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Lec. 6,7 1

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Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can’t resolve spacings Spacing is the distance between parallel planes of atoms. 2

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X-Ray Diffractıon Methods Von Laue Rotating Crystal Powder Orientation Single Crystal Polychromatic Beam, Fixed Angle single Lattice constant Single Crystal Monchromatic Beam, Variable Angle Varied by rotation Lattice Parameters Poly Crystal Monchromatic Beam, Variable Angle Many s (orientations) 3

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Laue Method The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal. The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out & diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d & θ involved. 4

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In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal. X-Rays Film Single Crystal Transmission Laue Method In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal. One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse. 5

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The Laue method is mainly used to determine the crystal orientation. Although the Laue method can also be used to determine the crystal structure, several wavelengths can reflect in different orders from the same set of planes, with the different order reflections superimposed on the same spot in the film. This makes crystal structure determination by spot intensity diffucult. The rotating crystal method overcomes this problem. Crystal Structure Determination by the Laue Method 6

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In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x-ray beam. A cylindrical film is placed around it & the crystal is rotated about the chosen axis. As the crystal rotates, Sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, & at that point a diffracted beam will be formed. Rotatıng Crystal Method 7

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The Lattice constant of the crystal can be determined with this method. For a given wavelength λ if the angle θ at which a reflection occurs is known, d can be determined by using Bragg’s Law. Rotatıng Crystal Method 8

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The reflected beams are located on the surfaces of imaginary cones. By recording the diffraction patterns (both angles & intensities) for various crystal orientations, one can determine the shape & size of unit cell as well as the arrangement of atoms inside the cell. Film Rotatıng Crystal Method 9

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For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is X-rays Hence, X-rays can be used for the study of crystal structures Beam of electrons Target X-rays An accelerating (/decelerating) charge radiates electromagnetic radiation 10

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Relationship of the Bragg angle (θ) and the experimentally measured diffraction angle (2θ). X-ray intensity (from detector) c d d n 2 sin c 11

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Intensity Wavelength ( ) Mo Target impacted by electrons accelerated by a 35 kV potential White radiation Characteristic radiation → due to energy transitions in the atom KK KK 12

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Target Metal Of K radiation (Å) Mo0.71 Cu1.54 Co1.79 Fe1.94 Cr

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BRAGG’s EQUATION d dSin The path difference between ray 1 and ray 2 = 2d Sin For constructive interference: n = 2d Sin Ray 1 Ray 2 Deviation = 2 14

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Note that in the Bragg’s equation: The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appears Change in position or spacing of atoms along the plane should not affect Bragg’s condition !! d Note: shift (systematic) is actually not a problem! 17

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Bragg’s equation is a negative law If Bragg’s eq. is NOT satisfied NO reflection can occur If Bragg’s eq. is satisfied reflection MAY occur Diffraction = Reinforced Coherent Scattering Reflection versus Scattering ReflectionDiffraction Occurs from surfaceOccurs throughout the bulk Takes place at any angleTakes place only at Bragg angles ~100 % of the intensity may be reflectedSmall fraction of intensity is diffracted X-rays can be reflected at very small angles of incidence 18

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n = 2d Sin , n= 1, 2, 3, … n is an integer and is the order of the reflection For Cu K radiation ( = 1.54 Å) and d 110 = 2.22 Å n Sin ºFirst order reflection from (110) º Second order reflection from (110) Also written as (220) 19

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In XRD n th order reflection from (h k l) is considered as 1 st order reflection from (nh nk nl) 20

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If a powdered crystal is used instead of a single crystal, then there is no need to rotate it, because there will always be some small crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample. Useful for samples that are difficult to obtain in single crystal form. The powder method is used to determine the lattice parameters accurately. Lattice parameters are the magnitudes of the primitive vectors a, b and c which define the unit cell for the crystal. The Powder Method 22

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For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line. The Powder Method 23

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If a monochromatic X-ray beam is directed at a single crystal, then only one or two diffracted beams may result. See figure For a sample of several randomly orientated single crystals, the diffracted beams will lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards. See figure For a sample of hundreds of crystals (powdered sample), the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. See figure The Powder Method 24

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THE POWDER METHOD Cone of diffracted rays 25

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Diffraction cones and the Debye-Scherrer geometry Film may be replaced with detector POWDER METHOD Different cones for different reflections 26

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A small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract X-Rays. The sample is placed in the Debye Scherrer camera and is accurately aligned to be in the center of the camera. X- Rays enter the camera through a collimator. The powder diffracts the X-Rays in accordance with Braggs Law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film. Debye Scherrer Camera 27

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When the film is removed from the camera, flattened & processed, it shows the diffraction lines & the holes for the incident & transmitted beams. Powder Diffraction Film 28

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Some Typical Measurement Results Laue - “white” X-rays –Yields stereoscopic projection of reciprocal lattice Rotating-Crystal method: monochromatic X-rays –Fix source & rotate crystal to reveal reciprocal lattice Powder diffraction - monochromatic X-rays –Powder sample to reveal “all” directions of RL 29

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Photograph of a XRD Diffractometer (Courtesy H&M Services.) 30

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(b) Diffraction Pattern from a sample of gold powder. (a) Diagram of a diffractometer showing a powdered sample, incident & diffracted beams. 31

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The results of a diffraction experiment using X-Rays with λ = Å (radiation obtained from a molybdenum, Mo, target) show that diffracted peaks occur at the following 2θ angles: Example (From the Internet) Find: The crystal structure, the indices of the plane producing each peak, & the lattice parameter of the material. 32

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First calculate the sin 2 θ value for each peak, then divide through by the lowest denominator, Example (Solution) 33

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Example (Solution Continued) Then use the 2θ values for any of the peaks to calculate the interplanar spacing & thus the lattice parameter. Picking Peak 8: 2θ = 59.42° or θ = 29.71° Ǻ So, for example: This is the lattice parameter for body-centered cubic iron. Ǻ 34

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Note: XRD is a nondestructive technique! Some uses of XRD include: 1. Distinguishing between crystalline & amorphous materials. 2. Determination of the structure of crystalline materials. 3. Determination of electron distribution within the atoms, & throughout the unit cell. 4. Determination of the orientation of single crystals. 5. Determination of the texture of polygrained materials. 6. Measurement of strain and small grain size…..etc. Applications of XRD 35

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Advantages X-Rays are the least expensive, the most convenient & the most widely used method to determine crystal structures. X-Rays are not absorbed very much by air, so the sample need not be in an evacuated chamber. Disadvantages X-Rays do not interact very strongly with lighter elements. Advantages & Disadvantages of XRD Compared to Other Methods 36

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Neutrons λ ~ 1 Ǻ E ~ 0.08 eV interact with nuclei, highly penetrating Electrons λ ~ 1 Ǻ E ~ 150 eV interact with electrons, less penetrating Diffraction Methods 37

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