Materials Structure, vol. 7, number 2 (2000) 43 DECONVOLUTION VERSUS CONVOLUTION – A COMPARISON FOR MATERIALS WITH CONCENTRATION GRADIENT David Rafaja Department of Electronic Structures, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, CZ-121 16 Prague, E-mail: [email protected] Abstract tional Bragg-Brentano geometry are the sample transpar- ency and the local sample displacement due to the flat spec- Efficiency of three deconvolution methods used in X-ray imen. powder diffraction analysis is compared for materials with There are two different approaches how to treat the ex- concentration gradient. The first deconvolution method perimental diffraction pattern affected by instrumental shown in this comparison is a modification of the classical broadening. The first one employs direct deconvolution. Stokes method. In two other methods, the re-convoluted in- The second one solves Eq. (1) through convolution of tensities are fitted to the measured data using the physical and instrumental profiles, assuming that the phys- least-square procedure. The deconvoluted profile is repre- ical profile can be approximated by a function of a limited sented either by a linear combination of harmonic functions number of parameters, which follow from the micro- or by coefficients describing the decomposition of mea- structure model of the respective material. Although the sured profile into the basis of instrumental functions. latter approach based on convolutions becomes more and The results of deconvolution methods are further com- more frequently used in the last time (see, e.g., Refs. [1-4]), pared with results of an alternative approach, which is a they are still numerous applications, for which the use of a convolution of the instrumental profile with an analytical deconvolution technique is inevitable. The reason is that a function describing the diffraction on samples with con- proper microstructure model cannot be built without an ap- centration gradient. In this case, tuneable parameters of the proximate knowledge of the deconvoluted profiles, an ex- analytical function following from the microstructure ample of them being materials with concentration gradient. model are varied to arrive at the best match between the Regarding the direct deconvolution of diffraction pro- measured and re-convoluted intensities. files, the excellent comprehensive overview of deconvo- Analysing diffraction profiles measured on materials lution methods by M. Èeròanský [5] can be recommended with concentration gradient, the most reliable information to the reader’s attention. Among the large number of on the concentration profile was obtained using a combina- deconvolution methods discussed in [5], the most popular tion of the above approaches. The direct deconvolution was methods work with the least-square fitting of re-convoluted used to get basic information on the shape of concentration profiles to the measured intensities, see Ref. [6] and the ref- profiles. The final form of the concentration profiles was erences therein. The least-square method is suitable to adjusted by refining free parameters of the diffusion model. solve integral equations of the first kind, which is also the case of the equation for convolution (1). Keywords: X-ray powder diffraction, convolution, In this paper, three procedures used for deconvolution deconvolution, and concentration gradient. of powder diffraction patterns are compared. The Stokes method [7] working with smoothed experimental and in- 1. Introduction strumental data, decomposition of physical profiles into a Fourier series [6] and decomposition of experimental data Decomposition of diffraction profiles is a typical task in the into a linear combination of instrumental profiles. Mate- diffraction profile analysis. The main problem is to sepa- rials with concentration gradients were selected for this rate the instrumental broadening from the measured dif- comparison, as the presence of concentration profiles im- fraction pattern in order to obtain the pure physical profile plies strongly asymmetrical physical broadening, which is containing information on microstructure of the material. not favourable for most deconvolution methods. Finally, a This means that the well-known equation for convolution microstructure model was built for samples with concen- tration gradient, in which the corresponding physical pro- ¥ file was calculated. Free parameters of the microstructure hx()=* f g =ò f ( ygx )( - ydy ) (1) model were refined to arrive at the best match between the -¥ experimental and the re-convoluted data. has to be solved for f (y), which represents the shape of the physical profile. Function h (x) describes the measured dif- fraction profile; g (x-y) is the instrumental function (re- 2. Competing deconvolution procedures sponse of the apparatus). The instrumental function covers usually the line broadening due to the spectral width of ra- 2.1. Stokes method with Gaussian smoothing diation, angular resolution of the diffractometer optics as well as a variety of aberrations. Typical examples for aber- The original Stokes method [7] utilises the well-known rations causing the instrumental broadening in the conven- property of the Fourier transformation – the Fourier trans- Ó Krystalografická spoleènost 44 Materials Structure, vol. 7, number 2 (2000) formation of convolution of two functions can be expressed Equation (8) specifies the amount of smoothing in the as multiplication of Fourier transforms of these functions: deconvoluted profile. It can simply be shown that the deconvoluted profile is smoothed by the Gaussian function FTh()=*= FTfg ( ) FTfFTg ( ). () (2) if both g and h are smoothed by Gaussian functions. =* =-1 × = Thus, the pure physical function f can be obtained from the sssFTFTsFTshfg[() f ()] g measured profile h and from the instrumental profile g us- é æ 2 ö æ 2 öù - t t ing the formula: =-FT 1 êexpç ÷×-expç ÷ú = ç s2 ÷ ç s2 ÷ ëê è f ø è g øûú æ ö æ ö --11ç FT() h ÷ H fFT= ç ÷ = FT ç ÷ (3) é æ ss22+ öù é æ 2 öù - fg - t è FT() g ø è G ø =-×FT12êexpç t ÷ú =-FT 1 êexpç ÷ú ç s2 ss2 ÷ ç 2 ÷ ëê è f ghøûú ë è øû –1 Throughout the text, the symbols FT and FT denote the (9) Fourier and the inverse Fourier transformation. The capi- tals F, G and H denote the Fourier transforms. Due to the Therefore, the smoothing parameter sf is given by recipro- ill-posed nature of deconvolution, which is caused mainly cal difference of the parameters s and s : by the truncation of diffraction profile and by the presence h g of noise in both the experimental and instrumental data, the ss22 formula (3) does not yield useful results. The solution is os- s2 = gh (10) f ss22- cillating as a rule; the amplitude of the oscillations being gh comparable with the maximum of the deconvoluted func- tion. The smoothing parameter sf describes the filtering of the In order to overcome problems with truncation of dif- Fourier transform of the deconvoluted profile: fraction profiles, the experimental data must be pre-pro- cessed. In the first step, the background is subtracted and æ t 2 ö the missing marginal data are filled by zeros. To avoid FT(') f== F ' F expç - ÷ (11) ç s2 ÷ problems with the noise, the input data are often smoothed. è f ø This means that typically the high-frequency noise is re- moved. Such a filtering of input data was used in the modi- To derive the exact form of the smoothing function for f,we fied Stokes procedure presented in this contribution. must perform the inverse Fourier transformation of Eq. Fourier transforms of both experimental and instrumental (11): profiles were multiplied by Gaussian functions: é æ 2 öù -1 ç t ÷ é æ 2 öù é æ 2 öù ffFT'exp=*ê - ú =*fs, (12) ç t ÷ ç t ÷ ç s2 ÷ f FH'=-ê expú êG exp - ú (4) ëê è f øûú ç ss2 ÷ ç 2 ÷ ë è hgøû ê è øú ë û where This corresponds to the following combination of the Fou- é æ 2 öù ¥ æ 2 ö =--1 ê ç tt÷ú =-1 ç ÷ = rier transforms: sFTf expò exp ç s2 ÷ p ç s2 ÷ ê è fføú 2 -¥ è ø ë û (13) FT() h× FT ( s ) FT() h* s FFTf'(')== h = h (5) æ x 22s ö × * =-1 s ç f ÷ FT() g FT ( sg ) FT( g sg ) f expç ÷ 2 p è 4 ø The functions sh and sg are the inverse Fourier transforms of the Gaussian functions multiplying the functions H and G It follows from Eq. (13) that the smoothing function is in Eq. (4). Further, it holds as a consequence of Eq. (5): Gaussian in form. The area below the function sf is equal to unity, *= × *= ** FT()(')()(') h shgg FT f FT g s FT f g s (6) ¥ s ¥ æ x 22s ö sdx=-ffexpç ÷ dx =1, (14) Applying inverse Fourier transformation on Eq. (6), we ò f ò ç ÷ -¥ 2 p -¥ è 4 ø will get a similar equation for convolutions: fgshshss'** =* =* * (7) if the inverse Fourier transformation is defined by the for- ghfg mula: The right hand of Eq. (7) was rewritten to arrive at formally ¥ - 1 same convolutions on the left and right side. The left and FT1 ( F )= F ( t )exp( itx ) dt (15) p ò right hands of Eq. (7) are equal if 2 -¥ =* ffs' f (8) In such a case, the smoothing has no effect on the total inte- grated intensity of the deconvoluted profile. Ó Krystalografická spoleènost Materials Structure, vol. 7, number 2 (2000) 45 Equation (10) has several important consequences. of this deconvolution routine implies also an automatic The deconvoluted profile is automatically smoothed if the smoothing of deconvoluted data. The amount of smoothing experimental data and the instrumental profile are is determined by a number of harmonic functions (m in Eq. smoothed. For correct smoothing, sh must be less than sg, (16)-(18)).