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SOLID STATE PHYSICS BY : GROUP 5, Laue Experiment, Why x-rays ? -…

- - - - Orientation Single Crystal Polychromatic Beam Fixed Angle
      - a) TRANSMISSION LAUE METHOD
        - the film is placed behind the crystal to record beams which are transmitted through 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.
        
        Can be used to orient crystals for solid state experiment.
        
        Most suitable for the investigation of preferred orientation sheet particularly confined to lower diffraction angles.
        
        Also used in determination of symmetry of single crystal.
      - b) BACK-REFLECTION METHOD
        - the film is placed between the x-ray source and the crystal. the beams which are diffracted in a backward direction are recorded.
        
        one side of the cone of Laue reflection is defined by the transmitted beam. the film intersects the cone, with the diffraction spots generally lying on an hyperbola.
        
        This method is similar to Transmission method however , black-reflection is the only method for the study of large- and thick specimes.
      - Used to determine the orientation of large single crystal.
        
        White radiation is reflected from, or transmitted through a fixed crystal.
        
        The diffracted beams form arrays of spots, that lie on curves on the film.
        
        Each curve therefore correspond to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone.
        
        Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis.
      - The incident rays comprises of various wavelength from λmin-0.2Å and λmax-3.0 Å
        
        TWO TYPES
        
        TRANSMISSION TECHNIQUE
        
        Film use for recording the diffraction pattern is placed behind the sample and records the forward diffracted ray.
        
        REFLECTED RAY TECHNIQUE
      - The Laue method is the oldest of the x-ray diffraction methods.
        
        A collimated beam of continuous spectrum falls upon a fixed single crystal.
        
        For each set of planes hkl, the spacing d(hkl) and the Bragg angle θ(hkl) are fixed.
      - In the Laue method the incident radiation covers a wide band of wavelengths. Correspondingly, all Ewald spheres with radii within the appropriate range of incident wavevectors are to be drawn in the reciprocal lattice; whenever the Ewald spheres intercept points of the reciprocal lattice, elastic diffractions become possible.
      - Ewald Construction for Laue Technique
        
        Two incident wave vectors in the same direction with their tails on the same reciprocal lattice point.
        
        |q2| = 2π/λmin and 2π/ λmax
        
        For each vector, draw an Ewald sphere centered at the head of the vector and its radius is the magnitude of each vector.
        
        All reciprocal lattice points that touch the surface of the Ewald sphere between the two limiting spheres will satisfy Laue requirement.
    - - Lattice Constant Single Crystal Monochromatic Beam Variable Angle
      - The incident radiation is monochromatic but the crystal is capable of rotational motion around given axis. The radius of the Ewald sphere is thus fixed, but the reciprocal space rotates following the crystal rotation in direct space, when points of the reciprocal lattice intercept the Ewald sphere, elastic diffractions become possible.
    - - Lattice Parameters Polycrystal ( powdered ) Monochromatic Beam Variable Angle
      - The incident radiation is monochromatic, but now the single (rotating) crystal is replaced by a poly crystalline specimen (powder) with random orientation of the composing crystallites. :
- - - - Measure the average spacings between layers or rows of atoms.
        
        Determine the orientation of a single crystal or grain.
        
        Find the crystal structure of an unknown material.
        
        Measure the size, shape and internal stress of small crystalline regions
    - - An X-ray diffraction pattern is a plot of the intensity of X-rays scattered at different angles by a sample
        
        The detector moves in a circle around the sample
        
        The detector position is recorded as the angle 2θ
        
        The detector records the number of X-rays observed at each angle 2θ
        
        The X-ray intensity is usually recorded as "counts" or as "counts per second"
        
        To keep the X-ray beam properly focused, the sample will also rotate.
        
        On some instruments, the X-ray tube may rotate instead of the sample.
        
        From the XRD pattern you can determine:
        
        What crystalline phases are in a mixture
        
        How much of each crystalline phase is in the mixture
        
        If any amorphous material is present in the mixture
        
        Production of X-rays
        
        Cross section of sealed-off filament X-ray tube
        
        X-rays are produced whenever high-speed electrons collide with a metal target. A source of electrons - hot W filament, a high accelerating voltage between the cathode (W) and the anode and a metal target, Cu, Al, Mo, Mg. The anode is a water-cooled block of Cu containing desired target metal.
        
        Different planes have different spacings. To satisfy Bragg’s Law, θ must change as d changes e.g., θ decreases as d increases.
        
        Bragg’s Law
        
        W. L. Bragg: Explained the diffracted rays from crystals
        
        Assumption :
        Specular reflection:
        θi = θr θi = incoming angle, θr = reflection/ dispersion angle.
        
        For elastic dispersion λ constant before and after dispersion.
        For a cubic unit cell (dimension a), dhkl
        
        Derivation of Bragg’s Law
        Bragg's law tells us about the spacing between atomic planes (d) base on the wavelength of X-ray (λ ) and the angle of incidence of the X-rays (θ).
        nλ = 2dSinθ
        
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      - XRD-Three basic elements
        
        X-ray tube
        
        Sample holder
        
        Detector
      - How are X-rays generated in Cathode ray tube?
        
        By heating a filament to produce electrons
        
        Accelerating electrons towards a target by applying the voltage
        
        Bombarding the target materials with electrons
        
        Principle : When electrons have sufficient energy to dislodge inner shell electrons of the target material, characteristic X-ray spectra are produced.
        
        Copper is the most common target material for single-crystal diffraction
    - - Powder diffraction
        
        Peak Positions: Determined by the size and shape of unit cell (d spacings and systematic absences)
        
        Peak Intensity: Determined by the atomic number and position of the various atoms within the unit cell
        
        Peak width: Determined by instrument parameters, temperature, crystal size, strain and inhomogeneity
      - Phase Identification
        One of the most important uses of XRD
        
        Obtain XRD pattern
        
        Measure d-spacings
        
        Obtain integrated intensities
        
        Compare data with known standards in the JCPDS file, which are for random orientations (there are more than 50,000 JCPDS cards of inorganic materials).
  - - - Structure of anything, as the framework of its body.
      - How Can We Determine Crystal Structures ?
        
        Measure the inter-planar spacing, in a particular direction
        
        How do we Measure this spacing ? X-RAY DIFFRACTION
        
        :pencil2: Diffraction gratings must have spacing comparable to the wavelength (λ) of diffracted radiation (X-rays)
        :pencil2: In crystal : diffraction are planes of atoms. Spacing is the distance btwn parallel palnes of atoms.
  - - - Planck's hypothesis :
      - X-Ray Generation
        
        Process 1 : Deceleration of electron in the metal target (Bremsstrahlung)
        
        Process 2 : Non-elastic excitation of core electrons in the target atoms. (X-ray fluorescence (XRF))
        
        How Diffraction Works ?
        
        Wave Interacting with a single particle
        
        Wave Interacting with a solid
        
        Crystalline Material
        
        Bragg's Law
      - Neutron
        
        de Broglie eqn.
        
        Electrons
        
        de Broglie
  - - - Used
        
        Bragg-Brentano geometry
        
        Function
        
        offering high-resolution and high beam-intensity analysis
        
        :star: The incident angle,ω- between X-ray source and the sample
        -always 1/2 of the detector angle 2θ
        :star: Diffraction angle,2θ-between the incident beam and the detector angle
        :star: ω:2θ or θ:2θ scan-the sample rotates at θ/min and the detector always at 2θ/min
        :star: θ:θ scan:the tube rotates at the same rate as the detector at θ/min.
        
        Reflection (signal)
        
        Process
        
        1) occurs when constructive interference between beams are met
        
        2 more items...
        
        Bragg's Law
        
        Inter-plannar Spacing
        
        BCC
        
        h+k+l=even
        
        FCC
        
        h,k,l (all even @ all odd)
      - Powder Diffraction
        a scientific technique using X-ray, neutron, or electron diffraction on powder.
- - - - The diffracted beam by the plane shown strikes the film S. The normal to this plane is CN.
        
        During rotation, the plane normal would cut the film in the straight line AB and the diffracted beam in the hyperbola HK.
        
        AB therefore the locus of plane normal intersections with the film and HK the locus of diffracted beam intersections.
        
        The plane with diffracts a beam to S, for example has a normal which intersects the film at N.
        
        Since the orientation of the plane in space can be described by its angular coordinates, the problem is to determine the angular coordinates from the measured coordinates x and y of the diffraction spots S on the film.
- - - - BRAGG'S LAW
        
        A monochromatic X-ray beam scatters off the randomly oriented powder crystallites, giving rise to diffraction peaks at detector angle 2θ that satisfy the Bragg's Law.