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

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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

ایا سلول واحد است؟ Crystal Structure

ایا سلول مثلثی سلول واحد است؟ Crystal Structure

Crystal Structure

پنج شبکه براوه دو بعدی Crystal Structure

یک ملکول هر نوع چرخشی را می تواند داشته باشد اما شبکه؟

Crystal Structure

1-CUBIC Crystal Structure

a- Simple Cubic (SC) Crystal Structure

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

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

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

2 - HEXAGONAL SYSTEM Crystal Structure Atoms are all same.

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 = g ¹ 90o Crystal Structure

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 و نحوه انتخابشان بگونه ای است که هماره صفحه انتخاب شده داخل سلول واحد قرار می گیرد. Crystal Structure

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

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

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

اندیس های میلر و جهتهای صفحات اتمی در بلور

اندیس های میلر و جهتهای صفحات اتمی در بلور

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

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

3–Hexagonal Close-Packed Str. Crystal Structure

Hexagonal Close-packed Structure a=b a=120, c=1.633a,  basis : (0,0,0) (2/3a ,1/3a,1/2c) Crystal Structure

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

First Brillouin Zone: Two Dimensional Oblique Lattice

First Four Brillouin Zones: Square Lattice

All Brillouin Zones: Square Lattice

Primitive Lattice Vectors: BCC Lattice

First Brillouin Zone: BCC

Primitive Lattice Vectors: FCC

Brillouin Zones: FCC

First Brillouin Zone BCC

First Brillouin Zone FCC

X-ray Diffraction Typical interatomic distances in solid are of the order of an angstrom. Thus the typical wavelength of an electromagnetic probe of such distances Must be of the order of an angstrom. Upon substituting this value for the wavelength into the energy equation, We find that E is of the order of 12 thousand eV, which is a typical X-ray Energy. Thus X-ray diffraction of crystals is a standard probe.

Wavelength vs particle energy

Bragg Diffraction: Bragg’s Law

Bragg’s Law The integer n is known as the order of the corresponding Reflection. The composition of the basis determines the relative Intensity of the various orders of diffraction.

Many sets of lattice planes produce Bragg diffraction

d Deviation = 2 Ray 1 Ray 2 dSin BRAGG’s EQUATION Deviation = 2 Ray 1 Ray 2     d  dSin The path difference between ray 1 and ray 2 = 2d Sin For constructive interference: n = 2d Sin

Bragg Spectrometer

Bragg Peaks

A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal The electrons oscillate under the influence of the incoming X-Rays and become secondary sources of EM radiation The secondary radiation is in all directions The waves emitted by the electrons have the same frequency as the incoming X-rays  coherent The emission will undergo constructive or destructive interference with waves scattered from other atoms Secondary emission Incoming X-rays

Sets Electron cloud into oscillation Sets nucleus (with protons) into oscillation Small effect  neglected

Oscillating charge re-radiates  In phase with the incoming x-rays

von Laue Formulation of X-Ray Diffraction

Condition for Constructive Interference

Bragg Scattering =K

The Laue Condition

Ewald Construction

Crystal structure determination Many s (orientations) Powder specimen POWDER METHOD Monochromatic X-rays Single  LAUE TECHNIQUE Panchromatic X-rays ROTATING CRYSTAL METHOD  Varied by rotation Monochromatic X-rays

THE POWDER METHOD Cone of diffracted rays

Different cones for different reflections POWDER METHOD Diffraction cones and the Debye-Scherrer geometry Different cones for different reflections Film may be replaced with detector http://www.matter.org.uk/diffraction/x-ray/powder_method.htm

Schematic X-Ray Diffractometer Detector X-Ray Source Powdered sample

Sample XRD Pattern

strong intensity = prominent crystal plane weak intensity = subordinate crystal plane background radiation

Determine D-Spacing from XRD patterns Bragg’s Law nλ = 2dsinθ n = reflection order (1,2,3,4,etc…) λ = radiation wavelength (1.54 angstroms) d = spacing between planes of atoms (angstroms) θ = angle of incidence (degrees)

strong intensity = prominent crystal plane nλ = 2dsinθ (1)(1.54) = 2dsin(15.5 degrees) 1.54 = 2d(0.267) d = 2.88 angstroms background radiation

d-spacing Intensity 2.88 100 2.18 46 1.81 31 1.94 25 2.10 20 1.75 15 2.33 10 2.01 1.66 5 1.71

The Bragg equation may be rearranged (if n=1) from to If the value of 1/(dh,k,l)2 in the cubic system equation above is inserted into this form of the Bragg equation you have Since in any specific case a and l are constant and if l2/4a2 = A pma 2010

Insert the values into a table and compute sin and sin2. Since the lowest value of sin2 is 3A and the next is 4A the first Entry in the Calc. sin2 column is (0.10854/3)*4 etc. d/Å Sin Sin2 Calc. Sin2 (h, k, I) 2.338 0.32945 0.10854 (1,1,1) 2.024 0.38056 0.14482 0.14472 (2,0,0) 1.431 0.53826 0.28972 0.28944 (2,2,0) 1.221 0.63084 0.39795 0.39798 (3,1,1) 1.169 0.65890 0.43414 0.43416 (2,2,2) 1.0124 0.76082 0.57884 0.57888 (4,0,0) 0.9289 0.82921 0.68758 0.68742 (3,3,1) 0.9055 0.85063 0.72358 0.72360 (4,2,0) The reflections have now been indexed. pma 2010