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

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


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


x, y, z 0, 0, 0 1, 0, 0

-x, -y, -z -0, -0, -0 -1, -0, -0

Inversion Center


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


1/2, 1/2, 1/2

Inversion Center


x, y, z 0, 0, 0 1/2, 1/2, 1/2

-x, -y, -z -0, -0, -0 -1/2, -1/2, -1/2


Unit Cell


1, 1, 1/2

Inversion Center


x, y, z 0, 0, 0 1, 1, 1/2

-x, -y, -z -0, -0, -0 -1, -1, -1/2


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

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


(shared with 2 cells)

NaCl


Ion on edge of cell = 1/4 per cell



NaCl



Ion at corner of cell = 1/8 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 0Ion within cell = 1 per cell

NaCl




Na+ Cl-

1 0

Total ions in cell: Na+ Cl-

1 3


NaCl




Na+ Cl-

1 0


1 3 3 1


NaCl




Na+ Cl-

1 0


1 3 3 1

Z = 4


Determining ionic radii using

crystal structures.

CsCl





CsCl




1 0

Cs+ Cl-


CsCl




1 0

Cs+ Cl-

Z = 1


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 1 2




CaCl2

Ca2+ Cl-

Ion within cell = 1 per cell 1 2




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

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



C

Diamond


Atom on face of cell = 1/2 per cell 6



C

Diamond



Atom on edge of cell = 1/4 per cell 0


C

Diamond




Atom at corner of cell = 1/8 per cell 8

C

Diamond




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-

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.

Symmetry in crystals

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