Crystals

1
Crystals

• Humans have held a long facination for crystals.
  What is it about them that we find facinating?
• e.g. why use diamonds for jewlery and not
  graphite?
• two samples of, e.g. quartz, are not identical
  externally different sized faces.
• early crystallographers discovered that there was
  a symmetrical relationship between the faces
  extend surface normals (-) to see symmetry. The
  angles between the faces are identical from
  crystal to crystal, although the sizes will
  differ and some faces will be missing.
• found rotation axes 360o/n n 1, 2, 3, 4, 6
• mirror planes m 2
• inversion 2/m i 2-fold rotation, followed
  by a mirror reflection - to 2-fold axis

2
Crystals

• After determining this external symmetry, it was
  infered that there must also be some symmetrical
  internal structure.
• Kepler (1611) reflected on the internal structure
  of snowflakes, asking why they always had 6
  corners. He proposed that snowflakes were made up
  of close-packed spheres of ice.

3
Crystals

• Hooke (1665) extended this idea to other crystals
  and showed how other geometries could be obtained
  from the packing of these spheres.
• note mostly cp, except L, which is sc.
• The connection between these packing types and
  the shape and the angular relationship was put
  forward by Haüy (1784), who connected the
  internal order to the external symmetry.
• Note that this is all long before atoms were
  accepted as existing.

4
Connection Between Crystals Crystallography
Symmetry

• Crystals have the properties they do (both
  internal and external) because of a regular,
  repeating pattern of atoms and/or molecules in
  their internal structure. This packing in 3-D is
  highly symmetric.
• Because of this we will spend a lot of time
  looking at symmetry first of individual objects,
  and then building up the packing of objects,
  first into 1-D arrays, and then into 2-D and 3-D
  patterns.
• This will lead us into the diffraction of light
  by crystals and, ultimately, x-ray
  crystallography.

5
Point Symmetry

• Which of these two compounds is more symmetrical?
• Symmetry is a concept familiar to us. Individual
  objects can be described/defined based on the
  amount and type of symmetry they possess.
• Can quantitate symmetry. Two molecules, which may
  appear at first to be very different, can even
  have identical symmetries.
• A symmetry operation is the act of physically
  doing something to an object so that the result
  is indistinguishable from the initial state.
• Even if we do nothing to the object, it still
  possesses a symmetry element, a geometrical
  property which generates the operation.
• There are 5 symmetry elements E, Cn, s, I, and
  Sn.

H COOH
CC HOOC H
H Cl CC H Br
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Point Symmetry Elements Operations

• E identity no change in the object (needed for
  mathematical completeness).
• Cn proper rotation axis n order of rotation,
  360/n
• for objects with more than one Cn axis, the one
  with the largest value of n principle axis. If
  different axes with same order, designate as C2,
  C2, C2, etc.

1 3 2
3 2 1
2 1 3
Each indistinguishable
C3
C3
2
? ?
?
3 E
C3
Note that there are 3 C2 axes.
2 1 3
2 1 3
1 2 3
C2
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Point Symmetry Elements Operations

• s reflection (aka a mirror plane)
• exchanges every point on one side of the mirror
  with every point on the other side.
• nomenclature sv plan includes principle axis
  (v vertical)
• sh plane is perpendicular to principle axis (h
  horizontal)
• sd plane in between C2 axes perpendicular to
  principle axis (d dihedral)
• all s2 E, so usually not considered.
• Rule for mirror planes in molecules A plane must
  either pass through an atom, or an atom must
  occur in pairs, symmetrically on either side of
  the plane.

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Point Symmetry Elements Operations

• i inversion (about a center).
• each point moves through the center of the object
  to an identical position opposite the original.
• cannot be physically carried out using models.
• again, i2 E
• an inversion center in molecules may be found
• in space
• at a single atom
• Rules
• if in space, then all atoms must be present in
  even numbers on either side of the center.
• if on an atom, then only that atom can be present
  in an odd number.

9
Point Symmetry Elements Operations

• Sn improper rotation (n order of axis).
• rotation by 1/n turn, followed by reflection in a
  plane perpendicular to the axis.
• neither rotation nor reflection need be present
  on their own.

10
Point Symmetry

• So, the symmetry element is present even if you
  do not perform the operation.
• Note when you perform any of the above
  operations, there is always one point in the
  object that remains unchanged/unmoved. That is
  why these are called Point Symmetries.

11
Point Symmetry Elements

• Example. What symmetry elements are present in
  the following molecules?

O H H
N
H H C-C H H
H H C-C HOOC
COOH
P F F F
12
Point Symmetry Elements

• Example. What symmetry elements are present in
  the following molecules?

O H H
E C2 sv sv
E C2 sv sv
N
H H C-C H H
E C2 C2 C2 sv sv sh i
Most symmetric
H H C-C HOOC
COOH
E C2 sv sv
P F F F
E C3 3sv
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Point Symmetry Elements

• Example. What symmetry elements are present in
  the following molecules?

O H H
E C2 sv sv
E C2 sv sv
N
Identical sets of symmetry elements.
H H C-C H H
E C2 C2 C2 sv sv sh i
Most symmetric
Instead of having to write out all symmetry
elements each time, use a short hand symbolism
for the sets these setsare called Point Groups.
H H C-C HOOC
COOH
E C2 sv sv
P F F F
E C3 3sv
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Determining Point Groups

• Find the highest order rotation axis n.
• Are there n C2 axes perpendicular to this
  principle axis?
• Is there a mirror plane perpendicular to the
  principle axis?
• Are there dihedral mirror planes? 4. Are
  there vertical mirror planes?

yes
no Dn set

Cn set
yes no Dnh
yes no Cnh
yes no Cnv
yes no Dnd
Dn
5. Is there a S2n?
yes no S2n
Cn
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Determining Point Groups. Examples.

• Determine the Point Group for each of the
  following molecules

Assume eclipsed configuration.
3 4
1 2
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Determining Point Groups Example 1.

• Find the highest order rotation axis n.
• Are there n C2 axes perpendicular to this
  principle axis?
• Is there a mirror plane perpendicular to the
  principle axis?
• Are there dihedral mirror planes? 4. Are
  there vertical mirror planes?

n 2
yes
no Dn set

Cn set
yes no Dnh
yes no C2h
yes no Cnv
yes no Dnd
Dn
5. Is there a S2n?
yes no S2n
Cn
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Determining Point Groups. Example 2.

• Find the highest order rotation axis n.
• Are there n C2 axes perpendicular to this
  principle axis?
• Is there a mirror plane perpendicular to the
  principle axis?
• Are there dihedral mirror planes? 4. Are
  there vertical mirror planes?

n 2
yes
no Dn set

Cn set
yes no Dnh
yes no Cnh
yes no C2v
yes no Dnd
Dn
5. Is there a S2n?
yes no S2n
Cn
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Determining Point Groups. Example 3.

• Find the highest order rotation axis n.
• Are there n C2 axes perpendicular to this
  principle axis?
• Is there a mirror plane perpendicular to the
  principle axis?
• Are there dihedral mirror planes? 4. Are
  there vertical mirror planes?

n 4
yes
no Dn set

Cn set
yes no D4h
yes no Cnh
yes no Cnv
yes no Dnd
Dn
5. Is there a S2n?
yes no S2n
Cn
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Determining Point Groups. Example 4.

• Find the highest order rotation axis n.
• Are there n C2 axes perpendicular to this
  principle axis?
• Is there a mirror plane perpendicular to the
  principle axis?
• Are there dihedral mirror planes? 4. Are
  there vertical mirror planes?

n 5
yes
no Dn set

Cn set
yes no Dnh
yes no Cnh
yes no Cnv
yes no D5d
Dn
5. Is there a S2n?
yes no S2n
Cn
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Special Cases of Low and High Symmetry
High Symmetry
Low Symmetry
C8v linear, with no perpendicualr C2 axes.
C1 no symmetry other than E.
D8h linear, with perpendicualr C2 axes and sh.
Cs only one mirror plane.
Td pure tetrahedral symmetry. NOT just geometry.
Ci only one inversion center.
Oh pure octahedral symmetry. NOT just geometry.
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Character Tables.

• The collection of symmetry elements represented
  by a given Point Group is listed as part of the
  Character Table.