Symmetry in crystals. Infinitely repeating lattices.

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Presentation transcript:

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

3-fold axis 4-fold axis 6-fold axis

Symmetry in Crystals

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. 2 1 screw = 180 o rotation + 1/2 cell translation

3 1 screw = 120 o 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

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 one must 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 ionic bond energy is 785 kj/mol.

NaCl CsCl

Na Å Cl Å Cs 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 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 Total ions in cell: Na + Cl 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 Total ions in cell: Na + Cl 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 Total ions in cell: Na + Cl Z = 4 Ion within cell = 1 per cell

Determining ionic radii using crystal structures.

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 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 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.

CaCl 2

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

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

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

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

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 -

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

Diamond

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

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

Diamond C - C covalent bond = 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.

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

V = Å 3 C 17 H 24 NO 2 + Cl -. 3 H 2 O FW = g/mol Z = 2 Density = g x cm 3 Density = g (2) x x 6.02 x = g/cm 3

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

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

Cl -

H2OH2O

H2OH2O Hydrogen bonds Cl OH 2

Hydrogen bond

Model with atoms having VDW radii.

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.

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 ions of similar radii.