Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi

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1 TEQIP WORKSHOP ON HIGH RESOLUTION X-RAY AND ELECTRON DIFFRACTION, FEB 01, 2016, IIT-K. Introduction to x-ray diffraction Peak Positions and Intensities Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi

5 Question 1: Why crystal have regular external shapes? Postulate of Kepler, Hooke et al.: Because they have regular arrangement of building blocks ( atoms in modern language)

6 1895 Wilhelm Röntgen Discovered x- rays First Nobel Prize in physics: 1901

7 First x-ray picture November 1895 Hand of Roentgen s wife

8 Question2: Are x-rays waves or particles? X stands for the unknown

$x-rays are waves Then Crystals should act as a 3D diffraction grating for$

9 Laue s Postulate If crystals are periodic arrangement of atoms And If x-rays are waves Then Crystals should act as a 3D diffraction grating for x-rays

12 Two GREAT results from a single experiment: 1. X-rays are waves 2. Crystals are periodic arrangement of atoms One of the greatest scientific discoveries of twentieth century

13 X-Ray Peak Positions Diffraction = + Peak Intensities Crystal = Lattice + Structure Motif: Atom Positions

14 X-Ray Diffraction Incident Beam Sample Transmitted Beam Diffracted Beam

16 X-Ray Diffraction Sample Bragg Reflection Incident Beam Transmitted Beam Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes.

17 X-Ray Diffraction Braggs recipe for Nobel prize? Call the diffraction a reflection!!! The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. W.L. Bragg

18 A father-son team that shared a Nobel Prize Nobel Prize (1915) William Lawrence Bragg ( ) William Henry Bragg (

20 X-Ray Diffraction Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes. i θ plane θ r Specular reflection: Angle of incidence =Angle of reflection (both measured from the plane and not from the normal) The incident beam, the reflected beam and the plane normal lie in one plane

21 X-Ray Diffraction i d hkl θ θ r n λ = 2d sinθ hkl Bragg s law (Part 2):

22 i r θ θ θ θ P R d hkl Q Path Difference =PQ+QR = 2d hkl sinθ

23 i r θ θ P Q R Path Difference =PQ+QR = 2d hkl sinθ Constructive inteference n λ = 2d sinθ hkl Bragg s law

24 Two equivalent ways of stating Bragg s Law n λ = 2d sinθ hkl 1 st Form d hkl λ = 2 sinθ n d nh, nk, nl = = ( nh) a + ( nk) + ( nl) d n hkl λ = 2d sinθ nhnk nl 2 nd Form

25 Two equivalent ways of stating Bragg s Law n λ = 2d sinθ λ = 2d sinθ hkl nhnk nl n th order reflection from (hkl) plane 1 st order reflection from (nh nk nl) plane e.g. a 2 nd order reflection from (111) plane can be described as 1 st order reflection from (222) plane

26 X-rays Characteristic Radiation, K α Target Mo Cu Co Fe Cr Wavelength, Å

27 Experimental Diffraction Settings Laue method Variable λ Fixed θ Rotating crystal method Fixed λ Variable θ Powder method Fixed λ Variable θ

28 Powder Method λ is fixed (Kα radiation) θ is variable millions of powder particles randomly oriented in space rotation of a crystal about all possible axes

29 Powder diffractometer geometry i θ plane Diffracted beam 2 Incident beam sample t θ θ r 2θ 1 Diffracted beam 1 Transmitted beam Intensity Strong intensity Zero intensity X-ray detector 2θ 1 2θ 2 2θ

30 Crystal monochromator detector X-ray tube X-ray powder diffractometer

31 The diffraction pattern of austenite Austenite = fcc Fe

32 Bcc crystal x λ λ/2 d 100 = a z y No 100 reflection for bcc No bcc reflection for h+k+l=odd 100 reflection= rays reflected from adjacent (100) planes spaced at d 100 have a path difference λ

33 Extinction Rules Bravais Lattice Allowed Reflections SC BCC FCC DC All (h + k + l) even h, k and l all odd, or h, k and l all even h, k and l are all odd Or if all are even then (h + k + l) divisible by 4

34 X-Ray Diffraction Peak Positions = + Peak Intensities Crystal = Lattice + Structure Motif: Atom Positions

35 Intensity of Powder Diffraction Peaks 1. Scattering by an electron 2. Scattering by an atom (Atomic scattering factor) 3. Scattering by a unit cell (Structure factor) 4. Polarization factor 5. Multiplicity factor 6. Lorentz factor 7. Absorption factor 8. Temperature factor B.D. Cullity, Elements of X-Ray Diffraction, 2 nd. Edn., Addison Wesley, 1978, Ch. 4.

36 Intensity of Powder Diffraction Peaks 1. Scattering by an electron Thompson s equation I P = I 0 K r 2! 1+ cos 2 2θ $ # & " 2 % (4-2) P 2θ O Location of an electron Direction of transmitted beam I p = Intensity of scattered beam at at P I 0 = Intensity of incident beam at O K = constant (7.94 x m 2 ) R = distance of P from O 2θ = scattering angle (angle between transmitted and scattered beam) Polarization factor! 1+ cos 2 2θ $ # & " 2 % Unpolarization factor!!

39 1 Polarization Factor θ

40 Intensity of Powder Diffraction Peaks 2. Scattering by an atom (Atomic scattering factor) Cu Cullity Fig. 4-5 Cullity Fig. 4-6 f = amplitude of the wave scattered by an atom amplitude of the wave scattered by one electron

41 3. Scattering by a unit cell (Structure factor) Phase difference between waves scattered from atom at the origin 000 k th atom at a fractional coordinate u i v i w i for waves reflected by (hkl) plane ϕ k = 2π (hu k + kv k + lw k ) Amplitude of this wave A k = f k where f k is the atomic scattering factor of the ith atom In the complex exponential notation this wave is represented by A k e iϕ k

42 Summation of all such scattered waves from the entire unit cell is called the STRUCTURE FACTOR, F F = n 1 A k e iϕ k n 1 F = f n e 2πi(hu k+kv k +lw k ) Intensity * I = FF = F 2

43 Example: Structure factor of a monatomic bcc unit cell Coordinate of atoms in the unit cell = ; + + = n lw kv hu i n n n n e f F 1 ) ( 2π ) ( 2 0) 0 0 ( 2 l k h i l k h i e f e f = π π [ ] ) ( 1 l k h i e f = π If, even l k h = + + then f F 2 = then odd l k h = = F

44 Intensity of Powder Diffraction Peaks 4. Multiplicity factor {100} cubic = (100), (010), (001), (100), (010), (001) {111} cubic = (111), (111), (111), (111) (1 1 1), (11 1), (111),(1 11) p{ 100} = 6 p{111} = 8 The ratio of intensities of 100 reflection to the intensity of 111 reflection, other things being equal, is expected to be: 6 = 8 3 4

45 Intensity of Powder Diffraction Peaks 5. Polarization Factor P = 1+ cos 2 2 2θ Due to the fact that incident wave is unpolarized

46 Intensity of Powder Diffraction Peaks 6. Lorentz Factor: (A) reflection at non Bragg angles Imax 1 sinθ B 1 cosθ Integrated intensity(a) 1 sin 2θ

47 Intensity of Powder Diffraction Peaks 4. Lorentz Factor: (B) Fraction of properly oriented crystals ΔN N = rδθ 2π r sin(90 2 4π r θ ) B = Δθ cosθ B 2 Figure 4-15 of Cullity Integrated intensity(b) cosθ

48 Intensity of Powder Diffraction Peaks 6. Fraction of diffraction cone recorded Figure 4-16 of Cullity Full length of the diffracted line = 2π Rsin2θ B 1 Integrated intensity(c) sin 2θ

49 Intensity of Powder Diffraction Peaks 6. Lorentz Factor: Integrated intensity(a) 1 sin 2θ reflection at non Bragg angles cosθ 1 sin 2θ Lorentz factor Fraction of properly oriented crystals Fraction of diffraction cone recorded L = 1 sin 2 1 cosθ ( cosθ ) = 2 θ sin 2θ sin 2θ

50 Intensity of Powder Diffraction Peaks 5. Polarization Factor P = 1+ cos 2 2 2θ 6. Lorentz Factor: L = cosθ 2 sin 2θ Lorentz-Polarization Factor L P = 2 1+ cos 2θ 2 sin θ cosθ ignoring the numerical constant

51 Figure 4-17 from Cullity

52 7. Absorption Factor A Absorption in the specimen Independent of θ: x x d Low θ: Low penetration depth, d Large irradiated area, x d High θ: High penetration depth, d Small irradiated area, x Effective irradiated volume constant and independent of θ.

$Thus there cannot be any periodicity of instantaneous positions Therefore no sharp diffraction pattern will be observed.$

53 Temperature factor Gist of a Discussion in a Coffee House in Munich before Laue s famous experiment Atoms are continuously vibrating with random amplitude Thus there cannot be any periodicity of instantaneous positions Therefore no sharp diffraction pattern will be observed. Reference: Kittel, Solid state Physics

54 Temperature factor 1. Lattice parameter increase -> Peaks shift to lower θ 2. Intensity of the diffraction lines decrease 3. Intensity of background scattering increases Rather surprisingly, there is no significant change in peak width.

55 Final Intensity expression for Diffractometer! I = F 2 p 1+ cos2 2θ $ # & e 2M " sin 2 θ cosθ %

X-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity

$X-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity$ X-ray Diraction Interaction o Waves Reciprocal Lattice and Diraction X-ray Scattering by Atoms The Integrated Intensity Basic Principles o Interaction o Waves Periodic waves characteristic: Frequency :