BRAVAIS LATTICE

1. BRAVAIS LATTICE BL describes the periodic nature of the atomic arrangements (units) in a X’l. X’l structure is obtained when we attach a unit to every lattice point and repeat in space Unit – Single atoms (metals) / group of atoms (NaCl) [BASIS] Lattice + Basis = X’l structure Infinite array of discrete points arranged (and oriented) in such a way that it looks exactly the same from whichever point the array is looked at.

2. R Q P 2-D honey comb net Not a BL Lattice arrangement looks the same from P and R, bur rotated through 180º when viewed from Q

3. Primitive Translation Vectors If the lattice is a BL, then it is possible to find a set of 3 vectors a, b, c such that any point on the BL can be reached by a translation vector R = n1a + n2b + n3c where a, b, c are PTV and ni’s are integers eg: 2D lattice (B) (A) (C) a2 a2 a2 a1 a1 a2 a1 (D) a1 (A), (B), (C) define PTV, but (D) is not PTV

4. 3-D Bravais Lattices (a) Simple Cube k PTV : a3 a2 a j a1 i

5. Face centered cubic C2 PTV B F5 F4 F2 F3 F1 a2 A C1 a3 F6 a1 For Cube B, C1&C2 are Face centers; also F2&F3 All atoms are either corner points or face centers and are EQUIVALENT

6. PTV (-1,-1,3) (-1,-1,2) (-1,0,2) (0,0,2) (0,-1,2) a3 (0,0,1) (0,1,1) a2 (0,0,0) (0,1,0) a1 (1,1,0) (1,2,0) (1,0,0)

7. Alternate choice of PTV a2 a3 a1

8. Only 2-fold symmetry 2 Oblique Lattice : a ≠ b, α ≠ 90

9. Rectangular Lattice : a ≠ b, α = 90

10. 2 Rectangular Lattice : a ≠ b, α = 90 : Symmetry operations b a mirror

11. Hexagonal Lattice : a = b, α = 120 b a

13. PUC and Unit cell for BCC Unit Cell Primitive Unit Cell

14. Body-centered cubic: 2 sc lattices displaced by (a/2,a/2,a/2) A is the body center B PUC A B is the body center All points have identical surrounding

15. PUC and Unit cell for FCC PUC Unit Cell

16. PUC and Unit cell for FCC : alternate PTV

17. P I P I F P I C F P (Trigonal) P P C P 7 X’l Systems 14 BL

18. c b a

19. a2 60º a1 A lattice which is not a BL can be made into a BL by a proper choice of 2D lattice and a suitable BASIS 2-D Lattice B A The original lattice which is not a BL can be made into a BL by selecting the 2D oblique lattice (blue color) and a 2-point BASIS A-B

20. BCC Structure

21. FCC Structure

22. NaCl Structure

23. Diamond Structure

24. No. of atoms/unit cell = 8 Corners – 1 Face centers – 3 Inside the cube – 4 (¾, ¼, ¾) (¼, ¾, ¾) (¾, ¾,¼) z (¼, ¼,¼) y x (0,0,0) DiamondStructure

26. Hexagonal Close Packed (HCP) Structure

27. HCP = HL (BL) + 2 point BASIS at (000) and (2/3,1/3,1/2)

28. The Simple Hexagonal Lattice

29. The HCP Crystal Structure

30. 4-circle Diffractometer

32. Reciprocal Lattice (000)

33. (201) plane k = k´- k = G201 k´ k θ201 2π/λ Incident beam (102) (002) (302) (202) (301) (201) (101) (001) (300) (200) (100) (000) (00 -1) (30 -1)

34. a* (-200) (000) (200) b* 2π/λ Incident beam Rotaion = 0º

35. (-200) (000) (200) a* 2π/λ b* Rotaion = 5º

36. a* (-200) (000) (200) b* Rotaion = 10º

37. a* (-200) (000) (200) 2π/λ b* Rotaion = 20º

38. (-200) (000) (200) 2π/λ 2π/λ Incident beam Incident beam Rotaion = 5º Rotaion = 20º a* b*

39. Schematic diagram of a four-circle diffractometer.

40. Scattering Intensities and Systematic Absence I 2θ

41. Diffraction Intensities • Scattering by electrons • Scattering by atoms • Scattering by a unit cell • Structure factors • Powder diffraction intensity calculations • – Multiplicity • – Lorentz factor • – Absorption, Debye-Scherrer and Bragg Brentano • – Temperature factor

42. Scattering by atoms • We can consider an atom to be a collection of electrons. • This electron density scatters radiation according to the Thomson approach (classical Scattering). However, the radiation is coherent so we have to consider interference between x-rays scattered from different points within the atom • – This leads to a strong angle dependence of the scattering – FORM FACTOR.

43. Form factor (Atomic Scattering Factor) • We express the scattering power of an atom using a form factor (f) • – Form factor is the ratio of scattering from the atom to what • would be observed from a single electron 30 29 Form factor is expressed as a function of (sinθ)/λ as the interference depends on both λ and the scattering angle Form factor is equivalent to the atomic number at low angles, but it drops rapidly at high (sinθ)/λ 20 fCu 10 0 0.6 1.0 0.8 0 0.2 0.4 sinθ/λ

44. X-ray and neutron form factor The form factor is related to the scattering density distribution in an atoms - It is the Fourier transform of the scattering density - Neutrons are scattered by the nucleus not electrons and as the nucleus is very small, the neutron form factor shows no angular dependence NEUTRON b F- X-RAY 3He 7Li f C 1H Li+ sinθ/λ sinθ/λ

45. (b) 1 3 3 2 2 (b) Scattering by a Unit Cell – Structure Factor The positions of the atoms in a unit cell determine the intensities of the reflections Consider diffraction from (001) planes in (a) and (b) If the path length between rays 1 and 2 differs by λ, the path length between rays 1 and 3 will differ by λ/2 and destructive interfe-rence in (b) will lead to no diffracted intensity c (a) b a 1 1 2 2 (a)