# Chapter Outline. How do atoms arrange themselves to form solids?

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1 Chapter Outline How do atoms arrange themselves to form solids? Fundamental concepts and language Unit cells Crystal structures Simple cubic Face-centered cubic Body-centered cubic Hexagonal close-packed Close packed crystal structures Density computations Types of solids Single crystal Polycrystalline Amorphous University of Tennessee, Dept. of Materials Science and Engineering 1 Types of Solids Crystalline material: atoms self-organize in a periodic array Single crystal: atoms are in a repeating or periodic array over the entire extent of the material Polycrystalline material: crystals or grains comprised of many small Amorphous: lacks a systematic atomic arrangement Crystalline Amorphous SiO 2 University of Tennessee, Dept. of Materials Science and Engineering 2

2 Crystal structure To discuss crystalline structures it is useful to consider atoms as being hard spheres with well-defined radii. In this hard-sphere model, the shortest distance between two like atoms is one diameter. We can also consider crystalline structure as a lattice of points at atom/sphere centers. University of Tennessee, Dept. of Materials Science and Engineering 3 Unit Cell The unit cell is the smallest structural unit or building block that can describe the crystal structure. Repetition of the unit cell generates the entire crystal. Example: 2D honeycomb net can be represented by translation of two adjacent atoms that form a unit cell for this 2D crystalline structure Example of 3D crystalline structure: Different choices of unit cells possible, generally choose parallelepiped unit cell with highest level of symmetry University of Tennessee, Dept. of Materials Science and Engineering 4

3 Metallic Crystal Structures Metals are usually (poly)crystalline; although formation of amorphous metals is possible by rapid cooling As we learned in Chapter 2, the atomic bonding in metals is non-directional no restriction on numbers or positions of nearest-neighbor atoms large number of nearest neighbors and dense atomic packing Atom (hard sphere) radius, R, defined by ion core radius - typically nm The most common types of unit cells are the facedcentered cubic (FCC), the body-centered cubic (BCC) and the hexagonal close-packed (HCP). University of Tennessee, Dept. of Materials Science and Engineering 5 The 14 Bravais ;attices grouped into 7 lattice types. The restrictions on the lattice parameters (a,b,c) and the angles of the unit cell are listed for each. What are the most comon? University of Tennessee, Dept. of Materials Science and Engineering 6

4 Miller indices- h,k,l- for naming points in the crystal lattice. The origin has been arbitrarily selected as the bottom left-back corner of the unit cell. University of Tennessee, Dept. of Materials Science and Engineering 7 (a) the coordinates of the six face centers and the center of the cube, (b) the location of the point ¼, ¾, ¼ which is found by starting at the origin and moving a distance a o /4 in the x-direction, then 3 a o /4 in the y-direction, and finally a o /4 in the z-direction. University of Tennessee, Dept. of Materials Science and Engineering 8

5 Positions, Directions, and Planes Z x, y, z u, v, w X Y Lattice Positions and Directions: 1) Always establish an origin 2) Determine the coordinates of the lattice points of interest 3) Translate the vector to the origin if required by drawing a parallel line or move the origin. 4) Subtract the second point from the first: u2-u1,v2-v1,w2-w1 5) Clear fractions and reduce to lowest terms 6) Write direction with square brackets [uvw] 7) Negative directions get a hat. University of Tennessee, Dept. of Materials Science and Engineering 9 Indices of Planes: 1) Identify the locations where the plane intercepts the x, y, z axes as the fractions of the unit cell edge lengths a, b, c. 2) Infinity if the plane is parallel. 3) Take the reciprocal of the intercepts. 4) Clear any fraction but do not reduce to lowest terms. 5) Example: 1/3,1/3,1/3 is (333) not (111)!!! 6) Use parentheses to indicate planes (hkl) again with a hat over the negative indices. 7) Families are indicated by {hkl} Remember Terminology: Defined coordinate system: x, y, z Respective unit cell edge lengths: a, b, c Direction: Denoted by [uvw] Family of direction(s): Denoted by: <uvw> Plane: Denoted by: (hkl) Family of Plane(s): Denoted by: {hkl} University of Tennessee, Dept. of Materials Science and Engineering 10

6 Directions are always perp. to their respective planes, i.e. [111] perp. (111) (for cubic systems) Families of equivalent planes are equal with respect to symmetrical structures, they do not have to be parallel. Equivalent planes must be translated to the correct atomic positions in order to maintain the proper crystal symmetry. Families of directions are equivalent in absolute magnitude. (222) planes are parallel to the (111) planes but not equal. Intercepts for the (222) planes are 1/2,1/2,1/2 Intercepts for the (333) planes are 1/3,1/3,1/3, remember this is in what we call reciprocal space. If you draw out the (333) plane it is parallel to the (111) plane but not equivalent. -b -a (-110) (110) b 0,0,0 a University of Tennessee, Dept. of Materials Science and Engineering 11 Representation of a series each of (a) (001), (b) (110), and (c) (111) crystallographic planes. University of Tennessee, Dept. of Materials Science and Engineering 12

7 Notation for lattice positions. University of Tennessee, Dept. of Materials Science and Engineering 13 Lattice translations connect structurally Equivalent positions (example shown is body centered) in various unit cells. University of Tennessee, Dept. of Materials Science and Engineering 14

8 Family of directions, <111>, representing all body diagonals for adjacent unit cells in the cubic system. University of Tennessee, Dept. of Materials Science and Engineering 15 z y x University of Tennessee, Dept. of Materials Science and Engineering 16

9 Miller-Bravis indicies, (hkil) for the hexagonal System. University of Tennessee, Dept. of Materials Science and Engineering 17 Simple Cubic Rare due to poor packing (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres atoms unit cell APF = volume 4 atom 1 3 π (0.5a)3 a 3 volume unit cell University of Tennessee, Dept. of Materials Science and Engineering 18

10 Face-Centered Cubic (FCC) Crystal Structure (I) Atoms are located at each of the corners and on the centers of all the faces of cubic unit cell Cu, Al, Ag, Au have this crystal structure (Courtesy P.M. Anderson) Two representations of the FCC unit cell University of Tennessee, Dept. of Materials Science and Engineering 19 Face-Centered Cubic Crystal Structure (II) R a The hard spheres or ion cores touch one another across a face diagonal the cube edge length, a= 2R 2 The coordination number, CN = the number of closest neighbors to which an atom is bonded = number of touching atoms, CN = 12 Number of atoms per unit cell, n = 4. (For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m). In FCC unit cell we have: 6 face atoms shared by two cells: 6 x 1/2 = 3 8 corner atoms shared by eight cells: 8 x 1/8 = 1 Atomic packing factor, APF = fraction of volume occupied by hard spheres = (Sum of atomic volumes)/(volume of cell) = 0.74 (maximum possible) University of Tennessee, Dept. of Materials Science and Engineering 20

11 Face-Centered Cubic Crystal Structure (III) Corner and face atoms in the unit cell are equivalent FCC crystal has APF of 0.74, the maximum packing for a system equal-sized spheres FCC is a close-packed structure FCC can be represented by a stack of close-packed planes (planes with highest density of atoms) University of Tennessee, Dept. of Materials Science and Engineering 21 Body-Centered Cubic (BCC) Crystal Structure (I) Atom at each corner and at center of cubic unit cell Cr, α-fe, Mo have this crystal structure The hard spheres touch one another along cube diagonal the cube edge length, a= 4R/ 3 The coordination number, CN = 8 Number of atoms per unit cell, n = 2 Center atom (1) shared by no other cells: 1 x 1 = 1 8 corner atoms shared by eight cells: 8 x 1/8 = 1 Atomic packing factor, APF = 0.68 Corner and center atoms are equivalent R a University of Tennessee, Dept. of Materials Science and Engineering 22

12 Hexagonal Close-Packed Crystal Structure (I) HCP is one more common structure of metallic crystals Six atoms form regular hexagon, surrounding one atom in center. Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes Cd, Mg, Zn, Ti have this crystal structure University of Tennessee, Dept. of Materials Science and Engineering 23 Hexagonal Close-Packed Crystal Structure (II) Unit cell has two lattice parameters a and c. Ideal ratio c/a = The coordination number, CN = 12 (same as in FCC) Number of atoms per unit cell, n = 6. 3 mid-plane atoms shared by no other cells: 3 x 1 = 3 12 hexagonal corner atoms shared by 6 cells: 12 x 1/6 = 2 2 top/bottom plane center atoms shared by 2 cells: 2 x 1/2 = 1 Atomic packing factor, APF = 0.74 (same as in FCC) All atoms are equivalent c a University of Tennessee, Dept. of Materials Science and Engineering 24

13 (b) Stacking of close-packed planes Normal to close-packed planes (a) Stacking of close packed planes Normal to close-packed planes A Close-packed planes A B C Closepacked planes B A A Face centered cubic Hexagonal close-packed Comparing HCP and FCC structures. The difference between the two structures lies in the different stacking sequences. University of Tennessee, Dept. of Materials Science and Engineering 25 Close-packed Structures (FCC and HCP) Both FCC and HCP crystal structures have atomic packing factors of 0.74 (maximum possible value) Both FCC and HCP crystal structures may be generated by the stacking of close-packed planes The difference between the two structures is in the stacking sequence HCP: ABABAB... FCC: ABCABCABC University of Tennessee, Dept. of Materials Science and Engineering 26

14 FCC: Stacking Sequence ABCABCABC... Third plane is placed above the holes of the first plane not covered by the second plane University of Tennessee, Dept. of Materials Science and Engineering 27 HCP: Stacking Sequence ABABAB... Third plane is placed directly above the first plane of atoms University of Tennessee, Dept. of Materials Science and Engineering 28

15 Density and Atom Packing Bulk Measurements Counting atoms within a unit cell Area density in plane # atoms/nm 2 Linear density along direction #atoms/nm University of Tennessee, Dept. of Materials Science and Engineering 29 Density Computations Since the entire crystal can be generated by the repetition of the unit cell, the density of a crystalline material, ρ = the density of the unit cell = (atoms in the unit cell, n ) (mass of an atom, M) / (the volume of the cell, V c ) Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP) Mass of an atom, M = Atomic weight, A, in amu (or g/mol) is given in the periodic table. To translate mass from amu to grams we have to divide the atomic weight in amu by the Avogadro number N A = atoms/mol The volume of the cell, V c = a 3 (FCC and BCC) a = 2R 2 (FCC); a = 4R/ 3 (BCC) where R is the atomic radius Thus, the formula for the density is: na ρ = V c N A Atomic weight and atomic radius of many elements you can find in the table at the back of the textbook front cover. University of Tennessee, Dept. of Materials Science and Engineering 30

16 Polycrystalline Materials In polycrystalline materials, grain orientations are random, so bulk material properties are isotropic Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation University of Tennessee, Dept. of Materials Science and Engineering 31 Polymorphism and Allotropy Some materials may exist in more than one crystal structure, this is called polymorphism. If the material is an elemental solid, it is called allotropy. An example of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon. Pure, solid carbon occurs in three crystalline forms diamond, graphite; and large, hollow fullerenes. Two kinds of fullerenes are shown here: buckminsterfullerene (buckyball) and carbon nanotube. University of Tennessee, Dept. of Materials Science and Engineering 32

17 Single Crystals and Polycrystalline Materials Single crystal: atoms are in a repeating or periodic array over the entire extent of the material Polycrystalline material: comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries. Grain Boundary University of Tennessee, Dept. of Materials Science and Engineering 33 Polycrystalline Materials Atomistic model of a nanocrystalline solid by Mo Li, JHU University of Tennessee, Dept. of Materials Science and Engineering 34

18 Polycrystalline Materials Simulation of annealing of a polycrystalline grain structure University of Tennessee, Dept. of Materials Science and Engineering 35 Anisotropy Different directions in a crystal have different packing. For instance, atoms along the edge of FCC unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals, for instance, the deformation depends on the direction in which a stress is applied. In some polycrystalline materials, grain orientations are random, so bulk material properties are isotropic Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties Polycrystals -Properties may/may not 200 µm vary with direction. -If grains are randomly oriented: isotropic. -If grains are textured, anisotropic. University of Tennessee, Dept. of Materials Science and Engineering 36

19 Anisotropy Properties of crystals may be different along different directions, because atomic periodicities are different. E.g. in single crystal cubic system: <111> <110> <100> Cube edges <110> Face diagonals <111> Body diagonals <100> Modulus of Elasticity (Gpa) Metal [100] [110] [111] Aluminum Copper Iron University of Tennessee, Dept. of Materials Science and Engineering 37 Non-Crystalline (Amorphous) Solids In amorphous solids, there is no long-range order. But amorphous does not mean random, in many cases there is some form of short-range order. Schematic picture of amorphous SiO 2 structure Amorphous structure from simulations by E. H. Brandt University of Tennessee, Dept. of Materials Science and Engineering 38

20 Summary Make sure you understand language and concepts: Allotropy Amorphous Anisotropy Atomic packing factor (APF) Body-centered cubic (BCC) Coordination number Crystal structure Crystalline Face-centered cubic (FCC) Grain Grain boundary Hexagonal close-packed (HCP) Isotropic Lattice parameter Non-crystalline Polycrystalline Polymorphism Single crystal Unit cell University of Tennessee, Dept. of Materials Science and Engineering 39

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