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Symmetry in crystals. Infinitely repeating lattices. An integral number of unit translations along any axis will arrive at an identical point. A unit translation along any axis will arrive at an identical point. The composition of each unit should be identical. - PowerPoint PPT Presentation
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Symmetry in crystals
Infinitely repeating lattices
An integral number of unit translations
along any axis will arrive at
an identical point.
A unit translation along any axis will
arrive at an identical point
The composition of each unit should
be identical.
A unit translation parallel to any axis will
arrive at an identical point
Face centered lattice
Unit Cell
3 axes, a, b, c and 3 angles , , and
4 3-fold axes along diagonals
4-fold axis
3 2-fold axes
1 6-fold axis
6-fold
A cube with 1 diagonal shortened or lengthened.
3-fold axis
1 2-fold axis
2-fold
2-fold axis
2-fold axis3-fold axis4-fold axis6-fold axis
Symmetry in Crystals
Rotational symmetry
Rotational symmetry
Possible: 2, 3, 4, 6 - fold axes
Rotational inversion
Mirror plane
Screw axes:
a combination
of rotation and
translation.
Screw axes:
a combination
of rotation and
translation.
21 screw = 180o rotation + 1/2 cell translation
31 screw =
120o rotation +
1/3 unit translation
Glide plane:
a combination
of mirror and
translational
symmetry.
Glide plane:
a combination
of mirror and
translational
symmetry.
1/2 unit translation
Given the 7 crystal systems
and various symmetry operations,
the number of ways a continuously
repeating lattice can be formed is
limited.
Theoretical studies of the
geometries of crystals
completed in 1890 demonstrated
that there are 230 ways to put
together an infinitely repeating
lattice.
Unit Cell
Unit Cell
Space group P1
Unit Cell
P1 = primitive cell + inversion center
Unit Cell
P1 = primitive cell + inversion center
Unit Cell
x, y, z = 0, 0, 0
Unit Cell
x, y, z = 1, 0, 0
Unit Cell
x, y, z = 0, 1, 0
Unit Cell
x, y, z = 0, 0, 1
Unit Cell
x, y, z = 1, 0, 1
Unit Cell
x, y, z = 1, 1, 1
Unit Cell
P1 = primitive cell + inversion center
Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0
-x, -y, -z -0, -0, -0
Fractional coordinates: the fraction onemust move along each axis to arrive at a point.
Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0 1, 0, 0
-x, -y, -z -0, -0, -0 -1, -0, -0
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.
Unit Cell
P1 = primitive cell + inversion center
1/2, 1/2, 1/2
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.
Unit Cell
P1 = primitive cell + inversion center
1, 1, 1/2
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.
What causes crystals to
form and take a particular structure?
Strong Forces:
Electrostatic forces in ionic crystals.
NaCl
NaCl+ -
NaCl+ -
+ +
+ +
++
- -
--
- -
NaCl ionic bond energy
is 785 kj/mol.
NaCl CsCl
NaCl CsCl
Na+ 1.00 Å Cl- 1.81 Å Cs+ 1.69 Å
NaCl CsCl
Two different cells; same charges;
same stoichiometry.
Determining the contents of
the unit cell.
NaCl
Ion within cell = 1 per cell
NaCl
Ion on face of cell = 1/2 per cell
Ion within cell = 1 per cell
(shared with 2 cells)
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)
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)
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 0Ion within cell = 1 per cell
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
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
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
Determining ionic radii using
crystal structures.
CsCl
CsCl
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
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
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
Ionic crystals are held together
by strong electrostatic forces.
The crystal unit cell is influenced
by ionic sizes.
CaCl2
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
CaCl2
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
CaCl2
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
CaCl2
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
CaCl2
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
CaCl2
Ca2+ Cl-
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
CaCl2
Ca2+ Cl-
Z = 2
Diamond
The strengths of chemical bonds: kJ/mol Weak < 200
Average <500>
Strong >800
Diamond
C - C covalent bond = 1.544 Å
Bond enthalpy 348 kJ/mol
Diamond
C - C covalent bond = 1.544 Å
Bond enthalpy 348 kJ/mol
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
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
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
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
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
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
Molecular Crystals
Molecular Crystals:
Consist of repeating arrays
of molecules and/or ions.
C17H24NO2+ Cl- . 3 H2O
V = 974.45 Å3
C17H24NO2+ Cl- . 3 H2O FW = 363.87 g/mol
Z = 2
Density =727.74 g
5866.19 x 10-1cm3
Density =363.87 g (2)
974.45 x 10-24 x 6.02 x 1023
= 1.241 g/cm3
C17H24NO2+ Cl- . 3 H2O
Although Z = 2, the unit cell containsportions of a number of molecules.
Cl-
Cl-
Cl-
H2O
Cl-
H2O
Hydrogen bondsCl OH2
Hydrogen bond
Model with atoms having VDW radii.
C17H24NO2+ Cl- . 3 H2O
Although this material is ionic, the + and - chargesare not close enough tocontribute to the formationof the crystal.
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.
A layer in an ionic solid with ionsof similar radii.