Symmetry in crystalline solids.

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1 Symmetry in crystalline solids. Translation symmetry n 1,n 2,n 3 are integer numbers 1

2 Unitary or primitive cells 2D 3D Red, green and cyano depict all primitive (unitary) cells, whereas blue cell is not a unitary cell Only P and R are primitive cells 2

3 Unitary or primitive cells 3

Point symmetry There can be point operation like: Rotation around an axis Inversion with respect to a point Reflection with respect to a plane Rotation and Reflection Rotation and

4 Point symmetry There can be point operation like: Rotation around an axis Inversion with respect to a point Reflection with respect to a plane Rotation and Reflection Rotation and Inversion Not all rotation axis are compatible with translational symmetry in 2D or 3D. For example 5-fold rotation as shown in the figure is not compatible with 2D and 3D lattice. 4

5 Impossibile visualizzare l'immagine. Notations: Schoenfliess and Hermann-Mauguin (International) Schoenfliss Hermann-Mauguin Rotation axis C n (n=1,2,3,4,6) n (n=1,2,3,4,6) Reflection plane σ v,h m Inversion Rotation and reflection Rotation and inversion i S n S n 1 n 5

6 Bravais Lattice. 2D: 5 Bravais Lattices 6

7 3D: 14 Bravais Lattices grouped in 7 classes Triclinic Monoclinic Monoclinic c Hexagonal Trigonal Orthorhombic Orthorhombic c Orthorhombic bc Orthorhombic fc Tetragonal Tetragonal bc Cubic Cubic bc Cubic fc Corso CFSSM. LM-Chim 8 7

8 Crystal System Axial Lengths and angles Space Lattice Cubic a = b = c, α = β = γ = 90 Sim ple Body- centered Face- centered Tetragonal a = b c, α=β=γ=90 Sim ple, BC Orthorhom bic a b c, α=β=γ=90 Sim ple,bc, FC, Base Cent ered Rhom bohedral a = b = c, α =β =γ 90% Sim ple Hexagonal a = b c, α=β=90, γ=120 Sim ple Monoclinic a b c, α=γ=90 β Sim ple, Base- C Triclinic a b c,α β γ 90% Sim ple 8

9 Orthorhombic Orthorhombic c Orthorhombic bc Orthorhombic fc P = primitive C = base centered BC = body centered (si usa anche il simbolo I da Innenzentrierte) FC = Face Centered The 7 crystal classes all possible point groups The 14 Bravais Lattices all possible space groups. The space group comprises both the point symmetry operations and the translational symmetry ones. 9

10 Global Crystal Structure Lattice Basis Crystal Addition of a base of atoms can lower the symmetry of the Bravais Lattice. Considering all possible combination of symmetric point groups compatible withthe the seven crystal classes with bases of lowering symmetry there are in total 32 Crystallographic Point Groups In order to find all the Space Groups two additional symmetry elements should be considered 10

11 Glide plane Screw Axis 11

12 Adding to the 32 crystallographic point groups all possible translational symmetry and the ones arising from glide planes and screw axis, the total number of crystalline space group is 230. Of these 73 groups are called symmorphic because the y arise from the addition of translational symmetry to the point group symmetry The other 157 arise from the presence of glide planes or screw axis and are named non-symmorphic 12

13 Numbers refer to an axis of rotation (2, 3, 4 or 6- fold) or rotoinversion. (3,4 or 6 -fold) The symbol "m" refers to a mirror plane (either a single mirror, or a set of mirrors related by a rotation or rotoinversion axis). A symbol written as a fraction (e.g. 4/m) refers to a rotation axis (in this case, 4-fold) perpendicular to a mirror plane. (*) Classes whose symbol is followed by an asterisk are centrosymmetric Corso CFSSM. LM-Chim 13 13

14 Examples: The large majority has bcc, cfc or hexagonal structure 14

15 Examples: NaCl Has an fcc structure with a basis formed by the pair Na + and Cl. The Cl is in the (0,0,0) position and the Na + in the (1/2,1/2,1/2) position 15

16 Examples: Diamond and Zincblende FCC lattice with a basis of 2 atoms at (0,0,0) and (¼, ¼, ¼). The FCC cell contains a total of 4x2 atoms The 2 atoms are identical in diamond, Si, Ge; different in ZnS, GaAs, etc. 16

17 Hexagonal close packed layers 17

18 Atomic packing in an FCC structure Atoms in crystal planes perpendicular to the body diagonal form hexagonal closed packed planes with an A-B-C-A- B-C- arrangement 18

19 Wiegner Seitz cells From a point in the lattice draw all the segments connecting it to all the nearest neighbor points. Draw lines(2d) or planes (3D) which are perpendicular to the segments and cut them in the middle. The area or volume inscribed in the lines or planes is the Wiegner Seitz cell. 19

20 Miller Indeces 2D 3D 20

$X-ray diffraction Von Laue method: The crystal is fixed and the wavelength λ of the X-ray is varied Rotating Crystal method: The crystal is rotated to change angle of$

21 X-ray diffraction Von Laue method: The crystal is fixed and the wavelength λ of the X-ray is varied Rotating Crystal method: The crystal is rotated to change angle of incidence of the X-rays and λ is fixed Powder method: X-rays of fixed λ are shined on the crystalline powder, with micrometric crystals oriented in every direction. 21

22 X-ray diffraction Von Laue Method: Lecture Notes 22

23 Ewald Sphere 2D 3D Draw a sphere with radius k (wavavector of the X-ray:red arrow) reaching a point in the reciprocal lattice. There will be scattering if another point falls on the surface of the sphere. r k r r k ' = G 23

$diffraction$ diamond

24 Von Laue method Von Laue diffraction from diamond oriented [100] 24

Physics of Materials: Symmetry and Bravais Lattice To understand Crystal Plane/Face. Dr. Anurag Srivastava

Physics of Materials: Symmetry and Bravais Lattice To understand Crystal Plane/Face Dr. Anurag Srivastava Atal Bihari Vajpayee Indian Institute of Information Technology and Manegement, Gwalior Physics