无机化学(7.1) The Crystalline Solid State (Part 1)
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– 14 possible crystal structures (Bravais lattices) – Spatial positions of the repeating units
Crystal Families, Systems, and Lattices
Crystal family Crystal system Triclinic Monoclinic Required symmetries of Point groups Space groups Bravais lattices Lattice system point group None 2 2 1 Triclinic One 2-fold axis of rotation or one 3 13 2 Monoclinic mirror plane Three 2-fold axes of rotation or one 2fold axis of rotation 3 59 4 Orthorhombic and two mirror planes One 4-fold axis of 7 68 2 Tetragonal rotation One 3-fold axis of 7 1 Rhombohedral 5 rotation 18 One 6-fold axis of 1 Hexagonal 7 27 rotation Four 3-fold axes of 5 36 3 Cubic rotation 32 230 14 7
Orthorhombic
Tetragonal Trigonal Hexagonal Hexagonal Cubic Total: 6 7
Formulas and Structures
• Unit Cell = simplest repeating unit of the regular crystalline array • Bravais Lattices = 14 possible basic crystal structure unit cell types
• Dense, ordered packing
Dense, ordered packed structures tend to have lower energies.
Crystalline Solid State • Many more “molecules” in the solid state.
Volume of atoms in unit cell* Volume of unit cell
APF =
*assume hard spheres
Cubic Structures
Primitive Cubic is the simplest type To fully describe: length of side, 90o angles, Lattice point: (0,0,0) 8 x 1/8 = 1 atom in the unit cell each atom surrounded by 6 others (Coordination Number = CN = 6) » Not efficiently packed; only 52.4% (??) of volume is occupied » Vacant space at center with CN = 8 0.73r (??) sphere would fit here – » » » »
Amorphous vs. Crystalline Solid
Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline
14
Lattice Points
– Lattice Points = positions of atoms needed to generate the whole crystal, for example » Body Centered Cubic: (0,0,0) = origin (½ , ½, ½) = center » All other atoms can be generated from these 2 by moving them exactly one cell
The Cubic Unit Cell (or Primitive) • 1 atom per unit cell (how?). • What is the coordination number? Volume occupied (Atomic Packing Factor)? • Let’s calculate the length of the edge. What size of sphere would fit into the hole?
Si
crystalline SiO2 Oxygen
noncrystalline SiO2
Amorphous vs. Crystalline Solid
• Non dense, random packing
Energy typical neighbor bond length typical neighbor bond energy r Energy typical neighbor bond length typical neighbor bond energy r
Metallic Crystal Structures
• How can we stack metal atoms to minimize empty space? 2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures
Metallic Crystal Structures
• Tend to be densely packed (Nature hates a vacuum!) . • Reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. - Electron cloud shields cores from each other • Have the simplest crystal structures. We will examine three such structures...
Formulas and Structures
• Unit Cell = simplest repeating unit of the regular crystalline array • Bravais Lattices = 14 possible basic crystal structure unit cell types
István Lagzi, et al. Langmuir 2014, 30, 9251–9255.
Crystalline Solid State
A Simple Description on Crystallography and Structural Chemistry
References
Advanced Structural Inorganic Chemistry Wai-Kee Li, Gong-Du Zhou and Thomas C. W. Mak 2008, Oxford University Press The Basics of Crystallography and Diffraction, 3rd Ed. Christopher Hammond 2009, Oxford University Press Crystals - Growth, Morphology, and Perfection Ichiro Sunagawa 2005, Cambridge University Press
ቤተ መጻሕፍቲ ባይዱ
Shared Atoms
– Atoms on corners and edges and in faces are shared between unit cells » Rectangular corners shared by 8 unit cells (8 x 1/8 = 1 total atom in cell) » Other corners shared unequally, but still contribute 1 atom to unit cell » Edges shared by 4 cells, contribute (4 x ¼ = 1 atom to unit cell) » Faces shared by 2 cells, contribute ½ atom to each
Maze Solving Using Acid-Base Chemistry
aqueous solution of KOH (0.05 M) containing 0.2 vol% 2-hexyldecanoic acid
S: phenol red dye powder (∼0.3 mg)
10s
60s E: agarose gel soaked in a solution of HCl (1.0 M) poly(dimethylsiloxane) (PDMS)
– We will focus on crystalline solids composed of atoms or ions.
• Unit cell – structural component that, when repeated in all directions, results in a macroscopic (observable) crystal.
a
– Rare due to low packing density (only Po has this structure)
Body-Centered Cubic (bcc)
One more atom is added to the center of the cube Size of unit cell must increase over simple cubic Diagonal across center = 4r (r = radius of one atom) Corner atoms not in contact with each other due to cell size expansion – Side = 2.31r (??), Not efficiently packed; only 68.0% (??) of volume is occupied – Unit cell contains 1(1) + 8(1/8) = 2 atoms – Lattice Points: (0,0,0) and ( ½ , ½ , ½ ) – – – –
Cubic Structures
– Primitive Cubic is the simplest type atoms unit cell R=0.5a close-packed directions contains 8 x 1/8 = 1 atom/unit cell APF = volume atom 4 3 (0.5a) 1 3 a3 volume unit cell
Crystal Families, Systems, and Lattices
Crystal family Crystal system Triclinic Monoclinic Required symmetries of Point groups Space groups Bravais lattices Lattice system point group None 2 2 1 Triclinic One 2-fold axis of rotation or one 3 13 2 Monoclinic mirror plane Three 2-fold axes of rotation or one 2fold axis of rotation 3 59 4 Orthorhombic and two mirror planes One 4-fold axis of 7 68 2 Tetragonal rotation One 3-fold axis of 7 1 Rhombohedral 5 rotation 18 One 6-fold axis of 1 Hexagonal 7 27 rotation Four 3-fold axes of 5 36 3 Cubic rotation 32 230 14 7
Orthorhombic
Tetragonal Trigonal Hexagonal Hexagonal Cubic Total: 6 7
Formulas and Structures
• Unit Cell = simplest repeating unit of the regular crystalline array • Bravais Lattices = 14 possible basic crystal structure unit cell types
• Dense, ordered packing
Dense, ordered packed structures tend to have lower energies.
Crystalline Solid State • Many more “molecules” in the solid state.
Volume of atoms in unit cell* Volume of unit cell
APF =
*assume hard spheres
Cubic Structures
Primitive Cubic is the simplest type To fully describe: length of side, 90o angles, Lattice point: (0,0,0) 8 x 1/8 = 1 atom in the unit cell each atom surrounded by 6 others (Coordination Number = CN = 6) » Not efficiently packed; only 52.4% (??) of volume is occupied » Vacant space at center with CN = 8 0.73r (??) sphere would fit here – » » » »
Amorphous vs. Crystalline Solid
Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline
14
Lattice Points
– Lattice Points = positions of atoms needed to generate the whole crystal, for example » Body Centered Cubic: (0,0,0) = origin (½ , ½, ½) = center » All other atoms can be generated from these 2 by moving them exactly one cell
The Cubic Unit Cell (or Primitive) • 1 atom per unit cell (how?). • What is the coordination number? Volume occupied (Atomic Packing Factor)? • Let’s calculate the length of the edge. What size of sphere would fit into the hole?
Si
crystalline SiO2 Oxygen
noncrystalline SiO2
Amorphous vs. Crystalline Solid
• Non dense, random packing
Energy typical neighbor bond length typical neighbor bond energy r Energy typical neighbor bond length typical neighbor bond energy r
Metallic Crystal Structures
• How can we stack metal atoms to minimize empty space? 2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures
Metallic Crystal Structures
• Tend to be densely packed (Nature hates a vacuum!) . • Reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. - Electron cloud shields cores from each other • Have the simplest crystal structures. We will examine three such structures...
Formulas and Structures
• Unit Cell = simplest repeating unit of the regular crystalline array • Bravais Lattices = 14 possible basic crystal structure unit cell types
István Lagzi, et al. Langmuir 2014, 30, 9251–9255.
Crystalline Solid State
A Simple Description on Crystallography and Structural Chemistry
References
Advanced Structural Inorganic Chemistry Wai-Kee Li, Gong-Du Zhou and Thomas C. W. Mak 2008, Oxford University Press The Basics of Crystallography and Diffraction, 3rd Ed. Christopher Hammond 2009, Oxford University Press Crystals - Growth, Morphology, and Perfection Ichiro Sunagawa 2005, Cambridge University Press
ቤተ መጻሕፍቲ ባይዱ
Shared Atoms
– Atoms on corners and edges and in faces are shared between unit cells » Rectangular corners shared by 8 unit cells (8 x 1/8 = 1 total atom in cell) » Other corners shared unequally, but still contribute 1 atom to unit cell » Edges shared by 4 cells, contribute (4 x ¼ = 1 atom to unit cell) » Faces shared by 2 cells, contribute ½ atom to each
Maze Solving Using Acid-Base Chemistry
aqueous solution of KOH (0.05 M) containing 0.2 vol% 2-hexyldecanoic acid
S: phenol red dye powder (∼0.3 mg)
10s
60s E: agarose gel soaked in a solution of HCl (1.0 M) poly(dimethylsiloxane) (PDMS)
– We will focus on crystalline solids composed of atoms or ions.
• Unit cell – structural component that, when repeated in all directions, results in a macroscopic (observable) crystal.
a
– Rare due to low packing density (only Po has this structure)
Body-Centered Cubic (bcc)
One more atom is added to the center of the cube Size of unit cell must increase over simple cubic Diagonal across center = 4r (r = radius of one atom) Corner atoms not in contact with each other due to cell size expansion – Side = 2.31r (??), Not efficiently packed; only 68.0% (??) of volume is occupied – Unit cell contains 1(1) + 8(1/8) = 2 atoms – Lattice Points: (0,0,0) and ( ½ , ½ , ½ ) – – – –
Cubic Structures
– Primitive Cubic is the simplest type atoms unit cell R=0.5a close-packed directions contains 8 x 1/8 = 1 atom/unit cell APF = volume atom 4 3 (0.5a) 1 3 a3 volume unit cell