Josef Buršík - assignment 4
1(a). In order to describe the crystal structure, a description of the contents of the cell unit plus the symmetry of lattice translation are needed. The selection of the unit cell is on the crystalllographers. Crystallographers usually choose an origin of the unit cell that is convenient to describe the contents of the unit cell (usually a point of high symmetry). There are other rules how to select the proper unit cell (it should has the same symmetry as the whole lattice, it should contain the smallest number of lattice points, the maximum of rigth angles, the smallest volume ...). (ii) should be correct.

1(b). Any crystal is characteristic by regular arrangement of its building units (complex molecules, group of atoms, individual ions). When we abstract from the real content of this unit and take it as a point then this regular arrangement builds lattice. The type of lattice is dictated by structure of the crystal (i) should be correct.

1(c). The Reciprocal Lattice exists in the reciprocal space and represents the scattering possibilities of a given crystal (it is used by crystallographers as a way of visualising the different diffraction possibilities available by re-orienting or rotating a single crystal). The Origin of the Reciprocal Lattice is given by the intersection of the straight line of incident X-ray beam with the Ewald sphere. I think that the Origin of the R.L. is given by the particular experimental arrangement (the mutual orientation of incident beam and the crystal). Because this experimental arrangnment is done by crystallographer is in this sense the Origin chosen by him. Once the arrangement is selected, the Origin of the Reciprocal Lattice emerge naturally from the chosen arrangement, independently from the will of crystallographer. I would say (iii) is correct.

1(d). I thin that the concept of the Origin of the crystal is not use in crystallography.  There is only one case where I would use the words the Origin of crystal: when crystallographers are depicted macroscopic symmetry of crystal they use different kinds of projections (e.g. stereographical projection). In this case the Origin of crystal is placed (together with the Origin of projection) in the centre of the crystal. I thin that the choice (ii) is correct.

1(e). The space-group symmetry of the crystal is given by its structure (= type of lattice + basic content, motive (of which representation the lattice point is)): (i) should be correct.

2. The Laue Classes are 11 centrosymmetric point groups which define at the same time the possible symmetry of the diffractiograms (which are according to the Friedel rule in every case (with the exception when anomalous scattering occurs) centrosymmetric). For example the single crystal diffraction of an substance belonging to monoclinic crystals system (3 possible crystal clases and 13 possible space groups) gives picture which is uniquely described by one Laue class (2/m) (the so called Laue check is done automatically by computer). At this moment it is not possible to distinquis between these three crystal classes (2, m, and 2/m). To detemine the space group of this substance is neccesary to deal e.g. with systematic absences. Situation is worse (in case of use of powder diffraction) when the unit-cell geometry corresponds to tetragonal lattice since there are two possible Laue classes which are indistinguishable (4/mmm and 4/m) in powder pattern.

3. The inversion point and mirror planes cannot be found in the crystal structure of an enantiomorphic  molecule. Crystals of these commpounds (sugar such as D-glucose) can occur in nature in both enanthiomorphic forms - right and left handed. With intruduction of e.g. mirror plane into 4 point group (becomming 4/m group) the crystal would lose this effect of "right and left hand".

4. I think that mor convenient is to start with answer on the second question. The Co atom (0.271, 0.250, 0.458) lies on the special position. From the picture is clear that it lies on the c-glide plane with symmetry operator (x, 1/2-y, 1/2+z).

Using the clickable pictures on P21/c space group I collected these symmetrically equivalent Co atoms: Co1(0.271, 0.250, 0.458); applying 1-x, 1/2+y, 1/2-z operator: Co2(0.729, 0.750. 0.042); applying x, 1/2-y, 1/2+z operator: Co3(0.271, 0.250. 0.958); applying 1-x, -y, 1-z operator: Co4(0.729, 0.750. 0.542).

5. In the molecule of 1,2-dichloroethane are present these symmetry elements: 1, 1(bar), 2, and m (perpendicular to 2). The last two elements build composed element -2 (twofold rotarya-inversion axis). The point group of  1,2-dichloroethane should be 2/m.

6. i) There are three a-glide planes, two mirror planes parallel to yz plane, nine twofold rotation axes parallel with c axis.

    ii) Consider the red with coordinates x,y,z; it is symmetry related to the points: (1) labelled Z by the mirror plane parallel to yz plane going through  x = 1/4, which has the symmetry operator 1/2-x, y, z; (2) labelled K by the two fold rotation axis going parallel to c axis trough the point a = 1/2, b = 1/2, which has symmetry operator 1-x, 1-y, z; (3) labelled J by the a-glide plane parallel to xz plane going through  y = 1/2, which has the symmetry operator 1/2+x,1- y, z.

    iii) the mirror plane parallel to yz plane changes the handedness of Z point to the left-handed (property of mirror planes); the two fold rotation axis going parallel to c axis does not changes the handedness of K point (remains right-handed, property of rotational axes); the a-glide plane parallel to xz plane changes the handedness of J point to the left-handed (property of glide planes)

    iv) the crystal system to which this space group belongs should be ortogonal. The reason is that (1) the a and b axis are in the picture perpendicular and the existence of twofold rotation axis perpendicular to xy-plane implies that also the c-axis is also perpendicular to xy-plane. Because there is no three or four fold axes, the crystal system should be orthorombic.