lect2_2007_xrd(1) - Crystals Crystal Structures&...

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Unformatted text preview: Crystals, Crystal Structures & Crystallography Crystalline Solids CaF2 Unit Cell • Crystal Lattice – A 3D array of points where each point has an identical environment. • Unit Cell – The repeating unit (a unit cell is to a crystal, like a “brick” is in a house). In a given crystal all unit brick” cells are identical. 1 Translational Symmetry in 2D b a Unit Cell Arbitrary Lattice Point 2a + 3b • a and b are the lattice vectors, every lattice point can be defined by and adding these two vectors • The unit cell is defined by the lattice vectors, all space can be filled be by tiling unit cells 2D Bravais Lattices b γ a Centered Unit Cell b a Rectangular (a ≠ b, γ = 90°) γ Primitive Unit Cell Centered Rectangular (a ≠ b, γ = 90°) b a b γ Square (a = b, γ = 90°) a γ Hexagonal (a = b, γ = 120°) Including Oblique (a ≠ b, γ ≠ 90°), shown on the previous slide, there are 4 crystal systems (oblique, rectangular, square and hexagonal) and 5 Bravais lattices in 2D. 2 Cubic 3D Crystal Systems (7) Cubic Tetragonal Hexagonal Rhombohedral Orthorhombic Monoclinic Triclinic 14 Bravais Lattices The crystal systems each have distinctive symmetry and unit cell dimensions X-ray Diffraction 3 Diffraction Demo Take home message • The diffraction pattern is related but not equal to the grid pattern • Diffraction is most effective for monochromatic light whose wavelength is similar to the spacing of “slits” slits” • For crystals X-rays have a Xwavelength comparable to spacings of atoms Bragg’s Law A B d The extra path length traveled by wave B (shown in red) is equal to 2d sin θ, where d is the spacing between planes. Constructive interference will only occur if this distance is an integer multiple of the wavelength (nλ). 4 X-Ray Powder Pattern 20 30 20 40 50 60 2-Theta (Degrees) There are many different planes of atoms in a crystal. In an X-ray powder diffraction pattern we see many peaks, each one corresponding to scattering from different planes of atoms. The numbers in the above diagram are called Miller Indices, they identify different planes of atoms in the crystal. Miller Indices in 3D Miller The distance between planes is given by the following formula (for an orthorhombic lattice): Plane that goes through the origin 1/d2 = h2/a2 + k2/b2 + l2/c2 For a cubic lattice this reduces to: 1/d2 = (h2 + k2 + l2)/a2 The next plane is the one used to calculate hkl In a 3D system there are three Miller Indices, h, k and l. The values of h, k and l are integers whose values are determined as follows: h = 1/(x-intercept) 1/(xk = 1/(y-intercept) 1/(yl = 1/(z-intercept) 1/(z- h = a/(1a) = 1 k = b/(1b) = 1 l = c/(∞) = 0 110 plane 5 X-Ray Powder Diffraction Pattern (Ex. Lead Sulfide, PbS) (Ex. Int. 350 020 planes 300 250 200 220 planes 150 100 50 0 20 30 40 2Theta 50 60 70 Each peak corresponds to scattering from a different set of lattice planes. Two planes are shown above for PbS, which has the same structure as NaCl. Bond Distance from XRD (Cont.) (Ex. PbS, Rock Salt Structure) PbS, 2. Use Bragg’s Law and the wavelength of radiation (typically λ = 1.541 Å) to calculate dhkl λ = 2dhkl sin θhkl dhkl = λ /(2 sin θhkl) dhkl = 1.541 Å /{2 sin (38.7°/2)} = 2.10 Å 3. The interplanar spacing, dhkl, is related to the unit cell size. For a cubic crystal: a = (h2 + k2 + l2)1/2 dhkl a = (22 + 22 + 02)1/2(2.10 Å) = 5.94 Å 6 Bond Distance from XRD (Cont.) (Ex. PbS, Rock Salt Structure) PbS, 4. Now that we know the unit cell size, the Pb-S distance can be determined from the unit cell using simple geometry. dist (Pb-S) = a/2 dist (Pb-S) = 5.94 Å/2 dist (Pb-S) = 2.97 Å Peak Positions Bragg’s Law: λ = 2dhkl sin θhkl The distance between different planes of atoms in a crystal, dhkl, where h, k and l are integers that correspond to different planes Cubic: 1/d2 = (h2 + k2 + l2)/a2 Tetragonal: 1/d2 = {(h2 + k2)/a2} + (l2/c2) Orthorhombic: 1/d2 = (h2/a2) + (k2/b2) + (l2/c2) Hexagonal: 1/d2 = (4/3){(h2 + hk + k2)/a2} + (l2/c2) 7 ...
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