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 بعدی سلول 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. Crystal Systems – Some Definitional information
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Fig. 3.4, Callister 7e.
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. 1-CUBIC CRYSTAL SYSTEM a- Simple Cubic (SC)
یک اتم در هر سلول واحد برای شبکه مکعبی ساده a b c
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 b c a
Crystal Structure

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

31. Atomic Packing Factor: BCC3 a a 2
length = 4R =
Close-packed directions:
3 a
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume 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
• APF for a face-centered cubic structure = 0.74 a
2 a
The maximum achievable APF!
Close-packed directions:
length = 4R =
2 a
(a = 22*R)
Unit cell contains:
6 x 1/2 + 8 x 1/8 =
4 atoms/unit cell
Adapted from
Fig. 3.1(a),
Callister 7e.
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

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

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):
conventional ¹ 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. -face centered cubic close pack
Packing
Close pack A B A B C
Sequence AAAA…
- simple cubic
Sequence ABABAB..
hexagonal close pack
Sequence ABAB…
- body centered cubic
Sequence ABCABCAB..
-face centered cubic close pack
Crystal Structure

44. Crystal Structure

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

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

47. We find that both FCC & HCP are highest density packing schemes (APF =
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
Triclinic (Simple) a ¹ ß ¹ g ¹ 90
oa ¹ b ¹ c
Monoclinic (Simple) a = g = 90o, ß ¹ 90o
a ¹ b ¹c
Monoclinic (Base Centered) a = g = 90o, ß ¹ 90o
a ¹ b ¹ c,
Crystal Structure

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

51. 6 – TETRAGONAL SYSTEM Tetragonal (BC) a = ß = g = 90o
Tetragonal (P) a = ß = g = 90o
a = b ¹ c
Tetragonal (BC) a = ß = g = 90o
a = b ¹ c
Crystal Structure

52. 7 - Rhombohedral (R) or Trigonal
Rhombohedral (R) or Trigonal (S) a = b = c, a = ß = 90,g ¹ 90o
Crystal Structure

53. (Truncated Octahedron)
Cubic Crystals
a = b= c  =  =  = 90º
Simple Cubic (P)
Body Centred Cubic (I) – BCC
Face Centred Cubic (F) - FCC
Vapor grown NiO crystal
[2]
Tetrakaidecahedron
(Truncated Octahedron)
Pyrite Cube
[1]
[1]
Fluorite
Octahedron
Garnet Dodecahedron
[1]
[1]
[2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.

54. Tetragonal Crystals a = b  c  =  =  = 90º Simple Tetragonal
Body Centred Tetragonal
Zircon
[1]
[1]
[1]
[1]

55. Orthorhombic Crystals
a  b  c  =  =  = 90º
Simple Orthorhombic
Body Centred Orthorhombic
Face Centred Orthorhombic
End Centred Orthorhombic
[1]
Topaz
[1]
[1]

56. Hexagonal Crystals a = b  c  =  = 90º  = 120º Simple Hexagonal
[1]
Corundum
[1]

57. 5. Trigonal Crystals Trigonal (simple) a = b = c  =  =   90º
[1]
[1]
Tourmaline
[1]

58. Monoclinic Crystals a  b  c  =  = 90º   Simple Monoclinic
End Centred (base centered) Monoclinic (A/C)
[1]
Kunzite
[1]

59. 7. Triclinic Crystals a  b  c      Simple Triclinic Amazonite
[1]
Amazonite
[1]

60. Miller Indices
اندیس های میلر نمادهایی هستند که جهت صفحات اتمی را در کریستال مشخص می کنند
این اندیس ها به گونه ای مشخص می شوند که مجوعه ای بی نهایت از صفحات بلوری را شامل مشوند
نحوه انتخابشان بگونه ای است که هماره صفحه انتخاب شده داخل سلول واحد قرار می گیرد.
Crystal Structure

61. Crystal Planes

62. Crystallographic Planes -- families

63. Example-1 Axis 1 ∞ 1/1 1/ ∞ Miller İndices (100) (1,0,0) X Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
Smallest Ratio
Miller İndices (100)
Crystal Structure

64. Example-2 Axis 1 ∞ 1/1 1/ 1 1/ ∞ Miller İndices (110) (0,1,0) (1,0,0)Y Z
Intercept points 1 ∞
Reciprocals
1/1
1/ 1
1/ ∞
Smallest Ratio
Miller İndices (110)
Crystal Structure

65. Example-3 Axis 1 1/1 1/ 1 Miller İndices (111) (0,0,1) (0,1,0) (1,0,0)Y Z
Intercept points 1
Reciprocals
1/1
1/ 1
Smallest Ratio
Miller İndices (111)
Crystal Structure

66. Example-4 Axis 1/2 1 ∞ 1/(½) 1/ 1 1/ ∞ 2 Miller İndices (210) (0,1,0)
(1/2, 0, 0)
(0,1,0)
Axis X Y Z
Intercept points
1/2 1 ∞
Reciprocals
1/(½)
1/ 1
1/ ∞
Smallest Ratio 2
Miller İndices (210)
Crystal Structure

67. Intercept in terms of lattice parameters Reciprocals Reductions
Determine the Miller indices for the plane shown in the sketch
x y z
Intercepts
Intercept in terms of lattice parameters
Reciprocals
Reductions
Enclosure
a b c/2
 /2
N/A
(012)

68. Example-5 Axis 1 ∞ ½ 2 Miller İndices (102) a b c 1/1 1/ ∞ 1/(½)
Intercept points 1 ∞
Reciprocals
1/1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102)
Crystal Structure

69. Example-6 Axis -1 ∞ ½ 2 Miller İndices (102) a b c 1/-1 1/ ∞ 1/(½)
Intercept points -1 ∞
Reciprocals
1/-1
1/ ∞
1/(½)
Smallest Ratio 2
Miller İndices (102)
Crystal Structure

70. Crystallographic Planes (HCP)
In hexagonal unit cells the same idea is used a2 a3 a1 z
example
a a a c
Intercepts  -1 1
Reciprocals / -1 1
Reduction -1 1
Miller-Bravais Indices
(1011)
Adapted from Fig. 3.8(a), Callister 7e.

71. Miller Indices [2,3,3] Plane intercepts axes at
Reciprocal numbers are:
Indices of the plane (Miller): (2,3,3)
Indices of the direction: [2,3,3]
(200)
(111)
(100)
(110)
(100)
Crystal Structure

72. Crystallographic Directionsz
Algorithm
1. Vector is repositioned (if necessary) to pass through the Unit Cell origin. 2. Read off line projections (to principal axes of U.C.) in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas
[uvw] y x
ex: 1, 0, ½
=> 2, 0, 1
=> [ 201 ]
Lecture 2 ended here
-1, 1, 1
where ‘overbar’ represents a negative index
[ 111 ]
=>
families of directions <uvw>

73. What is this Direction ????? x y z 0c a/2 b 1 1/2 1 2 [120]
Projections:
Projections in terms of a,b and c:
Reduction:
Enclosure [brackets]
x y z 0c
a/2 b 1
1/2 1 2
[120]

74. اندیس های میلر و جهتهای صفحات اتمی در بلور

75. اندیس های میلر و جهتهای صفحات اتمی در بلور

76. جهت های بلوری و صفحات اتمی عمود بر انها اندیس های میلر یکسانی دارند.

77. HCP Crystallographic Directions- a3 a1 a2 z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin. 2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas
[uvtw]
dashed red lines indicate
projections onto a1 and a2 axes a1 a2 a3
-a3 2 a 1
Adapted from Fig. 3.8(a), Callister 7e.
[ 1120 ]
ex: ½, ½, -1, =>

78. زاویه بین دو جهت بلوری دلخواه بر حسب اندیس های میلر [u1,v1,w1]

79. فاصله صفحات موازی با اندیس های میلر یکسان
فاصله صفحات موازی با اندیس های میلر یکسان
PH UNIT 4 LECTURE 2

80. h=n a/OA
k=nb/OB
l=nc/OC
با جایگزینی در رابطه مثلثاتی
d=n/{h2/ a2 + k2/ b2 + l2/ c2 }0.5

81. Crystal Structure

82. Example-7
Crystal Structure

83. Indices of a Family or Form
این {hkl} نماد کلیه اندیس های میلر مربوط به صفحات (hkl) را شامل می شود که بوسیله چرخش به همدیگر مر بوط می شوند
Crystal Structure

84. 3D – 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEM
تنها 14 شبکه براوه وجود دارد که فضای سه بعدی را می پوشاند.
این 14 شبکه نیز در هفت سیستم بلوری معرفی شده گنجانده می شوند.
Cubic Crystal System (SC, BCC,FCC)
Hexagonal Crystal System (S)
Triclinic Crystal System (S)
Monoclinic Crystal System (S, Base-C)
Orthorhombic Crystal System (S, Base-C, BC, FC)
Tetragonal Crystal System (S, BC)
Trigonal (Rhombohedral) Crystal System (S)
Crystal Structure

85. THE MOST IMPORTANT CRYSTAL STRUCTURES
Sodium Chloride Structure Na+Cl-
Cesium Chloride Structure Cs+Cl-
Hexagonal Closed-Packed Structure
Diamond Structure
Zinc Blende
Crystal Structure

86. 1 – Sodium Chloride Structure
نمک طعام NaCl بصورت شبکه مکعبی در سلول غیر بسیط ظاهر می شود
تعداد برابر از دو نوع اتم در سلول ان بطور تناوبی قرار گرفته اند.
هر یون بوسیله شش یون نوع دیگر محاصره شده است.
Crystal Structure