Diffraction - Crystal Structures Figuring out the structure of materials: an introduction to X-ray diffraction CeTi2O6 CeO6 c TiO6 a 1 X-ray Diffraction •  Light (X-rays), having wavelengths of Angstroms (10-10 m), can interact with the atoms and diffract •  Diffraction results in an interference pattern •  Pattern is specific to the crystal structure and contains information on composition 2 Bragg Diffraction Condition for Diffraction: Need Constructive Interference A specific angle of incidence exists for each family of planes (hkl) which allows for constructive interference of X-rays diffracted from adjacent planes Note: Technically, X-rays are “scattered” in all directions, but peak only seen where they constructively interfere 3 Bragg Diffraction For constructive interference, difference in path length (EF + FG) must be equal to an integral number of wavelengths (nλ) Trigonometry: EF + FG = dhklsinθ + dhklsin θ = 2dhklsinθ Bragg Equation: nλ = 2dhklsinθ or simply λ = 2dhklsinθ Powder X-ray Diffraction (pXRD) For powders, assume a random distribution of orientation of crystals (may not be random – but we’ll leave that for another day…) 4 Powder X-ray Diffraction (pXRD) For powders, assume a random distribution of orientation of crystals (may not be random – but we’ll leave that for another day…) (331) (400) (222) (311) (220) (200) (111) Nickel 5 Bragg Diffraction & Miller Indices Bragg Equation: λ = 2dhklsinθ Example: Nickel, fcc, a = 3.524 Å at 18oC, Cu Kα = 1.5418 Å Miller Indices dhkl θ 2θ {100} 3.524 Å 12.64o 25.27o {110} 2.492 Å 18.02o 36.04o {111} 2.035 Å 22.26o 44.52o {200} 1.762 Å 25.95o 51.89o {210} 1.576 Å 29.28o 58.57o {211} 1.439 Å 32.39o 64.78o {220} 1.246 Å 38.22o 76.44o {300} & {221} 1.175 Å 41.00o 82.00o how we usually report the diffraction angle 6 Systematic Absences Why aren’t all Miller Indices seen for Nickel pXRD pattern? Symmetry of unit cell (fcc) leads to destructive interference of certain “allowed” diffraction peaks i.e., diffraction from {200} peaks destructively interferes with {100} peaks {200} {100} For FCC lattices - indices must either be all odd or all even ( 0 considered even) For BCC lattices – indices must add up to even number 7 Four major effects: Peak Intensities Scattering amplitude related to “structure factor” which includes the “atomic form factor” 1. Diffraction will increase as you increase the atomic number (Z; more electrons) Therefore, planes with heavy atoms should have higher intensities 2. Scattering becomes poorer at high angles – X-rays diffracted by one part of atom out of phase with those from another part of atom 3. Preferred orientation – particles may not be randomly oriented (for example, think of what the distribution of orientations for cubic crystals would be like – most crystals would be lying flat on the surface) 4. Also may have destructive interference effects i.e. NaCl – planes of sodium interfere with (111) planes of chlorides (222) 8 Line width dependence: size effects •  Few atoms: scattering pattern will appear as a broad Guassian like peak •  Increase # of atoms and order: constructive and destructive interference •  More atoms; more diffraction; narrower peaks I II ΘΘ 9 Scherrer Line Broadening • Smaller crystals give broader powder XRD peaks • Scherrer Equation: Crystalline Diameter = 0.9 λ /(b cos θ) • Where b is the full width at half-maximum (FWHM; radians) and λ = 1.5418 Å (Cu) • 0.9 is a “form/shape” factor for spherical particles – can change 10 PXRD for Phase Identification Can be difficult to solve for a stucture by power x-ray diffraction However, every structure has an unique collection of peaks and peak intensities – we can use PXRD pattern as a “fingerprint” for identifying different crystal phases. (CSI uses this technique from time to time) JCPDS Database (Joint Commission for Powder Diffraction Standards) Can also identify relative amounts of several phases BUT Will not give information on amorphous materials, or phases present at less than 5% loading So why is diffraction useful? Phase Identification RE M Pn 11 So why is diffraction useful? Effects of substitution Ti/Zr Gd O Gd2Ti2-xZrxO7 12 Example of a PXRD pattern for a triclinic unit cell with no symmetry Less symmetry – more diffraction peaks! Nonane: a=4.088, b =4.630, and c =28.810 Å α =97.228°, β=91.004°, and γ=74.980°. Indexing an unknown Powder diffraction pattern d spacing from Bragg equation given Cu Kα radiation Indexing an unknown Powder diffraction pattern Given Re-scale as 1/d2 values, as these will be proportional to h2+k2+l2 term. Indexing an unknown Powder diffraction pattern Indexing an unknown Powder diffraction pattern Indexing an unknown Powder diffraction pattern Indexing unknown PXRD patterns Indexing unknown PXRD patterns For FCC lattices - indices must either be all odd or all even ( 0 considered even) Other Types of Diffraction Electron Diffraction • Often used to identify crystal phases during electron microscopy Neutron Diffraction • Intensity • Does NOT increase with atomic number • Scales with number of neutrons (can have isotope effects) • Often used to identify H atoms in structures and other light elements • Can be used to identify magnetic unit cells (why? neutrons carry a spin!) BUT, need good neutron source • Nuclear Reactor (Chalk River, Ontario – Canadian Site) • But! Spallation neutron source allows for better resolution 13 Comparison between Neutron and X-ray diffraction Neutron Diffraction • scattering lengths (b) are dependant on composition of nucleus not the # of e-’s! • b can be very different for neighbouring atoms and is isotope dependant X-ray (and e-) Diffraction • atomic form factor influenced only by the number of e-’s on the atom • increases with Z, but very similar when comparing Z-1, Z, and Z+1 14 Major differences between x-rays and neutrons: 1. Neutron Diffraction is from nuclei of atoms. •  λ = h/mv (de Broglie relation from velocity) •  Diffraction does not drop off at high angles (angle invariant) • Scattering lengths of nuclei do not scale with atomic number but vary irregularly instead • can differentiate between like atoms (e.g., Co vs. Fe – very difficult by X-rays as Z is very close) •  often complementary to diffraction obtained by x-rays 2.Isotopes of atoms have different scattering lengths •  1H has neutron scattering length of -3.74 x 10-15 m •  2H has neutron scattering length of + 6.67 x 10-15 m •  Partial deuteration can lead to large changes in neutron scattering – can “highlight” parts of molecule via deuteration 3. Neutrons can be used to extract magnetic structures as they have a magnetic moment!!!  Very important difference! 15 Elastic Scattering: Electron Diffraction Bragg Law: nλ = 2dsin θ λ = 0.00197 nm (1.97 pm) for 300 kV electrons. A typical value for the interplanar distance is d = 0.2 nm. If these values are put in the Bragg law, then the scattering angle is: θ = 0.28° Example: Electron Diffraction TiO2, Tetragonal, Rutile P42/mnm, a = b = 4.594 Å, c = 2.959 Å First ten reflections: (110), (101), (200), (111), (210), (211), (220), (002), (310), (221)