- State, with a very brief explanation, which of the following items are
(i) due to the packing of atoms/molecules in a crystal,
(ii) chosen by the crystallographer, and (iii) neither (i) or (ii):
-
(a) The Unit Cell,
(b) The Crystal Lattice,
(c) The Origin in Reciprocal Space,
(d) The Origin in the Crystal,
(e) The Space-Group Symmetry of the Crystal.
-
- Briefly explain, with two examples, what is meant by the
term Laue Class.
The Laue class describes the point group symmetry of the diffraction pattern. There
are 11 such classes and all are centrosymmetric. Each crystal system has either one Laue
class associated with it (triclinic, monoclinic and orthorhombic) or two Laue classes (tetragonal,
trigonal, hexagonal and cubic). Thus for the monoclinic system the Laue class is 2/m and for the
tetragonal class the possibilities are 4/m and 4/mmm.
-
Crystallographic symmetry operations may be grouped into 5 distinct groups,
namely, lattice translations, rotations, rotary-inversions, screw-rotations,
and glide planes.
Which of these symmetry operations cannot be found in the crystal structure of an enantiomorphic molecule, e.g. a naturally-occurring sugar such as D-glucose? Give brief reasons for your answer.
The enantiomorphic space groups are those that contain no operations that change the handedness of
an object. Operations that involve either a centre of inversion or a reflection will change the
handedness and are therefore forbidden. In the list above, lattice translations, rotations
and screw-rotations do not change the handedness, while rotary-inversions and glide planes
do change the handedness. Hence the forbidden operations are rotary-inversions and glide planes.
- An organometallic cobalt compound crystallises in space group P21c.
If the coordinates of the central metal atom, Co, in this compound are
0.271,0.250,0.458, determine the coordinates of any other Co atoms
within the unit cell, showing your workings.
Is the Co atom sited on a special position in this space group? Give brief reasons for your answer.
The symmetry related positions for this space group are:
x,y,z
-x,0.5+y,0.5-z
-x,-y,-z
x,0.5-y,0.5+z
Applying the last three of these to the point 0.271,0.250,0.458 (taken as x,y,z) gives:
-0.271,0.75,0.042
-0.271,-0.250,-0.458
0.271,0.25,0.958
Comment: Question asked for coordinates within the unit cell
2 Marks
The Co atom is on a general position, not a special position. An atom on a special position generates fewer than the maximum number possible of equivalent positions when the symmetry operations are performed on it. In the present case, four symmetry related points are possible and four are found. If the atom was on a special position, such as 0,0,0 then only two positions would have been found.
2 Marks - Three different views are shown below of 1,2-dichloroethylene in its most-likely (minimum-energy) conformation. What symmetry elements are present (if any)? Hence suggest the molecular point symmetry for this particular conformation.
- You are supplied with the diagram of the primitive space group shown below:
- Identify the three types of symmetry element present in the diagram.
- Given the position of the right-handed object shown in red, specify (using the letters A-Z, a-i) the position of symmetry-related objects within the unit cell, stating for each the symmetry element which relates the red and magenta objects.
- For each symmetry-related object specified in (ii) above, state its handedness giving a brief reason for your answer.
- Specify the crystal system to which this space group belongs giving a brief reason for your answer.
- The symmetry elements present are:
The identity operation.
Twofold rotation axes perpendicular to the plane of the page, identified by the solid ovals (nine present)
Two sets of orthogonal mirror planes perpendicular to the plane of the page, identified by the heavy solid lines and the heavy dashed lines respectively.
Comment: The dashed line is a glide and not a mirror plane
2 Marks - I define the origin as the twofold rotation axis at the bottom left of the diagram and consider each side of the
diagram to be of unit length. Thus the first vertical mirror plane (solid line) is at 0.25a.
The point is mapped to Z by reflection in the mirror plane at 0.25a.
1 Mark
The point is mapped to H by reflection in the mirror plane at 0.5b.
0 Mark
The point is mapped to K by rotation about the axis at 0.5,0.5.
1 Mark
Any other operation, eg rotation about the twofold rotation axis at 0,0.5 results in a point outside the unit cell, in this case point G. - A reflection changes the handedness of an object, a rotation does not. Hence points Z and H are
oppositely handed to the red point, K has the same handedness.
2 Marks - The only symmetry element allowed in the triclinic class is a centre of inversion, so this one may be
eliminated.
In the monoclinic class, if a twofold rotation axis and a mirror plane are present, the rotation axis must be perpendicular to the mirror plane, hence this class is also eliminated.
In the orthorhombic class, mirror planes and twofold rotation axes can be parallel, thus this is a possibility.
All the other crystal classes may be rejected since they require the presence of rotation axes of order greater than two. Thus by elimination, the crystal class must be orthorhombic.
2 Marks
PS From looking at the table of 230 crystallographic space groups , my best guess is that the space group is Pmm2.
(a) The Unit Cell
(ii) The unit cell is chosen by the crystallographer.
A crystal is built up by the addition, in three dimensions, of identical building blocks.
The contents of the first block may be the minimum necessary to build the crystal, but it is
often convenient to work with a larger cell that allows the symmetry relationships between the
atoms/molecules to be more clearly seen.
1 Mark
(b) The Crystal Lattice
(i) Due to the packing of atoms/molecules in a crystal
The crystal lattice is the arrangement of atoms that make up the solid. This is determined by the interatomic
and/or the intermolecular forces present. The arrangement of atoms does not depend on the viewpoint of the
observer.
1 Mark
(c) The Origin in Reciprocal Space
(ii) The origin in reciprocal space is chosen by the crystallographer.
The reciprocal lattice is defined in terms of the cell parameters, since the cell parameters depend on the
choice of unit cell which is an arbitrary construction, the reciprocal lattice must also be an arbitrary
construction, hence the origin may be placed anywhere. However, the usual choice is the straight through beam.
0 Mark
(d) The Origin in the Crystal
(ii) The origin in the crystal is chosen by the crystallographer.
Since every unit cell is identical and related by a translation, the origin may be placed anyway.
It may be convenient to choose the crystal centre to coincide with the laboratory reference frame
but it is not mandated.
1 Mark
(e) The Space-Group Symmetry of the Crystal.
(i) Due to the packing of atoms/molecules in a crystal
The physical arrangement of the atoms/molecules in the crystal lattice define the symmetry elements
and hence the space group present. (In writing this it occurs to me that that this is only true if the
definitions of symmetry elements we use are the only possible ones. If they are not, then there are
other space groups possible and the choice becomes the crystallographer's).
[See remarks in the model answers re the effects of decisions taken by the
crystallographer.]
1 Mark
Minor comment: Initial explanation is a bit brief, though it does contains the relevant key words (just!)
3 + 2 Marks
Comment: Bit too brief
4 Marks
In view A there is a twofold rotation axis about the mid-point of the C-C bond, perpendicular to the plane of the paper.
Minor comment: axis is not actually in the plane of the paper (at 30° to it to be precise!)
From view B, the mid-point of the C-C bond is a centre of inversion.
In view C, the plane containing the two C-Cl bonds is a mirror plane.
3 Marks
Thus, the symmetry elements are the identity (which is always present), a centre of inversion, a twofold rotation axis and a mirror plane perpendicular to it. This leads uniquely to the point group
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2 Marks