Crystallography : How do you do? From Diffraction to structure…. Normally one would use a microscope to view very small objects. If we use a light microscope we cannot look at objects smaller than about the wavelength () of light which is about m. Since the atom has dimensions of about m we cannot image an atom with a photon of light. X-rays, on the other hand, have a wavelength of about m and are suitable for imaging objects at the atomic scale. Unfortunately hard (<2x m) X-rays can not be lensed and a hard X-ray microscope can not be constructed, however hard X-ray diffraction can be employed to image objects at the atomic scale without a lens… X-ray Diffraction is.. X-rays scatter off of electrons, in a process of absorption and re- admission. X-ray diffraction is the accumulative result of the X-ray scattering of a group of electrons. A crystal is a solid material whose constituent atoms (molecules and ions) are arranged in an orderly repeating pattern extending in all three spatial dimensions. Describing Order in Space…. l Wave 1 Wave 2 l d θ θ l When l+l = n then the two waves are in phase Wave 1 Wave 2 l /d=sin  l =dsin  (given n = l+l= 2 l ) n =2 l =2dsin   1/d=2sin  / n The Miller Indices (h,k,l) (Grid) h, k, l. are the reciprocal of the intercepts of imaginary planes that dissect the unit cell with the crystallographic indices : h, k, l. phase Two parallel waves show constructive interference when they are in phase i.e. when n 1 l = n 2 (where n 1 and n 2 are integers) The planes are describe by Miller indices h,k,l d θ l Structure from crystals There are roughly atoms of Na and Cl in a single crystal of salt. To describe the structure of salt at the atomic level we would need to describe roughly 5 x atomic positions!! BUT there is an easier way. A crystal has long range three-dimensional order. If we could describe a small portion of that order (unit cell) we could then use SYMMETRY to describe the whole crystal! Crystal of Salt xyz  (xyz) \Electron Density Map x i y i z i Model : x i y i z i Model Fits the map !!! Report x i y i z i c a b    Make a model to approximately fit the map Refine the model with non-linear least squares to fit the map unit cell a, b, c, , ,  The unit cell is a parrallepiped that describes the smallest unique volume of the crystal (cell parameters a, b, c, , ,  ) space groupsarrange objects in the unit cell There are 230 ways (space groups) to arrange objects in the unit cell that brings a periodic arrangement of the objects back to their original positions. unit cell Data Collection, structure solution and refinement… The spatial arrangement of diffracted X-ray beams gives us the shape, size, and orientation of the unit cells within the crystal. The intensities of the diffracted X-ray beams are determined by the types of atoms present, their positions (and hence symmetry) within the unit cell and the angle of diffraction (X-ray intensity decreases with increasing angle). To find the structure we must collect all the diffracted X-rays, measure their intensities, convert the intensity to an electron density map, fit a model to that map and finally report the model. Collect Data and Integrate Electrons in atoms scatter high energy photons (particles) Ordered electrons (atoms) scatter orderly Crystals have long-range 3D ordered atoms (electrons) Interference Ordered pinholes 1D diffraction pattern 2D diffraction pattern d hkl 1(b) 1(a) 1(c) ½ ½ ½ k=2 h=2 l=2 [(hkl)] (222) planes [phase angle] electron density map with grid (x, y, z) y x z 1 l = 4 Where d* = reciprocal of d and a * is the recip. of a and b * is the recip. of b etc… structure solution: start with random phase angles or atomic positions and deduce initial phase angles Modify the parameters (p) and/or model and repeat p =p i +  p θ  +  2  The angle of the outgoing beam relative to the incident beam is  +  or 2  Crystals and Photons… Waves diffract off of planes (of electrons) Integrate Diffractometer Crysta l Detector X-ray source Cold Stream Cool crystal to hinder atom vibration Position crystal in space with mechanical circles Model Fits? yes no + y In this example move model atom (+ y direction) to fit the map better. XRDLab TAMU PER LUCEM VIDEMUS CHEMISTRY TEXAS A&M UNIVERSITY Bragg’s Law Sum over all (h, k, l) 1/d = d * Typical diffracted pattern (frame) 3D (set of merged 2D) diffraction patterns