Part 4-32 point groups

Transcription

1 Part 4-32 point groups 4.1 Subgroups point groups 4.2 Crystal forms

2 The 32 point groups The point groups are made up from point symmetry operations and their combinations. A point group is defined as a group of point symmetry operations whose operation leaves at least one point unmoved. Any operation involving translation is excluded. Those point groups derived from the space groups of the lattice are the highest symmetry possible for the particular crystal system.

3 Subgroups - 1 The point groups of highest symmetry in each crystal system all contain the symmetry elements of one or more point groups of lower symmetry (sub-groups). Triclinic: the only subgroup of 1 is 1. Monoclinic: 2/m has the subgroups 2, m, 1, 1. Space group of highest symmetry: P2/m: 2 normal to m b 0,0,0 X,1/2,z 1/2,y,1/2

4 Subgroups - 2 Orthorhombic: if inversion symmetry is removed from point group 2/m, 2/m, 2/m, each 2/m must be reduced either to 2 or to m. mmm 2/m2/m2/m 22m 2/m2/m2/m Space group of highest symmetry: P2/m 2/m 2/m (P4/mmm) a b c Subgroups: mm2(m2m, 2mm) and 222.

5 32 point groups All crystal systems result in 32 subgroups, called crystallographic point groups full symbol short symbol

6 Crystal forms A space group reveals the entire symmetry of a crystal structure. If the crystal is bounded by plane faces, the symmetry of its morphology will be the symmetry of that point group. A set of equivalent faces is called a crystal form. PbS-galena crystal: 4/m 3 2/m <a><111> <110> Stereographic projection of symmetry elements

7 Crystal form - general form The indices of a form are placed in braces: {hkl}. A general form is a set of equivalent faces, each of which has face symmetry 1. Or not laying on any of the symmetry elements in stereogram. General forms have general indices {hkl}. e.g. {hkl} tetragonal pyramid, point group 4.

8 Crystal form - special form A special form is a set of equivalent crystal faces which themselves have a face symmetry higher than 1. Or the poles of the faces of special form lie on at least one of the symmetry elements in a stereogram. e.g. {hhl} tetragonal pyramid, point group 4mm {hhl} The pole of the faces have a single degree of freedom The form will remain a tetragonal pyramid as long as the pole remains on the mirror plane..m

9 Crystal form - limiting form A limiting form is a special case of either a general or a special form. It has the same number of faces, each of which has the same face symmetry, but the faces are differently arranged. e.g. {hk0} tetragonal prism, point group 4mm. Limiting form of general form {110} Limiting form of special form

10 Crystal forms - point group 4/mmm - general form Each point group has characteristic forms. The asymmetric face unit of a point group is the smallest part of the sphere which, by application of the symmetry operations, will generate the entire surface of the sphere. An asymmetric face unit of a point group contains all the information necessary for the complete description of the crystal forms in this point group. Point group 4/mmm-general form m..,.m.,..m refer to the symmetry direction of tetragonal system: c, <a>, <110>

11 Crystal forms - point group 4/mmm - special form Ditetragonal prism {hk0} A pole of a face on A pole of face on vertices of asymmetry face unit

12 Crystal forms - stereogram 4/mmm Stereogram of the poles of the faces in all crystal forms of the point group of highest symmetry in tetragonal system, 4/mmm.

13 Crystal form - 4mm point subgroup Deriving subgroups from the general crystal form of the point group of highest symmetry. Ditetragonal pyramids 4/mmm Two ditetragonal pyramid 4mm subgroup

14 Crystal forms - Tetragonal system and face symmetries Point group Asymmetry face unit and face symmetry Special forms Limiting forms {hhl} {100} 4/m 2/m 2/m (4/mmm) 4mm 42m 422 Tetragonal dipyramid..m Tetragonal pyramid..m Tetragonal disphenoid..m Tetragonal dipyramid 1 Tetragonal prism m2m. Tetragonal prism.m. Tetragonal prism.2.

15 Crystal forms - other systems The crystal forms in other crystal systems can be developed in the same way as have done for the tetragonal system.

16 Characteristic symmetry elements of seven crystal systems