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Jana 2006 powder examples

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+∞G L ∫ GL,−∞Voigt function: V ( b , b , x) G( b , x′) L( b x − x′)= dx′is a convolution of Gaussian and Lorentzian function. For <strong>powder</strong> profile weuse a simpler analytical approximation of the Voigt function called pseudo-Voigt function:( H, x) = ηL( H,x) + ( 1−η) G( H x)pV ,the parameters η and H are functions of HGandHL5 543 22 34 5H = HG+ 2.69269HGHL+ 2.42843HGHL+ 4.47163HGHL+ 0.07842HGHL+ HLHL ⎛ HL ⎞ ⎛ HL ⎞η = 1.36603− 0.47719⎜⎟ + 0.11116⎜⎟H ⎝ H ⎠ ⎝ H ⎠23


The diffraction line broadening induced by the sample is usually dividedinto to crystalline size effect T and microstrain Y . In a firstapproximation they have the following angular dependence:H = b = X cosθ + Y tanθ =180λ πTLLLLX LSimilar equations are valid for Gaussian distribution but then an additionalfactor ln 2 is to be applied to get b from H .8 G22 2 2HG= 8ln 2bG= XGcos θ + YG2tan2θThese ideal equations are valid only if broadening , induced byexperimental parameters are neglected. This means we can use therefined values to make conclusions about crystalline size and microstrainonly if some type of fundamental approach is used.But usually we are using an additional terms to take into account otherexperimental effects.G


Stephens model for anisotropic broadeningThis phenomenological model is based on a general tensor expression in whichthe anisotropic strain is described by a symmetrical 4th order tensor:2σijmn( hkl) = D hihjhmhn= ∑SH , K , LHKLhHkKThe first term is just a general tensor expression where Landau summationconvention is used and which allows a simple derivation of symmetryrestrictions similar to those for 4th order ADP parameters. The secondterm has an explicit form as used by Stephens in which summation isrestricted to H + K + L = 4lLThis term is used to modify equations forbGandbLFor modulated structures this method has been generalized byA.Leineweber and V.Petříček, (2007). J.Appl.Cryst. ,40, 1027-1034.


The angular parameters (LX, LY,…) are measured in 0.01 deg, the squared onesin 0.0001 deg 2 .


Asymmetry options in <strong>Jana</strong><strong>2006</strong>Simpson’s method – Peak is combined with several shifted peaks having theidentical shape according to the formula:Pcorr16n2n[ ] ∑ +2θ=1i=1⎡ ⎛ i −1⎞kiP⎢2θ+ a⎜⎟⎣ ⎝ 2n⎠2⎤cot 2θ⎥⎦kkkiii= 1= 4= 2forforfori = 1or i = 2n+ 1i = 2mi = 2m+ 1 m ≠ 0m≠na is the only parameter to be refined


Asymmetry options in <strong>Jana</strong><strong>2006</strong>Berar-Baldinozzi method –P corr( θ ) = P( 2θ)( z) + p F ( z) p F ( z) p F ( ) ⎞ ⎟⎠⎛ p F12 2 3 1+4⎜1++⎝ tanθtan 2θ1 2z22where F ( z) = H ( 2z) exp( − z )F1212( z) = H ( 2z) exp( − z )3θ −θ0z =FWHM( ) z H nstand for Hermit polynomials


Asymmetry options in <strong>Jana</strong><strong>2006</strong>By axial divergence – according to Finger, Cox and Jephcoat, (1994)J.Appl.Cryst. 27, 892-900.Two parameters are used:HLSL“height” and “sample”These parameters are strongly correlated. Our recommendation is toestimate their ratio and keep it as a restriction during the refinement.


Asymmetry options in <strong>Jana</strong><strong>2006</strong>Fundamental approach – it follows the method introduced by Cheary andCoelho, (1998), J.Appl.Cryst. 31, 851-861.This method can estimate the profile asymmetry just on the base ofexperimental parameters. For Bragg-Brentano geometry it works very nicely.The method makes multiple convolution of several functions.


Background correctionLegendre polynomials – set of orthonormal polynomials defined on theinterval −1 , + 1. The measured interval have to befirst linearly projected into this interval. Then theirorthonormality considerably suppress their mutualcorrelations.Chebyshev polynomials – they are also orthogonal but in difference sense1∫−1w( x) pi( x) pj( x) = δijLegendre polynomials:Chebyshev polynomials:www( x) = 12( x) = 1 1−x2( x) = 1−xFor data collected uniformly along diffraction interval we should preferablyuse the unique weight.


Background correctionManual background – the background is expressed as a set of backgroundintensities over the diffraction interval. The actualvalue is calculated by a linear interpolation. This methodis can very effectively describe even very complicatedbackground profiles. But it need some user assistanceto select it properly. Moreover this first backgroundestimation can be combined with some of previouscontinuous functions.In the program <strong>Jana</strong><strong>2006</strong> there is a tool to select a first estimation ofmanual background but then you can interactively modify it.


Shift parameters


Shift parametersShift – it defines the zero shift (again in units of 0.01 deg). This value is to beadded to the theoretical peak position to get a position in experimentalprofilesycos – in analogy with the Fullprof: ∆2 θ = c.cosθis connected with a specimen displacementsysin – in analogy with the Fullprof: ∆2 θ = s.sin2θis connected with a transparency correctionThese three corrections are combined together to the actual peak position.


Profile refinement – le Bail techniqueBased on a peak decomposition and subsequent refinement of profileparameters.


Profile refinement – le Bail techniqueIn <strong>Jana</strong><strong>2006</strong> is also used to predict symmetry by comparing of profilefits for different space groups:


Profile refinement – le Bail techniqueIn <strong>Jana</strong><strong>2006</strong> is also used to predict symmetry by comparing of profilefits for different space groups:


Example 2.1: PbSO 4Application of <strong>Jana</strong><strong>2006</strong> to simple structure from <strong>powder</strong> data. Ideal case wheredetermination of symmetry and structure solution are simple.Powder data measured with laboratory diffractometerInput files:PbSO4.mac (<strong>powder</strong> profile data)PbSO4.txt (additional information)


Step 1: profile fittingWizard for profile fitting connects tools, which can be also started separatelyfrom the main <strong>Jana</strong> window


Step 2: symmetry determinationEach line contains ratio of extinct and all reflections and R pcorresponding to a profile with discarded extinct reflections. Thus we arelooking for case where number of extinct reflections is large withoutserious impact on R p .∑2y(obs)− y(calc)w(y(obs)− y(calc))R p=⋅100R wp=⋅1002y(obs)wy(obs)∑∑∑


Data importRefinement of profileparameters (with symmetry P1)Determination of symmetryCharge flipping (based onBragg intensities calculatedfrom the profile)Rietveld refinement


Step 3: structure solutionSuperflip uses intensities from profile decompositionStep 4: Rietveld refinementInstead of Le Bail fit the intensities are calculated from the structure.


Step 4: Completing the structure from difference Fourier map

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