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1. Detector 2. Crystal diffraction conditions 2-7-05

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Detector in X-ray crystallography: Image plates (IP) The imaging plate (IP) is a reusable 2-D X-ray detector. Imaging plates are sensitive not only to X-rays, but other types of radiation as well: gamma radiation, alpha and beta particles, neutrons and electron beams, and serve as the primary radiation detector in many fields Advantages: high sensitivity, low noise, wide dynamic range, good linearity, good spatial resolution, modest cost. It consists of a photostimulable phosphor powder in an organic binder with a thickness between 25 and 150 m deposited on a flexible polymer support film of about 250 m thickness. The family of compounds comprising BaFX:Eu2+ (X = Cl, Br) have been known to have high luminescence efficiency for X-ray excitation. A mar detector Dia. = 345mm Supporting film Phosphor

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Detector in X-ray crystallography: Image plates (IP) The IP is exposed to the x-ray source and stores the impinging X-rays as a latent image in the phosphor. This image is recovered by scanning a laser beam across the IP, causing photostimulated luminescence (PSL). This PSL is recorded by a photomultiplier that is scanned across the IP at the same time as the laser. The resultant signal is digitized and stored in a file for data processing.

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Image plates are commonly used in academic laboratories Disadvantage: decay, high background, slow readout. Advantages: high sensitivity, low noise, wide dynamic range, good linearity, good-to-high spatial resolution, modest cost.

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Detector in X-ray crystallography: CCD Charge Coupled Device (CCD) was invented in the late 1960s by researchers at Bell Labs, The CCD's superb ability to detect light has turned it into the industry- standard image sensor technology. CCD Basics - CCD imaging is performed in a three-step process: 1. Exposure, which converts light into an electronic charge at discrete sites called pixels 2. Charge transfer, which moves the packets of charge within the silicon substrate 3. Charge-to-voltage conversion and output amplification.

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Detector in X-ray crystallography: CCD The three main components of the Quantum CCD detector are phosphor screen (to convert X-rays to light), fiber-optic taper (to demagnify the light image down to the size of the CCD chip), and CCD chip to detect the light image as an electric charge image. The electric charge image is read out of the CCD chip and digitized (converted to binary numbers) then fed into a computer. An ADSC CCD detector used in many major synchrotron stations

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Detector in X-ray crystallography: CCD The principal advantages of CCDs are their sensitivity, dynamic range and linearity. –The sensitivity, or quantum efficiency, is simply the fraction of photons incident on the chip which are detected. It is common for CCDs to achieve a quantum efficiency of about 80%. Compare this figure with only a few percent for photographic plates. –A typical dynamic range (that is, the ratio of the brightest accurately detectable signal to the faintest) is about 10 5. The corresponding figures for a photographic plate are a range of less than about 1000. –Within this dynamic range the response is essentially linear: the size of the signal is simply proportional to the number of photons detected, which makes calibration straightforward. –Fast readout of digitized images Disadvantages: High costs and small areas

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A typical diffraction pattern

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Diffraction geometry: an outline Diffraction basics Laue equations Braggs law Reciprocal space and diffraction Diffraction methods – Laue photographs – Oscillation photographs

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X-ray is an electromagnetic wave The electric component: E=Acos( t + Three component: amplitude, frequency, initial phase angle

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Interference between waves Waves with the same frequency can interfere: (a) in phase (b) out of phase (c) partially out of phase

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Double slit experiment

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Diffraction at a single slit

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The envelop function

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Diffraction and sampling

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Diffraction from crystals A crystal is a three dimensional diffraction grating The lattice periodicity of the crystal determines the sampling regions of the diffraction pattern Where the peaks appear The unit cell contents (the distribution of electron densities) give you the envelope Function –The intensity of the peaks

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A wave can be represented as a vector in a Argand diagram Acos Asin A Real Imaginary The above vector corresponds to a wave cos + iAsin = Aexp[ia] The electric component of an electromagnetic wave is: A cos( t+ ) = A cos cos t – A sin sin t = A cos cos t + A sin cos( t+90°)

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Scattering for a system with two electrons where S is perpendicular to the imaginary reflecting plane What is S???

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Scattering for a system with two electrons: origin shift does not change wave intensity Origin 1 Origin 2 The amplitude and intensity of wave T does not change

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Scattering by an atom The electron cloud of an atom Atomic scattering factor is: Assuming centrosymmetric electron cloud distribution f for a carbon atom f is real if centrosymmetry is assumed

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Scattering by a unit cell A unit cell with three atoms For each atom, the scattering is: For the whole unit cells, we will have: Where F(S) is called structure factor Structure factor F is the sum of scattering by all atoms

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Scattering by a crystals For a general unit cell in a crystal, the structure factor is: The total wave scattered by the crystal is: A crystal contains a large number of unit cells

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Scattering by a crystals A crystal does not diffract X-ray unless This is the famous Laue condition Not constructive interference unless all waves have the same phase

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Laue conditions specify scattering conditions S is perpendicular to the imaginary reflecting plane What is S??? a, b, c are unit vectors of crystal unit cell axes For a particular crystal unit, X-ray will only be scattered along discrete directions What does Laue condition mean???

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Diffraction geometry: Bragg’s law This is the Bragg’s law A Crystal contain many lattice planes Lattice plane reflect incident X-ray Scattered X-rays interfere with each other Constructive interference results only when: 2dsin =n

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Diffraction geometry: Bragg’s law Diffraction angle for a particular lattice plane can be calculated from a particular lattice plane, or vice versa. From http://www.matter.org.uk

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Real space and reciprocal space

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Imaginary reciprocal lattices are created to help us understand diffraction geometry bcsin V a* = acsin V b* = absin V c* =

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Real space vs. reciprocal space a* a=1 V=1/V* In addition:

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The construction of a reciprocal lattice The reciprocal unit cell dimension is inversely related to the direct unit cell dimensions. Each reflection falls onto to a reciprocal lattice point The length of a reciprocal vector is inversely related to the distance between crystal lattice planes that give rise to that particular reflections.

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Physical meaning of reciprocal lattice If Laue conditions are met, the end point of the vector S(h,k,l) are located in the reciprocal lattice points. Proof: S can always be written as S = X a* + Y b* + Z c* Therefore, a S = X ( a* a) + Y (b* b) + Z (c* c) = X Because Laue condition states: a S = h, therefore, X=h

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The construction of Ewald sphere Ewald sphere is a geometric construction that allows one to visualize the diffraction directions and the properties of Bragg's law

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The construction of Ewald sphere Diffraction occurs when a reciprocal lattice point intersects the Ewald sphere. Ewald sphere is a geometric construction that allows one to visualize the diffraction directions and the properties of Bragg's law

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Conclusions for reciprocal lattice The reciprocal lattice rotates exactly as the crystal does. The Ewald sphere allows us to visualize a diffraction experiment The diffracted beam will "travel" from the center of the Ewald sphere through the point where its reciprocal lattice point intersects the sphere. The radius of the Ewald sphere is 1/ Diffraction only occurs when a reciprocal lattice point intersects the Ewald sphere

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EEE539 Solid State Electronics

EEE539 Solid State Electronics

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