond (schematic view) Single crystal of CaF 2

5 Formation of Polycrystals 1) During heavy deformation 2) During solidification from melt Solidification of a pure metal (T=Tm) t=t1 Solid crystals t=t2 Liquid Liquid and small crystals Liquid and relatively larger crystals t1 t2 t3 (Tm) t=t3 t=t3 Structure of the material seen under microscope Completely solid. Material contains many grains(polycrystalline material)

6 Describing Crystal Structure The atoms or ions are thought as solid spheres with their sizes defined. This is called atomic hard sphere model. All atoms are identical in this model. UNIT CELL: Smallest repeating group Unit cells can be imagined as the building block of the crystal structure. Unit cells in general are paralelepipeds or prisms having three sets of parallel faces, one is drawn within the aggregate of spheres.

7 Crystal Structures of Metallic Materials Atoms tend to be densely packed Atomic bonding is metallic and nondirectional. No restrictions as to the number and position of nearest neighbor atoms Four simple crystalline structures found in metallic materials: 1) SC (Simple cubic) crystal structure 2) BCC (Body Centered Cubic) crystal structure 3) FCC (Face Centered Cubic) crystal structure 4) HCP (Hexagonal Close-Packed) Crystal Structure

8 Simple Cubic (SC) Structure Rare in nature due to poor packing (only Po has this structure) Close-packed directions are cube edges. * Coordination number: number of nearest neighbor or touching atoms. (Arrangement of atoms in one SC unit cell) Coordination number = 6 The number of atoms per SC: (1/8)*8 = 1 atom/cell The relationship between unit cell edge length (a) and atomic radius (R); a=2r

9 Body Centered Cubic (BCC) Structure BCC unit cell --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. a (Arrangement of atoms in one BCC unit cell) Coordination number = 8 The number of atoms per BCC: (1/8)*8 + 1= 2 atoms/cell The relationship between unit cell edge length (a) and atomic radius (R); a=(4/ 3)R

10 Face Centered Cubic (FCC) Structure --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. FCC unit cell a (Arrangement of atoms in one FCC unit cell) Coordination number = 12 The number of atoms per FCC: (1/8)*8 + 6*(1/2)= 4 atoms/cell The relationship between unit cell edge length (a) and atomic radius (R); a=(2 2)R

11 Hexagonal Closed Packed (HCP) Structure The top and bottom faces of the unit cell have six atoms that form regular hexagons and a single atom in the center. Another plane provides three additional atoms is situated between top and bottom planes. HCP Unit Cell Coordination number = 12 The number of atoms per HCP: (1/6)*12 + 2*(1/2) + 3 = 6 atoms/cell Ideally c/a=1.633, but for some metals this ratio deviates from the ideal value.

12 Structure of Compounds Ionic bonding (NaCl) Compounds often have similar close-packed structures. Cl - Na + The number of atoms per SC: [(1/8)*8+(1/2)*6] Cl ions + [(1/4)*12+1] Na ions (4 Cl Na + ) ions/unit cell The relationship between unit cell edge length (a) and atomic radius (R); a=2r Cl- + 2R Na+

13 Polymorphism Atoms may have more than one type of crystal structure Example : Iron (Fe) FCC heating cooling BCC

14 Characteristics of Some Selected Elements

15 Note that cubic system is the most symmetric, while triclinic is the least one. Crystal Systems There are 7 different crystal structures; Cubic, tetragonal, orthorhombic, rhombohedral, monoclinical, triclinic, hexagonal

16 Crystal Systems The unit cell geometry: x,y,z coordinate system is established with its origin at one of the unit cell corners and axes coincide with the edges of the paralelepiped extending from that corner, the origin. There are six parameters to define the geometry of the unit cell: Three edge lengths : a, b, c Three interaxial angles : α, β, Also called lattice parameters.

17 Crystallographic Directions in Cubic Crystals Particular crystallographic direction is shown in a unit cell The direction is a line between two points or a vector as shown below:

18 Steps for defining a direction in a crystal system: 1) A vector is positioned such that it passes through the origin of the coordinate system. Then you can move the vector if you keep the parallelism. 2) The length of the vector projection on each of the three axes is determined in terms of the unit cell dimensions (a, b, c). 3) The three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. 4) Three indices are enclosed in square brackets as [uvw] 5) Remember to count for positive and negative coordinates based on the origin. When there is a negative index value, then show that by a bar over it, as [111] This vector has a component in y direction.

19 Example1: Example2:

20 FAMILY OF DIRECTIONS, <uvw> For cubic crystal structures, several nonparallel directions with different indices are equivalent. (the spacing of atoms along each direction is the same) (equivalent directions) <100> Family

21 Hexagonal Crystals: There is a four-axis (Miller-Bravais) coordinate system used for this type of structures. Three a1, a2, and a3 axes are placed within a single plane (basal plane) and at 120 angles to one another. The z axis is perpendicular to the selected basal plane.

22 Some directions in HCP crystal structure

23 Crystallographic Planes in Cubic Crystals Except HCP, crytallographic planes are specified using three MILLER INDICES (hkl). Any two planes parallel to each other are equivalent and have same indices. The determination of the h,k, and l index numbers are as follows: 1) If the plane passes through the selected origin, then construct a new parallel plane or change the originto a corner of another unit cell. 2) Plane intersects or parallels each of the axes: the length of each axis is determined by using lattice parameters; a,b, and c. 3) Take the reciprocals of the lattice parameters. Therefore a plane that parallels an axis has a ZERO index. (1/infinity=zero) 4) You may then change these three numbers to the set of smallest integers using a common factor. 5) Report the indices as (hkl). 6) An intercept on the negative side of the origin is indicated by a bar over that index.

24 Example1: Example2: Example3:

25 For cubic crystals: Planes and directions having the same indices are parallel to one another.

26 Family of Planes, {hkl} A family of planes is formed by all those planes that are crystallographically equivalent, {100}, {111}. (for cubic structure) {111} =

27 Planes in Hexagonal Crystals: Equivalent planes have the same indices as directions Four-index scheme is used (hkil) and the index i is calculated by the sum of h and k through i = - (h+k) This scheme defines the orientation of a plane in a hexagonal crystal.

28 Theoretical Density, ρ Theoretical density (ρ) of elements can be found by the use of unit cell and crystal structure Example: Copper has an atomic radius of nm, FCC crystal structure and an atomic weight of 63.5 g/mol. Compute its density and compare the answer with its measured density Result: theoretical Cu = 8.89 g/cm 3 Compare to actual: Cu = 8.94 g/cm 3

29 Theoretical Density of NaCl NaCl unit cell

30 Atomic Packing Factor (APF) Metals generally have high APF to maximize the shielding provided by electron cloud. Fraction of solid sphere volume in a unit cell

31 APF for a face-centered cubic(fcc) structure is 0.74 For FCC structure; a a=(2 2)R The number of atoms per SC: (1/8)*8 + 6*(1/2)= 4 atoms/cell APF for BCC structure = 0.68 APF for HCP structure = 0.74

32 Linear and Planar Density Linear and planar atomic densities are one and two dimensional analogs of atomic packing factor (APF). Linear density: Fraction of line length in a particular crystallographic direction that passes through atom centers Planar density: Fraction of total crystallographic plane area that is occupied by atoms. (The plane must pass through an atom s center for particular atom to be included in calculations) Equivalency of directions and planes is related to the degree of atomic spacing or atomic packing

33 Example1: z Atomic arrangement on [100] direction; [100] direction x y a = 4/ 3R a= R Linear density of [100] direction in BCC; = Line length of atoms within [100] direction / Length of [100] direction = 2R/4/ 3R = 0.866

34 Example2: (110) plane Atomic arrangement on (110) plane in FCC; a (110) plane Area of (110) plane; = 2.a 2 = 2(2 2)R 2 = 8 2R 2 Area of atoms inside (110) plane; = ¼*4* R 2 + ½*2* R 2 = 2 R 2 2.a = 4R Planar density of (110) plane in FCC = Area of atoms inside (110) plane / area of (110) plane Planar density of (110) plane in FCC = 2 R 2 / 8 2R 2 = 0.555

35 Anisotropy Dependency of properties on direction: ANISOTROPY It is associated with the variance of atomic or ionic spacing with crystallographic direction. Example: For a single crystal material, different mechanical properties are observed in [100] and [111] directions and also in some other directions. Substances in which measured properties are independent of the direction of measurement are ISOTROPIC

36 Determination of Crystal Structure Incoming X-rays diffract from crystal planes. Measurement of: Critical angles, q c, for X-rays provide atomic spacing, d. n = 2dSinq Bragg s Law

37 Summary Atoms may assemble into crystalline or amorphous structures. We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). Material properties generally vary with single crystal orientation (i.e., they are anisotropic),but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.