Presentation on theme: "More on symmetry Learning Outcomes:"— Presentation transcript:
1 More on symmetry Learning Outcomes: By the end of this section you should:have consolidated your knowledge of point groups and be able to draw stereogramsbe able to derive equivalent positions for mirrors, and certain rotations, roto-inversions, glides and screw axesunderstand and be able to use matrices for different symmetry elementsbe familiar with the basics of space groups and know the difference between symmorphic & non-symmorphic
2 The story so far… In the lectures we have discussed point symmetry: RotationsMirrorsIn the workshops we have looked at plane symmetry which involves translation = ua + vb + wcGlidesScrew axes
3 Back to stereograms and point symmetry AboveBelowExample: 2-fold rotation perpendicular to plane (2)
4 More examples Example: 2-fold rotation in plane (2) Example: mirror in plane (m)
6 Roto-InversionsA rotation followed by an inversion through the origin (in this case the centre of the stereogram)Example: “bar 4” = inversion tetradMore examples in sheet.
7 Special positionsWhen the object under study lies on a symmetry element mm2 exampleGeneral positionsSpecial positionsEquivalent positions
8 In terms of axes… Again, from workshop: Take a point at (x y z) Simple mirror in bc planex, y, z-x, y, zab
9 General convention Right hand rule (x y z) (x’ y’ z’) or r’ = Rr b(x y z)(x’ y’ z’)rr’cor r’ = RrR represents the matrix of the point operation
10 Back to the mirror… Take a point at (x y z) Simple mirror in bc plane
11 Other examples roto-inversion around z Left as an example to show with a diagram.
12 More complex casesFor non-orthogonal, high symmetry axes, it becomes more complex, in terms of deriving from a figure. 3-fold example:ba
13 3-fold and 6-foldetc.It is “obvious” that 62 and 64 are equivalent to 3 and 32, respectively.
14 32 crystallographic point groups display all possibilities for the symmetry of space-filling shapesform the basis (with Bravais lattices) of space groupsEnantiomorphicCentrosymmetricTriclinic1 *Monoclinic2 *2/mm *Orthorhombic222mmmmm2 *Tetragonal4 *4224/m4/mmm4mm *2mTrigonal3 *323m *Hexagonal6 *6226/m6/mmm6mm *Cubic23432mm m3m
15 32 crystallographic point groups EnantiomorphicCentrosymmetricTriclinic1 *Monoclinic2 *2/mm *Orthorhombic222mmmmm2 *Tetragonal4 *4224/m4/mmm4mm *2mTrigonal3 *323m *Hexagonal6 *6226/m6/mmm6mm *Cubic23432mm m3mCentrosymmetric – have a centre of symmetryEnantiomorphic – opposite, like a hand and its mirror* - polar, or pyroelectric, point groups
16 Space operationsThese involve a point operation R (rotation, mirror, roto-inversion) followed by a translation Can be described by the Seitz operator:e.g.
17 Glide planesThe simplest glide planes are those that act along an axis, a b or cThus the translation is ½ way along the cell followed by a reflection (which changes the handedness: ),ac,Here the a glide plane is perpendicular to the c-axis This gives symmetry operator ½+x, y, -z.
18 n glide n glide = Diagonal glide Here the translation vector has components in two (or sometimes three) directionsab-+,So for example the translations would be (a b)/2Special circumstances for cubic & tetragonal
19 n glide Here the glide plane is in the plane xy (perpendicular to c) a b-+,Symmetry operator ½+x, ½+y, -z
20 d glide d glide = Diamond glide Here the translation vector has components in two (or sometimes three) directionsab-+,So for example the translations would be (a b)/4Special circumstances for cubic & tetragonal
21 d glide Here the glide plane is in the plane xy (perpendicular to c) a b-+,Symmetry operator ¼+x, ¼+y, -z
22 17 Plane groupsStudied (briefly) in the workshop Combinations of point symmetry and glide planesE. S. Fedorov (1881)
24 Screw axes Rotation followed by a translation Notation is nx where n is the simple rotation, as beforex indicates translation as a fraction x/n along the axis /22 rotation axis21 screw axis
25 Screw axes - examples Looking down from above Note e.g. 31 and 32 give different handedness
26 ExampleP42 (tetragonal) – any additional symmetry?
27 Matrix4 fold rotation and translation of ½ unit cellCarry this on….
28 Symmorphic Space Groups If we build up into 3d we go from point to plane to space groupsFrom the 32 point groups and the different Bravais lattices, we can get 73 space groups which involve ONLY rotations, reflection and rotoinversions.Non-symmorphic space groups involve translational elements (screw axes and glide planes).There are 157 non-symmorphic space groups230 space groups in total!
33 Space Group exampleP21/c : note glide plane shifted to y=¼ because convention “likes” inversions at originEquivalent positions:
34 Special positions Taken from last example If the general equivalent positions are:Special positions are at:½,0,½ ½,½,00,0,½ 0,½,0½,0,0 ½,½, ½0,0,0 0,½,½
35 Space groups…Allow us to fully describe a crystal structure with the minimum number of atomic positionsDescribe the full symmetry of a crystal structureRestrict macroscopic properties (see symmetry workshop) – e.g. BaTiO3Allow us to understand relationships between similar crystal structures and understand polymorphic transitions
36 Example: YBCO Handout of Structure and Space group Most atoms lie on special positionsYBa2Cu3O7 is the orthorhombic phaseSpace group: Pmmm