1. Symmetry in crystals

2. Infinitely repeating lattices

4. An integral number of unit translations along any axis will arrive at an identical point.

5. A unit translation along any axis will arrive at an identical point

6. The composition of each unit should be identical.

7. A unit translation parallel to any axis will arrive at an identical point

10. Face centered lattice

12. Unit Cell

13. 3 axes, a, b, c and 3 angles , , and 

18. 4 3-fold axes along diagonals

20. 4-fold axis

22. 3 2-fold axes

24. 1 6-fold axis 6-fold

26. A cube with 1 diagonal shortened or lengthened.

27. 3-fold axis

29. 1 2-fold axis 2-fold

30. 2-fold axis

31. 3-fold axis 4-fold axis 6-fold axis

36. Symmetry in Crystals

37. Rotational symmetry

38. Possible: 2, 3, 4, 6 - fold axes

39. Rotational inversion

40. Mirror plane

41. Screw axes: a combination of rotation and translation.

42. Screw axes: a combination of rotation and translation. 2 1 screw = 180 o rotation + 1/2 cell translation

43. 3 1 screw = 120 o rotation + 1/3 unit translation

45. Glide plane: a combination of mirror and translational symmetry.

46. Glide plane: a combination of mirror and translational symmetry. 1/2 unit translation

47. Given the 7 crystal systems and various symmetry operations, the number of ways a continuously repeating lattice can be formed is limited.

48. Theoretical studies of the geometries of crystals completed in 1890 demonstrated that there are 230 ways to put together an infinitely repeating lattice.

49. Unit Cell

50. Space group P1

51. Unit Cell P1 = primitive cell + inversion center

52. Unit Cell P1 = primitive cell + inversion center

53. Unit Cell x, y, z = 0, 0, 0

54. Unit Cell x, y, z = 1, 0, 0

55. Unit Cell x, y, z = 0, 1, 0

56. Unit Cell x, y, z = 0, 0, 1

57. Unit Cell x, y, z = 1, 0, 1

58. Unit Cell x, y, z = 1, 1, 1

59. Unit Cell P1 = primitive cell + inversion center

60. Inversion Center Cartesian Coordinates: x, y, z 0, 0, 0 -x, -y, -z -0, -0, -0 Fractional coordinates: the fraction one must move along each axis to arrive at a point.

61. Inversion Center Cartesian Coordinates: x, y, z 0, 0, 0 1, 0, 0 -x, -y, -z -0, -0, -0 -1, -0, -0

62. Inversion Center Cartesian Coordinates: x, y, z 0, 0, 0 1, 0, 0 -x, -y, -z -0, -0, -0 -1, -0, -0 An integral number of unit translations results in an identical point in the lattice.

63. Unit Cell P1 = primitive cell + inversion center 1/2, 1/2, 1/2

64. Inversion Center Cartesian Coordinates: x, y, z 0, 0, 0 1/2, 1/2, 1/2 -x, -y, -z -0, -0, -0 -1/2, -1/2, -1/2 An integral number of unit translations results in an identical point in the lattice.

65. Unit Cell P1 = primitive cell + inversion center 1, 1, 1/2

66. Inversion Center Cartesian Coordinates: x, y, z 0, 0, 0 1, 1, 1/2 -x, -y, -z -0, -0, -0 -1, -1, -1/2 An integral number of unit translations results in an identical point in the lattice.

67. What causes crystals to form and take a particular structure?

68. Strong Forces: Electrostatic forces in ionic crystals.

69. NaCl

70. + -

71. NaCl + - ++ ++ + + -- -- --

74. NaCl ionic bond energy is 785 kj/mol.

75. NaCl CsCl

76. Na + 1.00 Å Cl - 1.81 Å Cs + 1.69 Å

77. NaCl CsCl Two different cells; same charges; same stoichiometry.

78. Determining the contents of the unit cell.

79. NaCl Ion within cell = 1 per cell

80. NaCl Ion on face of cell = 1/2 per cell Ion within cell = 1 per cell (shared with 2 cells)

81. NaCl Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion within cell = 1 per cell (shared with 4 cells)

82. NaCl Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell Ion within cell = 1 per cell (shared with 8 cells)

83. NaCl Ion on face of cell = 1/2 per cell 0 6 Ion on edge of cell = 1/4 per cell 12 0 Ion at corner of cell = 1/8 per cell 0 8 Na + Cl - 1 0 Ion within cell = 1 per cell

84. NaCl Ion on face of cell = 1/2 per cell 0 6 Ion on edge of cell = 1/4 per cell 12 0 Ion at corner of cell = 1/8 per cell 0 8 Na + Cl - 1 0 Total ions in cell: Na + Cl - 1 3 Ion within cell = 1 per cell

85. NaCl Ion on face of cell = 1/2 per cell 0 6 Ion on edge of cell = 1/4 per cell 12 0 Ion at corner of cell = 1/8 per cell 0 8 Na + Cl - 1 0 Total ions in cell: Na + Cl - 1 3 3 1 Ion within cell = 1 per cell

86. NaCl Ion on face of cell = 1/2 per cell 0 6 Ion on edge of cell = 1/4 per cell 12 0 Ion at corner of cell = 1/8 per cell 0 8 Na + Cl - 1 0 Total ions in cell: Na + Cl - 1 3 3 1 Z = 4 Ion within cell = 1 per cell

87. Determining ionic radii using crystal structures.

90. CsCl

91. Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell Ion within cell = 1 per cell

92. CsCl Ion on face of cell = 1/2 per cell 0 0 Ion on edge of cell = 1/4 per cell 0 0 Ion at corner of cell = 1/8 per cell 0 8 1 0 Cs + Cl - Ion within cell = 1 per cell

93. CsCl Ion on face of cell = 1/2 per cell 0 0 Ion on edge of cell = 1/4 per cell 0 0 Ion at corner of cell = 1/8 per cell 0 8 1 0 Cs + Cl - Z = 1 Ion within cell = 1 per cell

94. Ionic crystals are held together by strong electrostatic forces. The crystal unit cell is influenced by ionic sizes.

95. CaCl 2

96. Ion within cell = 1 per cell Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell CaCl 2

97. Ion within cell = 1 per cell Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell CaCl 2

98. Ion within cell = 1 per cell Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell CaCl 2

99. Ion within cell = 1 per cell Ion on face of cell = 1/2 per cell Ion on edge of cell = 1/4 per cell Ion at corner of cell = 1/8 per cell CaCl 2

100. Ion within cell = 1 per cell 1 2 Ion on face of cell = 1/2 per cell 0 4 Ion on edge of cell = 1/4 per cell 0 0 Ion at corner of cell = 1/8 per cell 8 0 CaCl 2 Ca 2+ Cl -

101. Ion within cell = 1 per cell 1 2 Ion on face of cell = 1/2 per cell 0 4 Ion on edge of cell = 1/4 per cell 0 0 Ion at corner of cell = 1/8 per cell 8 0 CaCl 2 Ca 2+ Cl - Z = 2

102. Diamond

103. The strengths of chemical bonds: kJ/mol Weak < 200 Average Strong >800

104. Diamond C - C covalent bond = 1.544 Å Bond enthalpy 348 kJ/mol

105. Diamond C - C covalent bond = 1.544 Å Bond enthalpy 348 kJ/mol

106. Diamond Atom within cell = 1 per cell Atom on face of cell = 1/2 per cell Atom on edge of cell = 1/4 per cell Atom at corner of cell = 1/8 per cell

107. Diamond Atom within cell = 1 per cell 4 Atom on face of cell = 1/2 per cell Atom on edge of cell = 1/4 per cell Atom at corner of cell = 1/8 per cell C

108. Diamond Atom within cell = 1 per cell 4 Atom on face of cell = 1/2 per cell 6 Atom on edge of cell = 1/4 per cell Atom at corner of cell = 1/8 per cell C

109. Diamond Atom within cell = 1 per cell 4 Atom on face of cell = 1/2 per cell 6 Atom on edge of cell = 1/4 per cell 0 Atom at corner of cell = 1/8 per cell C

110. Diamond Atom within cell = 1 per cell 4 Atom on face of cell = 1/2 per cell 6 Atom on edge of cell = 1/4 per cell 0 Atom at corner of cell = 1/8 per cell 8 C

111. Diamond Atom within cell = 1 per cell 4 Atom on face of cell = 1/2 per cell 6 Atom on edge of cell = 1/4 per cell 0 Atom at corner of cell = 1/8 per cell 8 C Z = 8

112. Molecular Crystals

113. Molecular Crystals: Consist of repeating arrays of molecules and/or ions.

116. C 17 H 24 NO 2 + Cl -. 3 H 2 O

117. V = 974.45 Å 3 C 17 H 24 NO 2 + Cl -. 3 H 2 O FW = 363.87 g/mol Z = 2 Density = 727.74 g 5866.19 x 10 -1 cm 3 Density = 363.87 g (2) 974.45 x 10 -24 x 6.02 x 10 23 = 1.241 g/cm 3

118. C 17 H 24 NO 2 + Cl -. 3 H 2 O

120. Although Z = 2, the unit cell contains portions of a number of molecules.

121. Cl -

123. H2OH2O

124. H2OH2O Hydrogen bonds Cl OH 2

126. Hydrogen bond

128. Model with atoms having VDW radii.

130. C 17 H 24 NO 2 + Cl -. 3 H 2 O Although this material is ionic, the + and - charges are not close enough to contribute to the formation of the crystal.

131. Molecular crystals tend to be held together by forces weaker than chemical bonds. van der Waal’s forces are always a factor. Hydrogen bonding is often present.

133. A layer in an ionic solid with ions of similar radii.