Q1.

(a) The Unit Cell is chosen by the crystallographer. This is an arbitrary choice as any unit cell may be described as primitive triclinic, but it is usually chosen so as to include all the symmetry elements relating the molecules within the crystal lattice, but not including the lattice translations.

(b) The Crystal Lattice is chosen by the crystallograhper. Again this may be described as a primitive triclinic lattice, but this will not always describe the full symmetry present such as face-centring, so higher order lattices would be chosen.

(c) The Origin in Reciprocal Space is neither chosen by the crystallographer or due to the arrangement of the atoms/molceules in the crystal. It is derived from the geometry of the Ewald sphere and the theory of diffraction.

(d) The Origin in the Crystal is chosen by the crystallographer. Certain space groups may be described with different origins, but usually an origin at a centre of symmetry is chosen.

(e) The Space-Group Symmetry of the Crystal is due to the packing of the atoms/molecules within the crystal. Although the unit cell may be chosen arbitrarily, the symmetry relating various moieties remains the same (hence the choice of unit cell should be chosen to reflect this symmetry).

Q2.

The term Laue Class refers to a set of space groups possessing a centre of symmetry. All translational elements removed from a space group will leave the Laue class of which it is a subgroup. Two examples are 4/m and 4/mmm.

Q3.

Any symmetry operation within a unit cell must by definition act on every molecule within that cell. Enantiomorphic molecules possess "handedness", ie. they are mirror images of the opposite enanitomorphs. This means any symmetry operation involving reflection cannot exist (except in a racaemic mixture). Lattice translations are inherent in a crystal so must exist. Rotations and screw-rotations do not involve reflection and so may exist within the crystal, but rotary inversions and glide planes do involve reflection and so cannot exist in the crystal structure of a pure enantiomeric molecule.

Q4.

The coordinates (0.271, 0.250, 0.458) refer to the position (x, y, z) within the crystal. Along with the position (x, y, z), the symmetry elements within the space group P2₁/c produce equivalent positions at (-x, -y, -z) , (-x, 0.5+y, 0.5-z), and (x, 0.5-y, 0.5+z). These positions produce coordinates at (-0.271, -0.25, -0.458), (-0.271, 0.75, 0.042) and (0.271, 0.250, 0.958).

These coordinates do not lie within the unit cell. Any translation of 1 along any unit cell axis produces the same position in the adjacent unit cell. Therefore, the coordinates may be translated to (0.729, 0.750, 0.542), (0.729, 0.750, 0.042) and (0.271, 0.250, 0.958).

The Co is situated on the c-glide plane at (x, 0.25, z).

Q5.

There is a centre of symmetry present lying between the two carbon (black) atoms. The molecular point symmetry is -1.

Q6. (i) The three types of symmetry element in the diagram are mirror planes, a-glide planes and two-fold rotation axes.

(ii) The symmetry related positions within the cell are at positions labeled

J : a-glide plane at (x, 0.5, z)

K : two-fold rotation axis at (0.25, 0.25, z)

Z : mirror plane at (0.25, y, z)

(iii) Position J is left-handed due to the reflection across the glide plane.

Position K is right-handed as a rotation axis does not involve reflection.

Position Z is left-handed due to the reflection in the mirror plane.

(iv) The space group is Pma2 with equivalent positions (x, y, z), (0.5+x, -y, z), (-x, -y, z) and (0.5-x, y, z). The crystal system is orthorhombic as all angles must be 90 but no axes have to be the same length.