If we build up into 3d we go from point to plane to space groups From the 32 point groups and the different Bravais lattices, we can get 73 space groups which involve ONLY rotations, reflection and rotoinversions. 31 and 32 give different handednessĢ6 Example P42 (tetragonal) – any additional symmetry?Ģ7 Matrix 4 fold rotation and translation of ½ unit cell Carry this on…. Notation is nx where n is the simple rotation, as before x indicates translation as a fraction x/n along the axis /2 2 rotation axis 21 screw axisĢ5 Screw axes - examples Looking down from above Fedorov (1881)Ģ3 Another example Build up from one point:Ģ4 Screw axes Rotation followed by a translation Here the translation vector has components in two (or sometimes three) directions a b - +, So for example the translations would be (a b)/4 Special circumstances for cubic & tetragonalĢ1 d glide Here the glide plane is in the plane xy (perpendicular to c) aĢ2 17 Plane groups Studied (briefly) in the workshop Combinations of point symmetry and glide planes E. Here the translation vector has components in two (or sometimes three) directions a b - +, So for example the translations would be (a b)/2 Special circumstances for cubic & tetragonalġ9 n glide Here the glide plane is in the plane xy (perpendicular to c) a It is “obvious” that 62 and 64 are equivalent to 3 and 32, respectively.ĭisplay all possibilities for the symmetry of space-filling shapes form the basis (with Bravais lattices) of space groups Enantiomorphic Centrosymmetric Triclinic 1 * Monoclinic 2 * 2/m m * Orthorhombic 222 mmm mm2 * Tetragonal 4 * 422 4/m 4/mmm 4mm * 2m Trigonal 3 * 32 3m * Hexagonal 6 * 622 6/m 6/mmm 6mm * Cubic 23 432 m m m 3mĮnantiomorphic Centrosymmetric Triclinic 1 * Monoclinic 2 * 2/m m * Orthorhombic 222 mmm mm2 * Tetragonal 4 * 422 4/m 4/mmm 4mm * 2m Trigonal 3 * 32 3m * Hexagonal 6 * 622 6/m 6/mmm 6mm * Cubic 23 432 m m m 3m Centrosymmetric – have a centre of symmetry Enantiomorphic – opposite, like a hand and its mirror * - polar, or pyroelectric, point groupsġ6 Space operations These involve a point operation R (rotation, mirror, roto-inversion) followed by a translation Can be described by the Seitz operator: e.g.ġ7 Glide planes The simplest glide planes are those that act along an axis, a b or c Thus the translation is ½ way along the cell followed by a reflection (which changes the handedness: ), a c, Here the a glide plane is perpendicular to the c-axis This gives symmetry operator ½+x, y, -z. 3-fold example: b aġ3 3-fold and 6-fold etc. Left as an example to show with a diagram.ġ2 More complex cases For non-orthogonal, high symmetry axes, it becomes more complex, in terms of deriving from a figure. Simple mirror in bc plane x, y, z -x, y, z a bĩ General convention Right hand rule (x y z) (x’ y’ z’) or r’ = Rrī (x y z) (x’ y’ z’) r r’ c or r’ = Rr R represents the matrix of the point operationġ0 Back to the mirror… Take a point at (x y z) Simple mirror in bc planeġ1 Other examples roto-inversion around z Rotations Mirrors In the workshops we have looked at plane symmetry which involves translation = ua + vb + wc Glides Screw axesĪbove Below Example: 2-fold rotation perpendicular to plane (2)Ĥ More examples Example: 2-fold rotation in plane (2)ĥ Combinations Example: 2-fold rotation perpendicular to mirror (2/m)Įxample: 3 perpendicular 2-fold rotations (222)Ħ Roto-Inversions A rotation followed by an inversion through the origin (in this case the centre of the stereogram) Example: “bar 4” = inversion tetrad More examples in sheet.ħ Special positions When the object under study lies on a symmetry element mm2 example General positions Special positions Equivalent positionsĨ In terms of axes… Again, from workshop: Take a point at (x y z) By the end of this section you should: have consolidated your knowledge of point groups and be able to draw stereograms be able to derive equivalent positions for mirrors, and certain rotations, roto-inversions, glides and screw axes understand and be able to use matrices for different symmetry elements be familiar with the basics of space groups and know the difference between symmorphic & non-symmorphicĢ The story so far… In the lectures we have discussed point symmetry:
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |