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Two-dimensional X-ray powder diffraction

1. rin DOF 21 Iterations 2 Chi2 0 043625579 Figure 106 3D line plot By clicking the 3D plot button the default 3D line plot is displayed Figure 106 The data range can be set as with the 2D film plot Rotation around X and Y axes can be controlled via slides Two preset views are stored and can be accessed via the buttons Top and Slant Mouse rotation is enabled by default pressing the appropriate radio buttons activates translation and scaling modes Finally the line plot can be immediately changed to a rendered surface plot by selecting the Surface radio button 13 2 3 3D surface plot The surface plot Figure 107 is identical to the line plot in handling As rendering the surface can be a slow process dependant on the amount data displayed the surface mode requires more patience while adjusting the view All images can be exported either by copying to the clipboard Edit gt Copy exporting to an image file File gt Export gt Image or printing File gt Print For the latter two operations the image is rendered from scratch and dependant on the resolution of the image file or printer can take a considerable amount of time Lights The 2nd tab on the 3D display co
2. I ul iF N N Yu LAS I N N un _ 7m 1 2 2 22 o no om Im Mia J Select outer ranges of data and press enter to crop Figure 77 Cropping bars On pressing the enter button the entire array is cropped to the new range Please note It is a good idea to save the data regularly to avoid possible data loss File gt Save 104 13 3 2 Single pattern visualization Figure 78 Zoom button Si Powder 3D File Edit Display Tools Plot Window Help tal Intensity TO S i N Lil re N Er mr A IL m Figure 79 Zoom tool The default tool in the 2D plot window is the zoom box Figure 79 Pull it over a region you wish to enlarge Once the enlarged window is displayed click in the borders around the plot to navigate A click to the right of the X axis moves the pattern s to the left and displays the pattern in a 20 region shifted 10 higher Clicking in the other borders works in analogous fashion A single click in the plot window resizes it to the maximum view Selecting multiple patterns from the list displays them up to 6 colour coded overlaid patterns Figure 80 105 Two dimensional X ray powder diffraction 106 Sl Powder 3D File Edit Display Tools Plot Window Help tal Intensity Pe O N IN t r MIN AURORA JAD LN Figure 80 Mult
3. ee TE En I RRO et a TE a TE TE TT 0 5 10 15 20 20 1 Figure 6 11 The Rietveld refinement plot ofthe high pressure triclinic phase showing the experimental data as open circles the calculated pattern as the upper solid line and the difference pattern as the lower solid line The positions of the reflections are marked by vertical lines The scale of the difference plot is identical to the upper plot 85 Two dimensional X ray powder diffraction 86 6 3 Conclusion Three new high pressure phases of tin sulphate SnSO could be characterized The unit cell compression seemed to be continuous when taking into account only the change in volume and the change in length of the unit cell axes The phase transitions could at first only be identified from well defined discontinuities in the unit cell angles extracted from a serial triclinic Rietveld refinement The highest pressure phase shows a dense layered structure which seems to support the idea that layered structures might be more favoured at high pressures than their dense three dimensionally connected counterparts Murakami et al 2004 This study shows the supreme importance of high quality data and data reduction methods to extract Rietveld quality powder diffraction patterns from two dimensional data with extreme intensity distributions 7 4 Precis The combination two dimensional detectors powder diffraction and synchrotron light sources has been staggeringly success
4. Murakami M Hirose K Kawamura K Sata N amp Ohishi Y 2004 Post Perovskite Phase Transition in MgSiO3 Science 304 855 858 Murnaghan F D 1944 The compressibility of Media under Extreme Pressure Proceedings of the National Academy of Sciences 30 244 247 Newman M E J 2005 Power laws Pareto distributions and Zipf s law Contemporary Physics 46 323 NIST 2006 Standard Reference Materials X ray Diffraction https srmors nist gov tables view_table cfm table 209 1 htm Norby P 1997 Synchrotron Powder Diffraction using Imaging Plates Crystal Structure Determination and Rietveld Refinement Journal of Applied Crystallography 30 21 30 Pareto V 1896 Cours d economie Politique Geneva Droz Paulus D amp Hornegger J 1995 Pattern Recognition and Image Processing in C Braunschweig Vieweg Poisson S D 1838 Recherches sur la probabilit des jugements en matieres criminelles et matiere civile Ponchut C 2006 Characterization of X ray area detectors for synchrotron beamlines Journal of Synchrotron Radiation 13 195 203 Poulsen H F Wert J A Neuefeind J Honkimaki V amp Daymond M 2005 Measuring strain distributions in amorphous materials Nature Materials 4 33 36 Press W H Teukolsky S A Vetterling W T amp Flannery B P 1992 Numerical Recipes in C 2nd ed The Art of Scientific Computing Cambridge University Press Rajiv P Hinri
5. This section will give you a run down on how to reduce your data sets effectively using the sample data 1 The data format is CHI the wavelength is roughly 0 9184 ngstroms These values should be used to import the data successfully The temperature ramp of the experiment was 298 gt 838 gt 293 K These values can be entered into Increments and phase ranges dialog available via the menu Edit gt Increments and phase ranges 2 A first step in the data reduction would be to crop the data to a sensible range 3 4 46 7 20 3 Next we define the background As the background lacks great relief high values for the smoothing box smoothing box 0 9 iterations 4 resulting in a flatter underground do the job well For the data sets from 80 to 140 slightly lower values were chosen due to the higher background at around 18 smoothing box 0 7 iterations 4 4 For the next step it is recommended to have the indexing package Crysfire installed Select the first pattern and do a peak search Save the found peaks to your working directory and then start Crysfire preferably by pressing the Crysfire button Using the peak refinement the profile can be refined and the position of the peaks determined more precisely 5 Should you have indexed your phase s using Crysfire you can import them back into Powder3D in the Le Bail fit dialog For this step to be successful Fullprof gt Version 3 3 has to be insta
6. Tin sulphate at high pressures Table 6 1 Structural parameters of the monoclinic low pressure structure of tin sulphate which crystallizes in the space group P2 a The parameters were refined to data collected at p 0 2 GPa Phase Et mW fy O Pressure GPa a aao aa e Fear fon DEE err DE Er Pe HL a ne VIZ TEE BEE BEE EEE ZI S 104 15 7 Rep 8 1 feo jsa Ts Rewp 13 914 12 014 B21 No of variables 1630 HO No of refined atoms 5732 2 12 eh ESE 3 2 24 3 2 24 3 2 24 rebinnino 7 A CH Su Figure 6 6 The low pressure monoclinic structure of SnSO viewed down the b axis The light grey atoms are tin the yellow tetrahedra represent the sulphate anions Bonds have been drawn where the Sn O distance is less than 2 6 A Initially the monoclinic structure was refined using centrosymmetric triclinic symmetry The refinement was stabilized by the use of rigid bodies for the sulphate anions After the correct symmetry P2 a was determined the 81 Two dimensional X ray powder diffraction structure was transformed and refined in that symmetry again with rigid body constraints The final refinement cycles were performed without positional restraints The isotropic atomic displacement factors were however constrained to be equal for identical elements The refinement converged well with no distinctive discrepancies in the difference plot figure 6 7 The monoclinic structure differs from orthorhom
7. 63 Two dimensional X ray powder diffraction 3 5 Conclusion A study on the effects of fractile filtering two dimensional powder diffraction images has been made taking into consideration optimal data and typical high temperature and high pressure data The applicability of the normal intensity distribution to optimal and high temperature data has been shown However high pressure data are quite differently distributed They follow a Pareto function convoluted with a normal distribution function As a direct cause this data can impossibly be used in a reliable fashion with least squares minimization analyses A method has been developed to circumnavigate this problem allowing the extraction of data describing a near normal distribution from high pressure data To do this the parameters of the normal Pareto distribution have to be determined this allows a numerical approximation of the normal distribution within the peak of the normal Pareto distribution to be made The dependence of this normal distribution on the parameters of the normal Pareto distribution were analyzed Inessence itis nowfeasible to extract fundamentally reliable intensities from high pressure powder data This task was until present unachievable Quality assessment 4 Quality assessment 4 1 How good are my data Suggestions for an image reliability value It has already been mentioned the two dimensional powder diffraction images often suffer from extreme spottine
8. E cos Z 1 55 L M 5 1 55 Equation 1 51 then becomes 2a f 82 6 So 27 2x 44 24 cee When reduced to the fundamental parameters the function takes the final form of 1 57 R 2W arcsin mits iw Ba an NEA w L arccos GE 2 tl R Ea W z n WNE 21 arccos 4 29 Two dimensional X ray powder diffraction 30 10 S 100a u S 200a u S 300a u S 400a u S 500a u 0 8 0 6 0 4 Relative Lorentz factor 0 2 0 100 200 300 400 Beam width a u In order to see the effect this correction factor has for differing beam sizes and capillaries a simple two dimensional plot has been made In figure 1 16 you see that as the beam size approaches zero so does R approach zero whereas when the beam size equals or is larger than the sample diameter the standard correction factor of unity is applied 1 3 1 2 Polarization correction When X rays are diffracted by a lattice plane they are partially polarized This leads to an intensity reduction that can be expressed as a function of the diffraction angle For a completely unpolarized primary beam this leads to the following correction Lipson amp Langford 1999 P 1 cos 20 Should the primary beam be polarized this changes the correction to Azaroff 1955 1956 Kahn et al 1982 Whittaker 1953 P R P P gt 1 cos 20 l P a OS 2a sin 20 where Figure 1 16 Relative Lorentz factor The image depicts the trend of th
9. Los Alamos National Laboratory Report 86 748 Le Bail A Duroy H amp Fourquet J L 1988 Mat Res Bull 23 447 452 Rodriguez Carvajal J 2001 Commission on Powder Diffraction Newsletter 26 12 19 Shirley R 2002 The Crysfire 2002 System for Automatic Powder Indexing User s Manual Sonneveld E J amp Visser J W 1975 Journal of Applied Crystallography 8 1 7 IDL IDL VM and iTools are trademarks of Research Systems Inc Boulder CO USA Windows is a registered trademark of Microsoft Corporation Redmond WA USA Written by B Hinrichsen 06 10 2004 Last updated by B Hinrichsen 30 05 2007 130 131 Two dimensional X ray powder diffraction 132 Powder3D IP 0 1 A Tutorial Bernd Hinrichsen Robert E Dinnebier and Martin Jansen Max Planck Institute for Solid State Research 133 Two dimensional X ray powder diffraction 134 Introduction Image loading Opening and saving Calibration Filters Fractile filter Mask growth Beam stop mask Intensity corrections Lorentz correction Polarization correction Incidence angle correction Background correction Display properties Intensity analyses Area selection Intensity displays Integration References 140 140 141 141 147 147 149 149 150 150 191 191 152 152 153 153 154 157 160 Introduction Powder3D IP is a program designed to integrate powder diffraction images from two dimensional detectors
10. gt 4 s s s wo 5 4x1 0 ZT heta D E Test Apply Done Chi2 2446 67 2x10 Drag mouse to zoom into region Figure 87 Smoothing window The correct selection of the function shape has great effect on the smoothing efficacy 13 3 8 Background O x Si Powder 3D File Edit Display Tools Plot Window Help S tal Sonneveld 1 Iterations ro 4 gt Secondary iterations CE AJ I u Curvature 5 00 6x 10 4 gt 8x 10 anchor points V Reduce anchor number V Keep anchors under data 4x10 V Magnetize anchors Generate Export Import Done 1h Il A A PL Intensity Ri ff M h u A hy Ah dan 10 20 30 40 20 Select outer ranges of data and press enter to crop Figure 88 Background reduction Background determination Edit gt Background can be done in two modes Either a pattern can be loaded XY format as for import or it can be calculated Figure 88 Should a pattern be loaded itis displayed asisthe calculated background The normalize button then interpolates the background giving it the same number of data points as the diffraction pattern Then a linear function fitted using the least absolute deviation method is added to correct the background Should the background be higher than the powder pattern at any point it is lowered The Calculate background method utilises a robust algorithm ba
11. is selected to portray data in a typical integration bin The comparison of the datasets via a histogram as is shown above is hindered by the fact that the histogram of the unfiltered data is singular Virtually the entire histogram density is collected in the first histogram bin with a constant further density of two pixels for each of the following intensity bins With such a distribution no model fitting can be undertaken Once the top 2 5 of the intensities per bin have been filtered from the data a histogram with a discernable normal distribution presents itself figure 3 7 Intensity distributions and their application to filtering A te u FR ML a dela vn Figure 3 6 A rendered image of a high temperature powder diffraction data set collected by a two dimensional detector in a study of the decomposition products of Bischofite Mg CI H O The fine monodisperse grains of cause the rings to have such an even intensity The high intensity spikes are caused by reflections of the high temperature sapphire capillary The normal probability distribution function is fitted to the data The parameters from the fit are then compared with the arithmetical values Now the use of the filtering becomes visible Whereas the mean and the variance of the unfiltered data are unrelated to the true values the arithmetic mean and variance computed from the filtered data are virtually identical to those values given by the fitting procedure T
12. 34 38 45 47 47 47 50 51 64 65 66 67 67 67 68 15 76 76 76 87 89 91 Two dimensional X ray powder diffraction Abbreviations LP CCD PSF FWHM 20 tilt rot If not temporarily defined otherwise within the text the following definitions Combined Lorentz and polarization corrections Charge coupled device Point spread function Full width at half maximum Angle of diffracted beam to primary beam Half the above angle Distance of the detector from the sample along the primary beam Maximum angle between the primary beam normal plane and the detector Rotation of the tilted plane Angle of azimuth Incidence angle of the diffracted beam on the detector Semi latus rectum Distance between focus and centre of an ellipse Semi major axis of an ellipse Semi minor axis of an ellipse Eccentricity of a conic section Radius Lorentz correction 0 Introduction With the general availability of high intensity parallel synchrotron radiation the use of two dimensional detectors like CCD detectors or on line image plate readers for fast high resolution data acquisition enjoys a growing popularity As a consequence the field of X ray powder diffraction has experienced a renaissance For the first time it is now possible to record the entire Debye Scherrer rings up to high angular range with high angular resolution within a few seconds or even less The field of applications
13. Display graphics Update Figure 125 The calibration wizard Refinement settings for the calibration image The parameters you wish to refine can be ticked using the tick boxes to the right of the parameter values Bi E alt xi Dss Bearbeiten Aceh Epfigen Format Extras Tabelle MathType Fenster 2 Adobe POF Acrobat Comments x Edi 7a a Ee ox CE tI a tY gde W Select the parameteres for Lear Sawa Hetnemert Fie ab 2 300 009 mn A ie O 00000000 Dian 12200000 P Morning ONE F Raekson 3007200 pies Flies no fee 000000000 p p Wavelength PAJ fO SOCCER Ta idapeea fo o SE Reding wadya No 40 of 72 pak No 13 tm Load Advanced lt lt Back Newt gt gt Cancer y Arani bal Histogram for azimuth 197 a aca TEE ATTEETEG 01 A Ry a ae n A A Sete 6 1 66 wS tS wu AN nw Endsch s os Ds go C hias fer bheridh PA Eneee men f 3 10t Ei ice mob ence gt ood rion Sirro wa E Atte aitat ro 20 20a zum Figure 126 The calibration wizard Refining the radial line intersections with the ellipses If you have chosen to have the graphics displayed while determining the intersection points you will see something like the image in figure 126 139 Two dimensional X ray powder diffraction Si Calibration x Sample Pixel Center and Fiotation Eccentricity Least square fit Superfine Select the parameteres for Least Square R
14. Of greatest significance are those corrections which are a function of the azimuth These include polarization and Lorentz corrections for which the experimental geometry has to be determined As the filter is applied sequentially to a small 28 range the effects of corrections which are only a function of the diffraction angle have little impact It should be further noted that for correct error estimation a precise intensity distribution is essential This can only be achieved by prior two dimensional corrections For reliable integrated intensities and variances only exclusive filters give good results 3 4 Filter applications 3 4 1 Ideal Data Calibration images are as close to perfect as two dimensional powder diffraction images get figure 3 5 To ascertain the distribution characteristic of a very monodisperse rotated sample of high scattering power and micro structural perfection a calibration image of a NIST LaB6 standard SRM660a was analyzed 51 Two dimensional X ray powder diffraction 52 5000 10000 15000 Intensity a u Figure 3 3 Two data samples showing the effect of fractile filtering on the intensities of a Bragg reflection as it would contribute to a two theta bin The intensity of the unfiltered sample is represented by the red histogram and the filtered data by the blue histogram Curves show the normal probability distribution function fitted to the data Parameters of the fitted distributions are given in
15. Refine gt Export cell data 116 Sl LeBail refinement O x Fhase information Selected phase ranges from set number O to 0 Curent set for refinement Pattern Zero offset 0 002750 refine Import Fullorot Reset Remove HEL Cell Profile FuvHh U 0 0731 fe yf 0 054403 WW 0 035902 MW refine Be ja 0oo1 OO y 0 035815 W refine Asummelne S L 0 001730 refine D L 0 001400 refine M Strain broad IAF file Load profile Global Chiz 34 3075 A Bragg 0 053547 Figure 97 Le Bail refinement profile parameters Sl Powder 3D File Edit Display Tools Plot Window Help tal 8x 10 6x 10 4x 10 Intensity Counts 2x 10 Der ee o ono LO LO N IET ee ER ET EI TER IE en BE I EI 5000 10 20 30 40 20 P Select fitting range by dragging mouse over peak Figure 98 Le Bail refinement 117 Two dimensional X ray powder diffraction 13 3 13 Peak analysis A new tool has been designed to assist in the sequential refinement of peak profiles To call it you should select the menu Tool gt Peak analysis play Tools Plot Window Help Le Bail Peak Analysis Export cell data Analysis Figure 99 Peak analysis menu The window in Figure 100 shall appear The pattern ranges that have been set in the main program interface are the initial ranges displayed by the peak progression tool It is therefore very usefu
16. The central valley is parallel to the sample rotation axis 2 Two dimensional X ray powder diffraction 28 1 3 1 1 1 Lorentz correction for highly collimated beams The rotational correction should be used if the powder sample is rotated within the beam in the single crystal sense i e all crystallites should complete their rotation within the beam Should the beam be collimated to dimensions below those of the sample containment then this further reduces the rotational impact on the cumulative Lorentz factor A term R can be introduced to quantify the rotational Lorentz factor from O for no rotational element to 1 for full rotation of all crystallites within the beam The introduction of this factor leads to equation 1 47 cosa tan 20 L sin R cos u 1l cos a tan 20 1 sin a tan 20 R has been deduced for the case of the rotational axis being normal to the primary beam The common Lorentz formulation is valid if a crystallite is rotated within the beam by w 2rr For a certain number of crystallites with a rotational radius less than the beam radius this is true Crystallites outside this radius experience a rotation weff which is dependent upon their rotational radius and the width of the primary beam It can be given as w y 4arcsin WA where Wis the beam width and R is the rotational radius As all crystallites between R W and R L where L is capillary diameter are affected differently by rotational radius
17. no filter 0 00000000 Figure 120 A calibration image has been loaded 13 4 Opening and saving The current data and settings can be saved using the save option File gt Save These files can become extremely large They can be loaded using the open option File gt Open Exporting Images and data can be exported using the File gt Export functionality Data is saved as a diffractogram the image is saved directly from the main image display into a graphical file format 13 5 Calibration The first step when analyzing two dimensional data is to calibrate the detector parameters To do that a carefully taken image of a well prepared sample is needed The calibration parameters are then determined as exactly as possible These are applied to all subsequent images to extract the standard powder diffractograms As this step affects all the subsequent data much care should be taken during this step Open the calibration dialog figure 121 by selecting Calibrate from the Edit menu You are prompted for information on experimental details The d spacings of the sample are required for the calibration These should be well known their precision is important for a successful calibration The wavelength and detector to sample distance measured along the primary beam not normal to the detector should also be entered in the first window 136 SA Calibration Ol x Sample Pixel Cente
18. to discard or OK to accept the normalization Si Normalize Patterns I O x Data sets to normalize Phase 1 Please select scaling method Scale to maximum Scale to minimum Scale to diffimas min Scale to average Subtract minimum Left click to zoom right to mark region OF Test Cancel Figure 81 Normalization function 13 3 4 Wavelength The wavelength dialog accessible via the menu Edit gt Wavelength allows you to alter the radiation wavelength Figure 82 Should you wish to recalculate the pattern select the radio button recalculate patterns The wavelengths for some standard anode elements have integrated in the top list SI Change Wavelength f x Operators conectvalue C recalculate patterns Wavelength Theta E Dispersive C T Fe Lambda 1 0 322400 Co Mi Lambda Z 0 000000 Cu ba w oa Lambda ratio 0 000000 K alpha doublet Polarisation 0 000000 Figure 82 Wavelength dialog 13 3 5 Phases and ranges The increments between the patterns can be entered into a table Figure 83 which is called by Edit gt Increments and phase ranges For unvarying increments the fields on the right of the window can be used to insert values into the table Once two of the first three fields are filled the other is calculated and filled automatically Should all fields contain values no updating takes place Pressing the Insert button f
19. to standard powder diffractograms In the following a short introduction is given to the functionalities of the software This software is similar in functionality to Fit2D Hammersley et al 1996 and uses the same projection functions Installation Powder3D IP has up to now no installation routine but it does have one general prerequisite the IDL virtual machine This is similar to the Java virtual machine and can be downloaded from the site of ITT Visual Information Solutions for free Once that software has been installed starting Powder3D IP is only a matter of unzipping and saving the program files to a convenient directory and double clicking on the Powder3DIP sav file 13 3 Image loading lol x File Edit Integrate Window Help Import Exit Figure 119 The initial view of Powder3D IP showing the file menu Loading an image can either be done using the Open or the Import function found in the File menu The Open function can only open files written by Powder3D IP whereas the Import function can import various different image types These include binary mar345 Stoe IPDS tiff and bmp image formats Once a file has been selected and successfully loaded the screen should resemble figure 2 135 Two dimensional X ray powder diffraction En Powder3D IP 10 x Fie Edit Integrate Window Help aas File lab6_92_180_009 mar2300 R im 0 00000000 Resolution 2300 2300 pixel R im
20. 16 the effect on the final diffraction pattern can be seen in figure 3 17 Even more pronounced is the effect of the filtering on the standard deviation of the integrated pattern figure 3 18 Figure 3 16 The effect of filtering on the diffraction image is shown The green mask represents the pixels which belong to the top 48 of the intensities per integration bin The blue mask shows the pixels which belong to the bottom 2 of the intensities per integration bin The yellow mask is the beam stop mask filtering the first 2 20 of the diffraction image Only the grey region of the image is used for the integration to a 1D diffractogram 62 Intensity distributions and their application to filtering Unfiltered diffractogram Filtered diffractogram 3000 2000 Intensity a u 1000 IDo 5 10 15 20 26 Figure 3 17 The effect of filtering on the final integrated and background corrected pattern is shown In red the integrated pattern of the unfiltered image is shown In blue the diffractogram of the image is shown that had 48 of the highest intensities and 2 of the lowest intensities removed 10000 1000 Intensity a u 100 20 F Figure 3 18 The effect of filtering on the standard deviation is shown In red the standard deviations of the unfiltered image are shown In blue the standard deviations of the image are shown that had 48 of the highest intensities and 2 of the lowest intensities removed
21. 289 479 Figure 144 The distribution curves computed during the analysis are displayed for closer perusal Information x ey These are the results from the normal pareto distribution analysis LD The pareto integral is 96612557 The normal integral is 451765 38 The normal Fraction of the entire data is 0 44692460 The ADDITIONAL high intensity fractile Filter should be set to 55 307520 Figure 145 The really nitty gritty information is displayed by an information message box In this case it recommends us to cut off an ADDITIONAL 55 off the top intensity to achieve a normal distribution This leads to a whopping 59 55 4 initial filter top fractile After clicking on the Fit NP context menu item the window shown in figure 144 is displayed The reason for this is to give you an idea of the reliability of the filtering suggestion If the normal Pareto curve the skew one describes your data well then the chances are good that the suggestion in the information box figure 145 is sensible Of course the Gaussian curve the automatic routine tries to fit into the normal Pareto curve should fit snugly as in figure 144 150 13 8 2 2 Azimuthal intensity display Azimuth plot 4E 003 2E 003 0 Figure 146 An azimuth plot of the intensities within the selected area Window gt Azimuth In order to see which and how azimuthal corrections should be made to the data the azimu
22. 2x 10 0 100 200 300 400 500 Pixels Figure 2 3 The intensity difference of the overlaid intensities upon folding along one pixel of figure 2 2 The strong dip in the centre is the mirror axis of the intensity array shown in figure 2 2 37 Two dimensional X ray powder diffraction 38 2 3 Parameter refinement 2 3 1 The radial line intersection method A successful refinement of the ellipse parameters requires that all ellipses are characterized by a number of points on the accessible ellipse arcs To do this the generally accepted method is to draw radial lines from the common focal point of the ellipses to the image edge The intensity extracted along these lines is very similar to that of a normal diffractogram figure 2 4 except that the axis normally representing the diffraction angle now represents the radial length The optimal ellipse intersection is then computed using peak profile fitting 10000 8000 6000 Intensity a u 4000 2000 50 100 450 Radius mm Figure 2 4 The intensities as traced along a 5 pixel wide line from the common focus to the edge of the image are shown The blue lines represent the Gaussian peak as it is automatically fitted to the data points around the region presumed to contain the intersection The presumption is based on the initial parameter estimates All the peaks are fitted and the details of the fit stored for an automated selection process in which ill fitting peaks are d
23. 46 120 2 0 24 46 120 2 0 24 46 120 Step size 26 ater rebinning a w lo Ie E Sr r ep u Su nn Be I I observed calculated intensity F F_ observed calculated structure factor w ighting per data point n number of data poi The powder patterns of phases Il and III contained sufficiently resolved diffraction peaks to allow for ab initio crystal structure determination as well as for Rietveld refinement The direct method program EXPO Altomare et al 2005 was used to determine the positions of the Iron and Antimony atoms Subsequent Rietveld refinements in combination with difference Fourier analyses then revealed the positions of the oxygen atoms in the asymmetric unit For the Rietveld refinements using the program GSAS the lattice and reflection profile parameters were first kept at the values as obtained from the LeBail fits Slack soft constraints for the four Fe O bond lengths of 2 1 1 Awere used to stabilize the refinements The atomic displacement parameters for the oxygen had to be restrained to be equal in the monoclinic phase to hinder some parameters having physically meaningless negative values For the same reason one oxygen atom in the low pressure phase and all oxygen atoms in the high pressure phase had to be refined with fixed isotropic atomic displacement parameters 5 3 3 Equation of State Lattice parameters as a function of pressure were extracted from each diffraction pattern The derive
24. 49 Two dimensional X ray powder diffraction 50 models are overlaid the blue line represents the normal distribution and the red line corresponds to the Poisson distribution Contrary to common belief Chall et al 2000 the Poisson distribution overestimates the variance of the data sample Intensity a u 100 200 300 Azimuth Figure 3 2 The distribution of the intensity contained in the histogram in figure 3 1 is shown as a function of the azimuth The intensity is the background between the first two Bragg reflections of the standard LaB calibration sample Air scattering is the main source of the background radiation The black symbols represent the intensity of a single pixel The main drop in intensity at 270 is due to shadowing by the primary beam stop arm The intensity reduction at 0 360 and 180 is due to shadowing by the sample capillary A normal distribution is clearly the most convincing model describing the signal statistics of a very small diffraction angle range and the complete azimuthal range This is the typical data entering a single bin when integrating the 2D image into a conventional 1D powder diffractogram The intensities are seldom so small as to make the Poisson distribution a sensible alternative 3 3 Filter models inclusive or exclusive filters Attempts to extract the most probable mean value from a set of observations can be divided into two general approaches The first is an inclu
25. I I I I I I I Fa N I aa a Ww u 4 5 is l HI pressure A lfa Figure 5 3 Dependence of the lattice parameters of FeSb O on pressure in the range of p 0 19 8 GPa 12 Schafarzikite FeSb O at high pressure K ty GPA k 7737 GPa K 4542 Gilta TLE ay 4 I Ih pressure Cl Figure 5 4 Dependence of the volume of FeSb O on pressure in the range of p 0 19 8 GPa Smooth solid lines correspond to the least square fits of the Vinet equations of state 5 3 4 Symmetry relations Translationengleiche group sub group relations can be found for all phases They are P 4 m bc 2142 P2 c gt t8 gt P4 m and P 4 m b c gt 12 gt P4 m The initial path leads over Pham a space group type known from the high pressure phase of Pb O but not observed for FeSb O Four possible paths lead back from P2 c to P4 m as can be seen in figure 5 5 Figure 5 5 Group sub group tree of the observed phases Grey lines are possible transition paths Symmetrical considerations would suggest the orthorhombic space group Pbam to bridge the higher and lower space groups figure 5 5 The space group type Pbam is a known low temperature high pressure space group type for minium Pb O A direct transition path from the ambient pressure space group P4 mbc to the high pressure space group P4 m is 13 Two dimensional X ray powder diffraction symmetrically plausible as P4 m is a non isomorphic maximal subgrou
26. I I i l Save Done l I I ji Al ilk AM P d Laat Vi Lely IN n A un Me we 0 10 40 26 peaks found Click to add shift click to remove peak Figure 91 Peak searching The peak positions and intensities can be saved to a Crysfire Shirley 2002 format file via the button Save on the Peak search tab Peaks can be added and removed manually by selecting the edit button as with the background points 113 Two dimensional X ray powder diffraction 13 3 10 Single peak fitting Once the peaks have been found their position can be Figure 92 using a pseudo Voigt function corrected for axial divergence Finger et al 1994 All refined values are stored for later use You can select the peak by dragging over it A light blue box fills the border of the selection Pressing enter starts the refinement with the default values The fit is overlaid in blue for visual inspection Please note The FWHM estimated using the Caglioti function has a profound effect on the convergence of the fit Should the procedure fail ensure the FWHM distribution is set to a realistic value Peak markers can be added and removed manually by returning to the peak search window Figure 90 the modus operandi is identical to the addition and removal of background points 13 3 10 1 Data export 13 3 10 2 By pressing the export button you are able to export the peak data of the pattern in a variety of methods The positions and heig
27. Traditionally Lorentz factors Buerger 1970 Zevin 1990 which originally only were only angular speed corrections have been conveniently combined in powder diffraction formalisms with diffraction probability factor sin6 and the polarization correction to the enigmatic LP correction As in two dimensions no current simple LP corrections exist or are even appropriate they are separated here The angular speed correction is called the single crystal correction here the angular speed is dependent the angle of the rotational vector to the primary beam generally 90 as well as the phase angle This angle is the projection of the rotational axis on the detector more simply the capillary shadow relative to the detector coordinates 0 in detector coordinates is at 3 o clock It should be mentioned that the single crystal Lorentz correction does become undefined in the region close to the capillary shadow This effect is well known in single crystal area detector reduction This region cannot be used for integration and should be excluded The powder Lorentz correction is the simple probability factor correction of sin 13 7 2 Polarization correction R Intensity corrections Lorentz Polarization Geometry Backgroun al W Apply polarization correction Polarization factor i Phase angle 90 Figure 135 The polarization factor and its phase angle which is dependent on th
28. an integration over that range has to be made 1 oy fo y dy 2 z 2 1 y 2yaresin y y Yy Interestingly the integration tends to a value of roughly 33 as the beam width to sample radius ratio tends to nought 1 im Je y dy 1 Now this rotation needs to be put into relation to the entire illuminated area The normalization takes the form of the average rotational angle of the entire illuminated area relative to the full 21 rotation of standard Lorentz correction 20 f QQ So R lt lt 2n f f Where f is the area in which the crystallites experience a complete rotation in the collimated primary beam and fw is the area in which the crystallites only experience a partial rotation within the beam figure 1 16 These are computed in the following manner 1 47 1 48 1 49 1 50 1 51 The experimental setup Figure 1 15 Relative Lorentz factor The image depicts a perpendicular section ofa capillary of diameter L being illuminated by a beam of width W Only crystallites falling in the light grey inner circle are rotated completely Zr within the beam Crystallites outside this region but still within the beam path only experience a limited rotation w thus reducing the effective single crystal Lorentz factor to be applied to them a y r cos Z w l w 1 52 f 2n W4 1 53 f 2n 44 f 24 1 54 Reforming equation 1 54 and simplifying leads to 1 55 W VY La W WW E W
29. diffraction patterns lattice parameters and cell volumes of Rb C O and Rb CO as a function of temperature in the range from 298 to 838 K 2 4 K min and back down to 298 K 4 8 K min Taken from Dinnebier et al 2005 13 2 3 Import One of the first things you shall want to do is to import data To do that press the import button Figure 14 or select File gt Import Figure 14 Import menu Six different formats are available although XY and Chi formats are identical You shall be prompted to select a file Figure 15 Standard versions of GSAS GDA Fullprof DAT Bruker UXD and DASH XYE data files are imported Select files to load 2x My Recent Documents s Desktop My Documents My Computer VAE File name 200 chi 001 chi 002 chi 003 chi 004 c Places Files of type chi Cancel Figure 15 Select import files Select all files you wish to import This is different to the previous version in which you only had to select a single file for the entire directory to be imported The files are sorted by the operating system so please ensure they are displayed in the correctly Should your files have a different suffix type lt your suffix here gt lt Enter gt in the File name field to display them Upon pressing enter the file names in the directory are read then the first file is loaded and the theta range read The next window promp
30. free parameter v which represents the mean and the variance Two free parameters are used to describe the normal distribution the variance o is free as is the mean Np 3 2 2 Arithmetic statistical values Fitting the population of each bin to a distribution function is computationally unfeasible This is due to instabilities when dealing with small populations and to the high computational cost involved with nonlinear fitting routines For standard integration computations arithmetic means medians and variances are calculated The arithmetic mean x of N number of observations x is given by equation 3 4 Ill le 5 X 3 4 i l The variance is defined by equation 3 5 l A bin is a container into which pixels are grouped It spans a 2D region of 28 which is identical to the 28 step width of the integrated pattern The intensities of the pixels within a bin determine the corresponding intensity of a step in the integrated pattern 48 Intensity distributions and their application to filtering x x 3 5 Q I z M II The median x is given by equation 3 6 for a population Y of the size N that has been previously sorted Y N 1 2 C z if N is odd gt Yy Yun if N is even In evaluating the filter performance we will compare the values returned by the arithmetic functions with those returned by fitting the model functions to the histograms 3 2 3 Model testing on ideal data To det
31. from the CCD chip which requires permanent cooling to reduce these effects The experimental setup 1 1 2 Imaging plate detectors Imaging plates were the first digital technology to replace films in synchrotron and laboratory equipment The concept is extremely simple A layer of BaF Br l Eu which contains colour centres is deposited on a robust film like base plate The plate is then exposed to X rays The image is later scanned by an online or a more cumbersome offline scanner Scanning the image comprises exciting the colour centres and then detecting the induced radiation Stimulating colour centres does not require much energy generally red lasers suffice The stimulated green light is detected by a photomultiplier following in the path of the laser The great advantages of imaging plates are their large size low cost and their high dynamic range The latter quality has made it the detector of choice for two dimensional powder diffraction Their major drawback is the high dead time associated with the time consuming scanning This can take the best part of two minutes for large images Rigaku MSC has developed a practical solution to this problem The detector system is comprised of two or even three detectors When one is being scanned the other can be exposed A precision rotation system transports the imaging plate from one position to the other The future for this detector type does seem rather bleak especially in view of fl
32. group of this space group is P112 a phase Il at p 0 2 GPa a 8 7022 9 A b 5 3393 5 A c 7 0511 6 A y 89 90 1 This space group type describes the crystal structure of the medium pressure phase The two high pressure phases both crystallize in the space group type P1 phase Ill at p 13 5 GPa a 8 067 3 A b 5 141 2 A c 6 609 2 A a 90 56 3 6 90 65 2 y 89 46 2 and phase IV at p 20 5 GPa a 7 889 5 A b 5 028 3 A c 6 462 3 A a 90 99 3 6 91 01 3 y 89 89 4 6 2 Experimental 6 2 1 Synthesis and X ray diffraction measurements Tin sulphate of a purity of 99 was used as bought from Acros Organics The starting substance was analyzed using laboratory X ray powder diffraction and was found to contain no detectable traces of impurities The sample was hand ground in an agate mortar for twenty minutes A small amount of sample was loaded in a Merrill Basset Merrill amp Bassett 1974 type diamond anvil cell DAC Silicon oil was used as a pressure medium The pressure was measured off line by the ruby line shift method Forman et al 1972 Mao et al 1982 Monochromatic radiation of a wavelength of 0 368194 A 33keV was used The diffraction pattern was recorded by a Marresearch Mar345 online image plate system A set of 34 images at pressures ranging from p 0 GPa to p 20 5 GPa were made Nine images were taken during decompression Exposure times ranged from 60s
33. gt K alpha stripping Select which wavelength you wish to keep Figure 85 108 lolx File Edit Display Tools Plot Window Help tal 2 00000 8x10 k ioj xi K alpha2 stripping Keep K alphal 6x10 Keep K alpha2 Cancel 4x10 2x 10 Raw Data K alphal K alpha2 Drag mouse to zoom into region Figure 85 Ka2 stripping The following dialog Figure 86 enables you to select the sets which should be stripped The raw data is overwritten and therefore this action cannot be undone a Apply K alpha stripping le 63 From set no fi To etno 200 Select phase Phasel Apply Cancel Figure 86 Batch stripping 109 Two dimensional X ray powder diffraction 110 13 3 7 Smoothing An advanced full width at half maximum FWHM optimised smoothing Figure 87 algorithm described by Dinnebier Dinnebier 2003 is implemented Edit gt Smoothing The variation of the FWHM of the peaks is generally described by the Caglioti Caglioti et a 1958 formula FWHM NU tan q V tang W This can be graphically set with aid of the function window on the left Here the function can be dragged to the desired shape with the aid of three red boxes Sl Powder 3D Oj x File Edit Display Tools Plot Window Help 4 EE iojxi a BE Order of polynomial aan ML No of iterations BE 8x 10 FHM factor 0 7 6x10 Ww E O O
34. images These include among others automatic detection and calibration of Debye Scherrer ellipses using pattern recognition techniques and signal filtering employing established statistical procedures like fractile statistics All algorithms are implemented in the freely available program package Powder3D developed for the evaluation and graphical presentation of large powder diffraction data sets As a case study we report the pressure dependence ofthe crystal structure of iron antimony oxide FeSb O p lt 21 GPa T 298K using high resolution angle dispersive X ray powder diffraction FeSb O shows two phase transitions in the measured pressure range The crystal structures of all modifications consist of frameworks of Fe O octahedra and irregular Sb O polyhedra At ambient conditions FeSb O crystallizes in space group P4 mbc phase Between p 3 2 GPa and 4 1 GPa it exhibits a displacive second order phase transition to a structure of space group P2 c phase Il a 5 7792 4 A b 8 3134 9 A c 8 4545 11 A p 91 879 10 at p 4 2 GPa A second phase transition occurs between p 6 4 GPa and p 7 4 GPa to a structure of space group P4 m phase Ill a 7 8498 4 A c 5 7452 5 A at p 10 5 GPa A non linear compression behaviour over the entire pressure range is observed which can be described by three Vinet equations in the ranges from p 0 52 GPa to p 3 12 GPa p 4 2 GPa to p 6 3 GPa and from p 7 5 GPa
35. is vast with current experiments including texture analysis Wenk amp Grigull 2003 and in situ powder diffraction measurements in dependence on pressure Hanfland et al 1999 temperature Norby 1997 chemical composition Meneghini et al 2001 electric and magnetic fields Knapp et al 2004 or external strains Poulsen et al 2005 The experimenters is faced with two major challenges Firstly large area detectors produce large numbers of two dimensional images which need to be reproducibly reduced to one dimensional powder patterns Secondly the sets of hundreds or even thousands of powder patterns need to be evaluated and graphically presented It is interesting to note that for both tasks only very few generally available programs exist These are keyed toward single powder patterns involving extensive manual interaction This approach is unsuitable for mass data analysis not only is the manual workload exorbitant but the induced subjectivity hinders reproducible results The key to solve the first problem lies in the reliable extraction of a powder diffraction pattern unaffected by graininess effects detector aberrations and scattering from other sources like reaction cells diamond anvils and gaskets The diffraction pattern from a two dimensional image can be reduced to a simple geometric figure the ellipse calling for the application of modern pattern recognition techniques Paulus amp Hornegger 1995 Theodori
36. is generally regarded as the model of choice for the analysis of counting statistics The discrete binomial probability distribution Abdi 2007 is given by equation 3 1 This returns the probability 47 Two dimensional X ray powder diffraction P of exactly n successes out N trials where each trial has the probability of success p and probability of failure g 7 p In two dimensional powder diffraction success would represent an intensity count at a pixel failure would represent no intensity P n N Noa 3 1 N 1 yo n N n The limits of this distribution are represented by the normal and the Poisson distribution Poisson 1838 When n and p remains unchanged the continuous normal Gaussian estimation 3 2 is valid de Moivre 1738 P n o 3 2 N Xp SZ l x oON2n i 20 Here o is the variance The discrete Poisson estimation is valid if n gt and p 0 with Np v gt 0 vie P n er 3 3 Established experience shows that the Poisson estimation holds well for mean values less than 20 and the normal estimation holds well for mean values above that threshold From a computational point of view testing the Poisson estimation is limited to n less than 170 as the factorial for IEEE standard floating point variables cannot be computed for higher values On the other hand the Poisson estimation offers an extremely reduced formalism for the distribution function containing only one
37. methods A simulated annealing algorithm is used to determine the beam centre The cost function utilized is the inverse normalized variance ofthe radial distribution function In effect this value gives the sharpness of the integrated peaks for a trial beam centre ignoring the effects of detector tilt The temperature parameter for the simulated annealing is the domain size of the trial beam centre In the second step the tilt and rotation of the detector are approximated Certain high intensity ellipses are selected and their inverse squared radii as a function of the azimuth are fitted to a summation of three exponentially damped complex sinusoids The fitting routine is a modification of the robust singular value decomposition SVD method called Hankel Lanczos SVD Press et al 1992 A Hankel matrix is filled with parameters from the summation and solved by SVD using the fast Lanczos bidiagonalization algorithm The refined parameters from the summation can be directly used for calibration The advantage of this method is that it returns reliable calibration values for a detector that has a near orthogonal setting to the primary beam It could become a standard method especially for laboratory experiments using radiation of a well defined wavelength in which the diffracted beam has a substantial half width It is however questionable whether the centre estimation suffices to locate the detector beam intersection with enough precision
38. non orthogonally to the primary beam This can enlarge the detectable q space in a very cost effective manner The down sides are the strongly elliptical conical projections and the loss of the entire azimuthal information of a diffraction cone Extraction of standard powder diffractograms from two dimensional images requires knowledge of the diffraction angle at each pixel These angles have to be known to a precision equal to or less than the detector resolution The detector resolution is mainly governed by the point spread function PSF In addition the calculation of air absorption would require the sample to pixel distance in each case The azimuthal angle is vital for the application of the Lorentz and polarization corrections as is the incident angle for a detector dependent correction The following chapter will deal with the derivation of all possible geometrical values which could be of importance during data reduction 1 2 1 Resolution and FWHM in two dimensional diffraction The resolution of a two dimensional detector is governed to a dominant extant by the PSF This can be very well observed by the behaviour of the FWHM distribution of reflections over an image plate The PSF of a standard image plate is roughly 300um The projection A of the diffracted beam width d on the image plate in the case of a fully parallel beam is given by d Al cos This would result in the projection of the diffracted beam leaving a large
39. seems like such a positive development does on closer inspection have its problems Two dimensional correction factors effectively do not exist for powder diffraction experiments All commonly used Lorentz and polarization LP corrections are meaningless outside the thin equatorial strip for which they were determined Furthermore various other detector and geometry dependent factors have to be considered should a high quality powder diffraction pattern be extracted from the image The first chapter of this thesis takes on this challenge and presents all applicable two dimensional correction factors as well as the basis for their application the experimental set up Determining the geometry to the highest possible precision is paramount to the quality of the experiment How can one achieve this goal without losing oneself in diverging refinements and renitent analysis software Pattern recognition methods and whole image refinement have been used to solve the two main problems of calibration and are presented in the second chapter The first global search gives sensible starting values for what is probably the most extreme refinement single pattern powder diffraction has to offer whole image refinement Here the entire two dimensional image is rebuilt based on the initial values and subtracted from the experimental image This residual is then minimized using a Levenberg Marquardt non linear least squares refinement algorithm This method lead
40. strong interactions with the tin atom 79 Two dimensional X ray powder diffraction The effect of the lone pair electron on the structure is quite visible figure 6 5 Channels down the b axis are forced open by the lone pair electrons from the only symmetrically independent tin atom The channels are reminiscent of those in the AB O type structures in which the lone pairs of B atoms caused large channels running through the structure Dinnebier et al 2003 Hinrichsen et al 2006 The edge linked octahedra of the latter structures are replaced by isolated tetrahedra in the sulphate These promise to be more dynamic under pressure Crichton et al 2005 Figure 6 5 The ambient pressure structure of SnSO viewed down the b axis The open channels down the b axis are a result of the lone pair electrons of tin The light grey atoms are tin the yellow tetrahedra represent the sulphate anions Bonds have been drawn for Sn O distances less than 2 6 Intensity a u l IO MiNI IO MILO MEO WALE NAI TE IDEE TUN NaN A Oaa dia 3 TRIBAL a a N I 5 10 15 20 20 I Figure 6 6 The Rietveld refinement plot of the orthorhombic ambient pressure phase of tin sulphate showing the experimental data as open circles the calculated pattern as the upper solid line and the difference pattern as the lower solid line The positions of the reflections are marked by vertical lines The scale of the difference plot is identical to the upper plot 80
41. the median value instead of the mean The median is well known to be a more robust estimator than the mean generally unaffected by strong outlier signals In a similar vein fractile filtering ensures by removing highest and lowest fractions of the data that the mean value becomes a sensible estimator The great advantage of fractile filtering is that the variance standard deviation become more meaningful to Fractile statistics is also referred to as quantile or percentile statistics E Ges preon geet peor bre ie fate etic fee oats bnin Coe Lad aise a ee ee Er Dtm muisie i eiF a y ee ES Ae AHEAASERSUEF FT TL ro nnd En gt Merl nn E u i Fractile Grow Beam Stop 1 50000 Converg criteria high ll 2 1 50000 c iteria ow onyerg cntena OW ll 0 0100000 Step width ep wi Al 0 100000 ted ncrement 4 m FT E toe ao maj I Fak Wo mar wa WE uci int Ge Delete mask Done Pem oe b Pov beter irira paa ge a Ba eee tumra fa 2 er Ptr Ie adets era ira EN Ce EET Deletemask Done Figure 131 The mask dialog and the effect of fractile masking on the image In Powder3D IP the filtering is done on the entire image with user set fractions and step width The step width should be similar to the step width of the integration Another alternative offered to the setting of a fixed fraction is the dynamic fraction filter This especially 142 interesting when o
42. the rotation of the sulphate anions and the volume reduction the tin atoms are hardly displaced throughout the entire compression process table 6 1 and table 6 2 From the polyhedral distortion of its surroundings it can be assumed that one of the symmetry independent tin atoms Sn1 see table 6 2 has its lone pair electrons facing the neighbouring layer The other tin atom Sn2 still fills a small channel parallel to 010 with its lone pair electrons 0 584 2 0 973 3 0 AB 0 070 8 0 TTE 0 984 6 0 889 1 0 504 2 0 213 2 0 er 0 542 4 0 185 3 0 638 2 0 458 4 0 685 3 0 G 0 527 8 0 147 8 0 SEIS 0 487 2 0 806 2 0 IG 0 674 10 0 380 6 0 67 0 074 8 0 822 6 Tin sulphate at high pressures Figure 6 10 The structure of the high pressure triclinic phase IV is shown viewed down the b axis The light grey atoms are tin the yellow tetrahedra represent the sulphate anions Bonds have been drawn where the Sn O distance is less than 2 6 A The structure has been refined using the identical methods to the previous triclinic structural refinement On the whole the refinement suffered from the noticeable peak broadening which can be attributed to the loss of hydrostaticity within the pressure medium at these pressures The deterioration of the diffraction pattern was the reason for beginning the decompression after this data collection 1000 I ij S 2 800 u m g 600
43. the step of refining the radial line ellipse intersection the intensities and the full width at half maximum FWHM of the reflections was determined These values are taken as starting parameters for the intensities and the peak widths For each reflection a profile is calculated these are added up to a final pattern figure 2 6 Now the one dimensional diffractogram has to be converted to a two dimensional image 41 Two dimensional X ray powder diffraction 500000 400000 300000 Intensity a u 200000 100000 20 40 60 20 F Figure 2 6 One dimensional diffractogram generated as starting pattern for the two dimensional image construction 2 3 2 2 Image construction To construct a two dimensional image two ingredients are required a calculated one dimensional diffractogram and the computed two dimensional array representing the 20 values for each pixel of the experimental image figure 2 7 One prerequisite is that the diffractogram resolves the peak profile sufficiently ideally with five data points per FWHM Constructing an image with the correct intensities on elliptical arcs is done by interpolation of the intensities of the one dimensional diffractogram to the diffraction angles of the two dimensional array This results in an image representing the ideal projection of the diffraction cones on the plane of the detector figure 1 5 2 3 2 3 Calculating detector characteristics The only detector characteristi
44. to p 19 8 GPa The extrapolated bulk moduli of the high pressure phases were determined to K 49 2 GPa for phase I K 27 3 GPa for phase Il and K 45 2 GPa for phase Ill The crystal structures of all phases are refined against X ray powder data measured at several pressures between p 0 52 GPa and p 10 5 GPa Introduction FeSb O also known as the mineral Schafarzikite Krenner 1921 belongs to a group of compounds crystallizing in space group P4 mbc with the general formula AB O A Pb Cu Sn Ni Zn Mn Fe B Pb As Sb where B represents ions with a stereochemically active lone electron pair Generally they are regarded as pseudo ligands that are able to replace one or more of the regular ligands in a given coordination sphere leading to irregular polyhedra of a low coordination number The resulting stereochemical implications have been discussed in depth Gillespie 1967 Gillespie amp Robinson 1996 67 Two dimensional X ray powder diffraction Table 5 1 Structural parameters of the ambient and low pressure structure of Schafarzikite which crystallizes in the space group P4 mbc The parameters were refined to data collected at p 0 5 GPa a o o osa osrem _foosat a 8 5758 1 A c 5 8983 1 A The crystal structure of Schafarzikite is characterized by the presence of edge sharing iron octahedra connected with corner sharing antimony tetrahedra leading to open channels containing the lone pairs F
45. to 180s 6 2 2 Data filtering and reduction The image plate orientation and position were determined using a nanocrystalline CeO sample As with the samples within the DAC no rotation during the exposure was performed This ensured the intensity of the diffracted beams to be normal Pareto distributed Hinrichsen et al 2007b An initial traditional calibration routine was run using the Powder3D IP software Hinrichsen Dinnebier amp Jansen 2006 The results obtained were later refined using the whole image refinement WIR To do this successfully the background had to be determined and Tin sulphate at high pressures the outlier intensities had to be filtered prior to the refinement All images were filtered before integration A fractile filter removing a fraction of the highest and lowest intensities from each bin was used The fraction to be removed was determined using the relation of the normal to the normal Pareto distributed intensities of a strong peak Hinrichsen et al 2007b This resulted in 58 of the highest intensity being removed from each bin before integration This filtering method led to approximately normally distributed intensities ideally suited for least squares refinement 15 20 Pressure GPa a S 77 0 1856 7 32 11 39 15 46 19 54 Figure 6 1 A simulated Guinier plot showing the progression of the intensity normalized powder pattern over the measured pressure range 6 2 3 Crystal s
46. unit cell angles figure 6 2 from the triclinic Rietveld refinement a phase transition at p 15 GPa seems to be supported by both the beta and gamma angle progression Both have sharp discontinuities at p 15 GPa and the gamma angle abruptly changes its inclination But this was not the only feature made evident by inspection of the angular progression From the third pattern p 0 2 GPa onward the gamma angle shows a deviation from the orthorhombic right angle It rises linearly crossing 90 at about p 2 5 GPa until at p 5 GPa it starts falling again The reason for this is another phase transition to the triclinic crystal system This can easily be followed from the beginning and steady increasing deviation of the alpha and beta angles with higher pressures The strongest deviation within this phase is shown by the gamma angle which reaches a maximum deviation of 0 6 from the original orthorhombic angle The phase transitions have only a very slightly effect on the diffraction patterns figure 6 1 but show interesting structural changes The ambient pressure orthorhombic structure is dominated by isolated sulphate tetrahedra that are bonded to tin atoms at a lengths of 2 25 A and 2 27 A Tin is 12 fold coordinated by oxygen in this position however the distances to the rest of the oxygen atoms is substantially higher than those of the first three The long distances range from 2 95 A to 3 34 A and cannot be regarded as having very
47. 1985 Structures isomorphes MeX O Evolution structurale entre 2 K et 300 K de l antimonite FeSb O elasticite et ordre magnetique anisotropes Journal of Solid State Chemistry 60 78 86 Cheary R W amp Coelho A 1992 A fundamental parameters approach to X ray line profile fitting Journal of Applied Crystallography 25 109 121 Cheary R W amp Coelho A A 1998 Axial Divergence in a Conventional X ray Powder Diffractometer Theoretical Foundations Journal of Applied Crystallography 31 851 861 Chotas H G Dobbins J T Ill amp Ravin C E 1999 Principles of Digital Radiography with Large Area Electronically Readable Detectors A Review of the Basics Radiology 210 595 Chupas P J Ciraolo M F Hanson J C amp Grey C P 2001 In Situ X ray Diffraction and Solid State NMR Study of the Fluorination of y Al O with HCF Cl J Am Chem Soc 123 1694 1702 Coelho A 2004 TOPAS3 Cole J M Mcintyre G J Lehmann M S Myles D A A Wilkinson C amp Howard J A K 2001 Rapid neutron diffraction data collection for hydrogen bonding studies application of the Laue diffractometer LADI to the case study zinc tris thiourea sulfate Acta Crystallographica Section A 57 429 434 Crichton W A Parise J B Sytle m A amp Grzechnik A 2005 Evidence for monazite barite and AgMnO distorted barite type structure of CaSO at high pressure and temperature American Miner
48. 7 5 0 BT 0 725 1 0 311 9 0 a 0 253 14 0 845 10 0 116 5 0 647 8 0 348 7 0 393 2 0 325 10 0 906 2 0 829 6 0 787 9 0 482 6 0 607 2 0 095 9 0 028 9 0 920 7 0 009 7 0 217 3 0 641 9 0 S5 0 711 9 0 305 7 0 ace 0 170 7 0 845 5 0 065 6 0 545 6 0 364 5 0 565 7 0 388 9 0 706 5 0 994 2 0 996 11 0 332 3 0 596 5 0 356 3 0 022 3 0 931 6 0 756 4 0 079 7 0 844 3 The phase transition causes a doubling of the number of parameters table 6 1 needed to describe the structure Two tetrahedral rigid bodies representing the sulphates were therefore used to stabilize the refinement The isotropic atomic displacement factors were constrained to be equal for identical elements For the final cycles the positional constraints were removed Probably the most interesting phenomenon to be observed in this study is the last phase transition which is an isostructural phase transition The indicators of this phase transition were mentioned earlier namely two angular anomalies in the lattice constants and the observed hysteresis effect in the volume on decompression The structural change is quite fascinating The evolution starts with sparse single layers of sulphate anions connected via tin atoms parallel to 101 It ends in a dense double layer structure with the layers nearly perpendicular to the original orientation namely parallel to 110 This change is brought about solely by
49. 9881 0 99999842 N 100 _ es 46 Intensity distributions and their application to filtering 3 Intensity distributions and their application to filtering 3 1 Why filter 2D powder diffraction images Many powder diffraction experiments impede the realisation of ideal circumstances namely the contribution of a very large number of equally sized and randomly oriented crystallites to the diffraction pattern This would lead to an ideal binomial intensity distribution Yao 2006 over the entire Bragg cone Despite the great experimental effort expended to ensure good quality data in situ experiments habitually suffer from weaker signal quality than standard diffraction experiments Of the in situ experiments high pressure powder diffraction poses probably the greatest challenge to data interpretation Sample rotation is confined to small if any angular rocking due to the diamond anvil cell s DAC opening angle the wish to limit the effect of gasket shadowing and the avoidance of diamond reflections This can lead to an extremely spotty diffraction cone the result of relatively few crystallites passing through a diffraction position an effect that is enhanced by highly parallel synchrotron beams Spotty cones are also often found due to recrystallization in the course of a phase transition To remedy this effect the intensity of the entire diffraction cone is integrated This method generally resul
50. Applied Crystallography 15 330 337 Knapp M Baehtz C Ehrenberg H amp Fuess H 2004 The synchrotron powder diffractometer at beamline B2 at HASYLAB DESY status and 99 capabilities Journal of Synchrotron Radiation 11 328 334 References Krenner J A 1921 Schafarzikit ein neues Mineral Zeitschrift f r Kristallographie 56 198 Kumar A 2006 thesis Austin College Larson A C amp Von Dreele R B 1994 GSAS General Structure Analysis System Los Alamos National Laboratory Report 86 748 Lei Y amp Wong K C 1999 Ellipse detection based on symmetry Pattern Recogn Lett 20 41 47 Letoullec R Pinceaux J P amp Loubeyre P 1988 High Pressure Research 1 77 90 Lipson H amp Langford J I 1999 Trigonometric intensity factors International Tables for Crystallography Vol C edited by A J C Wilson amp E Prince pp 590 591 Mao H K Xu J amp Bell P M 1982 Calibration of the ruby pressure gauge to 800 kbar under quasi hydrostatic conditions Journal of Geophysical Research 91 4673 4676 Meneghini C Artioli G Balerna A Gualtieri A F Norby P amp Mobiliob S 2001 Multipurpose imaging plate camera for in situ powder XRD at the GILDA beamline Journal of Synchrotron Radiation 8 1162 1166 Merrill L amp Bassett W A 1974 Miniature diamond anvil pressure cell for single crystal x ray diffraction studies Review of Scientific Instruments 45 290
51. Dumas C amp Andre D 1982 Journal of Applied Crystallography 15 330 337 Lipson H amp Langford J 1999 Trigonometric intensity factors International Tables for Crystallography Vol C edited by A J C Wilson amp E Prince pp 590 591 Tanaka K Yoshimi T amp Morita N 2005 Acta Crystallographica Section A 61 C146 Whittaker E 1953 Acta Crystallographica 6 222 223 Wu G Rodrigues B L amp Coppens P 2002 Journal of Applied Crystallography 35 356 359 Zaleski J Wu G amp Coppens P 1998 Journal of Applied Crystallography 31 302 304 Zevin L 1990 Acta Crystallographica Section A 46 730 734 Written by Bernd Hinrichsen 20 02 2007 Last updated by Bernd Hinrichsen 30 05 2007 154 159 Two dimensional X ray powder diffraction 156 Bernd Hinrichsen Burgstra e 46 70569 Stuttgart b hinrichsen fkf mpg de 49 711 689 1506 49 177 553 5318 Geboren am 07 01 1974 in Tsumeb Namibia Verheiratet zwei Kinder Ausbildung Vortr ge 2006 2005 01 2004 jetzt Doktorand Max Planck Institut f r Festk rperforschung Stuttgart Betreut von PD Dr Robert E Dinnebier 04 1994 03 2002 Diplom Mineraloge Universit t zu K ln Institut f r Kristallographie Thema Strukturelle Untersuchungen und Kristallzuchtung von azentrischen Alkali Monoboraten Betreut von Prof Dr Ladislav Bohaty 04 1993 03 1994 Studium der Mathematik an der Universitat zu K l
52. E Axes nterpolation E Axis 0 E Axis 7 i Lights n Annotation Layer Image transparency Z value for image plane Figure 114 Correcting the height of the image Now our image is complete All that remains to be done is to improve the lighting and export a high resolution image You might have become aware of the Lights branch on the left in the Visualization Browser Figure 114 Open this branch and select the directional light You should observe something like Figure 115 126 Figure 115 Lighting in iTools Now you can manipulate the geometric coordinates with the mouse and alter the intensity in the properties window This you can do three different light types ambient positional and directional Add further lighting via Insert gt Light The mouse has two modes while manipulating lights positional and rotational Change between the two by selecting the pointer or the circled arrow Figure 116 alm oz es SIRE ee AN ES Figure 116 Mouse modus buttons This should be the last fine tuning the image needs When complete export the image via File gt Export Select To a file in the following dialog When questioned whether Window View or Data are to be exported either Window or View shall suffice Next specify your export file name and type and resolution 127 Two dimensional X ray powder diffraction 13 2 5 Recipe
53. M Schwarz U Syassen K amp Takemura K 1999 Crystal structure of the high pressure phase silicon VI Physical Review Letters 82 1197 Heiney P A 2005 Datasqueeze A Software Tool for Powder and Small Angle X Ray Diffraction Analysis Commision on Powder Diffraction IUCr Newsletter 9 11 Hinrichsen B Dinnebier R E amp Jansen M 2004 EPDIC IX edited by R Kuzel E J Mittemeijer amp U Welzel pp 231 236 Prague Zeitschrift fur Kristallographie 23 Hinrichsen B Dinnebier R E amp Jansen M 2006 EPDIC X Geneva Hinrichsen B Dinnebier R E amp Jansen M 2007a EPDIC X Geneva Hinrichsen B Dinnebier R E amp Jansen M 2007b Zeitschrift fur Kristallographie submitted Hinrichsen B Dinnebier R E Rajiv P Hanfland M Grzechnik A amp Jansen M 2006 Advances in data reduction of high pressure x ray powder diffraction data from two dimensional detectors a case study of schafarzikite FeSb O Journal of Physics Condensed Matter 18 S1021 Hulsen G Broennimann C Eikenberry E F amp Wagner A 2006 Protein crystallography with a novel large area pixel detector Journal of Applied Crystallography 39 550 557 ITT VIS 2006 Interactive Data Language IDL Kahn R Fourme R Gadet A Janin J Dumas C amp Andre D 1982 Macromolecular crystallography with synchrotron radiation photographic data collection and polarization correction Journal of
54. Minimierungsalgorithmus den zu reduzierenden Kostenfaktor dar Diese Methode f hrt zu Kalibrierungen die mindestens eine Gr enordnung genauer sind als solche die man mittels traditioneller Methoden errechnet Dieses ist f r die effektive Nutzung zuk nftiger hoch aufl sender Detektoren von fundamentaler Bedeutung Eine perfekte Kalibrierung reicht allerdings nur in der seltensten F llen f r eine gelungene Integration Gerade in situ Experimente welche die St rke der zwei dimensionalen Detektoren sind verursachen stark abweichende Intensit ten Diese lassen sich nach Ihrem Ursprung unterscheiden Ent Two dimensional X ray powder diffraction 90 weder sie stammen von der Probenumgebung oder von der Probe Nat r lich sind die ersteren leichter mit dem Auge zu erkennen und auch leicht mit Filter zu entfernen die in Kapitel drei vorgestellt werden Schwieriger gestaltet es sich bei den letzteren Eine neue Verteilungsfunktion die normale Pareto Verteilung beschreibt die Intensit tsverteilung von kleinen Probenmengen die kaum im Strahl rotiert werden Diese Verh ltnisse werden gerade bei Hochdruckuntersuchungen realisiert Der gro e Vorteil dieser Beschreibung ist Sie er ffnet die M glichkeit vern nftige Filtereinstellungen zu berechnen die dann eine normale Verteilung wieder herstellen k nnen Strukturanalysen werden immer von einer Vielzahl von R reliability Werten begleitet die die Rohdatenqualit t sowie die Verfe
55. The high density of diffraction lines in the bottom left is caused by the extreme tilt 45 of the detector D z tan 20 FWHM cos tilt cosa D z tan20 PSF 1 9 The change in distance is given by the following equation z rsin tilt 1 10 The radius has been derived and is given by equation 1 37 Substituting all values of z and r in equation 1 10 and equation 1 9 solving for the FWHM and simplifying leads to the fundamental formulation of equation 1 9 FWHM arccot x sec zilt cosa 20 en num D D cosa cos tilt tan tilt tan 20 1 11 D Dsin tilt tan 20 den PSF tan 20 i cosa tan ti t PSF D tan 260 Such a situation can be simplified if only a thin Debye Scherrer type strip along an azimuthal angle is considered Norby 1997 The effects of diffracted beam projection and the point spread function on the image can be seen in figure 1 5 This figure displays a theoretical image of LaB setting the detector at a tilt as well as a rotation angle of 45 At large incident angles the beam projection contributes most to the line broadening while at low incident angles both contribute to the line broadening The projection of this image into reciprocal space Figure 1 6 is enlightening When compared to the initial pattern used to generate the image it can be clearly seen how the line width alters for different peaks Instead of the typical orthogonal setup behaviour in which the line width dim
56. Two dimensional X ray powder diffraction Der Fakultat fur Geo und Biowissenschaften der Universitat Stuttgart zur Erlangung der Wurde eines Doktors der Naturwissenschaften Dr rer nat genehmigte Abhandlung Vorgelegt von Bernd Hinrichsen aus Tsumeb Namibia Hauptberichter Prof Dr R E Dinnebier Mitberichter Prof Dr P Keller Tag der mundlichen Prufung 7 11 2007 Max Planck Institut f r Festk rperforschung Stuttgart 2007 To Christiane Aurel and Darius Contents Abbreviations 0 Introduction 1 The experimental setup 1 1 Two dimensional detectors 1 2 Diffraction geometry 1 3 Corrections 2 Calibration 2 1 The calibration image 2 2 Starting parameter estimation 2 3 Parameter refinement 2 4 Comparison of methods 3 Intensity distributions and their application to filtering 3 1 Why filter 2D powder diffraction images 3 2 Detector signal distribution 3 3 Filter models inclusive or exclusive filters 3 4 Filter applications 3 5 Conclusion 4 Quality assessment 4 1 How good are my data Suggestions for an n image reliability value 4 2 Comparison of reliability values originating from different data 5 Schafarzikite pane O at high pressure 5 1 Abstract R a a 8 5 2 Introduction 5 3 Experimental 5 4 Conclusion 6 Tin sulphate at high pressures 6 1 Introduction 6 2 Experimental 6 3 Conclusion 7 1 Precis 1 2 Zusammenfassung 8 References 12 26 34
57. aks cover many neighbouring pixels The mask might however not cover the full extent tails of the outlier peak Growing the mask ensures that this happens 143 Two dimensional X ray powder diffraction 13 6 3 Beam stop mask a EC Fia md DE TER a a Oa Ko Pein EICHE pet FR mh ee COC Figure 133 The mask dialog and the effect of the primary beam stop mask on the image Large intensity aberrations exist close to the primary beam stop Masking this region is done by simply selecting the first angles of the diffraction image to be disregarded figure 133 13 7 Intensity corrections The intensity corrections of two dimensional diffractograms are slightly more complex than the equatorial correction functions It is sensible to apply these corrections before integrating the image as they reduce the data variance as well as ensuring more accurate integrated data These corrections then need not be applied by the Rietveld software 13 7 1 Lorentz correction Sil Intensity corrections 7 ol x Lorentz Polarization Geometry Background Appl powder Lorentz correction Apply single crystal Lorentz correction Angle between vector of rotation and primary beam 0 000000 Phase angle 000000 Figure 134 The separate Lorentz corrections can be made The statistical powder correction and the angular speed single crystal correction can be computed separately 144 Edit gt Corrections
58. alogist 90 22 27 Dammer C Leleux P Villers D amp Dosiere M 1997 Use of the Hough transform to determine the center of digitized X ray diffraction patterns Nuclear Instruments and Methods in Physics Research Section B Beam Interactions with Materials and Atoms 132 214 Darambara D G 2006 State of the art radiation detectors for medical imaging Demands and trends Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment In Press Corrected Proof 91 Two dimensional X ray powder diffraction de Moivre A 1738 The Doctrine of Chances Dinnebier R E Carlson S Hanfland M amp Jansen M 2003 Bulk moduli and high pressure crystal structures of minium Pb O determined by X ray powder diffraction American Mineralogist 88 996 1002 Donaldson J D amp Puxley D C 1972 The crystal structure of tin Il sulphate Acta Crystallographica Section B 28 864 867 Fauth F Patterson B Schmitt B amp Welte J 2000 The future powder diffraction station at the swiss synchrotron facility SLS Acta Crystallographica Section A 56 s223 Finger L W Cox D E amp Jephcoat A P 1994 A correction for powder diffraction peak asymmetry due to axial divergence Journal of Applied Crystallography 27 892 900 Fischer R amp Pertlik F 1975 Verfeinerung der Kristallstruktur des Schafarzikits FeSb O Mineralogy and Petro
59. and emitted light within the image plate layer The corresponding incident angles are v k and Y The vector of the incident X ray is given by z 33 Two dimensional X ray powder diffraction 2 Calibration The calibration of a two dimensional diffraction image is the most fundamental step in the process of the reduction of two dimensional powder diffraction data The distance of the sample to the detector the position of the primary beam on the detector the parameters of the detector tilt and the X ray wavelength have to be known to the highest possible precision Only if these values have been determined well can the single pixels of the image be tagged with reliable diffraction angle values Ifthe calibration has been done optimally the resulting integrated diffraction pattern will not suffer from unnecessary reflection broadening Especially those reflections contributing large arcs to the images are sensitive to precise calibration values In addition filtering routines pose high demands on the calibration quality 2 1 The calibration image All calibrations of powder diffraction images begin with the exposure of a calibration image This image should be of the highest attainable quality as it is the foundation for the data quality of the final diffractogram It is advisable to utilize as much of the dynamic range provided by the detector as possible and to achieve high contrast between the diffraction ellipses and the background intensit
60. arameters method Cheary amp Coelho 1992 Cheary amp Coelho 1998 to a conic section functional Zuev 2006 have been published Calibration 2 2 39 Two dimensional X ray powder diffraction 40 One of the most difficult problems for calibration procedures is that of highly tilted detectors It is not problematic determining the intersection of a radial line with an ellipse it is however difficult to determine all effects of line broadening in detector space that affect highly eccentric ellipses differently in different parts of the arc The angle of intersection between a radial line and an eccentric ellipse can be very acute again causing the trace of the peak along the line to become broadened Peak broadening considered on its own is not a serious problem in calibration However weighting schemes and automatic rejection algorithms aimed at eliminating outliers are generally based on the positional error consistence of the Gaussian peak this value correlates strongly with the width Should the width of an ellipse vary along its arc so does the estimated error directly leading to the elimination of good data points Most disadvantageous are the removal of data points of an intersection through an ellipse arc with a high incident angle The width of the peak in length scales is high but is extremely narrow in reciprocal space thus potentially contributing important information to the minimization algorithm Switching off
61. ase II is observed to occur between p 2 2 GPa and p 3 1 GPa which is stable to at least p 8 3 GPa Between p 8 3 GPa and p 9 5 GPa a second phase transition occurs to another tetragonal phase P4 m phase Ill which stays stable until at least p 19 8 GPa figure 5 1 For all data sets lattice parameters as a function of pressure were obtained by Le Bail type fits using the programmes FULLPROF Rodriguez Carvajal 2001 and GSAS Larson amp Von Dreele 1994 15 10 Presure GPa 10 15 20 20 P Figure 5 1 Simulated guinier plot showing the progression of the powder pattern over the measured pressure range The background was modelled using the program Powder3D figure 5 1 Hinrichsen et al 2004 The peak profile was described by a pseudo Voigt function The phenomenological microstrain model of Stephens Stephens 1999 as implemented in GSAS was used to model the anisotropy of the FWHM Four parameters were refined for the tetragonal phase The quality of the powder patterns of all phases was sufficient to extract lattice parameters and to verify the crystal structures via Rietveld refinement tables 1 2 and 4 69 Two dimensional X ray powder diffraction 70 Table 5 2 Details of the refinement of phases I II and Ill of FeSb O 42 P20 5 945 Roy b Ro b Ro h No of variables No of refined atoms 7 Temperature K Wavelength A 0 413251 0 413251 20 range counting time 2 0 24
62. at panel detectors and the single photon counting hybrid pixel detectors 1 1 3 Flat panel detectors Thin film transistor TFT arrays are produced inexpensively and in large numbers for use in modern computer monitors and televisions This readout system can be combined with amorphous hydrated Silicon or amorphous Selenium which is deposited over the large surface of the TFT array and acts as the X ray conversion layer Having established themselves firmly as an X ray detector for medical imaging already they have until now failed to make an impact on the field of crystallography Ross et al 1997 their high noise level being the main drawback It can be surmised with some confidence that this type of detector will become standard equipment in the near future Avery general categorization into direct and indirect conversion types can be made for these detectors Chotas et al 1999 Within direct converters X rays are transformed to electrons in a single step for example by a layer of amorphous Selenium One further step is required for the indirect sensors here a scintillating layer photoconductor converts the X rays into visible light which is in turn converted to an electronic charge by a further amorphous Silicon layer This brings with it the inevitable resolution loss associated with the radial diffusion of photons and again their interaction with the amorphous Silicon As with all detection layer systems they can be optimized usin
63. ata as well as the refinement quality Powder diffraction images completely lack any numerical estimation of their quality Functions giving universally comparable detector independent reliability values for images can be found in chapter four The final chapters represent the application of the tools developed in the foregoing chapters to the most challenging data produced by modern in situ powder diffraction experiments high pressure powder diffraction images Two substances have been described undergoing in total 5 phase transitions under elevated pressures All substances possess lone pair electrons which have a profound effect on the structures at ambient conditions as well as under pressure The data collected has undergone data reduction using software developed by the author All patterns extracted in this manner were of such high quality that they directly could undergo full Rietveld refinement giving excellent residuals and sensible atomic displacement parameters Manuals to the author s software Powder3D and Powder3D IP are appended to this work Precis 7 2 Zusammenfassung Die Kombination von Pulverdiffraktometrie gro en Fl chendetektoren und Synchrotronquellen ist ph nomenal erfolgreich und erm glicht viele neue Experimente Die gro en Vorteile solcher Datengewinnung liegen in den kurzen Belichtungszeiten sowie in der enormen Redundanz Ein gro er Winkelbereich des Brechungskegels wird in einer einzigen Bel
64. bic in the orientation of the sulphate tetrahedron They are rotated moving the oxygen and sulphur atoms out of the orthorhombic symmetry The Wyckoff positions split according to table 6 2 This causes all atoms within the sulphate anion to be symmetrically independent The effect of the symmetry breakdown is visible in the comparison of sulphate orientation in figure 6 5 and figure 6 6 1400 1200 1000 800 Intensity a u 600 HoT PETE PE TEE PIE TE ERE TEE TEE E E TE I BIT ET E77 17 7117 7 5 10 15 20 20 1 Figure 6 7 The Rietveld refinement plot of the monoclinic low pressure phase of tin sulphate showing the experimental data as open circles the calculated pattern as the upper solid line and the difference pattern as the lower solid line The positions of the reflections are marked by vertical lines The scale of the difference plot is identical to the upper plot The following triclinic phase table 6 2 emerges at p 5 GPa Most impressive in comparison to the latter phase is the increase in the density of oxygen atoms surrounding tin Eleven oxygen atoms are closer than 2 6 to the two symmetry independent tin atoms When these distances are drawn as bonds figure 6 8 it gives the impression of a highly interconnected structure 82 Tin sulphate at high pressures a in all UN k Figure 6 8 The structure of the intermediate pressure triclinic phase Ill is shown viewed down the b axis The light grey ato
65. ble 3 1 Parameters of the fit of the normal distribution function to filtered and unfiltered data from a 0 02 bin of the first LaB peak The values are equal within the standard deviation CC Unfiltered data Filtered data Mean oseon 048100 Those values for the filtered and unfiltered datasets can be seen in table 3 2 Here the deviation from the ideal values that have been determined by the fitting of the normal distribution function to the data is evident The ideal intensity of ca 9500 a u is not attained by either the filtered or the unfiltered dataset The large average value of the filtered dataset is caused in part by the aggressive low intensity filtering The most fundamental reason for the large discrepancies between the true and the arithmetic values is the small sample size The population of the raw data was 693 pixels after filtering only 636 pixels remained This small data base is the main cause of the strong local deviations noise from the fitted distribution although the histogram follows the normal distribution function well This phenomenon is also the cause for the attenuation of the noise level towards higher angle data as higher angle bins contain more pixels Table 3 2 Statistical values of the filtered and unfiltered data from a 0 02 bin of the first LaB peak The values show a small difference to those from the fit to the normal distribution in table 3 1 Only the variance of the unfiltered data is unrealis
66. branch on the left that is where the data is and assigned the Z value to the pixels Y values to the Y axis and X values to the X axis To do the assigning mark the data value on the left and click on the small right arrow in front of the fields you wish to be associated with the data Leave palette empty and press OK So that is done but where is the image Find your way back to the Visualization Browser or properties window as described above It shall have a new entry for the image Highlight the image and alter the Z value to well above the maximum intensity in your data array In our case it is 85000 Figure 114 Further set the transparency to 40 Si Untitled Oj x File Edit Insert Operations Window Help ael poz x SS Click to select items click and drag to select multiple items fl 72 446 Figure 112 Insert visualization 125 Two dimensional X ray powder diffraction S Insert Yisualization Data Manager Image 8 Image Parameters IMAGEPIXELS oof PALETTE PALETTE 4 Z a I Surface parameters PALETTE z P 010 X 010 Y x xx Plot Profile KO an xf Import Variable Import File Type IDLPALETTE Description BYTE 3 256 Figure 113 Define the type and variables for the visualization Sj IDL iTool Yisualization Browser Ei Window BO Yiew_1 Visualization Layer GE Data Space BA alate Image palette cies ere iS
67. c affecting the spatial signal that requires consideration is the PSF All other aberrations are corrected by modern detector recording software or firmware The point spread function PSF has a rather complex projection into reciprocal space but in detector space it reduces to a convolution with a Gaussian function This effect also known in image processing as a Gaussian blur can be easily applied to the image after the initial Bragg ellipses have been computed The half width of the Gaussian convolution corresponds directly to the point spread of the detector Further two dimensional corrections that can be applied are the incident angle correction polarization correction and if applicable a Lorentz correction In these cases two dimensional correction arrays of equal dimensions to the image can be calculated and applied pixel by pixel 42 Calibration 20 100 x O 100 200 300 400 500 Pixels Figure 2 7 Acolour coded image of the computed two theta values which will be used for the interpolation of the one dimensional diffractogram to the two dimensional image is shown Overlaid are the interpolated intensities from the pattern in figure 2 6 The centre lies in the lower left quarter of the image The extreme detector tilt of 45 combined with the rotation of the tilt plane of 45 causes the highest diffraction angle to be in the bottom left corner Figure 2 8 The observed two dimensional diffract
68. chsen B Dinnebier R Jansen M amp Joswig M 2007 Automatic calibration of powder diffraction experiments using two dimensional detectors Powder Diffraction 22 Rigaku 2004 High resolution X ray detector for protein structural analyses R AXIS HR The Rigaku Journal 21 39 42 Rodriguez Carvajal J 2001 Recent Developments of the program FULLPROF Commission on Powder Diffraction Newsletter 26 12 19 Ross S Zentai G Shah K S Alkire R W Naday I amp Westbrook E M 1997 Amorphous silicon semiconductor X ray converter detectors for protein crystallography Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment 399 38 Schmidt C L Dinnebier R Wedig U amp Jansen M 2007 Crystal Structure and Chemical Bonding of the High Temperature Phase of AgN Inorg Chem Schulze C Lienert U Hanfland M Lorenzen M amp Zontone F J 1998 Microfocusing of Hard X rays with Cylindrically Bent Crystal Monochromators Journal of Synchrotron Radiation 5 77 Stephens P W 1999 Phenomenological model of anisotropic peak broadening in powder diffraction Journal of Applied Crystallography 32 281 289 93 Two dimensional X ray powder diffraction Sugimoto K amp Dinnebier R 2007 Acta Crystallographica Section B accepted Tanaka K Yoshimi T amp Morita N 2005 IP slant incidence correction for accurate s
69. cribed by a single pixel in parameter space over the real image The value of all image pixels touched by the traced pattern are summed and then written to the pixel in parameter space It is immediately obvious that this process scales with the volume of the parameter space Reducing the number of dimensions the required resolution and the range can greatly improve the processing time Computing all five parameters simultaneously using a Hough transformation is to computationally expensive Bennett et al 1999 In order to diminish the size of the required parameter space the complete process of ellipse detection is decomposed into different steps The Hough transformation is used for the calculation of the parameter b alone Hence we require only a one dimensional parameter space 8000 6000 Intensity 4000 I 2 2000 NN if fe 0 100 200 300 400 500 Pixels Figure 2 2 The intensity along a grid line of a powder diffraction image made for calibration purposes The original image and the overlaid grid are shown in the top right corner 36 Calibration 2 2 2 1 1 Approximate centre determination Many methods have been used for the determination of the centre of an ellipse Dammer et al 1997 Lei amp Wong 1999 Most of the methods suffer either a lack of accuracy or inefficient memory usage We propose a generic two step approach to determine the centre co ordinates of an ellipse In the first step we find the a
70. d volume pressure dependence is represented by the equation of state EoS An EoS is typically fit to a model based either on series expansion of Eulerian strain Murnaghan 1944 or on cohesive energies in a condensed system Vinet et al 1986 The programme EOSFIT 5 2 Angel 2002 was used to fit the Vinet EoS defined as 0 4 P 3K exp 34 K 1 1 f 5 1 Schafarzikite FeSb O at high pressure 1 ABI MIN IL I RN III Nam N A A a AN A MAAS pa pia d er OY eee Intensity arbitrary units 2 4 6 8 10 12 14 16 18 20 22 24 20 7 Figure 5 2 Scattered X ray intensity for the low pressure phase of FeSb O at p 0 52 GPa as a function of diffraction angle 20 Shown are the observed pattern crosses the best Rietveld fit thick black line the difference curve thin black line and the reflection markers vertical bars The wavelength was 0 41325A where with volume at zero pressure V the bulk modulus K and its pressure derivative at zero pressure K In all calculations K was selected 4 The experimentally obtained values for the bulk modulus K presented in table 5 3 show good agreement with calculated values for the low pressure tetragonal phase being identical within experimental error Higher compressibility expressed by the lower bulk modulus for the intermediate phase Il of FeSb O is a feature shared by the intermediate phase Il of Pb O The increase in compressibility of the monoclinic phase is rou
71. d y values Image a shows the construction used to calculate the relation between x and x The similarity of the triangle spanned by D and x small triangle and the triangle spanned by D x sin tilt and x cos tilt is used to set up the relation Image b shows the construction used to elucidate the relation between y and y Here again the similarity of the smaller and larger triangles are used to set up the relation Adding a rotation angle around the normal to the focus of the ellipse leads us to the general ellipse represented in figure 1 11 Figure 1 11 The general description of an arbitrary tilt The added rotation angle suffices to describe any possible detector tilt This adds a cosine and a sine term of the rotation to the x and y values in the following form Again a separate notation is used to denote the rotated values x and y 21 Two dimensional X ray powder diffraction x x cos rot y sin rot 1 16 y y cos rot x sin rot On inserting these rotational equations into equation 1 15 one obtains the equation 1 17 x cos rot y sin rot cos tilt y cos rot x sin rot 1 17 D x cos rot y sin rot sin tilt tan 20 Equation 1 6 can be rearranged to the initial form equalling equation 1 12 x cos rot y sin rot cos tilt y cos rot x sin rot 20 arctan 5 D x cos rot y sin rot sin tilt 1 18 1 2 3 Inci
72. dent angle and ray distance calculations The incident angle of the reflected beam onto the detector is utilized in a factor often described as correcting for the flatness of a detector The diffracted beam penetrates into the image plate or fluorescent layer of the detector The penetration length is dependent on the angle of incidence and the linear attenuation factor for the utilised wavelength and fluorescent material Figure 1 12 Incident angle calculation The small triangle containing the complementary angle to the incident angle Y and tilt is used to deduce the formula for the calculation of the incident angle As can be seen from figure 1 12 the minimum and the maximum incident angle are given by the following equation 1 19 22 The experimental setup y 26 tilt We 20 tilt vn The tilt can be replaced with the effective tilt tilt y according to tilt y tilt sin a rot 1 20 Resulting in an effective incident angle of Y 20 tiltsin a rot 1 21 To determine the distance of the diffracted ray from the sample to the each point on the detector we use the construction presented in figure 1 12 The application of the sine rule results in the following equation sin 90 W _ sin 180 20 90 1 22 D ray distance This simplifies to ray distance D cos 20 sin 20 tan Y 1 23 1 2 4 General transformations As most pattern recognition algorithms use conventional geometric pa
73. dis amp Koutroumbas 1999 to determine the parameters describing their arcs As to the representation and evaluation of the integrated powder patterns a high level graphically powerful programming language offers the basis for an efficient solution We decided on the Interactive Data Language IDL ITT VIS 2006 to develop a general program for automatic data reduction and the evaluation of two dimensional powder diffraction data called Powder3D Various aspects necessary to solve problems encountered during the investigation of crystal structures at high pressure are described here The geometrical basis of two dimensional diffraction has been assimilated and transformed to a useful coordinate system This was most useful as most diffraction corrections functions for two dimensional data were devised for single crystal diffraction The effective Lorentz correction for a highly collimated beam was deduced A calibration method based on pattern recognition was developed to determine the detector position and orientation automatically These parameters were refined using a new approach termed whole image refinement in which the entire diffraction image is reconstructed and this used as a basis for the calculation of a least squares residual This method improves the accuracy of the calibration parameters by more than one order of magnitude Filtering methods based on fractile filtering have been introduced The intensity distribution of high pressure
74. distribution is the raw data The blue distribution is the masked data The two distributions do not share the same x axis Window gt Histogram The histogram provides insight the intensity distribution of the selected region as well as the effect that filtering has on the intensities in that region Generally the intensities describe a normal Gaussian distribution Sometimes different distribution models are better A more careful analysis of this data is possible by simply clicking the right mouse button within the histogram window and selecting Analyze The data is then transferred to iTools a software provided by ITTVIS free of charge One the most useful tools Hinrichsen et al 2007b within the histogram display is accessed via the context menu as well This is the normal Pareto distribution fitter which estimates the high intensity fractile setting to achieve a more or less normal intensity distribution To start it is generally a good idea to filter a few percent off the top Analyze Fit NP Figure 143 A histogram showing an exemplary normal Pareto distribution Using the context menu item Fit NP an analysis of the distribution is made 149 Two dimensional X ray powder diffraction IDL iPlot Untitled lAIx File Edit Insert Operations Window Help Des eel tele IR Se a for AlN lolol eo 9x10 4x10 Click on item to select or click amp drag selection box
75. e 139 A direct appreciation of the effect of the intensity corrections applied to the image can be previewed by selecting the appropriate tick box Edit gt Display To estimate the correctness of the background determination as well as the polarization correction these can be applied to the displayed image The default setting is not to display intensity corrections 13 8 Intensity analyses A few tools are provided to allow a closer look at the intensity distributions over the image 147 Two dimensional X ray powder diffraction 13 8 1 Area selection Theta Azimuth 300543 00 E ft g Position O 0200000 gt Width Apply Done Figure 140 The area selection interface Selecting the theta region The area selection tool is the most fundamental setting for intensity analysis The central theta value is set in the top field while the width of the theta region is set in the lower field The selected area then covers a region from theta width 2 to theta width 2 Theta Azimuth 0 000000 al gt Stark 360 000 IE Apply Done End Figure 141 The area selection interface selecting the azimuth region 13 8 2 Intensity displays Once the region of interest has been selected it is possible to view data within this region as a histogram or as a function of the azimuth 148 13 8 2 1 Intensity histogram IM Figure 142 A histogram of two intensity distributions The black
76. e axis of rotation perpendicular to the primary beam and reducing the vertical displacement to nought y is then the diffraction angle L sin20 1 44 The reduction from the two dimensional form to the one dimensional equatorial form was a requirement of the equatorial diffractometer geometries utilizing point detectors or at most linear position sensitive detectors It is important to note that this correction is neither applicable to the Bragg Brentano nor to the flat transmission geometry but is only valid for the Debye Scherrer geometry With the onset of area detection in powder diffraction a two dimensional correction has to be applied A formulation of 1 43 in dependence on the more accessible scattering angle 28 and the azimuthal angle a has been derived cosa tan 20 I cos u A 1 45 V1l cos a tan 20 1 sin a tan 20 Combining this with the statistical factor leads to the general formulation of the Lorentz correction for powder samples rotated within the beam cosa tan 20 sin cos u et A AE NT 1 46 1 cos a tan 20 1 sin a tan 20 gt no Lorenz correction N II WE gus Figure 1 14 The two dimensional single crystal Lorentz correction for an ideally aligned detector Note the zero values in the central valley These cause divergent intensities as they are multiplied with the inverse of the Lorentz correction Therefore the intensities in that region have no meaning
77. e correction factor for a primary beam that is collimated to below the sample size which is given by S Should the primary beam be larger than the sample the factor is unity 1 58 1 59 1 60 1 61 The experimental setup Polarization correction Figure 1 17 The two dimensional polarization correction for an ideally aligned detector The polarization factor has been set to 0 1 resulting in the black dome shaped surface and to 0 99 which results blue saddle shaped surface The correction is applied by dividing the intensities by the displayed correction values _ zul u l 1 Ban Here and _ are the vertical and horizontal intensities respectively The monochromator angle is related to u by the following equation u cos 20 1 63 With knowledge of the monochromator angle 20 and that of the initial polarization of the X ray beam one can calculate the effective polarization factor However it is possible to refine this parameter against two dimensional reflection intensities and thereby retrieve a reliable value In figure 1 17 one can see the effect of the polarization correction for two different polarization factors 1 3 1 3 Incident angle correction An intensity correction based on the angle at which the reflected beam strikes the detector plane was initially proposed by Gruner Gruner et al 1978 in his work on CCD detectors Since then the development of this correction has been mainly driven by el
78. e detector setup relative to the polarization plane Edit gt Corrections The polarization factor Azaroff 1955 1956 Kahn et al 1982 Lipson amp Langford 1999 Whittaker 1953 is determined by the intensity relations of the horizontally and vertically polarized radiation as well as the monochromator angle For polarization factors close to O the polarization correction has little or no azimuthal variation therefore the phase angle becomes unimportant For values close to 1 as is typical for synchrotron radiation there is a very strong azimuthal dependence The correct polarization phase angle is then very important Normally it is at right angles to the detector coordinates 0 90 145 Two dimensional X ray powder diffraction 13 7 3 Incidence angle correction Sil Intensity corrections i x Lorentz Polarization Geometry Background Figure 136 The incident angle correction can be applied using this tick box This is similar to the correction in the Fit2D software Edit gt Corrections Various incidence angle correction functions are in use in single crystal diffraction Tanaka et al 2005 Wu et al 2002 Zaleski et al 1998 The only implemented one currently is the rather simple cos incident angle correction that is known as geometry correction in Fit2D Hammersley et al 1996 13 7 4 Background correction S Intensity corrections O x Lorentz Polarization Ge
79. ectron density researchers striving to attain high quality intensity data from area detectors From figure 1 18 it becomes clear how the incident angle affects the path of the beam through the detecting layer The distance traversed by the beam in the detecting layer is d cosY where Y is the incident angle and d is the thickness of the detecting layer 31 Two dimensional X ray powder diffraction 32 X ray beam d P Detecting layer Figure 1 18 Incident angle correction The path of the incident ray through the detecting layer of thickness d depends upon the incident angle Y The length traveled within the layer is then d cosW This correction is necessary should the beam not be completely absorbed by the layer Reflections at high incident angles would have a falsified higher intensity because of the greater detection length This correction attempts to normalize the intensity to an incident reflection normal to the plain Ifthe absorption of the visible light generated within the layer is disregarded then the correction Zaleski et al 1998 is independent of the detector type CCD or IP l e l e 4 1 64 In 1 ic 1 7 with Lis K 1 65 and T being the transmission of the detector layer at normal incidence This leads to the complete correction function of 1 lt 7 115 zen 1 66 cos An empirical correction that is used by some single crystal diffractometers equipped with CCD cameras ta
80. efinement D mm 141 15251 M Rotation degrees 76 033152 m w yf Wavelength 4 0 92336796 iv Tilt degrees 0 20307021 a Refine Select fan Completed successtully Figure 127 The calibration wizard The refined parameters of the detector alignment Once the parameters have been refined the values will be updated with those from the least square refinement If you are satisfied with the values you can now press Done The refined values will then be used to compute the diffraction azimuth and incident angle for each pixel of the image This can take some time with large images 140 8 Calibration 0 x Sample Pinel Center and Rotation Eccentricity Least square fit Superfine Select the parameteres for Whole Image Refinement Dimm 141 19622 M Rotation degrees 78 320702 WM bal wi Wavelength ral 0 32346956 a Tilt degrees 0 20574914 M Feak properties Corrections wr retin image 4 Image 4 gt Refine Wax iter 10 Completed successfully Save Load Advanced lt lt Back W Display graphics Update Lee oF Cancel Done 1 mee o Boe of 1 Je oF 1 781 2 EF vn cokukted om g 10 Weighted ras dunl Figure 128 The calibration wizard A converged whole image refinement A whole image refinement WIR is a rather complex refinement of an entire synthetic image against the measured image Outliers in the measured image
81. ermine the statistical characteristics of the data an ideal image was selected This image is a typical calibration image using a well defined standard substance in our case LaB Contrary to what might be expected the distribution of intensities within a Bragg peak are not ideally suited for the statistical characterization Although polarization and Lorentz effects can be corrected a substantial azimuthal intensity deviation probably stemming from sample absorption mars a rational model selection Background intensity caused by air scattering is however ideal for statistical analysis Polarization correction results in a perfectly flat intensity distribution along the azimuth A histogram of such intensities contributing to a 0 02 bin is shown in figure 3 1 400 300 200 small peaks Density pixels per bin 60 Intensity a u Figure 3 1 The black line shows a histogram of the intensities contributing to one bin of 0 02 in 28 of the final integrated background intensity Two distribution models have been overlaid in blue the normal Gaussian probability distribution function and in red the Poisson probability mass function The Poisson distribution is not continuous as might be suggested by the line but discrete Two small peaks can be seen at lower intensities these originate from the primary beam holder and the capillary shadowing as can be seen in the azimuthal plot in figure 3 2 Two solid lines representing the statistical
82. es have been plotted against experimental data from a calibration image see Figure 1 4 The image had a minimal tilt of 0 2 the sample to detector distance was 141 mm the diffracted beam was 0 2 mm and the point spread was estimated to be 0 32 mm Dtan20 PSF d sec20 a D FWHM arctan 20 1 7 15 Two dimensional X ray powder diffraction 16 0 22 0 20 0 18 0 16 FWHM 0 14 0 12 en E 0 20 30 40 50 Diffraction angle Figure 1 4 Calculated full width at half maximum FWHM and observed FWHM for a calibration image of LaB are plotted against one another The solid line represents the calculated values whereas the crosses represent the measured values To calculate the theoretical values the diffracted beam width was taken to be 0 2 mm the point spread of the detector was set to 0 32 mm and the sample to detector distance was taken as refined by the calibration at 141 mm It is possible to estimate the line width of an orthogonally aligned diffraction experiment by knowing the width of the diffracted beam the PSF of the detector and the distance to the detector Adding tilt to the two dimensional detector makes the situation a lot more complicated Essentially an azimuthal factor is added to the incident angle as well as to the angular resolution This results in an azimuth dependent FWHM of the diffraction image The dependence of the line width on the detector orientation is given in its comple
83. experiments which in many aspects deviate from standard powder diffraction experiments was found to follow a somewhat blurred due to the detector point spread Pareto distribution This distribution Introduction Two dimensional X ray powder diffraction model was used to develop a method of extracting normally distributed intensities from the image a fundamental necessity of all minimization routines based on least squares An image reliability value was devised for quantifying the quality of a powder diffraction image Two applications of the methodological development are presented The structural evolution of FeSb O and SnSO under extreme pressures It has been mentioned that high pressure experiments using diamond anvil cell techniques are the most challenging experiments in two dimensional X ray diffraction not only in sample preparation but also in the data analysis Many results are published without structural Rietveld refinements as a direct result of the difficulties in extracting reliable intensities from the raw images That this is routinely and efficiently possible has been shown exemplarily in the presented studies The tin sulphate experiment shows which precision this method is capable of Most subtle changes in the symmetry of the crystal structure become apparent after the raw data has been treated accordingly The experimental setup 1 The experimental setup 1 1 Two dimensional detectors The first two dimensi
84. ful opening doors to many new experiments The great advantages of such data collection lie in the short exposure times as well as in the huge redundancy A large angular region of the Bragg cone is recorded in a single exposure indeed most detectors are set up perpendicular and centrally to the primary beam intercepting the Bragg cone over the entire azimuthal range The standard practice is to integrate the image along the ellipses described by the intersection of the cone with the planar detector to a conventional powder pattern This commonly reduces the amount of information by the square root of the number of pixels Does this represent the gamut of information contained in a powder diffraction image A glance at an image from a calibration standard might lend itself to such a conclusion Less perfect samples as well as sample environments leave distinctive artefacts on images How can they be extracted filtered or interpreted Methods offering answers to these questions are introduced The origins of powder diffraction were based on diffraction images however with the onset of equatorial electronic point detectors all high quality powder diffraction experiments switched to this method It has remained the experimental doctrine to this day Only recently have powder diffraction scientists rediscovered the allure of diffraction images Indeed high pressure powder diffraction experiments are unthinkable without two dimensional detectors What
85. g various layer thicknesses and scintillating substances which can be selected according to the X ray wavelength and the photon detection properties of the lower layer Currently vapour grown Csl TI as an extremely popular material It grows in columnar structures and can act as a guide in a similar fashion to fibre optics thereby reducing lateral scattering Its high atomic number secures a high X ray absorption and thus good conversion Other materials under study as possible photoconductors are Hgl Pbl and CdZnTe CZT Especially CZT grown using a high pressure Bridgeman technique has been rapidly implemented in a wide variety of medical detectors Darambara 2006 11 Two dimensional X ray powder diffraction 12 1 1 4 Hybrid pixel detectors Silicon pixel array detectors are based asthe name suggests on Silicon as the primary detecting layer The photoelectric effect causes one electron hole pair to be created for each 3 65eV of incident X rays This leads to 3220 electrons from each X ray photon at 12keV in a one millimetre layer which absorbs 98 of that radiation In contrast a CCD would only produce roughly 10 electrons The readout time of a few nanoseconds also contrasts impressively with all other detector systems Further no distortions are to be expected as no intermediary tapering or disconnected readout systems are involved In spite of these overwhelming advantages the price of prototyping and the expense of the readou
86. gen cryogenic loading as pressure medium The DAC had 300um culet and 125um hole diameters The pressure was determined by the ruby luminescence method using the wavelength shift calibration of Mao Mao et al 1982 High pressure X ray powder diffraction data were collected at room temperature at beamline ID9 of the European Synchrotron Radiation Facility ESRF using an experimental configuration following that described by Schulze Schulze et al 1998 Monochromatic radiation for the high pressure experiment was selected at 30 0keV 0 41325A The beam size was 30 x 30um Diffracted intensities were recorded with a Marresearch Mar345 online image plate system A set of 22 images at selected pressures between p 0 56 GPa and p 19 8 GPa were recorded 3 of the 22 images were taken during the decompression of the sample An exposure time of 120 seconds was chosen 68 Schafarzikite FeSb O at high pressure 5 3 2 Crystal structure determination and refinement Following the successful filtering and integration of the two dimensional images to conventional powder diffraction patterns the scattering profile of FeSb O in dependence of pressure figure 5 1 gives evidence for a second order phase transition followed by a first order phase transition The phase with tetragonal symmetry P4 mbc phase which is stable at ambient conditions is retained up to at least p 2 2 GPa A transition into a phase with monoclinic symmetry P2 c ph
87. ghly by a factor of two No difference within experimental error is registered for the EoS between the low pressure and high pressure phases It should be noted that the EoS of the first and second phases have been determined using only four and three data points respectively Thus the interpretation of these values is somewhat speculative 71 Two dimensional X ray powder diffraction Table 5 3 The Vinet equation of state for the three phases of FeSb O and related compounds so il 11 1 4 fix 2 PbO NT a Gavarri and Chater 1939 calculated values Chater et al 1987 measured at T 63K and T 240K NO NI An anisotropy in the change of lattice parameters induced by pressure has also been observed in the high pressure study of Pb O The monoclinic phase II of FeSb O shows a strong distortion of the lattice The highest observed discontinuities of the lattice constants relate however to the re entry of the tetragonal symmetry above p 7 4 GPa figure 5 3 and figure 5 4 A moderate compression of the c axis until this pressure can be noted For the remaining range up to p 19 8 GPa the c axis remains virtually unchanged The entire compression in this range takes place within the ab plane and is directly related to the constriction of the open channels containing the lone electron pairs 3 a3 S Faini 4 4 Om D a4 b T Katie PALIT A T K il a8 I I I I I I I I I I Yin
88. gration Integrate l Files Fiter Comections Binning Azimuth Method IV Apply intensity corectiong Settings Cancel UK Figure 149 The intensity corrections that have been computed are applied to each image before integration These are computed only once after the calibration and are applied quickly Integrate Files Filter Corrections Binning Azimuth Method 2300 E p Number of bins Step width 0 026167235 Cancel UK Figure 150 In this tab the number of bins is set As an aid the effective step width is computed and displayed 152 Integrate Figure 151 The integration can be divided into any number of wedges An offset angle can be set if required 0 is 3 o clock Integrate Figure 152 The intensity of each bin can be computed as mean or as median value The median is known to be a more robust estimator than the mean However the default setting is the mean 153 Two dimensional X ray powder diffraction 13 10 References Azaroff L 1955 Acta Crystallographica 8 701 704 Azaroff L 1956 Acta Crystallographica 9 315 Buerger 1970 Contemporary Crystallography McGraw Hill Hammersley A P Svensson S O Hanfland M Fitch A N amp Hausermann D 1996 High Pressure Research 14 235 248 Hinrichsen B Dinnebier R E amp Jansen M 2007 Journal of Applied Crystallography submitted Kahn R Fourme R Gadet A Janin J
89. he function File gt Import Figure 72 The pattern is added to the end of the pattern array Should the pattern differ with regard to the step width or position the intensities are interpolated to match the range and step width of the data already loaded A dialog requiring you to enter the wavelength for this pattern is displayed If one or more patterns need to be deleted select them in the Main controls window and press Delete pattern in the Edit menu Figure 73 Adding a single patter Main controls Pattern Range Info Figure 74 Pattern selection tool 102 Sl Powder 3D File Edit Display Tools Plot Window Help j A lOlx 6x 10 4x10 Intensity N x OS ji AL ll ld ot IR AJ U SU La im JU hat N i nn a j WN Im A Figure 21 The first powder pattern 13 3 Data reduction 13 3 1 Cropping tool The limits of the powder patterns can often be reduced This is conveniently done by means of the cropping tool A click on the crop button Figure 76 draws two vertical lines which you can drag to define the desired range Figure 77 he S Figure 76 Crop button 103 Two dimensional X ray powder diffraction Sl Powder 3D File Edit Display Tools Plot Window Help su 10 x 8x10 6x 10 Intensity 4x10 In Ly j nf Lo LS
90. he median is an excellent estimation of the true value 1000 100 Density pixels per bin Density pixels per bin 100 1000 10000 100000 1000000 1200 1400 1600 1800 2000 Intensity a u Intensity a u Figure 3 7 Two data samples showing the effect of fractile filtering on the intensities of a Bragg reflection in a high temperature experiment Left the histogram of the unfiltered data is shown using double logarithmic axes On the right the histogram of the filtered data is shown The normal distribution function is fitted to the data and the fitted and arithmetic parameters are listed in table 3 3 55 Two dimensional X ray powder diffraction 56 1000000 100000 5 v gt 10000 wo Cc E 1000 100 100 200 300 Azimuth Figure 3 8 The intensity is shown as a function of the azimuth for the data depicted in the histograms of figure 3 7 The green symbols represent the highest fraction the blue symbols represent the lowest fraction of the data These fractions are removed by the filter The barely visible sinusoidal perception of the intensity is due to sample absorption effects that have not been corrected It is clear that simple filtering algorithms have a substantial and positive effect on the quality of the resulting integral mean values and arithmetic variances The effect of the filter on the resulting diffractogram can be seen in figure 3 9 Table 3 3 The table compares ideal and arithmetic sta
91. hts can be exported to Crysfire format file cdt The refined values can be exported to text file Should you have refined more than six peaks an instrument resolution file in Fullprof format irf can be exported olx EL lolx Find Fit Sl Powder 3D File Edit Display Tools Plot Window Help tal V Refine profile Func Eta 0 0904 j sv 0 0140 7 HL 0 014077 Peak 1 Peak 2 IV Refine pos int Intensity 1210 7 17 Position 9 3312 17 15x 10 FWHMI0 1455 fv Intensity ar IV Individual FWHM IV Asymmetrical peaks Cycles Bo 50x10 Multi Refine Figure 92 Peak refinement 13 3 11 Peak Indexing Powder3D has no own indexing capabilities but does provide a simple interface to the powerful indexing Shirley 2002 Two very similar N methods can be chosen for interaction 1 Save the peak file in the Crysfire format and start Crysfire manually 114 2 Should Crysfire be installed only Windows operating systems according to the installation instructions in the Crysfire manual it can be called by pressing the Crysfire button The file is exported to the current working directory and Crysfire is started in there A message is displayed with the name of the file that has been exported DI a z0 410 JE 2T hei f Ovenwite Append Lrystire Search Save Done Figure 93 Crysfire button 13 3 12 Le Bail refinement assistant Select
92. ichtung er fasst h ufig wird sogar der gesamte Kegelschnitt detektiert da die meisten Kameras rechtwinklig und mittig zum Prim rstrahl positioniert sind Es istdann gebr uchlich die gesamten Ellipsen die die Kegelschnitte beschreiben zu einem konventionellen Pulverdiffraktogramm zu integrieren blicherweise wird die Datenmenge um eine Quadratwurzel der Anzahl der Pixel reduziert Gibt dieses jedoch die komplette Information aus dem Streuungsbild zufrieden stellend wieder Ein Blick auf ein ideales Kalibrierungsbild k nnte einen solchen Schluss nahe legen Minder perfekte Proben sowie Effekte von Probenumgebungen hinterlassen jedoch starke Spuren auf den Bildern Hier liegt das Problem Wie k nnen auch solche Daten gefiltert und interpretiert werden In dieser Arbeit werden Methoden vorgestellt die dar ber Aufschluss geben und die Ergebnisse von verbl ffender Qualit t erm glichen Urspr nglich basierte die Pulverdiffraktometrie auf Streubilder Mit der Ankunft der quatorialen Punktz hlrohren bernahm diese Methode die Messherrschaft ber qualitativ hochwertige Pulverdiffraktogramme Bis heute setzt sich diese experimentelle Doktrin durch K rzlich haben Wissenschaftler aber die Vorz ge der Streubilder wiederentdeckt So ist beispielsweise die gesamte Hochdruckpulverdiffraktometrie ohne den Einsatz von gro en zweidimensionalen Detektoren undenkbar Was auf dem ersten Blick als Segen daherkommt bringt allerdings auch einige Prob
93. ighted its properties are displayed in the right part of the window We shall change the surface values to the following Color 145 145 145 Fill shading Gouraud Draw method Triangles The last two changes make the rendering slower but the picture better We open up the Axes branch and delete the Z axis by a right click and delete Figure 111 124 4 IDL iTool isualization Browser EM Window BO view 1 B m Visualization Layer E Data Space E Surface Gl iE Axes E Axis 0 E Axis 1 Axis 2 Name Axis 2 Description Axis Show True Color 0 0 0 Number of major ticks a Lights Cut Number of minor ticks 1 m Annotation Layer Copy Tick layout Axis plus labels Tick direction Right Above Use logarithmic axis False Text show True I Text nnsitinn Below left x Grouping gt Order Axis Figure 111 Deleting an element We also alter X axis properties to Titel 2Q Font Symbol And the Y axis to Titel Scan number Font Times These changes give us a 20 on the X axis and a similar font for the Y axis Now we create that semi transparent 2D visualization hovering over the surface Select the menu item Insert gt Visualization Figure 112 You shall then be prompted for the type and variables defining the item We chose an Image type top right box opened the Surface
94. ighted residual figure 2 10 which is the difference between the observed and calculated image figures 2 8 and 2 9 represents the cost This operation can lead to exorbitant computational costs in both time and memory and is a challenge for currently available personal computers Once the tilt becomes appreciable the necessity becomes apparent for an alternative weighting scheme The weighting scheme based on an idealized circular diffraction pattern is no more appropriate A value that can improve the quality of a tilted refinement appreciably is that of the incident angle Adding a weighting factor that is based on the incident angles figure 2 11 improves the refinement notably 1 0 500 0 8 400 0 6 30 0 4 200 100 0 2 0 0 100 200 300 400 500 Pixels Pixels Incident angle weighting factor Figure 2 11 An image showing the weighting factors based on the incident angle weighting scheme The detector orientation is identical to that of figure 2 7 and figure 1 5 2 4 Comparison of methods It is quite surprising to see how successful the current standard software package Fit2D is in determining very slight deviations from a perfectly or nearly perfect orthogonal experimental setup and how unsuccessful it is refining strongly tilted detector setups The reason is that the software was never devised for this kind of refinement The initial tilt values are always 0 Calibration images from detectors at different angle cann
95. ills the table with the calculated values These values are used for labelling purposes only 107 Two dimensional X ray powder diffraction Sf Set increments and phase ranges lO x Increments Phase ranges Set increment values 1 00000 154 IK Poo a 2o0000 158000 Increment 3 00000 162 000 i J150 4 00000 166 000 First vue x 500000 170 000 174 000 Last value 466 000 X 7 00000 178 000 182 000 rom set no 136 000 130 000 A To set 1 F Insert Done Cancel Figure 83 Insert increments Adding phase ranges can either be done manually by editing the fields of the range table Figure 84 Please note while editing you have to leave the cell you are editing for the values to be read properly by the Insert command The assistant for entering the values at the bottom of the window works in the same fashion as the one used in the increment window These values are necessary for the Le Bail assistant of which a first version is included 1 Set increments and phase ranges 10 x Increments Phase ranges 278 000 33 282 000 syz 34 286 000 syz E 35 290 000 Milz ZU 36 244 000 ZUR 4 Phase2 z Mame Zz From set no 35 To set no E Insert Done Cancel Figure 84 Define phase ranges 13 3 6 Ka stripping Should you have collected laboratory data using Ka and Ka rays these can be separated using the menu Edit
96. iltering values figure 3 14 for 2D powder diffraction images Under the main peak of the normal Pareto distribution a normal distribution fits very snugly This distribution tells us the characteristics of the largest and weakest portion of the population To estimate arithmetically the values of this distribution the following steps were taken Both the integrals of ideal normal and the Pareto distribution were computed Their difference is an estimation of the normal fraction to the entire Pareto distribution according to 3 10 This is equivalent to estimating the high intensity fraction to be filtered under the approximation that the high intensity slope of the normal distribution is infinite and the low intensity slopes of both distributions are equal fR x dx f Po x dx 1 Frac 3 10 normally distributed fraction 5 10 15 Pareto parameter a Figure 3 15 As the high intensity tail is directly related to the a parameter of the Pareto function the fraction of the integral of the normal probability density function to the integral of the entire normal Pareto function can be plotted as a function of parameter a Different plots have been made to show the influence of varying the sigma of P The lines from the red through to the blue represent the normal results for sigma values of 0 5 0 7 0 9 1 1 1 3 times JT respectively For high values of parameter a the normal fraction approaches 1 asymptotically For a filter setting which w
97. inerungsqualit t bewerten Pulverstreuungsbilder sind in dieser Beziehung bisher leer ausgegangen In Kapitel vier werden Funktionen f r universell vergleichbare Detektor unabh ngige R Werte vorgestellt In den letzten Kapiteln werden exemplarisch zwei Beispiele beschrieben die mit den vorgestellten Werkzeugen analysiert wurden Es handelt sich um die schwierigsten Daten die moderne in situ Pulverdiffraktometrie zu bieten hat Hochdruckstreubilder Zwei Substanzen durchlaufen insgesamt f nf Phasen bergange unter erh htem Druck Alle besitzen freie Elektronen paare die gro e Auswirkungen auf die Kristallstruktur unter Normaldruck sowie unter hohem Druck haben Alle Pulverdiffraktogramme wurden mit der eigenen Software und den vorgestellten Methoden reduziert Sie waren von solch hoher Qualit t dass sie sofort volle Rietveldverfeinerung durchgehen konnten Die Ergebnisse waren durchweg berzeugend sogar die der thermischen Auslenkung Gebrauchsanleitungen zu der vom Autor entwickelten Software Powder3D und Powder3D IP sind angeh ngt References 8 References Abdi H 2007 Binomial Distribution Binomial and Sign Tests Encyclopedia of Measurement and Statistics edited by N J Salkind pp 87 89 Thousand Oaks Sage Altomare A Camalli M Cuocci C da Silva I Giacovazzo C Moliterni A G G amp Rizzi R 2005 Space group determination improvements in EXPO2004 Journal of Applied Crystallograph
98. ing Refine gt Le Bail via the menu opens the following window Figure 96 Here you can set up the starting values for a Le Bail refinement Le Bail et al 1988 using Fullprof Should a single pattern format PCR file exist you can load it by pressing the button Import Fullprof All the values displayed in the window are then imported n Tools Plot Help Le Bail Peak Analysis Export cell data Figure 94 Le Bail menu The cell dimensions can alternatively be imported from a Crysfire summary file Pressing Import in the cell frame does this You can select a SUM file and the cell dimensions are displayed in a table Select the desired cell by marking the entry Figure 95 The lengths and angles are imported on clicking OK 115 Two dimensional X ray powder diffraction Si Select the values you wish to import alpha beta gamma 27 17 0000 11 6400 35 221 0 750000 11 2755 11 2857 6 38560 90 0000 100 305 90 0000 z 29 17 0000 11 6200 506 60 O6F0000 11 3758 12 5650 4 13870 30 0000 94 4550 90 0000 f 29 17 0000 11 4500 55 463 0 740000 10 5946 11 2501 6 63480 90 0000 107 679 90 0000 x 17 0000 8 00000 891 009 0 870000 5 89660 8 89660 11 2574 30 0000 90 0000 90 0000 i 1 16 0000 132 000 257 910 0 250000 32 16 0000 16 4500 1025 51 1 00000 11 2997 12 5672 Ye 30 0000 30 0000 30 0000 li F Cancel UK Figure 95 Crysfire import The peak shape descriptors can also be imported if a peak
99. inishes with higher diffraction angle because of its equivalence to the incident angle a non orthogonal detector can give peak widths that broaden at higher diffraction angles 17 Two dimensional X ray powder diffraction 100000 10000 1000 Intensity a u 100 10 Figure 1 6 A comparison of the initial pattern black line used to generate a two dimensional image and the integrated pattern from the image in red The detector was highly tilted leading to a large difference in the incident angle between the low angle peaks which had large parts of their arc at high incident angles and the high angle peaks had low incident angles leading to greater peak widths The reason why the peaks at low diffraction angles are still so much narrower than those at high angles although parts of their ellipses are in a region of low incidence is simple The fraction of their ellipse arc at low incidence angles is relatively small the majority of the arc lies in a region of high incidence and therefore this dominates the signal The difference in peak width becomes quite clear when the peak width at different azimuthal angles is viewed figure 1 7 The high incident angle region in this case lies at an azimuth of 45 The low incidence angles lie diametrically opposite When using focusing optics von Dreele et al 2006 the detector distance to the optics is fixed and the focal spot of the beam well below the PSF of the detector The reso
100. inrichsen R E Dinnebier and M Jansen Two dimensional diffraction Powder Diffraction edited by R E Dinnebier and S Billinge London Royal Society of Chemistry B Hinrichsen R E Dinnebier P Rajiv M Hanfland A Grzechnik and M Jansen Advances in data reduction of high pressure X ray powder diffraction data from two dimensional detectors A case study of Schafarzikite FeSb O Journal of Physics Condensed Matter 18 1021 1037 B Hinrichsen R E Dinnebier M Jansen Powder3D A software tailored for in situ powder diffraction studies CPD Newsletter 32 12 22 R Paneerselvam B Hinrichsen M Joswig R E Dinnebier Detection of ellipses in powder diffraction patterns using Hough Transformation CPD Newsletter 32 27 30 B Hinrichsen F Hergert R E Dinnebier M Jansen R Hock Two dimensional intensity corrections for in situ X ray powder diffraction DGK Proceedings Zeitschrift f r Kristallographie 24 132 R Paneerselvam B Hinrichsen M Joswig R E Dinnebier M Jansen Detection of ellipses in powder diffraction patterns using Hough transformation DGK Proceedings Zeitschrift f r Kristallographie 24 132 B Hinrichsen R E Dinnebierand M Jansen Powder3D towards automatic image plate analysis UCr XX Proceedings Acta Crystallographica A61 C163 B Hinrichsen R E Dinnebier M Jansen Powder3D An easy to use program for data reduction and graphical presentation of large numbers of powder diff
101. ion pattern The intensities have been translated into heights above the base plain A colour gradient from red at low intensities to white at higher ones has been added to improve their visualization Notice the effect of the polarization and absorption on the equatorial intensities A shadow of the primary beam stop is visible at the lower edge 43 Two dimensional X ray powder diffraction Figure 2 9 The calculated two dimensional diffraction pattern The polarization correction leads to the dip in the equatorial intensities There is no scatter in the intensites and obviously no shadowing can be observed ud N i at hp il N Il Sf iia il kin ie Figure 2 10 The two dimensional diffraction residual Correct weighting of unexposed areas leads to a sensible residual The remaining intensity at the poles is caused by the uncorrected sample absorption This image resembles the one dimensional residual seen in standard powder diffraction fitting analyses The remaining noise scales directly to the intensities Calibration 2 3 2 4 Least squares refinement Essentially there is no difference in the least squares refinement strategy between WPPF and WIR The parameters of the detector calibration its PSF and optionally two dimensional polarization and Lorentz correction factors are added to the standard parameter set The main computational challenge is the number of data points which have to be simulated The we
102. iple pattern display An interesting feature is the ability to export an AVI film of the single patterns cycling through the selected sets and the zoomed range To do this press Export gt AVF in the context menu To make this export as versatile as possible it creates a film only of the currently selected data range 20 intensity and selected data sets You will have to select an appropriate compression algorithm for the video In our experience DiVX4 high motion gives the best results This feature is only available on the Windows operating system Please note the installation of the library for this function are described above see Installation 13 3 3 Normalizing Patterns Data collected using image plates generally lacks a beam decay correction This can be partially alleviated by normalization Two or more patterns can be corrected using this function Menu Edit gt Normalize patterns Select the patterns via the list or select patterns belonging to a phase by choosing the appropriate phase Next ensure the method of normalization is correct All patterns that have been selected are superimposed over one another It is possible to zoom by dragging open a zoom box with a left click Select the region you wish to normalize the patterns with by dragging over a range with a right click The range selected in this manner is used to normalize the data By pressing Test you can preview the results of the normalization Press Cancel
103. iscarded 2 3 1 1 Peak recognition and peak refinement As the starting parameters for the calibration are known it is possible to compute the resulting radius of the peak position for the given azimuthal angle a using equation 1 37 2 1 D tan 20 r lt r focus elipse 1 cosa tan tilt tan 20 For a delta equivalent to 0 5 28 around the speculative peak position a peak profile refinement is made using a Gaussian profile function This method enables sub pixel precision in the determination of the ellipse intersection on the radial line The positional statistical uncertainty returned by the covariance matrix of the least squares procedure can be used to weight the individual data points in the final parameter refinement Refined peaks from the same diffraction cone with strong aberrations in intensity or position can be automatically extracted from the data pool The polar coordinates are transformed to Cartesian coordinates for use within the refinement routine 2 1 A F et ea Figure 2 5 Here the intersection points of 90 lines with the diffraction cones of a LaB sample are shown Some points have been automatically removed as their positional errors were outside the 1 sigma limit imposed on the positional error It can be seen that this eliminates the peaks that would have been within the shadow of the primary beam holder As can be expected the intersections with weak cones are slightly more affected by this removal tha
104. ischer amp Pertlik 1975 The open channel structure see table 5 1 for the ambient and low pressure structural parameters and the high polarizability of the cations exhibiting the lone pair make it highly susceptible to pressure induced phase transitions A previous high pressure investigation of the related compound Minium Pb Pb O Dinnebier et al 2003 showed two phase transitions towards two phases of higher density at pressures of p 0 11 GPa 0 3 GPa and p 5 54 GPa 6 6 GPa respectively where the lone pair of the latter phase almost vanished tending towards an s state character While minium also shows several phase transitions upon cooling Gavarri et al 1978 no phase transitions of Schafarzikite on cooling to T 2K have been observed Chater et al 1985 Gonzalo et al 1966 In this work we have investigated the pressure dependence of the crystal structure of Schafarzikite up to a pressure of p 19 8 GPa For this purpose in situ X ray powder diffraction measurements were performed at room temperature and elevated pressures using a diamond anvil cell DAC 5 3 Experimental 5 3 1 Synthesis and X ray diffraction measurements FeSb O has been prepared according to procedures as described in the literature Chater et al 1985 For the X ray powder diffraction experiments the hand ground sample of Schafarzikite was loaded in a membrane driven diamond anvil cell DAC Letoullec et al 1988 using nitro
105. it peaks Function Peak i FC Eta 0 500 ity 0 000 Flot Pseudo Voigt Intensity a a Pearson yl sl 0 002 E Position 0 000 Normalize H L 0 002 a FWHM 0 010 u D Espot Refinement run Mas cycles per run 30 Refine Cancel Plot Export Done Refinement progress Figure 101 Peak analysis window initial display S Analyze peak progression l 15 x Select pattern range and peak position i Zoom f Select ranges Edit peaks Functions Peak f FCJ Eta 0 500000 f Pseudo Voigt a f Pearson Will SL 0 00200000 u Position 10 3451 C I Mormalize Plot Intensity 2851 58 HL 0 00200000 Export FH ja 200504 Refinement run Mas cycles per run 30 Refinement progress en Refine Cancel Flot Export Done Figure 102 Peak analysis window range selected and peaks marked 119 Two dimensional X ray powder diffraction Si Analyze peak progression p 10 x Select patter range and peak position Zoom Multipattern Single Pattern 2000 15 Select ranges Edit peaks 1300 L500 L700 1600 1500 LO L 102 10 3 LO 4 10 5 10 6 Function Peak FCJ Remove Eta ID 0476025 55 107 Flot piper 7 mens or 3 f Pearson Wl sL 0 00200000 T Position 10 3581 m Normalize H L 0 00200000 FWHM 0 186643 fe Espot Refinement run Refinement progress 115 Max cycles per run 30 i Cancel P
106. kes the following form K 1 m l cosY 1 67 Calibration Correction factor K 0 20 40 60 80 Incident angle Figure 1 19 Incident angle correction factor The correction factor of Zaleski et al 1998 has been calculated for an incident angle range from 0 to 90 and for a vertical transmission ratio from T 0 to T 1 The correction is applied by dividing the observed intensities by correction factor K The correction factor remains unity for complete X ray absorption T 1 as well as for incident angles of 0 Here m is a coefficient that parameterizes a detector wavelength combination For example a CCD detector optimized for copper radiation used m 0 1763 One optimized for Molybdenum radiation had m 0 3274 No deduction of this formulation has to date been published The correction implemented in the Fit2D package Hammersley et al 1996 is given by 1 7 Again no formal deduction of this correction has been published K cos W 1 68 A more complex function for imaging plates taking into consideration of the additional absorption of the excitation and emitted light has been proposed Tanaka et al 2005 K X y ZV Kk P kKGI ie a 1 69 ki x y exp as eos Yess A This was reported to provide a better normalization when applied to single crystal data collected for electron density studies of CeB In this equation Ms ML and u represent the linear absorption coefficients of the excitation X ray
107. l to select the interesting range using the film plot display and the zoom function before starting the tool The rendering of a couple of hundred patterns might be slow on older hardware so it is in your own interest to reduce the amount of information The aim was to create a module which does sequential peak profile refinement in a robust manner The tools that you can select on the left are 1 a zoom tool works in very much the same manner as the normal zoom tool 2 select ranges Drag this tool across a peak to select it and have the peaks displayed or found should none have been determined 3 a peak editor Add remove peaks using this tool Once you have selected a range with the range tool you can start refining those peaks The refinement will start with the strongest peak and work its way down to the smallest peak fixing the peak position automatically once the peak intensity drops below a threshold of 301 A maximum of two ranges can be refined together This is equivalent to refining them separately should there be no overlap Should you wish to plot the peak development first select the peaks with the Select peaks Figure 101 tool Now press the plot button The main program window is brought forward and a rudimentary plot of the refined parameters is displayed 118 Si Analyze peak progression 10 x Select patter range and peak position it Zoom Multipattern Single Pattern Select ranges Ed
108. leme mit sich Es existieren effektiv kaum zweidimensionale Korrek turfunktionen f r die Pulverdiffraktometrie Alle gebr uchlichen Lorentz und Polarisationskorrekturen LP sind au erhalb des d nnen quatorialen Bereichs f r den sie entwickelt wurden bedeutungslos Weiterhin m ssen viele geometrische und detektorische Eigenschaften in die Korrekturen einflie en um ein qualitativ hochwertiges Pulverdiffraktogramm aus dem Bild zu extrahieren Diesem Problem nimmt sich das erste Kapitel dieser Arbeit an Es stellt alle n tigen Korrekturfunktionen dar zusammen mit ihrer Basis der geometrischen Beschreibung des experimentellen Aufbaus DieBestimmungallergeometrischenParameterzurh chstm glichenPr zision ist bedeutsam f r die Qualit t des Experiments Wie erreicht man dieses Ziel ohne sich in den Wirren divergierender Verfeinerungen und Analysesoftware zu verlieren Hier werden Methoden der Mustererkennung und das so genannte whole image refinement also die Komplettbildverfeinerung eingesetzt um die beiden Hauptprobleme der Kalibrierung zu l sen Methode eins die Mustererkennung ist eine globale Optimierung gibt vern nftige Startwerte f r die zweite Methode eine lokale Optimierung der Parameter durch whole image refinement Hier wird das gesamte zwei dimensionale Bild basierend auf den Startwerten und den Probeneigenschaften berechnet und vom experimentellen Bild abgezogen Die Differenz wird gewichtet und stelltf rden
109. lled Sl LeBail refinement lEIxI Phase information Selected phase ranges from set number 0 to 0 Ranges Current set for refinement Patter Zero offset 0 01576C refine Import Fullpraf Reset Remove HEL Cell Profile Spacegroup Pham 7 refine cell 11 2244 alpha 30 0000 6 25995 20 0000 amp b 6 25995 beta 90 0000 7 3 60243 gamme 30 0000 Import Refinement information Cancel Refine Figure 117 Cell data for Le Bail fit On the phase tab press import and choose a Crysfire summary file then select the appropriate cell from the table and the values shall be inserted into the cell fields 128 1 Select the values you wish to import alpha beta gamma oo W MUNTE 14 0700 1047 08 1 00000 11 9363 12 7550 7 835930 106 569 34 6490 113 695 i 1 20 0000 13 6600 1196 53 1 13000 11 7274 12 5680 6 64290 103 657 101 374 75 7330 1 2 20 0000 11 3600 282 010 0 270000 6 47640 11 2638 4 01720 94 2700 102 733 59 0730 3 20 0000 11 3200 924 399 0 880000 9 47710 11 3328 8 74800 95 3320 92 6370 64 3320 4 20 0000 10 8400 913 302 0 870000 10 9177 12 3414 745370 103 038 92 1970 109 762 5 20 0000 10 7700 53 911 0 720000 10 0779 11 2517 6 396730 91 0250 104 252 92 1500 Sa a an mann Fre 4 4 a n dada mn Et en tue ea E En aoa nie Cancel OF Figure 118 Import Crysfire summary file i LeBail refinement l O x Phase information Selected phase ranges from set number 0
110. llthe coordinates are given in the cone coordinate system wherein z is the cones axis and x and y describe the plane perpendicular to it The position of the plane is at a distance D from the cones apex along the cones axis x y D tan 20 1 13 In our case the cones axis is synonymous with the primary beam For the conic section to change from a circle to an ellipse the angle between the plane normal and the cones axis has to be greater nought Dx cos tilt J Dy 2 nn D tan 20 1 14 D x s n tilt D x s n tilt This is realized in figure 1 9 and figure 1 10 by rotating the cones axis around the horizontal plane axis this is equal to the x axis For sake of compatibility with the established formulation Hammersley et al 1996 the detector is tilted around the y axis The coordinate system changes to that of the tilted detector and is denoted by x and y figure 1 10 Figure 1 9 The elliptical conic section resulting from a tilting of the detector around a horizontal axis This results in an ellipse which is mirror symmetrical along the central vertical axis The equation 1 14 then simplifies to 1 15 x cos tilt y D x sin tilt tan 20 1 15 20 The experimental setup 5 x s n tilt x sin tilt N a x costzilt b y Figure 1 10 Constructions after Kumar Kumar 2006 to deduce the tilted x and y values in terms of the orthonormal x an
111. logy 22 236 241 Fisker R Poulsen H F Schou J Carstensen J M amp Garbe S 1998 Use of Image Processing Tools for Texture Analysis of High Energy X ray Synchrotron Data Journal of Applied Crystallography 31 647 653 Fitzgibbon A Pilu M amp Fisher R B 1999 Direct Least Square Fitting of Ellipses IEEE Transactions on Pattern Analysis and Machine Intelligence 21 476 480 Forman R A Piermarini G J Barnett J D amp Block S 1972 Pressure Measurement Made by the Utilization of Ruby Sharp Line Luminescence Science 176 284 285 Gavarri J R Weigel D amp Hewat A W 1978 Oxydes de plomb IV E volution structurale de l oxyde Pb O entre 240 et 5 K et mecanisme de la transition Journal of Solid State Chemistry 23 327 339 Gillespie R J 1967 Electron Pair Repulsions and Molecular Shape Angewandte Chemie 79 885 896 Gillespie R J amp Robinson E A 1996 Electron Domains and the VSEPR Model of Molecular Geometry Angewandte Chemie 108 539 560 Gonzalo J A Cox D E amp Shirane G 1966 The magnetic structure of FeSb O Phys Rev 147 415 418 Gruner S M Milch J R amp Reynolds G T 1978 IEEE Trans Nucl Sci 25 562 565 Hammersley A P Svensson S O Hanfland M Fitch A N amp Hausermann D 1996 Two dimensional detector software From real detector to idealised image or two theta scan High Pressure Research 14 235 248 Hanfland
112. lot Export Dione Figure 103 Peak analysis window peak refinement in progress o File Edit Display Tools Plot Window Help S tal v Peak Progression PEAKINT Set No DOF 21 Iteralions 2 Chi2 0 043625579 Figure 104 Peak analysis a rudimentary initial plot of peak development 13 1 120 13 2 Graphics 13 2 1 2D film plot This module simulates a Guinier film Figure 105 Two sliding bars enable you set the brightness and contrast of the plot Colour inversion and square root scaling can be set A tick box Interpolate allows you to do a bicubic interpolation between your patterns This smoothes plots of small data sets Of course the data range displayed can be set via the 20 and Data range fields Needless to say background corrected data has a far higher contrast The zoom and pan tools are available for this display Sl Powder 3D P fal x Fie Edit Display Tools Plot Window Help E gt lel Data set range S a pa 0 40 im gt Brightness 4 00 Pe w Contrast IV Display square root IV Invert display I Interpolate Figure 105 2D film plot 13 2 2 121 Two dimensional X ray powder diffraction 13 2 2 3D line waterfall plot Sl Powder 3D File Edit Display Tools Plot Window Help tal Orientation Lights Rotation I im El x Axis Y Axis Top Slant C Surface Line
113. lta function stimulus Ideally the point spread is also a delta function Experimentally this is seldom the case as detector characteristics generally give the signal a Gaussian spread It is the point spread function which is the main cause of the limited resolution in powder diffraction experiments The size of the detector is an important factor determining the size of the accessible reciprocal or q space Larger detectors offer a greater area and thus a greater q space which can be imaged in one exposure Greater size also opens the possibility of moving the detector further from the sample to improve resolution Speed is of ultimate importance when acquiring data at a synchrotron beam line The readout time should be minimal to ensure a high time resolution for in situ experiments and a most efficient use of the costly synchrotron rays The dynamic range of the detector limits the intensity differences that are recordable on one image The higher the dynamic range the better one is able to characterize signals having a strong contrast Two dimensional X ray powder diffraction X ray interaction Conversion to Charge readout X rays Scintillator Photo conductor a Selenium silicon Photodiode a Silicon Excitation laser amp photo multiplier TFT Array TFT Array CCD D ge amp Sg o amp gt lt Phosphor Seintillator Fibre optics Figure 1 1 Detector types Five different de
114. lution is governed solely by the PSF of the detector To decrease the resulting line width the only solution is to increase the distance between the sample and the detector as tilting the detector would move it out of the focal point 40000 Azimuth range 45 75 30000 ne oe 135 165 195 225 20000 Intensity a u 10000 U pea 19 0 Pr 20 0 Figure 1 7 The effect of extreme detector tilt on the line width at different azimuthal angles A point spread of 300um and a detector tilt of 45 has been the basis of this computation 18 The experimental setup 1 2 2 Diffraction angle transformation The fundamental function relating the non orthogonality of the detector to the primary beam and the sample to detector distance into diffraction angles is given by 1 12 This equation can easily be deduced from two rotations of the plane out of its position orthogonal to the cones axis The first rotation is performed around the x axis As can be seen in figure 1 8 it causes the conic section to become elliptical The axis of the cone now intersects the plane at a focal point of the ellipse What was the radius in the circular conic section now has become the semi latus rectum figure 1 13 A second rotation is performed around the plane normal centred on the focal point which is the intersection of the cones axis The effect of this rotation is shown in figure 3 This transformation provides a general formulation of a c
115. lves of the undulations of background intensity visible in Figure 26 A value of 1 smoothing box and 3 iterations worked well for this data set It is convenient to decide on with part of the pattern array you wish to use by previewing it in the 3D surface mode in Powder3D Once you are happy with your selection a click on Presentation graphics copies the data over to the iSurface programme It is always possible to further reduce the displayed data in the iTools programme suite should you find it necessary A right click on the data opens a context menu which lets you select the properties of the view Figure 110 ioi xi Fie Edit Insert Operations Window Help Deals lele fro x oela Al ololele Surface target Z Figure 109 The primary iTools display Open the properties window to the full extent by pressing an unobtrusive left arrow in the top left corner of the window Figure 110 left J IDL iTool isualization Browser lol x 3 225 184 0 Use surface color for bottom Tue Imag lor Bottom col mE Minimum val lue 0 Maximum value 804043 4 Surface style Filled Fill shading Flat Draw method Quads Line style Line thickr Rer ove hidder It JISE L Show skirt False hd v Surface Figure 110 The iTools property window folded and unfolded In the left side of Visualisation Browser Figure 110 You see the various elements that make up the image Once an element is highl
116. ly IDLtoAVI dll IDLtoAVI dlm p3d dlm and p3d dll should be in the same directory as Powder3D The functions only work on the Windows operating system Other operating systems are not supported UNC unmapped network paths will cause the program to fail ij Powder 3D l0lx File Edit Display Tools Plot Window Help alu Figure 12 Welcome screen 13 2 2 The sample data set The data set that is the basis for the graphics displayed in the manual is available for download from http www fkf mpg de xray The data stems from an experiment carried out at the X7B beamline of the NSLS Dinnebier et al 200 et al 2005 The wavelength was 0 9224 A The initial substance was 5 Rb C 0 Seven phases can be identified and are tabulated below Table 1 Phases identified in the sample data set The temperature ramp was set to 298 gt 838 gt 298 K The sample was heated at a rate of 2 8 Kmin and cooled at a rate of 4 8 Kmin Considering the exposure and development time this leads to a heating rate of 4 2 Kframe and a cooling rate of 7 62 Kframe 99 Two dimensional X ray powder diffraction 100 Rb C O Il Rb CO N co oO N gt oO Volume A N oO oO D on Lattice parameters A ee ee c 2 l 8 me they wibeeaicceneceoeroceneces Pnma P jmme Pnma P2 c 300 400 500 600 700 8007 N 500 700 600 500 400 300 Temperature K Figure 13 Powder
117. ment is displayed Fitted intensity distributions of the normal the Pareto and proposed normal Pareto distribution are overlaid in colour The best fit is given by the normal Pareto distribution 0 forx lt b Porre X ab 3 8 ici PEER 3 8 X The two parameters of the Pareto distribution 3 8 describe the smallest possible value of x which is b and a form parameter a The Pareto probability density function is sharp containing a singularity at x b This is not compatible with the experimental data in which mainly the detector causes a Gaussian blur of the Pareto distribution see figure 3 12 To make allowance for this effect the Pareto distribution is folded with a normal distribution function leading to what will be called a normal Pareto function 3 9 This convolution gives the best Chi of any distribution model tested Table 3 4 Pyp x T ma x amp as x 3 9 This formalism for the intensity distribution is most attractive as when the parameter a in equation 3 8 tends toward infinity the Pareto distribution tends toward a Dirac delta function thereby reducing the convoluted function to the normal distribution of perfectly monodisperse grains The influence of the parameter a especially on the tail of the function can be seen in figure 3 13 Here one can quite clearly see that infinity is effectively reached at the modest value of a 20 Table 3 4 The table displays the chi squared statistical test resul
118. mi minor axis can be described using the scattering angle the tilt and the detector to sample distance as given in equation 1 32 Figure 1 13 Common ellipse parameters The centre is denoted by C one focal point by F the semi major axis by a the semi minor axis by b and the semi latus rectum by 1 1 27 1 28 1 29 1 30 1 31 1 2 4 0 1 The experimental setup D Jjcos tilt Vsin 20 vtan20 _ p cos 2tilt cos 49 Now a and c can be inserted into the well known identity e c a to denote the eccentricity in calibration parameters 1 32 tan tilt tan20 e 1 33 Detector coordinate transformations We will now attempt to deduce the detector coordinates from the calibration values and the reflection parameters Some well known elliptical identities 1 34 and 1 35 shall be used in these calculations 1 34 c ya b 1 35 The ellipse radius as measured from the focus can be described in terms of eccentricity semi major axis and the angle of the azimuth 1 36 da 1 36 r a focus ellipse 1 e cos a Inserting equations 1 34 and 1 35 into the equation 1 36 leads to D tan 20 T r m 00 7 focus elipse 1 cosa tan tilt tan 20 g The Cartesian coordinates take the following values Ya T P raazampe sin q rot 1 38 Xq ATE E cos a rot 1 39 Here x and y are the x and y positions relative to the focus The ellipse has been made non parallel to the axes by subtracting the ro
119. ms are tin the yellow tetrahedra represent the sulphate anions Bonds have been drawn where the Sn O distance is less than 2 6 A 5000 5 4000 gt T Cc g 3000 0 II eee em me et ER 1 ER U EEE U EN HL OT TOT a a 5 10 15 20 26 I Figure 6 9 The Rietveld refinement plot of the first triclinic phase phase Ill showing the experimental data as open circles the calculated pattern as the upper solid line and the difference pattern as the lower solid line The positions of the reflections are marked by vertical lines The scale of the difference plot is identical to the upper plot 83 Two dimensional X ray powder diffraction Table 6 2 A B rnighausen tree showing the relations between the phases of the ambient and high pressure modifications of tin sulphate All of the transitions are translationengleich without an origin shift making the comparison of atomic positions straightforward oT mie 0 ERT 0 755 2 0 707 1 0 HET 0 714 2 0 650 1 0 HID 0 732 4 0 630 2 0 212 1 0 245 1 0 207 1 0 som 0 259 2 0 194 1 0 5 0 303 3 0 211 2 S 4c O1 4c 03 8d 1 0 889 1 0 004 2 0 213 2 0 st 0 055 3 3 re 0 402 3 O 4e 4e 1 1 0 051 3 0 449 3 0 808 2 0 692 2 0 916 2 0 655 3 0 345 3 0 824 3 0 176 3 0 027 3 0 409 3 0 909 3 0 468 2 0 968 2 0 247 5 0 sa 0 771 5 0 316 1 0 581 1 0 229 5 0 816 2 0 74
120. muth Figure 3 11 The intensity is shown as a function of the azimuth The displayed intensity originates from a typical high pressure experiment in this case of SnSO at a pressure of P 20GPa The filtered pixels have been coloured green the high intensities blue the low intensities The effect of the filters on the intensity distribution can also be seen in the histogram in figure 3 12 The high number of high intensity pixels is caused by the lack of sample rotation the highly collimated and highly parallel beam used in such experiments as well as from large grain size differences within the sample Strong peaks resulting from larger grains lie within a ring of moderate intensities generated by small crystallites In most cases the number of high intensity pixels tends to be a couple of orders of magnitude lower than the number of low intensity pixels On account of the small number of intense spots they inevitably fail to ensure a statistical distribution Due to their intensity which often lies orders of magnitude above the low intensity pixels they have a great effect on the integrated pattern falsifying the intensities considerably This effect is not alleviated by a mere integration Incorrectly filtered images can result in peaks that cannot be fitted by conventional peak profile models as well as error bars that are completely meaningless To obtain an acceptably resolved histogram from an integration bin of such highly dis
121. n 11 1992 Abitur Deutsche Internationale Schule Kapstadt S dafrika European Powder Diffraction Conference 10 Geneva Two Dimensional Diffraction Does Delving Deeper Deliver DGK Koln Digitale Mustererkennung angewandt auf zweidimensionale Pulverdiffraktogramme Workshop Watching the Action Powder Diffraction at non ambient conditions Stuttgart Powder3D Software tailored for in situ studies Departamento de Fisica de la Materia Condensada Bilbao Banning tedium from in situ powder diffraction Modern solutions to a modern challenge International Union of Crystallographers Florenz Powder3D freely available software a program for multi pattern data reduction and graphical presentation 157 Two dimensional X ray powder diffraction Publizierte Artikel und Konferenzbeitr ge 2007 2006 2005 2004 158 B Salameh A Nothardt E Balthes W Schmidt D Schweitzer J Strempfer B Hinrichsen M Jansen and D K Maude Electronic properties of the organic metals Theta BEDT TTF I and ThetaT BEDT TTF I Physical Revue B 75 054509 B Hinrichsen R E Dinnebier and M Jansen Intensity distributions in 2D powder diffraction and their application to filtering Journal of Applied Crystallography submitted B Hinrichsen R E Dinnebier and M Jansen Two dimensional powder diffraction EPDIC X Proceedings Zeitschrift fur Kristallographie submitted B H
122. n are the intersections with intense cones 2 3 1 2 Least squares weighting The final parameter refinement is in the best case a straightforward matter Some hundred data points have been collected by the peak recognition and refinement routine with the majority of them being of very high quality The weighting scheme is based initially on the statistical uncertainty of the position of the ellipse markers This is a good choice aiding the stability and reliability of the refinement An additional weighting factor is added to take into consideration that the inner ellipses cover a smaller area than the outer ellipses The large arcs of the outer ellipses contain more highly resolved calibration information than the inner rings the resolution scales proportionally to the arc area of the ellipse The weighting for an orthogonal detector can take the simple form 2 2 describing the area of an idealized circle Wear x sr sin 20 2 3 1 3 Errors and pitfalls One source of systematic errors stems from the difference between the estimated fitted peak positions on the ellipse and the true intersection of the radial line with the central arc of the ellipse This error is effectively identical to the physical effect of axial divergence and equatorial aberration which are experienced by equatorial point detectors Numerous methods dealing with the profile ranging from an asymmetrical Pseudo Voigt function Finger et al 1994 to the fundamental p
123. nly very few striking artefacts need to be filtered This is the case in some high temperature diffraction images in which only a few sapphire peaks stemming from an enveloping capillary need to be removed Here the upper two text fields are now the convergence criteria This is amore computationally expensive filter which works in the following fashion For each step width bin the filter fraction is enlarged incrementally by the value entered in the last field The variance of the filtered population is compared to the previous value and if the change is less than the convergence criteria the filtering of that bin is complete The advantage of this filtering method is that if the values have been chosen well it is less aggressive than fixed fractile filtering which removes a lot of the good signal when filtering only a few strong outliers 13 6 4 Mask growth fete Dear da Erie Pee Dba Cael ake fee P aia A Aroba Cer AJ 4 JBBE awn F dj Barail Trnju ir t E HE pe i QM Ada eA th A se FP EII D Op pe af BE ey jaa Fim 1 DZ F A emk TOTS Fjo CO ealm acs IE rr SS e H mat b nenn Pets Mes ee eis Le ee ee im Figure 132 The mask dialog and the effect of mask dilatation on the image Edit gt Mask Mask growth is a useful method to ensure a good coverage of outlier peaks or instrument shadow figure 132 Generally the peak spread of the detector ensures that outlier pe
124. nsional X ray powder diffraction 66 4 2 Table 4 1 Image reliability values for different images before and after filtering are listed The first two lines compare high quality calibration images with good and bad calibrations with one another The bad calibration was created by merely changing the sign of the tilt value The experimental datasets are relatively standard high temperature sapphire tube and high pressure DAC images The high temperature image has peaks from the enveloping sapphire capillary and the high pressure image has an intrinsically high intensity distribution 4 5 where is the background intensity of the kth pixel Comparison of reliability values originating from different data Image reliability values are not only a matter of interest for the quality of the diffraction pattern but also are very sensitive to the calibration quality In table 4 1 various images from different in situ experiments have been analyzed before and after filtering The image reliability values not only prove useful in the quality estimation but also show prove the great improvements filters offer in the image reduction process Rim filter applied 100 Rim no filter 100 good calibration 0 414 25 9 Schafarzikite FeSb O at high pressure 5 Schafarzikite FeSb O at high pressure 5 1 5 2 Abstract Methods have been developed to facilitate the data analysis of multiple two dimensional powder diffraction
125. nting the apex Four of these pairs point inward to the channel resulting in a large unoccupied space Thus the channels are by trigonal pyramids of SbO the closest Sb Sb distance being 3 53 A In the monoclinic phase II all atoms are on the general 4e position table 5 4 The special positions and the high tetragonal symmetry are recovered at higher pressures The mechanism of the first phase transition can be interpreted as an initial continuous shearing toward the monoclinic symmetry The shearing presents itself in the growing monoclinic angle corresponding to the angle of the c axis to the ab plane in the tetragonal phases and Ill The distortion of the iron octahedra increases with pressure All changes are however continuous characteristic of a second order phase transition The second transition is different Sharp discontinuities of the lattice constants speak for a first order phase transition The iron octahedra remain distorted however the orientation has changed substantially to the phases and Il 74 Tin sulphate at high pressures Fe Sb O Figure 5 6 Low pressure tetragonal FeSb O phase I viewed down the c axis green spheres represent Fe grey spheres Sb The change in the antimony environment is substantial for the second phase transition Here one of the symmetry independent antimony atoms takes on a fourfold coordination in contrast to the dominant threefold coordination for the remaining phases It
126. ntrol board is Lights Figure 107 Four light sources can be controlled One ambient and three positional light sources can be manipulated 122 Sl Powder 3D File Edit Display Tools Plot Window Help su loxi Orientation Lights Ambient Positional Positional2 Positional3 0 000000 all gt x Position oso Ki 2 Y Position 1 00000 Ei Jr Z Position on a I u Intensity i 143 10 12 i 7 na 29 DOF 21 Iterations 2 Chi2 0 043625579 Figure 107 3D surface plot showing the lights control panel 13 2 4 iTools presentation graphics Version 2 0 For the very ambitious there is the extremely powerful data visualisation and manipulation software iTools that is called by pressing the button Presentation Graphics Figure 108 iTools has been developed by RSI Inc and is a completely independent of Powder3D Powder3D passes on the data to iTools and is free for user interaction again pelala sliseag SR Loli Alnlelsietel Cech aed Feren be Br ind Ze Fate LF Figure 108 iTools The iTools do represent an excellent set of programmes and contain complex architecture For this reason 123 Two dimensional X ray powder diffraction shall extend the tutorial to cover the creation of an image like that in Figure 26 First we shall have to subtract the background from all the data sets so that we rid ourse
127. oints Fitzgibbon et al 1999 and the extracted calibration parameters are used to provide the initial estimation Hammersley et al 1996 An automatic method has been proposed by Cervellino Cervellino et al 2006 They have presented a method to reduce 2D diffraction data providing both a mechanism to calibrate as well as to integrate such data sets Another approach is to use traditional pattern recognition methods to estimate the initial parameters Figure 2 1 Three calibration images of varying quality are shown The left image is a good example of a perfect calibration image utilizing a fine grained rotating sample of LaB The central image is one of a stationary sample of CeO Here shadowing causes a slight deviation in the background intensity and the lack of rotation leads to a non normal intensity distribution within the rings However the rings are continuous and intense and the detector can still be well calibrated using this image The right image is again one of a stationary sample this time of Si The coarseness of the sample results in an extremely discontinuous intensity distribution along the rings In addition there is a strong signal at low angles from the capton foil holding the primary beam stop These factors results in a less stable and less reliable calibration 2 2 1 The Cervellino method The calibration routine presented requires the sample to detector distance and wavelength to be well defined using alternative
128. ometry Background vr Subtract background image Figure 137 The background can be computation can be configured and started using this dialog Edit gt Corrections A median algorithm is used for the background determination figure 137 The structure size is the diameter of the circle which is sampled to calculate the median The background pixel in the centre of this circle then takes the value of the median It is clear that the larger this value is the higher the computational cost of the background calculation is The reduction factor is a simple method to keep this cost low The image is initially reduced by this factor then the background is determined the background is then scaled up to the original image size again 146 13 7 5 Display properties Sil Display Properties loj x Image depth Apply Comections Intensity histogram san 1000 1500 200G Mininum 7 Maximum 2000 Apply Cancel LIE Figure 138 The display dialog for setting the optimal display range Edit gt Display To aid in selecting an appropriate intensity range for the display of powder diffraction images an interactive histogram is made available showing the currently selected intensity distribution By pressing the Apply button the user can see the effect on the image fj Display Properties jol x Image depth Apply Corrections iW Intensity correction on i Background correction on Figur
129. onal detector in X ray diffraction was conventional film It remained for decades the detector of choice for both single crystal as well as powder diffraction experiments In the field of two dimensional detection it has been surpassed initially by image plates and later by CCD cameras Today virtually no film is in use with perhaps the exception of Polaroid used for single crystal images To be able to compare various detectors with one another and to select the most appropriate detector for a specific experiment certain key technical qualities are important These are in general the detective quantum efficiency the spatial response characteristics the size soeed and dynamic range Westbrook 1999 The detective quantum efficiency DQE Gruner et al 1978 is a measure of the signal to noise degradation caused by the instrument It is defined in equation 1 1 oe O ou l a ony DOE in 2 in O I and o represent the input and output 7 intensities and the input o and output o standard deviations of the signal intensities N is the number of incident X ray photons and R is the relative variance of the output signal A detector with a DQE of 50 has to count twice the time a detector with a DQE of 100 has to count to record a signal of equal variance The spatial response characteristics Ponchut 2006 are normally characterized by the point spread function PSF the detectors signal to a de
130. onic section using experimentally accessible parameters D sin silt xcos rot y sin rot The parameters x y tilt rotation and D are depicted in figure 1 8 to figure 1 11 A deduction of this equation has been given by Hammersley et al 1996 and is described with the aid of figure 1 8 to figure 1 11 Kumar 2006 gave a good overview of the transformations involved in calculating the conic section He however chose to use three angles to describe the detector orientation relative to the scattering cone As any detector orientation can be described by two angles alone this is the transformation we chose to use Figure 1 8 A circular conic section resulting from an orthogonal detector to primary X ray beam setting On the left is a view perpendicular to the detector on the right is a side view showing the primary beam entering from the right The primary and diffracted beams from the sample S intersect the detector on the detector plane The primary beam intersects the detector at the centre of the circle For clarity only one diffraction cone has been drawn The distance from the sample to the detector along the primary beam is given by D The detector coordinate system is denoted in x and y relative to the beam centre 19 Two dimensional X ray powder diffraction In figure 1 8 a conic section normal to the cones axis is displayed The diffraction angle which is half the cones opening angle is given by equation 1 13 A
131. ot be processed This causes a possible substantial loss in resolution should no other reliable method be available to determine the exact detector position 45 Two dimensional X ray powder diffraction In order to ascertain the value of the method it has been decided against using real data but rather using the methods described to create datasets computationally This has the distinct advantage in that the exact calibration parameters are known enabling a more simple comparison The results can be seen in table 1 Here the Chi values of the refined calibration parameters compared to the true values are given It can be seen that whole image refinement is capable of determining the calibration parameters to at least one order of magnitude more exactly than the traditional methods Table 2 1 Calibration parameters extracted form a synthetic image of a perfectly aligned detector tilt 0 Calibration has been performed using standard methods in Fit2D and Powder3D as well as using whole image refinement WIR Software method 72 Kopie eal Values 150 OO it2D traditional 150 owder3D traditional 150 0007 owder3D WIR 10 10 binning 149 9971 owder3D WIR 2 2 binning 150 0001 owder3D WIR no binning 150 avelength A eal Values it2D traditional 000053 owder3D traditional 100 00384 ____ 0 99998171 owder3D WIR 10 10 binning 99 999813 _ a 999989 owder3D WIR 2 2 binning 99 999886 _ _ 0 999999 owder3D WIR no binning 99 99
132. otation degrees 0 01 606980 Find Center and Rotation Save Load Advanced lt lt Back Display graphics Update Cancel Done Figure 123 The calibration wizard Setting the centre and rotation These can be found automatically in most images The value of the detector tilt is then given in the following dialog This is the tilt of the detector out of the ideal orthogonal setting Normally this value is rather small Si Calibration O x Sample Pixel Center and Rotation Eccentricity Least square fit Superfine Initial Tilt value Tilt degrees 0 00000000 Find Tilt Status Save Load Advanced lt lt Back Display graphics Update Cancel Done Figure 124 The calibration wizard Setting the detector tilt The last step is to refine the starting values using the intersection of radial lines with the diffraction ellipses These are refined automatically using the values determined so far Once all the intersections have been computed they are used to refine the calibration parameters 138 gi Calibration loj x Sample Pixel Center and Fiotation Eccentricity Least square fit Superfine Select the parameteres for Least Square Refinement D mm 142 00000 Iv Rotation degrees o n 606980 m wi e Wavelength A o 52000000 g Tilt degrees 0 000000 M Refine Select 7 Status Save Load Advanced lt lt Back
133. ould reduce the Pareto distribution to a roughly normal distribution 1 normal fraction of the highest intensities should be removed Frac from equation 3 10 is then the high intensity filtering fraction to be set In the present case it leads to a high intensity filter fraction of 0 43 As the histogram already has been filtered by 5 of the high intensities these have to be added This leads us to a high intensity filter setting of 48 The effect on the arithmetic statistical values can be seen in table 3 5 The arithmetic standard deviation and the mean correspond excellently to the model values In general the fraction to be filtered is not merely dependent on the parameter a of the Pareto distribution but also on the width of the normal distribution P with which it is convolved normal The effect of both parameters is shown in figure 3 15 It is quite surprising that the removal of such a substantial amount of the 61 Two dimensional X ray powder diffraction data leads to such a dramatic improvement in the least squares sense of the data quality This underlines once more the absolute necessity of large 2D detectors for the success of these types of experiments Table 3 5 The table displays the fitted mean of the normal Pareto distribution and the arithmetic mean from the filtered data and the associated errors The values are virtually identical within error 4014 103 4244 124 Once applied to the entire image figure 3
134. ows the greatest deviation It breaks the orthorhombic symmetry at the third pattern leading to a monoclinic phase Following the p 5 GPa mark only a triclinic cell can successfully describe the cell Another phase transition is visible at ca p 15 GPa Pnma P2 la P 1 P 1 m 8 4 7 6 7 2 c 6 8 6 6 6 4 5 4 5 2 5 1 0 5 10 15 20 25 Pressure GPa Figure 6 3 Three plots showing the progression of the unit cell axes over the measured pressuare range The black open circles are those values refined from data measured during compression the red open circles originate from data measured during decompression 18 Tin sulphate at high pressures 340 Pnma P2 la P 1 P 1 K 48 1 320 oo So So Ko 56 2 Volume 280 K 51 13 260 0 5 10 15 20 Pressure GPa Figure 6 4 The volume is shown as a function of pressure The solid lines are those of a Vinet equation of state function fitted to the data points Above the lines the bulk modulus K is given The open red circles represent data measured during decompression The unit cell axes and the volume show a marked phase transition on decompression The decompression values are designated by open red circles in figure 6 3 and figure 6 4 Taking into account the hysteresis effect well known from temperature dependent experiments this can be taken to imply a phase transition on compression at higher pressures When looking at the
135. p of P4 mbc The second transformation from P2 c to P4 m would require two further space groups figure 5 5 Table 5 4 Wyckoff splitting for the phase transitions P 4 m b c gt P2 c gt P4 m Xa JO 1751 N yb 0164 Ben SOC EEE FRE FB 0 158 2 EEE VER ES 4j Site symm _ 2 m m __ m m J1 xla JO 0 164 2 0 328 2 0 681 5 0 113 8 0 483 8 ze 0 222 5 O JO 220 8 oO JJO __ i D i N D Ol IO D o O Ny IR D Co ANININ oO D O O N Sad ad Dad oloolw ololol o er Je ololol o oo er I Sl le d D CO b b oo K oO SIND NCS oO DJO K co O 2181 oO oololo x OJAN WINI oO NIS N AS ar o OJ lo on O O oO OK o IN Of IOL IN 00 N 3 3 3 The structure of FeSb O figure 5 6 is dominated by infinite chains of edge linked distorted Fe O octahedra The chains project down the c axis and lie centred on the a and b axes akin to the orientation of the TiO octahedra in the rutile structure All Sb ions are located in the planes spanned by the shared edges of neighbouring octahedral chains Their polyhedra link the FeO l chains Sb is coordinated by three O atoms two representing the apex of neighbouring FeO octahedra from one chain and one equatorial oxygen from a neighbouring chain This results in a slightly irregular SbO pyramid with oxygen forming the base and the lone pair electrons represe
136. persed intensities an initial fractile filter of the upper 5 was required figure 3 11 The resulting histogram can be seen in figure 3 12 The distribution of the intensities is markedly different to those distributions studied before Fitting a normal distribution which has been so successful describing normal and HT intensity distributions to the HP data set leads to a most unconvincing result the magenta line in figure 3 12 and the Chi value in table 3 4 The intensity distribution is exclusively a sample characteristic no sample environment contamination affects them As the distribution has many low intensity and few high pixels the power law distributions seem the appropriate descriptive choice Of these the most promising is the Pareto distribution Pareto 1896 equation 3 8 which was initially devised to model the wealth distribution among individuals of a society This has become generally known as the 80 20 rule Eighty percent of the wealth is owned by twenty percent of the population Wide applications of this distribution in the fields of biology geology and physics Newman 2005 have been found to date 58 Intensity distributions and their application to filtering Filtered Histogram Normal distribution Pareto distribution Normal Pareto distribution Density pixels per bin 4000 Intensity a u Figure 3 12 The intensity distribution of a filtered bin from a high pressure experi
137. pproximate centre by using the intensity patterns of vertical and horizontal grids drawn on the image All pixel intensities along a grid are copied to an array figure 2 2 The mirror plane of this distribution is detected by finding the absolute difference of the mirrored intensities This absolute difference is calculated for each possible mirror plane position and copied to another array figure 2 3 The lowest point in this difference plot is the position of maximum peak overlap and represents the approximate mirror plane Each grid line results in one such point Two lines one for the points from the vertical grids and the other from the horizontal grid are fitted using the robust least absolute deviation Press et al 1992 method The approximate centre of the ellipse is the point of intersection This algorithm for centre determination is both robust against the outliers in the image and against the position of the ellipses with respect to the image centre This method requires the centre of the ellipses to be on the image but not necessarily in or even close to the middle The further parameters of rotation and tilt are found using the Hough transform as has been described by Rajiv and by Fisker Fisker et al 1998 Rajiv et al 2007 The approximate distance and wavelength have to be entered manually as to date no viable method has been found to estimate these parameters reliably 8x 10 6x10 Difference 4x10
138. r and Fiotation Eccentricity Least square fit Superfine Select Sample LaBE Wavelength A 0 32000000 Reset Selection Enter d Spacings Sample to detector distance mm 142 000 2 93938 al J 2 40000 2 07046 1 85303 1 69705 aj Save Load Advanced 4 l Display graphics Update Mek gt gt Cancel Done Figure 121 The calibration wizard Setting d spacings wavelength and detector distance Press Next to enter the effective pixel size This value is sometimes contained in the file header and if it was read out it will be given here SA Calibration lEIx Sample Pixel Center and Rotation Eccentricity Least square fit Superfine Pixel Dimensions Pixel Length microns 150 00000 Pixel Height microns 150 00000 Save Load Advanced lt lt Back Nert gt gt Cancel Done Display graphics Update Figure 122 The calibration wizard Setting the effective pixel size In the next dialog you can enter the intersection point of the primary beam with the detector An automated function can help locate the centre precisely should it be on the image The rotation will then be set to a sensible starting value 137 Two dimensional X ray powder diffraction lo x Sample Fisel Center and Rotation Eccentricity Least square fit Superfine Center and Rotation Center Pixels fi 162 3567 Y Center Pixels fi 143 6357 R
139. r footprint on the image plate at higher incident angles we should therefore expect higher FWHM of the diffracted beams at higher incident angles Experimentally we find an inverted relation How can this seemingly aberrant behaviour be explained The answer lies in the PSF 13 Two dimensional X ray powder diffraction of the detector Detectors do not resolve differences in the half width of the incident beam if they lie well below the point spread of the detector The increasing footprint of the incident beam is overshadowed by the detectors point spread thus leading to no discernable effect So it is no surprise that with changing incident angle and sample distance the number of points pixels across the peak is not changed Norby 1997 The reduction in the FWHM of the diffracted X rays at higher incident angles is more intrinsically connected to the angular resolution per pixel figure 1 2 X ray ae 20 Diffractogram re 20 Figure 1 2 The effect of the incident angle on the sharpness of the final angular projection The point spread of the detector does not change The difference in the angular resolution between the perpendicular and the tilted detector is the cause of the sharper peak Taking into account both the effect of the incident angle and the distance of the detector from the sample the resolution of an experimental set up can be calculated as the half width FWHM of the diffracted beam by its diffrac
140. raction of a second Most probably this type of detector will establish itself in the field of two dimensional diffraction 1 2 Diffraction geometry Early powder diffraction experiments relied mostly on the Debye Scherrer experiment to record a diffractogram A broad film strip set into a cylindrical chamber produced the first known two dimensional powder diffraction data In contrast to modern methods the thin equatorial strip was the only part of interest and intensities merely optically and qualitatively analysed This changed drastically with the use of electronic scintillation counters Intensities were no longer a matter of quality but quantity Inevitably the introduction of intensity correction functions long known to the single crystal metier i e Lorentz and polarization corrections see page 22 made their way into the field of powder diffraction Continuous detector development brought about the next revolution in the field of powder diffraction Large area detectors made their debut in powder diffraction at synchrotron beam lines in the beginning of the nineties having first been used in the field of single crystal diffraction First experiments only utilized thin equatorial strips Norby 1997 of the image but with the introduction of freely available software Hammersley et al 1996 the integration of the entire image to a standard one dimensional powder diffraction pattern became commonplace The experimental setup The te
141. raction patterns EPDIC IX Proceedings Zeitschrift fur Kristallographie 23 Sprachen Englisch Deutsch Afrikaans Niederl ndisch Muttersprache Muttersprache sehr gut flie end 159
142. rameters of ellipses namely semi major and semi minor axes and eccentricity the following chapter will deduce all the necessary transformations between the crystallographic and standard system Further transformations are needed to calculate the exact Cartesian coordinates of a reflection on the detector This corresponds to the determination of x y 29 a D tilt rot X Y This information is important for calculating and plotting theoretical ellipse positions The semi latus rectum is independent of the tilt and can be given in terms of the scattering angle and the sample to detector distance as in equation 1 24 Dtan 20 1 1 24 From figure 1 12 and the sine rule we can deduce the following relation sin 20 _ sin 90 20 tilt 1 25 a c D l This can be reformed to c a Dsec tilt 20 sin 20 1 26 23 Two dimensional X ray powder diffraction 24 However we also know from figure 1 12 and the sine rule that the following relation holds s n 20 u sin 90 20 tilt a c D This can be reformed to c a Dsec tilt 20 sin 20 Setting equations 1 26 and 1 28 equal and solving for a leads to the following formulation D cos tilt sin4 __ cos 2tilt cos 40 The same method can be used to find an expression of c in alignment variables This leads to the very similar formulation 2D sin tilt sin 20 u cos 2tilt cos 40 al b Because of the well known identity 1 31 the se
143. refinement has been done for the pattern Otherwise enter your desired starting values Figure 97 The background correction is integrated into the refinement if a background was defined using the background function The space group should be entered and then you can start the refinement by pressing Refine If the refinement is successful some statistics of the refinement are displayed at the bottom of the window The button Reset sets the phase information back to zero The button Remove HKL removes all HKL files for this pattern and forces Fullprof to recalculate them Please note The assistant expects sequentially numbered Fullprof compatible data sets Please make sure that you export your patterns immediately before attempting a Le Bail fit All the data is written to that directory LeBail refinement D x Phase information Selected phase ranges from set number 0 to Ranges Current set for refinement Pattern Zero offset O 015760 refine Import Fullprof Reset Remove HEL Cell Profile Spacegroup Pbam 7 refine cell a 11 2244 alpha 30 0000 b 6 25995 beta 30 0000 c 3 602438 gamma 80 0000 Import Refinement information Cancel Refine ive Figure 96 Le Bail refinement cell parameters Once the refinements for all the data sets have been completed close the window by pressing cancel The cell dimensions can be exported to a text file by selecting the menu
144. rm two dimensional powder diffraction does not imply any specific geometry it merely states the two dimensionality of the detected signal It could be conceived that this detector be cylindrical as in a Weissenberg camera Such detectors are still common in modern single crystal diffractometers in both standard laboratories Rigaku 2004 as well as at neutron beam lines Cole et al 2001 however the concept has never gained great popularity in the modern powder diffraction field The ubiquity of large flat image plate detectors their unparalleled dynamic range as well as a speedy read out time are the reasons for their current prominence in the field A precise determination of the experimental geometry is a prerequisite for highly accurate and well resolved diffraction angles peak profiles absorption effects or even good filtering Especially the separation of micro structural effects from the instrumental contribution to the peak profile needs exact 20 values An accurate calibration remains the single most significant factor in the extraction of high quality powder diffractograms from two dimensional images Generally the detectors are set up perpendicularly to the primary beam with the intersection of the primary beam at the detector centre This setting has some advantages the entire Bragg cones are detected and the deviation of the cone projection from an ideal circle is usually small sometimes a detector can be placed off centre and
145. rumental function in X ray powder diffraction Journal of Applied Crystallography 39 304 314 95 Two dimensional X ray powder diffraction 96 Powder3D 1 2 A tutonal Bernd Hinrichsen Robert E Dinnebier and Martin Jansen Max Planck Institute for Solid State Research Heisenbergstra e 1 Stuttgart Germany 97 Two dimensional X ray powder diffraction 98 13 1 Introduction Powder3D is a program for data reduction and publication quality visualisation aimed specifically atlarge data sets collected in time resolved powder diffraction experiments The program is in ongoing development so there shall be regular updates and extensions to the present functionality For comments or suggestions please contact 13 2 Getting Started 13 2 1 Installation The latest version of Powder3D can be downloaded from http www fkf Impa de xrayl As Powder3D is written in the programming language IDL you will need to install the IDL virtual machine IDL VM before being able to run Powder3D Virtual machines for various platforms can be downloaded from the RSI website free of charge When you have installed IDL VM unpack PowdersD into a directory of your choice and double click on the file Powder3D sav After dismissing the IDL VM current version 6 3 only splash screen the following window Figure 12 should welcome you Please note the libraries necessary for the AVI export and peak refinement name
146. s to calibrations that are at least one order of magnitude more precise than traditional calibration routines This is of fundamental importance for the effective use of future high resolution area detectors Aperfect calibration does not suffice to ensure a successful data reduction Especially in situ experiments the forte of two dimensional detectors cause intensity aberrations that need to be removed before the image can successfully be integrated to a conventional powder diffractogram The source of deviations can be sorted into two camps those originating from the sample environment and those emanating from the sample itself Of course the former is both more easily recognized visually and also removed more simply by the fractile filters presented in the third chapter Precis 87 Two dimensional X ray powder diffraction 88 When intensity deviations originate from the sample the matter becomes far more complex A new distribution function the normal Pareto function has been shown to describe the intensity distribution that results from small sample amounts without substantial sample rotation as is the case in high pressure powder diffraction The great benefit of this function is that it opens the possibility of extracting a fractional filtering setting which ultimately leads to normally distributed intensities Structural analysis from diffraction data is always connected to a plethora of reliability values describing the raw d
147. sed Hinrichsen et al 2007a This method is able to compute all detector and geometry genetic peak broadening as well as intensity effects such as polarization and Lorentz corrections Using this method the entire image is reconstructed and subtracted from the measured image This difference is weighted and used as the residual for the refinement process Conventional image calibration compared to whole image refinement WIR is in many ways similar to the comparison between the indexing of a powder diffraction pattern and the following whole powder pattern fit WPPF Results obtained from indexing programmes are never used as is but are further refined against the raw data using WPPF methods In conventional image calibration ellipse positions are given by points extracted from an image These points are used to calculate and refine the calibration parameters However the values obtained from an image calibration are taken at face value without further refining these against the raw data This gap is closed by WIR The values obtained from the refinement routine are used to calculate a theoretical two dimensional image much the way a WPPF is calculated from indexing results The parameters used to create the image are then refined until the fit is optimal The steps required to construct such an image and the parameters that determine various aspects of the image shall be discussed in detail 2 3 2 1 Whole pattern construction In
148. sed on an low pass filter as proposed by Sonneveld and Visser Sonneveld amp Visser 1975 Select the number of iterations curvature value and number of background points to attain an optimal background Please note that every iteration costs computing power for large data sets many iterations can make the automatic background reduction time consuming The apply button greets you with the following dialog Figure 89 Two dimensional X ray powder diffraction Si Apply background From set na fi To set no 200 Select phase Fhasel r Figure 89 Batch background reduction To edit background points manually select the edit tool Figure 90 This can become necessary if the background varies strongly from pattern to pattern You can now remove background points with a shift click left click while pressing the shift button and add background points with a click Figure 90 The edit tool You can cycle through the patterns in the usual fashion either by selecting the next and previous buttons in the context menu or by pressing the arrow buttons on the Main tab If a background has been calculated for the pattern it will be displayed Should phases have been defined it is possible to select the data sets associated to the phase by choosing the appropriate phase The Fullprof Rodriguez Carvajal 2001 format is a simple XY ASCII file containing 20 values in the first col
149. should already be masked and the background should be well determined Then the starting values from the initial refinement can be used to improve the calibration as well as refine polarization values and the detectors point spread function Of course each peak s intensity and width has to be refined as well This can be set by pressing the Peak properties button figure 129 oix PSF um 225 02 M refine FHM L 0 00233 W 0 00267 We 0 0067 I refine Intensities Wew I refine Dore Figure 129 The calibration wizard Setting and refining peak and detector properties 141 Two dimensional X ray powder diffraction The intensities can be viewed but not altered by pressing the View button R Corrections lOlx Polarization Lorentz Factor 1 00000 refine Phase 90 00000 refine Figure 130 The calibration wizard Setting and refining Polarization factor and phase The polarization factor and the phase angle can be refined to determine these to a higher precision using WIR The Lorentz refinement is not yet functional 13 6 Filters One of the most interesting features of the software is the provision of simple but powerful filters for diffraction images In the following pages the controls will be introduced 13 6 1 Fractile filter Edit gt Mask A fractile filter removes a fraction of the data in an attempt to eliminate outlier data The concept is similar to the use of
150. should be kept in mind that the refinement of atomic position of weak X ray scatterers such as oxygen in the vicinity of heavy atoms such as antimony which are strong scatterers is inherently difficult 5 4 Conclusion The general applicability of two dimensional signal filtering to powder diffraction data has been demonstrated In the presented case study of FeSb O high pressure data has been analysed successfully identifying two new phases at non ambient pressures All applied filters have been implemented in the freely available software Powder3D 19 Two dimensional X ray powder diffraction 76 6 Tin sulphate at high pressures 6 1 Introduction The structure of tin sulphate can be considered to have a highly distorted Barite structure Donaldson amp Puxley 1972 Unbonded lone pair electrons of the sp hybridized Sn orbitals can explain the high degree of distortion Tin sulphate was studied as part of a systematic survey of effects of high pressure on the structures of substances containing lone pair electrons High pressure powder diffraction experiments were performed at the High Pressure Collaborative Access Team at the Advanced Photon Source Argonne National Laboratories Argonne Illinois USA Three phase transitions were observed one between p 0 15 GPa and p 0 2 GPa one at p 5 GPa and one at p 15 GPa The initial soace group type Pnma phase only remained stable at near ambient conditions A sub
151. sive model whereby the value is selected based on the entire set A popular method of this type is the median value equation 3 6 intensity estimation The exclusive model selects only a subset to act as the basis for determining the descriptive value for the set Both methods can lead to similar intensities however they differ strongly in one respect the estimation of the statistical distribution The inclusive filters invariably estimate the distribution too liberally as all outliers are incorporated into this estimation Alternative Poisson based estimations of the distribution do not portray the distribution correctly A promising filter is a robust type of band pass filter based on fractile Statistics Hinrichsen et al 2006 A fraction x of the low intensity data and a fraction y of the high intensity data are removed equation 3 7 Intensity distributions and their application to filtering I nin x Lra In L ieii bee 1 y Tinos Iin 3 7 Unlike mean based filters the fractile method is insensitive to strong statistical aberrations and has been successfully applied to series of high temperature Schmidt et al 2007 and high pressure Hinrichsen et al 2006 powder diffraction data A defined fraction of the highest or lowest intensities within a 20 range is masked For the application of such intensity sensitive filtering procedures it is important to have previously applied all two dimensional intensity corrections
152. ss There is however no really comparative measure of spottiness The presented function provides a single value that can describe the quality of a two dimensional powder diffraction image with respect to spottiness The underlying idea is to use the variance of the pixel intensities associated with a single step in the one dimensional powder histogram A20 to estimate the quality of the image The unbiased variance is given by the well known relation I x o s Danilo 4 1 i l This is calculated for each step of the associated histogram and is averaged over the total image Be 1 n o 30 4 2 2 4 2 The average is normalized to the average pixel intensity making the value independent of the beam intensity and detector sensitivity resulting in the following expression Q Rina e m 4 3 I total A complete single expression of the value is Dim a ey Where is the total number of pixels in the image n is the number of bins used to integrate the image and m is the number of pixels within each bin R Lm 4 4 Du This reliability value has shown itself a robust estimator of the powder diffraction image quality however it is biased in the case of elevated background intensity This is especially prevalent in diamond anvil cell experiments Replacing the intensities with background reduced intensities resolves this weakness leading to the following equations 65 Two dime
153. t electronics design has inhibited the speedy development of this detector type Nevertheless a few groups have been working on realizing a detector specifically for crystallography Workers around Christian Broennimann Broennimann et al 2006 Hulsen et al 2006 have succeeded in building a one mega pixel detector for protein crystallography with a pixel size of 217x217 um Although this might be larger than the pixels of a CCD the perfect point spread function of a single pixel still represents a marked resolution improvement over the CCD detector types The detector was built up of an array of 18 modules covering a total area of 210 x 240 mm A full frame readout time of this detector takes 6 7 ms allowing a continual rotation single crystal data collection without the shutter closing between frames Detectors of this type are already in operation Fauth et a 2000 in the field of powder diffraction Large manufacturers of diffraction equipment have these detectors among their products disappointingly reduced to point detectors and not implemented as area detectors An installation at the material science beam line at the Swiss Light Source SLS of the Paul Scherrer Institute PSI in Switzerland is an equatorial type detector covering a fixed angle of 60 in 20 This is again no real two dimensional detector It has a faster readout time than the two dimensional detector mentioned earlier and can acquire entire diffractograms in a f
154. table 3 1 Arithmetic statistical values for this data are presented in table 3 2 15000 Intensity a u 100 200 300 Azimuth Figure 3 4 The intensity contained in the histograms of figure 3 3 a single bin of 0 02 in 20 as a function of the azimuth The coloured crosses at high and low intensities represent the data which is removed by the fractile filter The intensity drop at 270 is the primary beam stop arm shadow sample absorption is the cause of the dips in intensity at 0 360 and 180 Intensity distributions and their application to filtering The histograms in figure 3 3 show the effect of masking on the intensity distribution for a 0 02 20 bin of the first LaB peak 001 The red histogram represents the unfiltered data the blue the filtered data The fractile filter was set to remove 4 5 of the lowest intensities and 4 5 of the highest intensities within the bin The red line is a fit of the normal distribution density function to the unfiltered data the blue line is a fit of the same function to the filtered data The extracted values from the fit are identical within the standard deviation as can be seen from table 1 In general standard arithmetical values of the mean or median and variance are computed The intensities as a function of the azimuth can be seen in figure 3 4 Little scattering and no outlier data are discernable only uncorrected sample absorption causes a slight sinusoidal trend Ta
155. tation from the azimuth angle 25 Two dimensional X ray powder diffraction 26 1 3 Corrections 1 3 1 Intensity corrections As important as the diffraction angles are to the exactlattice parameters the intensities are for the precise determination of atomic position elemental species and their occupation and displacement parameters The great popularity of equatorial point detectors and later one dimensional position sensitive detectors in laboratory diffractometers has hindered the spread of generally applicable correction formula and canonized equatorial specific corrections These are often incorrectly applied to data collected from two dimensional detectors Important experimental factors influencing the intensity of a diffracted beam are discussed and the corresponding two dimensional correction functions are given 1 3 1 1 Lorentz corrections Lorentz correction applied to powder diffraction data are slightly different to those applied to single crystal data Whereas the single crystal correction only comprises a rotational factor the powder correction contains an additional statistical factor Zevin 1990 This corrects for the likelihood of a crystallite being in diffraction position This factor has a simple sin dependence and is found in the common Lorentz correction 1 40 U sin20 sin The well known correction for the speed of the transition of a reflection through the Ewald membrane is attributed to a lec
156. te form by equation 1 8 This equation is deduced in the same manner as equation 1 6 however starting from the more complex formulation of a tilted detector given by equation 1 12 FWHM Xpsp cos rot Ypsr sin rot cos tilt Y psp COS FOL X pop sin rot arctan 2 D xps COS rot Ypge Sin rot sin tilt x cos rot y sin rot cos tilt ycos rot x sin rot arctan 5 D xcos rot ysin rot sin tilt KH KL PLO Vig PEL OL For a precise explanation of the terms rot and tilt please refer to figure 1 9 and figure 1 11 The line width can also be expressed in the more general terms of diffraction angle 20 azimuthal angle a detector orientation D rot tilt and the detector point spread PSF To deduce the formula we start with equation 1 3 but alter it to fit a tilted detector This implies adding a distance z to the sample to detector distance It represents the change of the distance to the reflection point on the detector projected onto the primary beam vector This change is brought about by the tilt and can easily be derived as is shown in Figure 1 10a The factor narrowing the effective width of the tilted beam figure 1 2 has to be added leading to modified form of equation 1 5 The experimental setup Figure 1 5 A theoretical two dimensional diffraction pattern Notice the effect ofthe broadening by the incident angle and the point spread function on the pattern
157. tector designs are shown in their fundamental units This is an adaptation of an image in Chotas Chotas et al 1999 1 1 1 CCD Detectors 10 Probably the widest spread detector type utilized today in X ray crystallography is the CCD camera These detectors are in use in multifarious fields and their general ongoing development has been of benefit to the relatively small X ray detector segment The greatadvantages of these detectors are their high resolution and short readout times This is of importance especially in single crystal diffraction in which dead times can make up a great part of the measurement time The drawbacks stem from three basic elements of the detector The fluorescent screen has to be optimised for the required wavelength The higher energy radiation requires a thicker layer to fully absorb the incident rays Thicker layers are disadvantageous as the PSF increases with the strength of the layer This is due to the spherical dissipation of excited electrons within the fluorescent layer Fibre optical tapers channel the light from the large fluorescent layer to the smaller CCD chip The tapering often leads to an imperfect representation of the original image onto the CCD This has to be corrected as much as possible within the detector electronics firmware Some detectors have a CCD area of equal size to the fluorescent layer and can circumvent this source of errors Finally one major drawback is the substantial dark current
158. thal plot is most useful The coloured intensities are those masked by the filters Here similarly to the histogram window the data can be analyzed with iTools by a right click 13 9 Integration The integration is the final step of the two dimensional powder data reduction In general a large number of images need to be reduced to powder diffractograms These can be named as well as the output directory and output format Filters and intensity corrections can be applied to all images in the process An arbitrary number of integration bins can be selected One image can be reduced to a number of diffractograms using wedge integration Lastly the method of intensity extraction can be switched between mean and median Integrate Files Filter Corrections Binning Azimuth Method Image files Output directory labe_ 92 780_009 mar2300 D mgel New Folder a a C XY Lar C GSAS Fullprof Cancel UK Figure 147 The integration interface On the left the input files can be entered On the right the output directory and the output format are set 151 Two dimensional X ray powder diffraction Integrate Files Filter Corrections Binning Azimuth Method W Beamstop IW Fractile Settings Figure 148 On the second integration tab the applied filters can be set By pressing the Settings button the details of the filters can be set These are applied individually to each image before inte
159. these checks in turn also has a detrimental effect on the quality of the calibration refinement Other sources of errors are calibration images of less than perfect quality and sub optimal user interaction when selecting the starting parameters When the initial estimation of the ellipse intersections is not ideal then the algorithm designed to refine the ellipse intersections from the starting parameters might fail The fitting of peaks to the data presumed to be an intersection of an ellipse is again a local optimization The algorithm cannot see the ellipse and therefore cannot differentiate between data containing a peak or data containing only background The quality of such a refinement can deteriorate very strongly should not enough care have been given to ensure good starting values Here restarting the refinement with better starting parameters remains the only option for a successful calibration It poses a great challenge to extract good calibration data from a poor calibration image The quality of an image can be imperfect for numerous reasons The first is the contamination by single crystallite reflections Combining a highly parallel beam with little or no sample rotation as is the case for high pressure or high temperature experiments can lead to these very intense spots These spots are in general not numerous enough to be normally distributed thereby strongly biasing the base powder signal on integration They can severely
160. tically high Hi Unfiltered data Filtered data Mean orea 079303 In conclusion it can be said that the filtering of ideal data such as that provided by standard samples figure 3 5 is not necessary or even beneficial to the overall data quality Outliers cause hardly any aberrations as can be seen by the proximity of the median value to the mean and the small variance Filters reduce the scattering of the data as is shown by the reduced variance of the filtered data 53 Two dimensional X ray powder diffraction Figure 3 5 A rendered image of a calibration powder diffraction data set collected by a two dimensional detector The fine monodisperse grains of LaB cause the light red rings to be of such an even intensity The high absorption of the sample at this wavelength of 0 92 A causes the intensity reduction along the horizontal axes 3 4 2 High temperature data As an example of a realistic data set an image from a HT experiment Sugimoto amp Dinnebier 2007 has been chosen These were performed in a sapphire capillary Chupas et al 2001 resulting in images figure 3 6 containing large high intensity single crystal peaks The intensity of these peaks tends to be an order of magnitude higher than those of the sample To estimate the value of filtering an exemplary Bragg cone is selected it has a high intensity a relatively high angle 35 33 and intersects the tails of two sapphire reflections The bin size of 0 02
161. tion angle 20 the PSF of the detector and the sample to detector distance D For simplicity we assume that the detector tilt is negligible The radius is then related to the sample detector distance and the diffraction angle by the following equation see figure 1 8 Dtan20 r 1 3 Adding the FWHM and the point spread contribution would lead to Dtan 20 FWHM r PSF 1 4 As equation 1 3 still holds 1 4 would become D tan 20 FWHM D tan 20 PSF 1 5 Solving for FWHM in terms of 20 and the PSF then leads to 14 The experimental setup FWHM arctan Dtan29 PSF es 1 6 To show the effect of the PSF on the FWHM of a diffracted beam a surface spanning a 20 range from O to 70 and a detector distance range from 100mm to 1000mm has been calculated estimating the point spread to be 300um figure 1 3 FWHM Figure 1 3 The effect of the point spread of a two dimensional detector upon the FWHM of a diffracted beam is represented The detector is assumed to be ideally aligned normal to the primary beam The point spread is taken to be 300um Sample contributions to the peak width have not been considered The effect of the diffracted beam projection onto the image plate 1 2 has not yet been included into the FWHM calculation Adding this factor to the broadening leads to the following estimation 1 7 again for an idealized non tilted detector To verify the applicability of this estimation the theoretical valu
162. tistical values within an integration bin The ideal values have been extracted from fitting a normal probability density function to the histogram of the filtered data The arithmetic values are based on the entire data set Dr Unfiltered data Filtered data Mean fit N A 1619 14 Variance fit N A Mean _ 6896 310 1604 13 Intensity distributions and their application to filtering 3500 Unfiltered diffractogram Filtered diffr actogram S B 2000 2 S 500 10 20 F Figure 3 9 The dramatic effect of the fractile filter on the final integrated pattern from figure 3 6 is shown The red line is the pattern extracted from the unfiltered image the blue pattern is the pattern extracted from the image which has 5 of the highest and 5 of the lowest intensity filtered from it 3 4 3 High pressure data As already mentioned the quality of high pressure data poses probably the greatest challenge to filtering techniques An example of such data can be seen in figure 3 10 p is i a Mint N ll Ich rit a A iy Figure 3 10 A rendered image of a high pressure powder diffraction data set collected by a two dimensional image plate detector The white spikes are high intensity peaks originating from larger grains within a fine grained matrix The fine grains contribute the light red rings visible at the base 5 Two dimensional X ray powder diffraction Intensity a u Azi
163. to 0 Curent set for refinement Patter Zero offset 0 015Y760 refine Import Fullprot Reset Remove HEL Cell Profile FWHM constant variable U 0 133323 Mi 0 05645 wt 0 031660 refine ks 0 0001 OO 0 026704 W refine Asymmetry S L 0 000370 refine D L 0 001780 refine Import Should you have refined profiles you can load these to the profile mask by pressing the import button on the profile tab Data from an existing Fullprof file pcr can be loaded using the Import Fullprof button On pressing Refine a Fullprof file is written and should Fullprof be installed on the system a LeBail refinement is started The refined data are read back to the fields 129 Two dimensional X ray powder diffraction 13 2 6 References When publishing please give reference to Powder3D in the following manner Hinrichsen B Dinnebier R E and Jansen M 2004 Powder3D An easy to use program for data reduction and graphical presentation of large numbers of powder diffraction patterns Z Krist 23 231 236 Caglioti G Paoletti A amp Ricci F P 1958 Nucl Instr 3 223 228 Dinnebier R 2003 Powder Diffraction 18 199 204 Dinnebier R E Vensky S Jansen M amp Hanson J C 2005 Chemistry A European Journal 11 1119 1129 Finger L W Cox D E amp Jephcoat A P 1994 Journal of Applied Crystallography 27 892 900 Larson A C amp Von Dreele R B 1994
164. to calibrate synchrotron data sets The doubts stem from the difference between the ellipse centre and the confocus of the ellipses which is the true intersection of the primary beam with the detector Norby 1997 Calibration 35 Two dimensional X ray powder diffraction 2 2 2 Pattern recognition To overcome the first hurdle of finding precise starting parameters we have developed a robust method for the automatic detection and characterization of ellipses The method involves no mathematical complexity and exhibits excellent overall efficiency 2 2 2 1 Ellipse detection An ellipse can be described by five parameters the major axis a minor axis b centre co ordinates x and y and angle of rotation A widely used pattern recognition technique used in image analysis is the Hough transformation This has been applied lately Bennett et a 1999 Dammer et al 1997 Fitzgibbon et al 1999 Lei amp Wong 1999 to ellipse detection in an attempt to facilitate computer vision and biometrics The first step in a Hough transformation is setting up a parameter space The dimensionality of the space corresponds to the number of parameters that are to be determined The size of a dimension depends on the resolution and range required by the corresponding parameter Parametrical requirements dictate the volume of the parameter space The next step is to transform the image into the parameter space This is done by tracing the pattern des
165. tructure determination and refinement Following successful reduction the scattering profile gave little direct evidence of a phase transition The first patterns showed excellent convergence using the starting model The third pattern converged less well the fit getting progressively worse up to p 15 GPa where only a triclinic setting lead to an acceptable correspondence to the experimental data To ensure the structures were described using the correct symmetry a Rietveld refinement of all datasets in the triclinic setting Pl was performed with the idea to deduce the true symmetry of the structure from its triclinic pendant All patterns were refined in P 1 leading to excellent convergence low residuals and meaningful atomic displacement coefficients in every case All refinements were performed using the TOPAS3 Coelho 2004 software A closer look at the lattice parameter evolution figures 6 2 and 6 3 with increasing and decreasing pressure did however hold some surprises 71 Two dimensional X ray powder diffraction Pnma P2i 4 p 1 91 6 er a 91 2 _ 90 8 90 4 90 89 6 89 2 91 2 89 4 0 5 10 15 20 25 Pressure GPa Figure 6 2 Three plots showing the progression of the unit cell angles as a function of pressure For better clarity only data points from compression have been shown Lines have been drawn for some regions these are merely guides for the eyes and not the result ofa fit The y angle sh
166. tructure factor measurements Acta Crystallographica Section A 61 C146 Theodoridis S amp Koutroumbas K 1999 Pattern recognition Academic Press Vinet P Ferrante J Smith J R amp Rose J H 1986 Journal of Physics C Solid State Physics 19 467 473 Vogel S C amp Knorr K 2005 Two2One Software for the analysis of twodimensional diffraction data Commision on Powder Diffraction IUCr Newsletter 23 26 von Dreele R B Lee P L amp Zhang Y 2006 Protein polycrystallography Zeitschrift fur Kristallographie 23 3 8 Wenk H R amp Grigull S 2003 Synchrotron texture analysis with area detectors Journal of Applied Crystallography 36 1040 1049 Westbrook E M 1999 Detectors for Crystallography and Diffraction Studies at Synchrotron Sources p 2 Denver CO USA SPIE Whittaker E 1953 The polarization factor for inclined beam photographs using crystal reflected radiation Acta Crystallographica 6 222 223 Yao W M 2006 Review of Particle Physics Journal of Physics G Nuclear and Particle Physics 33 1 Zaleski J Wu G amp Coppens P 1998 On the correction of reflection intensities recorded on imaging plates for incomplete absorption in the phosphor layer Journal of Applied Crystallography 31 302 304 Zevin L 1990 Lorentz factor for oriented samples in powder diffractometry Acta Crystallographica Section A 46 730 734 Zuev A 2006 Calculation of the inst
167. ts goodness of fit of the distribution models to the filtered intensities contained in a single bin from a high pressure experiment The best fit is given by the smooth Pareto distribution Mode sis CR S o Normal distribution 1431 Pareto distribution Normal Pareto distribution 59 Two dimensional X ray powder diffraction u 2 Q E Mi el o 0 Mi 0 200 400 600 800 1000 Intensity a u Figure 3 13 The effect of parameter a is shown on the convoluted Pareto probability density function At values of a below 20 a shift of the peak to higher intensities can be seen The slower falloff of the distribution function to higher intensities is the most pronounced characteristic of small a values in the distribution function Filtered Histogram Normal Pareto distribution 80 Normal distribution am s 60 0 Pas 2 gt 40 wo DD O 20 0 4000 5000 000 Intensity a u Figure 3 14 A normal Pareto distribution is compared to a normal distribution of an ideal sample To the low intensity side of the functions an almost perfect correspondence of the two functions is observed To higher intensities the greyed area represents the additional high intensity pixels of a normal Pareto distributed signal 60 Intensity distributions and their application to filtering Some practical use of the otherwise rather academic knowledge of the distribution function is the calculation of optimal f
168. ts in powder patterns with reliable intensities The situation is not quite as daunting in high temperature HT experiments as generally larger amounts of sample can be used Spotty rings can be avoided by carefully preparing a finely ground sample Artefacts originating from the sample environment do however have to be filtered from the image a task that can be accomplished with the presented filters 3 1 1 Current standard method The accepted manner of filtering such data is to mask the high intensity peaks manually using software such as Fit2D Hammersley et al 1996 Manual masking is time consuming lacks reproducibility and relies too heavily on visual inspection to produce reliable results Two exceptions known to the authors are Two2One Vogel amp Knorr 2005 which contains a filter based on Poisson statistics and Datasqueeze Heiney 2005 which contains an averaging filter aimed at removing bad pixels These methods are extremely useful when the average intensity is not affected strongly by outliers once the outliers dominate the mean value these can no longer be used as a filter criterion and the methods inexorably fail Hinrichsen et al 2006 3 2 Detector signal distribution The general goal of signal filtering is the separation of the required signal from artefacts or noise To do this successfully knowledge of the statistical distribution of the data is required 3 2 1 Distribution models The binomial distribution
169. ts you for the wavelength and the two theta range and increment The increment entered cannot be altered later The range can only be cropped Should the range of your files differ from the set values the intensities are interpolated using the selected function linear recommended quadratic or spline The entries correspond to the values of the sample data provided H Imported files zi Radiation i monochromatic white wavelength Theta E Dispersive Lambda 1 1 54060 l i Lambda 2 0 000000 K alpha singlet Lambda ratio 0 000000 K alpha doublet Polarisation 0 300000 New polarisation 0 000000 Figure 16 Import settings After the files have been read a message displays a few details on the current database Figure 17 There is no set limit to the number of files that can be read the only limitation is the available memory 101 Two dimensional X ray powder diffraction Database information x Fa j The database now contains 200 datasets J Each patterns is comprised of 2299 datapoints 2Theta range 0 011956500 to 54 964087 degrees Intensity range 0 0967 683 to 56706 1 counts Figure 17 Import report As soon as you acknowledge the message you are displayed the first of your powder patterns Figure 21 13 2 4 Later pattern loading and deletion Should you wish to load a single pattern only or wish to compare a pattern to already loaded data this can be done by using t
170. ture given by Lorentz Azaroff 1968 In its form applicable to a perfect single crystal it normalizes the intensity of a single reflection to the shortest traversal of the Ewald sphere This motion is brought about by the rotation of the crystal in direct space A consequence is that the correction is not only dependent upon the rotation vector of the crystal but also on the detection method The general formulation Mcintyre amp Stansfield 1988 takes the form A dz ds f I s ds Here I is the reflection intensity measured as a function of a scan variable s z IS the direction normal to the Ewald sphere at the reflection position Integrating over s for a typical four circle diffractometer Busing amp Levy 1967 and approximating sine and cosine values for the small angular range of a reflection leads to the following formalism i Aw sin y cosv Ay sinw siny L Ag cos x siny cosv cosw sin x sinv When regarding rotation around a single axis an experimental set up most commonly used in two dimensional powder diffraction the function reduces to Buerger 1970 L cosu siny cosv 1 40 1 41 1 42 1 43 The experimental setup Where x angle between axis of sample rotation and the primary beam normal plane y angle of horizontal reflection displacement and v angle of vertical reflection displacement The well known equatorial form of equation 1 43 is obtained when setting th
171. umn and intensities in the second The GSAS Larson amp Von Dreele 1994 format contains four columns the first contains a single i the second contains 20 values the third the intensities and the fourth the standard errors 13 3 9 Peak hunting By selecting the menu Edit gt Peak search and clicking the search button you shall see the following window Figure 91 Changing the mouse to edit mode enables you to remove peaks with a right click and add peaks with the left click You can drag the borders of the peak search to encompass all important areas of the powder pattern by selecting the range tool in the context menu right click on the image Peaks are searched by a multiple pass variable FWHM second derivative method The convolution range is set via the Caglioti diagram the threshold and minimum distance between peaks can be set via the sliding bar The radio buttons under the Caglioti diagram determine if a new peak list is written with every run or the current peak list appended with the newly found peaks 112 Sl Powder 3D File Edit Display Tools Plot Window Help tal 5 x Find Fit T ol 3 0 ET E Threshold 1 26 0 10 7 7 8x10 l l Minimum peak distance x l l l Search between 4 76 I l I l and 30 14 I I l I i 6x10 x N I I lt gt I l g a 2 4x10 I u Fi Overwrite Append lt gt lt gt Crystire Search
172. undermine the calibration routine by causing the peak position to be removed from the ideal centre of the ellipse arc Masking these peaks is therefore a prerequisite to the successful use of such an image for calibration purposes Another reason for an image to be of inferior quality is poor signal to noise contrast What might seem to the eye to be an acceptably resolved image is not necessarily ideal for calibration algorithms Low signal to noise ratios tend to have detrimental effects on the quality of the peak fitting algorithms designed to return exact ellipse positions Should a weak signal be combined with non continuous intensity along the ellipse arc then the peak refinement algorithm might be refining a peak profile against background data returning highly suspect positional values Statistical methods can be used to recognize such damaging data points however even if the set is filtered successfully less data remains for the parameter refinement than in the ideal case Generally higher angle and therefore larger elliptical arc intensities are weaker merely as a cause of the atomic scattering factor Regrettably it is exactly these intensities that generally contain the most highly resolved Calibration calibration information There is no substitute for a carefully and diligently made calibration image 2 3 2 Whole image refinement A way to overcome most of the mentioned problems related to image calibration has recently been propo
173. y The second step is the manual selection of the inner ellipse at least five points are required to determine the ellipse parameters Hammersley et al 1996 The diffraction image should show well defined solid uninterrupted ellipses with if possible no aberrant intensities Images from standard samples as those distributed by the NIST NIST 2006 produce satisfactory results However the preparation of an own standard sample is relatively simple There are a few properties that determine the applicability of a crystalline powder as a calibration standard The compound should possess a high symmetry leading to highly intense and well distributed peaks The composition should comprise strongly scattering high atomic number elements so that even a sample of small volume can produce a strong signal Asmall monodisperse grain size as well as only small contributions from micro structural crystallite size and micro strain effects guarantee well defined peak profiles Substances that fulfil these requirements and that are often used for calibration images are LaB CeO and Si figure 2 1 2 2 Starting parameter estimation The calibration refinement is a local minimization For the refinement to produce a reliable calibration the starting parameters need to be close to the final true values The most widely used method is the manual selection of at least five points on the inner ellipse of the calibration image An ellipse is fitted to the p
174. y 38 760 767 Angel R J 2002 EOSFIT Version 5 2 Azaroff L 1955 Polarization correction for crystal monochromatized X radiation Acta Crystallographica 8 701 704 Azaroff L 1956 Polarization correction for crystal monochromatized X radiation Acta Crystallographica 9 315 Azaroff L 1968 Elements of X ray crystallography New York McGraw Hill Bennett N Burridge R amp Saito N 1999 A Method to Detect and Characterize Ellipses Using the Hough Transform IEEE Transactions on Pattern Analysis and Machine Intelligence 21 652 Broennimann C Eikenberry E F Henrich B Horisberger R Huelsen G Pohl E Schmitt B Schulze Briese C Suzuki M Tomizaki T Toyokawa H amp Wagner A 2006 The PILATUS 1M detector Journal of Synchrotron Radiation 13 120 130 Buerger 1970 Contemporary Crystallography McGraw Hill Busing W R amp Levy H A 1967 Angle calculations for 3 and 4 circle X ray and neutron diffractometers Acta Crystallographica 22 457 464 Cervellino A Giannini C Guagliardi A amp Ladisa M 2006 Folding a two dimensional powder diffraction image into a one dimensional scan a new procedure Journal of Applied Crystallography 39 745 748 Chall M Knorr K Ehm L amp Depmeier W 2000 Estimating intensity errors of powder diffraction data from area detectors High Pressure Research 17 315 323 Chater R Gavarri J R amp Hewat A W

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