Hansen, Per Christian

Poulsen, Henning Friis

Schmidt, Søren

Sørensen, Henning Osholm

Wright, Jonathan

Online Research Database In Technology

  General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.   Users may download and print one copy of any publication from the public portal for the purpose of private study or research.  You may not further distribute the material or use it for any profit-making activity or commercial gain  You may freely distribute the URL identifying the publication in the public portal  If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.         Downloaded from orbit.dtu.dk on: Oct 25, 2019Integrated intensities based on grain orientation distribution functionsSørensen, Henning Osholm; Wright, Jonathan; Schmidt, Søren; Hansen, Per Christian; Poulsen,Henning FriisPublication date:2009Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Sørensen, H. O., Wright, J., Schmidt, S., Hansen, P. C., & Poulsen, H. F. (2009). Integrated intensities based ongrain orientation distribution functions. Paper presented at Total Cryst Workshop, Grenoble, France.Integrated intensities based on grain orientation distribution functionsHenning Osholm Sørensen , Risø DTUJonathan Wright, ESRFSøren Schmidt, Risø DTUPer Christian Hansen, IMM, DTUHenning Friis Poulsen, Risø DTU  2 Risø DTU, Technical University of DenmarkAcknowledgments• Zsuzsanna Sokusd, summer student• Gavin Vaughan, ESRF, Grenoble • Anette Frost Jensen, H. Lundbeck A/S• Heidi Lopez de Diego, H. Lundbeck A/S• and all project participantsNew and Emerging Science and Technology - Adventure 3 Risø DTU, Technical University of DenmarkPeak overlap No of irradiated grainssingle crystal powderpolycrystal1 ∞ 4 Risø DTU, Technical University of DenmarkIntensity integration/separation• Having known reflection profiles can help extracting the individual intensities – A simple fit can made of the intensities of the overlapping peak profiles• Other integration programs do this, by “learning” the peak profile in different parts of detector space1 – This is not a viable route with a high number of overlaps1Kabsch, W. (1988) J. Appl. Cryst. 21, 916. 5 Risø DTU, Technical University of DenmarkPeak profiles and shapesA convolution of several factors• Some are related to the crystalline grains– Morphology– Orientation distribution• Others are instrumental in nature– beam divergence– beam profile– detector point spread– goniometer geometryTopo-tomography (W. Ludwig) 6 Risø DTU, Technical University of DenmarkPeak profiles and shapesGrain properties – Morphology– Orientation distribution• If the grain size is smaller than the detector pixel size the crystal morphology does not contribute to the peak profile• Hence the peak profile is determined mainly by the orientation spread of the grain• The aim will then be calculate the grain orientation distribution from a few non-overlapping reflections. 7 Risø DTU, Technical University of DenmarkOrientation Distribution Function (ODF)The ODF will be discretized in Rodrigues spacer = tan(θ /2) nxrnx’zyz’y’θr1  r2 r3Ω  Properties:Perfect single­crystal => single point in ΩMosaic crystal  => a distribution function in Ω  Pro:  Euclidian at small angles φCon:  For many space groups Ω  extents to infinity 8 Risø DTU, Technical University of Denmarky = y0+ dy,  dy is parameterized inu,v coordinatesFrom data to the ODFωk0k G0y0Poulsen, H.F. (2005) MST, 21, 1397 ûv ^ dyEwalds spherey 9 Risø DTU, Technical University of Denmarky0Poulsen, H.F. (2005) MST, 21, 1397 “reciprocal” sphere ω ,ηRelation between ODF and “data” is:       dy = 2(r × y0) From data to the ODF 10 Risø DTU, Technical University of DenmarkData integration routeg hdetydetzω   (2 0 0)       (0 2 0)      (0 0 2)   (1 1 1)         (­1 1 1)        (­1 1 ­1)ODFuv mapsDataReflection list with integrated intensitiesTransformReconstructProfileFitting 11 Risø DTU, Technical University of DenmarkProjection method• ODF voxels can be listed in 1D array, x• The transformed intensities in the uv map is listed as 1D array, b • A describes the geometrical relation between uv-maps (b) and ODF (x)Ax = b• Fast voxel traversal algorithm (3D-DDA) – Amanatides in Eurographics '87,(Elsevier)uv -pixel (b)ODF voxel grid (x) 12 Risø DTU, Technical University of DenmarkSolve:  Ax = bWe want to solve Ax = b, but A is ill-conditioned and b is noisy,henceAx = Axexact + e,      e being noiseIt follows thatxnaive = A-1b = xexact + A-1e ,    where     || A-1e|| >> |xexact|Not very useful results will be obtainedAim: Find a solution for which ||Ax – b|| is small (good fit) and x ~ xexact •To be solved by iterative methods as ART, CGLS etc. 13 Risø DTU, Technical University of Denmark Algebraic Reconstruction Techniquex−xbi−aiT x∥a∥22ai , i=1,.... , mOne approach to compute an approximate soulution  to Ax = bART does (by projections on hyperplanes):During the initial iterations the convergence is fastLater the convergence slows downGood method if only few iterations can be affordedInitial guessA-1b  14 Risø DTU, Technical University of DenmarkCGLS algorithm||Ax-b||2 15 Risø DTU, Technical University of DenmarkIteration       1             4           7             10            13         16           19preconditioned reconstructionsSince the border pixels generally are transversed by less rays than more central  this can lead to ripples at the borderBy “constraining” the solution to go towards zero at the border these types of effects can be suppressed CGLSPCGLS  16 Risø DTU, Technical University of DenmarkPreconditioning• Further smoothing and assert zero boundary conditions• Can be done by introducing a derivative operator DVariable transformation of Ax=bξ  = Dxminimize minx || (AD-1)ξ  – b ||2 back transformx = D-1ξ 17 Risø DTU, Technical University of DenmarkInvestigate quality of reconstruction methods by simulation• 5 reconstruction methods– ART– CGLS– P1ART– P1CGLS– P2CGLS• Alu grain ODF simulated by 3 Gaussians• uv maps calculated out sinθ /λ  =  0.45 Å-1 (29 reflections)• Added Poissonian noise (6 SNR levels)• 3 to 18 reflections used in reconstruction – randomly chosen• All calculations repeated 10 times 18 Risø DTU, Technical University of DenmarkSimulations with ”inverse crime” 19 Risø DTU, Technical University of DenmarkError histories of the iterative methods Iteration           1                4                  7                10                13              16               19 20 Risø DTU, Technical University of DenmarkStopping Criterion:  Stop the iterations when x(k) captures the desired informationHow do we measure this?• No way to measure if x(k) is close to xexact• Fit to noise level: ||A  xexact ||2 ~ ||e||2 • Information criterion: residual behaves statistically like eFit to noise level can sensitive to the estimate of  ||e||2 If  e is white noise, then the normalized cumulative periodogram can be used to measure the ”white-ness” of the residual.  21 Risø DTU, Technical University of DenmarkNCP = Normalized Cum. PeriodogramDominated bylow frequenciesDominated byhigh frequenciesQuite whiteThe NCP measures the frequency content in a signal s.Let the power spectrum of s (of length n) be given by p = |dft(s)|2then the NCP is a plot of the vector c with elementsThe closer c is to a straight line, the ”whiter” the signal s cl=∑i−1lpi∑i−1qpi, i=1,2,. .... , q , q=[n /2 ]Exampleswith n = 256and q = 128 22 Risø DTU, Technical University of DenmarkUse of the NCP stopping criteria• For each uv-map (i = 1,2,3,....,Nuv)– For each iteration step k• Compute the NCP– Return the iteration step giving the optimal NCP• Choose the iteration step k askopt = median (k1,k2,k3,....,kNuv)The median was choosen to minize the influence of outlier uv-maps 23 Risø DTU, Technical University of DenmarkQuality of reconstructed ODF as a func. of projections 24 Risø DTU, Technical University of DenmarkExtremely noissy dataSlice through ODF 25 Risø DTU, Technical University of Denmarkg h   (2 0 0)       (0 2 0)      (0 0 2)   (1 1 1)         (­1 1 1)        (­1 1 ­1)reconstructODFuv mapsSimulated  dataSimulationtransformSimulations without  ”inverse crime” 26 Risø DTU, Technical University of Denmark• For a test sample we have used pre-strained Al 1050 samples• Mean grain size 75 microns• Strained to 2, 4, 6, 8 and 10 %• Beam energy 25.514 keV• Beam vertical 100 micronsTensile direction400 micronsStrained Al samples 27 Risø DTU, Technical University of DenmarkAl sample - strain 6% uv-mapODF 28 Risø DTU, Technical University of DenmarkCompare experimental and reconstructed maps 29 Risø DTU, Technical University of DenmarkAl sample - strain 6% Winther et al. Acta Mater. (2004),  51, 286309/04/0930 Risø DTU, Technical University of DenmarkHLU1 kindly provided by H. Lundbeck A/SNClNNarrow (a few microns in height) and wide beam (>1 mm) to optimize flux for small crystalsHigh energy 46.837 keV (λ  = 0.2647 Å)Frelon4m detector (2k by 2k) Mounted samples on cryoloopsTemperature ~122 K Omega rotation 180 degress in 0.3 deg. stepsTest on an organic compound 31 Risø DTU, Technical University of DenmarkOrientation spread 32 Risø DTU, Technical University of DenmarkHLU1  - Not quite single crystal 33 Risø DTU, Technical University of DenmarkIndexingFirst grain indexed(fft_index.py, J. Wright)● Unit cell● a =  9.076 Å● b =  6.050 Å● c = 43.922 Å● Space gr. ? 34 Risø DTU, Technical University of DenmarkComparison of exp. and recontructed uv-mapsReflections on detector uv-mapsReconstructed maps•s 35 Risø DTU, Technical University of DenmarkIntegration by fitting the profile• Reflection profiles calculated from ODF + point-spread function• calculate CMS and maximum for both reflection box and profile• compare their positions•If far apart make initial fit a center•If not move profile grid to have same CMS as peak for intial fit• Calculate residual moving profile 1 grid point in all directions (26) •if not move center to position with lowest residual and do another round•If residuals all higher – stay•Decrease steps to ¼ grid points and do the same analysis until minimum residual found 36 Risø DTU, Technical University of DenmarkIntegration by fitting the profile3084  Unique reflections, of which   2886  observed (5982  Reflections read) Rint =  0.1055No structure yet .....  37 Risø DTU, Technical University of DenmarkSummery• Developed a method to reconstruct orientation-distribution functions of single grains.• Applied this in a procedure for extraction of integrated intensities.   The python program Fabric (not in GUI yet) is made to  reconstruct ODF's and do intensity integration. 

Integrated intensities based on grain orientation distribution functions

https://orbit.dtu.dk/ws/files/185820088/27389.pdf

Integrated intensities based on grain orientation distribution functions

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology