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

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

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سلول واحد در شبکه دو بعدی

سلول واحد در شبکه دو بعدی NaCl Crystal Structure

انتخاب دلخواه سلول واحد(حجم یکسان) Crystal Structure

چینش اتمها در سلول واحد مهم نیست Crystal Structure

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

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

سلول واحد دو بعدی سلول واحد سه بعدی صحیح است؟ چرا؟ Crystal Structure

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

Five Bravais Lattices in 2D Crystal Structure

Unit Cell in 3D Crystal Structure

Unit Cell in 3D Crystal Structure

سلول واحد

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

Wigner-Seitz Method Crystal Structure

Wigner-Seitz Cell - 3D Crystal Structure

Crystal Structure

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

1-CUBIC Crystal Structure

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)

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

a- Simple Cubic (SC) Crystal Structure

ضریب به هم پکیدگی در sc Crystal Structure

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

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

Atomic Packing Factor: BCC 3 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

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

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)

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

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

Primitive and conventional cells of FCC Crystal Structure

Primitive and conventional cells of BCC Primitive Translation Vectors:

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

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

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

3–Hexagonal Close-Packed Str. Crystal Structure

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

-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

Crystal Structure

2 - HEXAGONAL SYSTEM Crystal Structure Atoms are all same.

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 = 1.633 (ideal)

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”

Crystal Structure

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

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

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

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

(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] http://www.yourgemologist.com/crystalsystems.html [2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.

Tetragonal Crystals a = b  c  =  =  = 90º Simple Tetragonal Body Centred Tetragonal Zircon [1] [1] [1] [1] http://www.yourgemologist.com/crystalsystems.html

Orthorhombic Crystals a  b  c  =  =  = 90º Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic [1] Topaz [1] [1] http://www.yourgemologist.com/crystalsystems.html

Hexagonal Crystals a = b  c  =  = 90º  = 120º Simple Hexagonal [1] Corundum [1] http://www.yourgemologist.com/crystalsystems.html

5. Trigonal Crystals Trigonal (simple) a = b = c  =  =   90º [1] [1] Tourmaline [1] http://www.yourgemologist.com/crystalsystems.html

Monoclinic Crystals a  b  c  =  = 90º   Simple Monoclinic End Centred (base centered) Monoclinic (A/C) [1] Kunzite [1] http://www.yourgemologist.com/crystalsystems.html

7. Triclinic Crystals a  b  c      Simple Triclinic Amazonite [1] Amazonite [1] http://www.yourgemologist.com/crystalsystems.html

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

Crystal Planes

Crystallographic Planes -- families

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

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

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

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

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  -1 1/2 0 -1 2 N/A (012)

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

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

Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used a2 a3 a1 z example a1 a2 a3 c 1. Intercepts 1  -1 1 2. Reciprocals 1 1/ 1 0 -1 1 3. Reduction 1 0 -1 1 4. Miller-Bravais Indices (1011) Adapted from Fig. 3.8(a), Callister 7e.

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

Crystallographic Directions z 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>

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]

اندیس های میلر و جهتهای صفحات اتمی در بلور

اندیس های میلر و جهتهای صفحات اتمی در بلور

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

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, 0 =>

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

فاصله صفحات موازی با اندیس های میلر یکسان فاصله صفحات موازی با اندیس های میلر یکسان PH 0101 UNIT 4 LECTURE 2

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

Crystal Structure

Example-7 Crystal Structure

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

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

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

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