X-Ray Diffraction Dr. T. Ramlochan March 2010

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Crystals A crystal is a solid material where the constituent atoms are arranged in an orderly repeating pattern extending in all three spatial dimensions CaSO4·2H2O SrTiO3

Crystallography Crystals are divided into 7 lattice systems → all crystalline materials must fit in one of these unit cells lengths of edges (a, b, c) of unit cell and the angles (α, β, γ) between them are the lattice parameters The space group of a crystal is a description of the symmetry of the crystal → the unit cells do not just repeat side-by-side Space groups in three dimensions are made from combinations of different symmetry operations (reflection, rotation and improper rotation, the screw axis and glide plane) 230 unique space groups

Crystallography The atoms in a crystal lattice form planes (described by Miller indices) that repeat

X-rays and diffraction X-rays were discovered in 1895 by Röntgen X-rays are electromagnetic radiation with wavelengths in the range of 0.5-2.5 Å As with visible light X-rays will undergo diffraction when they encounter an obstacle If the diffracting obstacle is on the order of the size of the wavelength, the propagating waves will have interference due to different waves having travelled different path lengths X-ray diffraction image of DNA by Rosalind Franklin (1952)

X-rays and diffraction Differences in the length of the path travelled lead to differences in phase The introduction of phase differences produces a change in amplitude → summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes

Scattering of X-rays Atoms (or their electrons) will scatter X-rays in all directions If atoms are arranged in space in a regular periodic fashion, as a crystal, some of the scattered X-rays will undergo reinforcement in certain directions and cancellation in other directions producing diffracted beams Diffraction is essentially reinforced scattering

Bragg’s Law For a particular condition of scattering where the angle (θ) of the incident beam and the ‘reflected’ X-rays are the same, the scattered X-rays will be completely in phase and undergo reinforcement if the path difference is equal to a whole number of n wavelengths, such that nλ = 2d sin θ This was first identified by W.L. Bragg and is called Bragg’s Law

Bragg’s Law For a fixed wavelength (λ) and value of d, there will be an angle theta (θ) where diffraction (complete reinforcement) occurs Diffractogram is a plot of the intensity of the diffracted X-rays vs. 2θ over a range of angles Each peak represents a plane in the crystal lattice with a given ‘d-spacing’ Basis for powder diffraction

X-ray production X-ray are produced when electrically charged particles (e.g., electrons) with sufficient kinetic energy give up some energy Non-characteristic (continuous) X-rays → electrons decelerated in an electromagnetic field (Bremsstrahlung) Characteristic X-rays → if electrons have high enough kinetic energy can knock electrons out of their shells → when an electron moves from an outer shell to an inner one it is ‘excited’ and releases excess energy directly as X-rays with eV/wavelength characteristic of the atom released from

X-ray production X-rays named according to shell being filled and number of shells changed (e.g., K shell filled by L shell (Kα radiation) or M shell (Kß radiation)) Each peak represents a transition; more than one peak (‘family of X-rays’); Kα (highest probability) is ~5 times stronger than Kß Kα is a doublet (Kα1 and Kα2) → different spin states Kα1 always about twice the intensity of Kα2 For Cu Kα1 1.540598 Å Kα2 1.544426 Å Kα 1.541874 Å Kß 1.392250 Å

X-ray production For XRD we want monochromatic X-rays (i.e., X-rays of a single wavelength travelling in the same direction/plane) Can filter the beam by passing through a material with an absorption edge between Kα and Kß wavelengths For Cu radiation use Ni filter → Kß reduced to 1/500; Kα reduced by 1/2

X-ray generation To generate X-rays → a) source of electrons, b) high accelerating voltage, and c) a metal target Use a water-cooled X-ray tube Evacuated glass tube with an anode (Cu target) and cathode maintained at high negative potential (HT transformer) Filament is heated to emit electrons → accelerated towards target X-rays emitted through (X-ray-transparent) beryllium windows

X-ray diffractometer Diffractometer has two parts: Generator → to generate X-rays Goniometer → to scan sample through a range of angles

Diffraction optics/geometry X-rays diverge from source → pass through Soller slits and divergence slits to define and collimate incident beam Incident beam diffracted by ‘flat’ powder specimen Diffracted beam passed through receiving slits Secondary monochromator reduces background radiation from sample X-rays collected by detector (proportional, Geiger, scintillation, semiconductor)

Diffractograms Gives information about peak positions, intensity, and shape