Chapter 3. The structure of crystalline solids 3.1. Crystal structures 3.1.1. Fundamental concepts 3.1.2. Unit cells 3.1.3. Metallic crystal structures 3.1.4. Ceramic crystal structures 3.1.5. Silicate ceramics 3.1.6. Carbon 3.1.7. Polymorphism 3.2. Crystallography 3.2.1. Crystal systems 3.2.2. Crystallographic directions and planes 3.2.3. Linear and planar density 3.2.4. Closed-packed crystal structures 3.3. Crystalline and noncrystalline materials 3.3.1. Single crystal 3.3.2. Polycrystalline materials 3.3.3. X-ray diffraction 3.3.4. Noncrystalline solids
3.1. Crystal structures 3.1.1. Fundamental concepts Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another. A crystalline material is a material that the atoms are situated in a periodic array over large atomic distances. A noncrystalline or amorphous material is a material that does not have a long-range atomic order. Crystal structure is a manner in which atoms, ions, or molecules are spatially arranged. Some properties of crystalline solids depend on the crystal structure. Atoms or ions are considered as atomic hard sphere model (the atoms touch one another). Lattice is a [3] array of points coinciding with atom positions or sphere centers.
3.1. Crystal structures 3.1.2. Unit cell Unit cell is the basic structural unit or building block of the crystal structure. The unit cell defines the crystal structure by virtue of its geometry and the atom positions within Most of unit cells are parallelepipeds of prisms having 3 sets of parallel faces. A unit cell is a representation of the symmetry of the crystal structure which is the highest level of geometrical symmetry. The most relatively simple crystal structures are : Face-centered cubic (FCC), Body-centered cubic (BCC), Hexagonal close-packed (HCP).
3.1. Crystal structures 3.1.3. Metallic crystal structures FCC structure has a cubic geometry with atoms located at each of the corners and the centers of all the cube faces. Many metals have this FCC structure.
The relation between the cube edge length,a and the atomic radius, R: a = 2R 2 The FCC structure has 4 whole atoms [= (8 x 1/8) + (6 x ½)] The coordination number (CN) is a number that an atom touches the nearest neighbour atoms. The coordination number of FCC is 12 The atomic packing factor (APF) vol.of atoms in a unit cell APF = = total unit cell volume V V S C For FCC: V S 4 R 3 16 = (4) π = π R 3 and V 16 R 3 c = 2 3 3 APF = 0.74; which is the maximum packing possible
BCC structure has a cubic unit cell with atoms located at each of the corners and a single atom at the cube center.
The relation between the cube edge length,a and the atomic radius, R: a = 4R 3 The BCC structure has 2 whole atoms [= (8 x 1/8) + (1 x 1)] The coordination number of BCC is 8 For BCC: 4 3 V s = (2) πr = 3 8 3 πr 3 and Vc = 3 64R 3 3 APF = 0.68
HCP structure is the common metallic crystal structure. It has 6 atoms in regular hexagones at the top and the bottom faces and surround a single atom in the center. Three additional atoms are on a plane located between the top and the bottom planes.
The ideal c/a ratio is 1.633 The HCP structure has 6 whole atoms [= (2 x 6 x 1/6) + (2 x ½) + (1 x 3)] The coordination number of HCP is 12 and the APF is 0.74 Metal density ρ = na N A V C where: n = number of atoms associated with each unit cell, A = atomic weight, V c = volume of the unit cell N A = Avogadro s number (6.023 X 10 23 atoms/mol)
Example Copper has an atomic radius of 0.18 nm, an FCC crystal structure, and an atomic weight of 63.5 g/mol. Compute its theoretical density and compare the answer with its measured density. Answer FCC: n = 4 atoms/unit cell; A cu = 63.5 g/mol; and V 16R 3 2 ρ = = n V A C Cu N A [16 (1.28x10 = 8.89 g/cm 3 (4 atom/unit cell)(63.5 g/mol) -8 cm) 3 /unit cell The theoretical density is 8.94 g/cm 3 2](6.02x10 23 c = atoms/mol)
3.1. Crystal structures 3.1.4. Ceramic crystal structures Ceramics are more complex than metals because ceramics are consisted of at least two elements. The atomic bonding is in the range of ionic to covalent. The degree of ionic character depends on the electronegativities of the atoms. (the electronegativity is usually shown in the Periodic Table) For ceramics with predominantly ionic, the atoms are considered as ions. The metallic ions or cations are positively charged The nonmetallic ions or anions are negatively charged. The crystal structure depends on two characteristics: the magnitude of the electrical charge on each of the component ions, the relative sizes of the cations and anions
The 1 st characteritic: the magnitude of the electrical charge on each of the component ions. The cations must be in balance with the anions. The chemical formula of a compound indicates the ratio of cations and anions or the composition that achieves this charge balance. Example: A calcium ion has a +2 charge (Ca 2+ ) CaF A fluorine ion has a 1 charge (F - ) 2 The 2 nd characteristic: the relative sizes of the cations and anions. The sizes of ionic radii of the cations (r C ) and anions (r A ) determined the configuration of a compound. Cations are ordinarily smaller than anions, r C /r A 1
3.1.4. Ceramic crystal structures Most ceramics are composed by an equal number of cations and anions and identified as AX compounds (A refers to the cation and X refers to anion). 3.1.4.1. AX-type crystal structures A. Rock salt structures The most common AX crystal structures is the sodium chloride (NaCl) or rock salt. Two interpenetrating FCC lattices, one composed of the cations (Na + ), and the other of anions (Cl - ). The coordination number of both cations and anions is 6. The r C /r A is between 0.414 0.732.
Some common rock salt structures are NaCl, MgO, MnS, LiF, and FeO.
B. Cesium chloride structures The anions are situated a each of the corners of a cube, whereas the cube center is a single cation. This is not a BCC crystal structure!! The coordination number of both cations and anions is 8.
C. Zinc blende or sphalerite structure (Mineralogical term for zinc sulfide (ZnS) S atoms are located in all corners and face positions of a cubic, while Zn atoms fill the interior in tetrahedral positions in covalent bonding. The coordination number for both cations and anions is 4 (tetrahedrally coordinated). If the Zn and S atom positions are reversed, the same structure is obtained. Some common zinc blende crystals are ZnS, ZnTe, and SiC
3.1.4.2. A m X p -type crystal structures This structure has dissimilar charges of cations and anions. The chemical formula is A m X p where m and/or p 1. Example : CaF 2 The r C /r A is 0.8 and the coordination number is 8. Calcium ions are located at the center of a cube and fluorine ions at the corners. One unit cell consists of 8 cubes. Some other compounds are UO 2, PuO 2, and ThO 2.
3.1.4.3. A m B n X p -type crystal structures or perovskite crystal structure Some compounds have more than one type of cations, they are identified as A and B. The chemical formula is A m B n X p - Example: BaTiO 3. Ba2+ ions are located at eight corners of a cube, a single To4+ is at the cube center and O2- ions are situated at the center of each of the six faces.
Ceramic density ρ = n' ( A + A ) V C C N A A where: n = the number of formula units within the unit cell, ΣA C = the sum of the atomic weights of all cations in the formula unit, ΣA A = the sum of the atomic weights of all anions in the formula unit, V C = the unit cell volume, N A = Avogadro s number = 6.023 x 10 23 formula units/mol.
Example Calculated the theoretical density for sodium chloride. The atomic weight of sodium and chloride, A Na and A Cl are given as 22.99 g/mol and 35.45 g/mol respectively. The ionic radii of sodium and chloride, r Na+ and r Cl- are given as 0.102 nm and 0.181 nm respectively. Compare the result with its measured density? n = 4 ΣA C = A Na = 22.99 g/mol, ΣA A = A Cl = 35.45 g/mol, a = (2 r Na+ + 2 r Cl- ) m V C = a 3 = (2r Na+ + 2r Cl- ) 3 m 3 = [(0.204 x 10-9 )+0.362 x 10-9 )] 3 m 3 = 559.276 10-9 m 3 ρ ρ ρ = = = (2r n' ( A + A ) Na Cl 3 Na+ + 2rCl- ) NA 4(22.99 + 35.45) (559.276X10 2.14g/cm 3 23 )(6.023X10 23 )
3.1. Crystal structures 3.1.5. Silicate ceramics Silicates are materials composed primarily of silicon and oxygen. The basic unit of silicates is SiO 4 4- tetrahedron, which is each atom of silicon (located in the corners of tetrahedron) is bonded to four oxygen atoms (situated at the center of tetrahedron). Some common silicates are rocks, clays, sand, and a bulk of soils. Silicates are not considered to be ionic because interatomic covalent Si-O bonds are very strong. Silicate structures vary in different arrangements as each oxygen atom requires an extra electron to achieve a stable electronic structure. The arrangements could be [1] or [2] or [3] dimensional structures.
3.1.5.1. Silica Silica or silicon dioxide (SiO 2 ) is the most simple silicate. Oxygen atom at every corner of each tetrahedron shares the oxygen atoms with the adjacent tetrahedral and this configuration builds a [3] network structure. This material is electrically neutral and all atoms have stable electronic structures.
A crystalline structure is formed if these tetrahedra are arrayed in a regular and ordered manner. There are 3 primary polymorphic crystalline forms of silica, they are: quarts, cristobalite, and tridymite. The structures are complicated and relatively open so that they have low densities. (eg ρ quartz 2.65 g/cm 3 ). The Si-O interatomic bonding is strong so that they have relatively high melting temperature. (eg T mquartz 1710 ºC). A noncrystalline solid or glass structure can also be formed from silica.
3.1.5.2. The Silicates Oxygen atom at every corner of each tetrahedron shares the other tetrahedral to form complex structures. Some positively charged cations such as Ca 2+,Mg, and Al can be part of the structures. These cations compensate the negative charges of SiO 4 4- to form a neutral compound and to build an ionic bonding between the SiO 4 4- tetrahedral together. A. Simple silicates
B. Layered silicates A [2] sheet or layered structure can also be produced by the sharing of 3 oxygen ions in each of tetrahedral. The repeating unit formula is (Si 2 O 5 ) 2- The net negative charge is associated with the unbound oxygen atoms ( layer). The excess of cations is associated with the 2 nd layer. Net charge: neutral. Basic structure of clays and other minerals.
One of the most common clay minerals and simple two-layer silicate sheet structure is kaolinite. Its formula is Al 2 (Si 2 O 5 )(OH) 4 The OH - and the O 2- ions in the anion midplane has a strong ioniccovalent bonding, whereas the adjacent sheets have weak van der Waals bondings.
3.1. Crystal structures 3.1.6. Carbon Carbon is an element that exsist in various forms, from polymorphic to amorphous sates. Carbon is not classified as metal, ceramic or polymer, however sometimes it is classified as a ceramic polymorphic 3.1.6.1. Diamond Diamond is a metastable carbon polymorph at RT and atmospheric pressusre. The crystal structure is a variant of the zinc blende. It is called the diamond cubic crystal structure.
3.1.6.2. Graphite Graphite is more stable than diamond at ambient temperature and pressure. The crystal structure is composed of layers of hexagonally arranged carbon atoms. In the layers, each carbon atom is bonded to three coplanar neighbour atoms by strong covalent bonds. Between layers the electron is bonded by a weak van der Waals bond.
3.1.6.3. Fullerenes and carbon nanotubes Fullerenes is a polymorphic of carbon (1985). The structure consists of a hollow spherical cluster of 60 carbon atoms and is called a C60 molecule. A molecule is composed of groups of carbon atoms that are bonded to One another to from both hexagon (six-carbon atom) and pentagon (five-carbon atom) geometrical configurations. Its commercial name is buckminsterfullerene. Invented by R. Buckminster Fuller (1985) Soccer ball symmetry 20 hexagon and 12 pentagon are arrayed in such way that no two pentagons share a common side.
Carbon nanotube structure consists of a single sheet of graphite, rolled into a tube & both ends are capped with fullerene hemispheres. The nano represents that the tube diameter is on the order of a nanometer. It has a very good aspect ratio; which the length of the molecule is much greater than its diameter. Special properties of CNT: - Extremely strong and stiff, relatively ductile - Relatively low density - Unique and structure-sensitive electrical characteristics
Some properties: Tensile strength 50 200 GPa Elastic modulus 10 3 GPa Fracture strain 5% - 20% Another form: Multiple-walled CNT (consists of concentric cylinders) 3.1.7. Polymorphism Polymorphism is a phenomenon when some metals and nonmetals may have more than one crystal structure. (In elemental solids, it is called allotropy). Example: Graphite is stable at ambient condition, whereas diamond is formed at extremely high pressures. Iron has a BCC structure at RT and FCC structure at 912 C.