# 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 Face-centered cubic Body-centered cubic Hexagonal close-packed Close packed crystal structures Density computations Types of solids Single crystal Polycrystalline Amorphous Crystallography Not Covered / Not Tested 3.15 Anisotropy Not Covered / Not Tested 3.16 Diffraction Not Covered / Not Tested 1

2 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: disordered lack of a systematic atomic arrangement Crystalline Amorphous SiO 2 2

3 Crystal structures Why do atoms assemble into ordered structures (crystals)? Energy of interatomic bond 0 Interatomic distance Let s consider nondirectional bonding (like in metals) Energy of the crystal < Energy of the amorphous solid 3

4 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 of the hard sphere. 2R - hard-sphere model We can also consider crystalline structure as a lattice of points at atom/sphere centers. 4

5 Unit Cell The unit cell is a 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, we will consider parallelepiped unit cell with highest level of symmetry 5

6 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 Atomic (hard sphere) radius, R, defined by ion core radius - typically nm The most common types of unit cells are faced-centered cubic (FCC) body-centered cubic (BCC) hexagonal close-packed (HCP). 6

7 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 Two representations of the FCC unit cell 7

8 Face-Centered Cubic Crystal Structure (II) R The hard spheres 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 1/2 = 3 8 corner atoms shared by eight cells: 8 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) a 8

9 Face-Centered Cubic Crystal Structure (III) Let s calculate the atomic packing factor for FCC crystal R a = 2R 2 a APF = (Sum of atomic volumes)/(volume of unit cell) 4 4 πr 3 Volume of 4 hard spheres in the unit cell: 3 Volume of the unit cell: a 3 = 16 R APF = πr 16R 3 2 = π 3 2 = 3 maximum possible packing of hard spheres

10 Face-Centered Cubic Crystal Structure (IV) 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) 10

11 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 11

12 Body-Centered Cubic Crystal Structure (II) 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 a 12

13 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 13

14 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 14

15 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 15

16 FCC: Stacking Sequence ABCABCABC... Third plane is placed above the holes of the first plane not covered by the second plane 16

17 HCP: Stacking Sequence ABABAB... Third plane is placed directly above the first plane of atoms 17

18 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: ρ = na 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. 18

19 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. 19

20 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 20

21 Polycrystalline Materials Atomistic model of a nanocrystalline solid by Mo Li, JHU 21

22 Polycrystalline Materials Simulation of annealing of a polycrystalline grain structure from (link is dead) 22

23 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 23

24 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 24

25 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 25

26 Homework #1: 2.14, 2.15, 2.20, 3.7, and 3.17 Due date: Monday, September 6. Reading for next class: Chapter 4: Imperfections in Solids Point defects (vacancies, interstitials) Dislocations (edge, screw) Grain boundaries (tilt, twist) Weight and atomic composition Optional reading (Parts that are not covered / not tested): Microscopy 4.11 Grain size determination 26

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