Übungsaufgaben zur Kristallographie Serie 4

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1 Übungsaufgaben zur Kristallographie Serie HS ) Symmetrie Im Folgenden sind Abbildungen.6 und.7 aus dem Skript gegeben. Nummerieren Sie die eingezeichneten Objekte nach der Reihenfolge der Erzeugung wie in den zwei Beispielen (blau) für alle Graphen durch. Walter Steurer, Thomas Weber, Julia Dshemuchadse, Laboratorium für Kristallographie

2 Übungsaufgaben zur Kristallographie Serie HS Walter Steurer, Thomas Weber, Julia Dshemuchadse, Laboratorium für Kristallographie

3 Übungsaufgaben zur Kristallographie Serie HS ) Raumgruppen Tragen Sie einen Punkt allgemeiner Lage in die unten abgebildeten Symmetriegerüste verschiedener Raumgruppen ein und lassen Sie alle Symmetrieoperationen darauf wirken. a) Welche Zähligkeit besitzt die Punktlage? b) Gibt es spezielle Positionen? Welche? c) Wie lauten die Raumgruppen? Walter Steurer, Thomas Weber, Julia Dshemuchadse, Laboratorium für Kristallographie

4 International Tables for Crystallography (006). Vol. A, Chapter., pp Graphical symbols for symmetry elements in one, two and three dimensions BY TH. HAHN... Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions) Symmetry plane or symmetry line Reflection plane, mirror plane Reflection line, mirror line (two dimensions) Graphical symbol Glide vector in units of lattice translation vectors parallel and normal to the projection plane None m Glide line (two dimensions) lattice vector along line in projection plane a, b or c lattice vector along line in figure plane g lattice vector normal to projection plane a, b or c Double glide plane* (in centred cells only) Diagonal glide plane Diamond glide plane (pair of planes; in centred cells only) along line parallel to projection plane and normal to projection plane One glide vector with two components: along line parallel to projection plane, normal to projection plane along line parallel to projection plane, combined with normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) e n d *Forfurtherexplanationsofthe double glideplanee see Note (iv) below and Note (x) in Section... See footnote x to Section Symmetry planes parallel to the plane of projection Symmetry plane Graphical symbol* Glide vector in units of lattice translation vectors parallel to the projection plane Reflection plane, mirror plane None m lattice vector in the direction of the arrow a, b or c Double glide plane (in centred cells only) in either of the directions of the two arrows e Diagonal glide plane One glide vector with two components n in the direction of the arrow Diamond glide plane (pair of planes; in centred cells only) in the direction of the arrow; the glide vector is always half of a centring vector, i.e. one quarter of a diagonal of the conventional face-centred cell d *Thesymbolsaregivenattheupperleftcornerofthespace-groupdiagrams.Afractionh attached to a symbol indicates two symmetry planes with heights h and h above the plane of projection; e.g. 8 stands for h ˆ 8 and 5 8.Nofractionmeanshˆ0and (cf. Section..6). Forfurtherexplanationsofthe double glideplanee see Note (iv) below and Note (x) in Section... Seefootnotex to Section... 7 Copyright 006 International Union of Crystallography

5 . SYMBOLS AND TERMS USED IN THIS VOLUME... Symmetry planes inclined to the plane of projection (in cubic space groups of classes m and mm only) Symmetry plane Graphical symbol* for planes normal to Glide vector in units of lattice translation vectors for planes normal to [0] and 0Š [0] and 0Š [0] and 0Š [0] and 0Š Printed symbol Reflection plane, mirror plane None None m lattice vector along [00] 9 lattice vector along 00Š >= a or b lattice vector along 0 Š or along [0] lattice vector along 0 Š or along 0Š >; Double glide plane [in space groups Im (7) and Imm (9) only] Diagonal glide plane along [00] and along 0 Š or along [0] One glide vector: along Š or along [] along [00] and along 0 Š or along [0] One glide vector: along Š or along [] e n Diamond glide plane (pair of planes; in centred cells only) 8 >< >: along Š or along []x along Š or along Šx along Š or along Šx along Š or along Šx 9 >= >; d *Thesymbolsrepresentorthographicprojections.Inthecubicspace-groupdiagrams,completeorthographicprojectionsofthesymmetryelementsaroundhigh-symmetry points, such as 0, 0, 0;,0,0;,,0, are given as inserts. Forfurtherexplanationsofthe double glideplanee see Note (iv) below and Note (x) in Section... In the space groups Fm (6), Fmm (5) and Fdm (7), the shortest lattice translation vectors in the glide directions are t,, or t,, and t,, or t,,,respectively. x The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional body-centred cell in space groups Id (0) and Iad (0). Seefootnotex to Section Notes on graphical symbols of symmetry planes (i) The graphical symbols and their explanations (columns and ) are independent of the projection direction and the labelling of the basis vectors. They are, therefore, applicable to any projection diagram of a space group. The printed symbols of glide planes (column ), however, may change with a change of the basis vectors, as shown by the following example. In the rhombohedral space groups Rc (6) and Rc (67), the dotted line refers to a c glide when described with hexagonal axes and projected along [00]; for a description with rhombohedral axes and projection along [], the same dotted glide plane would be called n. The dash-dotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description; cf. footnote y to Section... (ii) The graphical symbols for glide planes in column are not only used for the glide planes defined in Chapter., but also for the further glide planes g which are mentioned in Section.. (Note x) and listed in Table...; they are explained in Sections..9 and... (iii) In monoclinic space groups, the parallel glide vector of a glide plane may be along a lattice translation vector which is inclined to the projection plane. (iv) In 99, the International Union of Crystallography introduced the double glide plane e and the graphical symbol.... for e glide planes oriented normal and inclined to the plane of projection (de Wolff et al., 99); for details of e glide planes see Chapter.. Note that the graphical symbol! # for e glide planes oriented parallel to the projection plane has already been used in IT (95) and IT (95). 8

6 .. GRAPHICAL SYMBOLS FOR SYMMETRY ELEMENTS..5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure Symmetry axis or symmetry point Graphical symbol* Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis Identity None None Twofold rotation axis None Twofold rotation point (two dimensions) Twofold screw axis: sub Threefold rotation axis Threefold rotation point (two dimensions) None Threefold screw axis: sub Threefold screw axis: sub Fourfold rotation axis None () Fourfold rotation point (two dimensions) Fourfold screw axis: sub Fourfold screw axis: sub Fourfold screw axis: sub Sixfold rotation axis Sixfold rotation point (two dimensions) None 6 (,) Sixfold screw axis: 6 sub 6 6, Sixfold screw axis: 6 sub 6, Sixfold screw axis: 6 sub 6, Sixfold screw axis: 6 sub 6, Sixfold screw axis: 6 sub , Centre of symmetry, inversion centre: bar Reflection point, mirror point (one dimension) Inversion axis: bar None, Inversion axis: bar None Inversion axis: 6 bar None 6 =m None (partial elements in parentheses) Twofold rotation axis with centre of symmetry None =m Twofold screw axis with centre of symmetry =m Fourfold rotation axis with centre of symmetry None =m,, sub screw axis with centre of symmetry =m,, Sixfold rotation axis with centre of symmetry None 6=m 6,,,, 6 sub screw axis with centre of symmetry 6 =m 6,,,, *Notesonthe heights h of symmetry points,, and6: () Centres of symmetry and, as well as inversion points and6 on and6 axes parallel to [00], occur in pairs at heights h and h. In the space-group diagrams, only one fraction h is given, e.g. stands for h ˆ and.nofractionmeansh ˆ 0and.Incubic space groups, however, because of their complexity, both fractions are given for vertical axes,includingh ˆ 0and. () Symmetries =m and 6=m contain vertical and6 axes; their and6 inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams. () Symmetries =m and 6 =m also contain vertical and6 axes,buttheir and6 inversion points alternate with the centres of symmetry; i.e. pointsath and h interleave with or6 pointsath and h. In the tetragonal and hexagonal space-group diagrams, only one fraction for andonefor or6 is given. In the cubic diagrams, all four fractions are listed for =m; e.g. Pmn (No. ): : 0, ; :,. 9

7 . SYMBOLS AND TERMS USED IN THIS VOLUME..6. Symmetry axes parallel to the plane of projection Symmetry axis Graphical symbol* Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis (partial elements in parentheses) Twofold rotation axis None Twofold screw axis: sub Fourfold rotation axis None () Fourfold screw axis: sub Fourfold screw axis: sub Fourfold screw axis: sub Inversion axis: bar None Inversion point on bar -axis point * The symbols for horizontal symmetry axes are given outside the unit cell of the space-group diagrams. Twofold axes always occur in pairs, at heights h and h above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and h.nofractionstandsforh ˆ 0and.Theruleofpairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including h ˆ 0and. This applies also to the horizontal axesandthe inversion points located on these axes...7. Symmetry axes inclined to the plane of projection (in cubic space groups only) Symmetry axis Graphical symbol* Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis (partial elements in parentheses) Twofold rotation axis None Twofold screw axis: sub Threefold rotation axis None Threefold screw axis: sub Threefold screw axis: sub Inversion axis: bar None, * The dots mark the intersection points of axes with the plane at h ˆ 0. In some cases, the intersection points are obscured by symbols of symmetry elements with height h 0; examples: Fd (0), origin choice ; Pnn (), origin choice ; Pmn (); Imm (9); Iad (0). 0

Figure.1. The conventional unit cells (thick black outline) of the 14 Bravais lattices. [crystallographic symmetry] 1

Figure.1. The conventional unit cells (thick black outline) of the 14 Bravais lattices. [crystallographic symmetry] 1 [crystallographic symmetry] The crystallographic space groups. Supplementary to { 9.6:324} In the 3-D space there are 7 crystal systems that satisfy the point (e.g., rotation, reflection and inversion)