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by: Kavon Huel


Kavon Huel
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Robert McKenna

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Robert McKenna
Class Notes
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This 13 page Class Notes was uploaded by Kavon Huel on Friday September 18, 2015. The Class Notes belongs to BCH 6744 at University of Florida taught by Robert McKenna in Fall. Since its upload, it has received 24 views. For similar materials see /class/206958/bch-6744-university-of-florida in Biochemistry and Molecular Biology at University of Florida.

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Date Created: 09/18/15
L10 Fourier transforms From reciprocal space to real space FS Fp Diffraction space Electron density STRUCTURE FACTOR ltgt ELECTRON DENSITY I 1 Lecture 10 Imaging ofa structure in the microscope Diffracted beams are recombined in the image plane by a Fourier synthesis di rm tvun FOURIER FOURIER 39 v swru i S tnqlx 5 OF DH TED incident m1 Ion BEAMS di mcunn T tll mucd a mm m bcuxm pcnadw mmqu image objecl sszmuAly planr into 06 Lecture 10 A simple wave such as Xrays can be described by a periodic function fxFcos21hx on or fxFsin2nhx on in one dimension fx is the vertical height of the wave at any horizontal position x variable x is in fractions of wavelengths F is the amplitude h is the frequency the constant on specifies the phase of the wave the position of the wave with respect to the origin of the coordinate system Lecture 10 3 Examples of simple waves Fc052nhxot 6 O 14 Figure 215 Graphs of four slmple wave cquauons 1m r cos 2nlu a a r I h39 VbF3 Inn 3quot F 39 gt I v n r l h 1 a Hm cos 2m 5 Changing a changts ma posilion nr phaVa of the wm Lecture 10 Fourier series Fourier showed that even the most complex periodic functions can be described as a sum of simple sine and cosine functions whose wavelengths are integral fractions of the wavelength of the complicated function fX a0 a1cos 2713X a2cos 272x ah cos 27EhX b1sin 2739EX bzsin 21t2X bh sin 275hX General onedimensional Fourier h fx a0 Z ahcos 2713hX bhsin 27hx 1 where h is an integer ah and bh are constants and X is a fraction of a period Lecture 10 5 General onedimensional Fourier h fX 30 Z ahcos 21thx bhsin 21thx 1 In complex numbers General form a ib Where i l 1 c080isin9 ei6 9 21thX h x 2 Fh e 21tihx 0 Lecture 10 Geometric ways of depicting phase angles Vector F W Phase angle 1th Amplitude F01 Two components Ahkl and Bhkl a0 cl90n 45 122539 Phase angle tan X39th EM Bhkl Ehkl B hkl B W A hkl E Emma tkn A MILL1 InteIISity Ahli2 Bhkl2 EFULM E Fhkl 177m E FEM Fhkl2 amplitude amplitude amplitude amplitude Lecture 10 7 Contributions to terms in a Fourier synthesis Individual terms Bragg reflections 000 to 1000 hkl Combined terms to give function Electron density Comments Importance of phase Missing terms 800 Lecture 10 8 Visualization of Bragg s re ections to density sou Bragg rellumon 7 l 0 0 2 0 Drug re ection Bragg re ection 1 u a density wave 2 o a density wave 3 o a density wave sumnmim uIdonsiLy waves electrondensity map Lecture 10 Calculation of an electrondensity map Crystals precisely ts the de nition of a periodic function Hence we can combine Bragg re ections in a Fourier synthesis to calculate an electron density map pXyz cos 2n hxky lz och ZZZ h kl 1 V pXyz electron density at a point Xyz in a unit cell of volume V Falkl structure factor amplitude and octh phase angle of each Bragg re ection hkl Lecture 10 10 Calculation of an electrondensity map complex numbers F amplitude 1 PXaYaZ Z Fhkl e 4 1 0c phase V hkl a Where I 27 hxkylz F a ib 1 pxyz Z V hkl e 2 11i hxkylzoc Fhkl Where OL 21t0t Lecture 10 ll Concept of crystallography Phase problem ml Tm I u myulm39 mmuycmml ualunm n y u 4 Mm m A l m a Crystals iuo g 1500 45 Diffraction data 100 m 700 M 4 u u 59 a u n n 5 u o m m u u s r m mum plmxr39 ufrm l zlrrlmndznnly wnvz w r snquot In11mm Mm mum m mlmlmd mm M Mm ntnmxc amngcment C Phase assignment D l 5 6 U D 15 znn 57 7110 a H 3 o 0 96 0 u uxmbs d 4 n n 759 a 0 0 17 5 o 0 710 10 D 0 6 D Electron density 3 o o o o o E Map interpretation Lecture 10 12 Draw a graph of the two waves A and B of the form fX F cos 27 hx or where for wave A F1 h1 oc0 and wave B F2 h2 oc0 Draw what the wave function C A B looks like Lecture 10 13


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