بلور شناسی در فیزیک حالت جامد سلول واحد (مفاهیم پایه ) انواع شبکه های دو بعدی و سه بعدی

1. بلور شناسی در فیزیک حالت جامد سلول واحد (مفاهیم پایه ) انواع شبکه های دو بعدی و سه بعدی اندیس های میلر

2. مواد بی شکل (نامنظم) توزیع کاتوره ای از اتمها، یونها، ملکولها

3. تفاوت؟

4. سلول واحد در شبکه دو بعدی

5. سلول واحد در شبکه دو بعدی NaCl Crystal Structure

6. انتخاب دلخواه سلول واحد(حجم یکسان) Crystal Structure

7. چینش اتمها در سلول واحد مهم نیست Crystal Structure

8. - or if you don’t start from an atom Crystal Structure

9. This is NOT a unit cell even though they are all the same - empty space is not allowed! Crystal Structure

10. سلول واحد دو بعدیسلول واحد سه بعدی صحیح است؟ چرا؟ Crystal Structure

11. Why can't the blue triangle be a unit cell? Crystal Structure

12. Five Bravais Lattices in 2D Crystal Structure

13. Unit Cell in 3D Crystal Structure

14. Unit Cell in 3D Crystal Structure

15. سلول واحد

16. Fig. 3.4, Callister 7e. Crystal Systems – Some Definitional information Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems of varying symmetry are known These systems are built by changing the lattice parameters:a, b, and c are the edge lengths , , and  are interaxial angles

17. Wigner-Seitz Method Crystal Structure

18. Wigner-Seitz Cell - 3D Crystal Structure

19. Crystal Structure

20. Crystal Systems Crystal structures are divided into groups according to unit cell geometry (symmetry).

23. 1-CUBIC Crystal Structure

24. Simple Cubic Structure (SC) • Rare due to low packing density (only Po – Polonium -- has this structure) • Close-packed directions are cube edges. • Coordination No. = 6 (# nearest neighbors) for each atom as seen (Courtesy P.M. Anderson)

26. b c a 1-CUBIC CRYSTAL SYSTEM a- Simple Cubic (SC) یک اتم در هر سلول واحد برای شبکه مکعبی ساده Crystal Structure

27. a- Simple Cubic (SC) Crystal Structure

28. ضریب به هم پکیدگی در sc Crystal Structure

29. b-Body Centered Cubic (BCC) سلول غیربسیط 8 همسایه نزدیک • (Fe,Li,Na..etc) are BCC c b a Crystal Structure

30. 2 (0,433a) ضریب به هم پکیدگی در BCC Crystal Structure

31. Atomic Packing Factor: BCC a 3 a 2 Close-packed directions: R 3 a length = 4R = a atoms volume 4 3 p ( 3 a/4 ) 2 unit cell atom 3 APF = volume 3 a unit cell a Adapted from Fig. 3.2(a), Callister 7e. • APF for a body-centered cubic structure = 0.68

32. Face Centered Cubic (FCC) • 4 اتم در سلول واحدش وجود دارد • (Cu,Ni,Pb..etc) ساختار fcc. دارند Crystal Structure

33. Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12 Adapted from Fig. 3.1, Callister 7e. 4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8) (Courtesy P.M. Anderson)

34. Atomic Packing Factor: FCC Close-packed directions: 2 a length = 4R = 2 a Unit cell contains: 6 x1/2 + 8 x1/8 = 4 atoms/unit cell a atoms volume 4 3 p ( 2 a/4 ) 4 unit cell atom 3 APF = volume 3 a unit cell • APF for a face-centered cubic structure = 0.74 The maximum achievable APF! (a = 22*R) Adapted from Fig. 3.1(a), Callister 7e.

35. 3 - Face Centered Cubıc Atoms are all same. Crystal Structure

36. Primitive and conventional cells of FCC Crystal Structure

37. Primitive and conventional cells of BCC Primitive Translation Vectors:

38. Primitive and conventional cells Body centered cubic (bcc): conventional ¹primitive cell مختصات نقاط مختلف 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½ Simple cubic (sc): primitive cell=conventional cell :مختصات نقاط مختلف 000, 100, 010, 001, 110,101, 011, 111 Crystal Structure

39. Primitive and conventional cells Face centered cubic (fcc): conventional¹ primitive cell مختصات نقاط مختلف : 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1 Crystal Structure

40. 2 - HEXAGONAL SYSTEM • سه اتم در سلول واحدش وجود دارد. Crystal Structure

41. 3–Hexagonal Close-Packed Str. Crystal Structure

42. Hexagonal Close-packed Structure He, Be, Mg, Hf, Re (Group II elements) ABABAB Type of Stacking  a=b a=120, c=1.633a,  basis : (0,0,0) (2/3a ,1/3a,1/2c) Crystal Structure

43. Packing A A A A A B B B B C C C A A A A A A A B B B B B C C C C A A A A A A A A B B B B C C C B B A A A A A A A Close pack Sequence AAAA… - simple cubic Sequence ABABAB.. • hexagonal close pack Sequence ABAB… - body centeredcubic Sequence ABCABCAB.. -face centered cubic close pack Crystal Structure

44. Crystal Structure

45. 2 - HEXAGONAL SYSTEM Atoms are all same. Crystal Structure

46. Hexagonal Close-Packed Structure (HCP) A sites Top layer c Middle layer B sites A sites Bottom layer a ex: Cd, Mg, Ti, Zn • ABAB... Stacking Sequence • 3D Projection • 2D Projection Adapted from Fig. 3.3(a), Callister 7e. 6 atoms/unit cell • Coordination # = 12 • APF = 0.74 • c/a = 1.633 (ideal)

47. We find that both FCC & HCP are highest density packing schemes (APF = .74) – this illustration shows their differences as the closest packed planes are “built-up”

48. Crystal Structure

49. 3 - TRICLINIC 4 - MONOCLINIC CRYSTAL SYSTEM • تری کلینیک کمترین میزان تقارن را داراست Monoclinic (Simple)a = g = 90o, ß ¹ 90o a ¹ b ¹c Monoclinic (BaseCentered) a = g = 90o, ß ¹ 90o a ¹ b ¹ c, Triclinic (Simple) a ¹ ß ¹ g ¹ 90 oa ¹ b ¹ c Crystal Structure

50. 5 - ORTHORHOMBIC SYSTEM Orthorhombic (FC) a = ß = g = 90o a ¹ b ¹ c Orthorhombic (Base-centred)a = ß = g = 90o a ¹ b ¹ c Orthorhombic (BC)a = ß = g = 90o a ¹ b ¹ c Orthorhombic (Simple) a = ß = g = 90o a ¹ b ¹ c Crystal Structure