DECONVOLUTION VERSUS CONVOLUTION – a COMPARISON for MATERIALS with CONCENTRATION GRADIENT David Rafaja

Home , Deconvolution

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 concentration 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 diffraction 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 function. 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)). The lower m, the higher is the degree of smooth- which means that the experimental function h must be fil- ing. tered to lower frequencies than the instrumental function g. The related computer code was written in the ® If sh and sg have approximately the same value, sf is very MATLAB environment. The convolutions in Eq. (18) high; thus, the physical function f is not smoothed at all. If were calculated with the aid of the fast Fourier transformation and the inverse fast Fourier transformation (see Eq. sg is much larger than sh, the smoothing of f is extreme – it is the same as the smoothing of h. (2)). The least-square refinement of the coefficients Cj and The computer code for deconvolution routine based on Sj is performed successively for increasing number of har- the modified Stokes method was written in the MATLAB® monic functions, m. Because of the large expected asym- environment. Fourier transforms were calculated using the metry of the deconvoluted profiles in samples with ® concentration gradient, the harmonic functions up to the internal MATLAB function for the fast Fourier transfor- th mation and the inverse Fourier transformation. 40 order were tested. The best set of Cj and Sj was selected according to the lowest sum of residuals between the original data h and the re-convoluted data h’ obtained from the 2.2. Decomposition of experimental profiles using back convolution: Fourier expansion N N 22 The second procedure tested in this paper is the decomposi- Shxhxfgh=-=*-=å ['() ()]å [ ] min (19) tion of diffraction profiles using expansion of the physical n=1 n=1 functions into the Fourier series. This approach is espe- cially advantageous if the subsequent method of data re- The deconvoluted profile is calculated from Eq. (16). The duction works with the Fourier coefficients, for instance if summation in Eq. (19) is performed over all experimental the Warren-Averbach analysis is applied. The computing data. routine used in the test is completely based on the technique, published by Sánches-Bajo and Cumbrera [6]. The 2.3. Experimental profile regarded as a linear only exception is that we took the instrumental profile in its combination of instrumental profiles measured form, instead of approximating it by an expansion into the Hermite polynomials like in [6]. Employing The last deconvolution procedure compared here solves the the expansion of the pure physical profile into a Fourier se- equation for convolution (1), rewritten into the form used ries, function f takes the form: for discrete data:

m m p ¥ =+www =2 = fx()åå Cj cos( j0 x ) Sj cos( j0 x )with 0 hfgmnmnå - (20) j==0 j 1 D -¥ (16) In the matrix representation, Eq. (20) takes the form: The symbol D in Eq. (16) defines the interval, in which the æ gg g gKöæ f ö æ h ö experimental profile was measured. Regarding the equa- ç 0123--- ÷ç 1 ÷ ç 1 ÷ K tion for convolution (1), the experimental profile can be ç gg10 g-- 1 g 2 ÷ç f2 ÷ ç h2 ÷ ç ÷ç ÷ ç ÷ written as ggggK f = h (21) ç 21 0- 1÷ç 3 ÷ ç 3 ÷ ¥ ç gg ggK÷ç f ÷ ç h ÷ é m m ù 32 1 0 4 4 =-×ww + = ç ÷ç ÷ ç ÷ hx()ò gx ( y )êåå Cj cos( j0 y ) Sj sin( j0 y )údy è MMM MOøè MMø è ø -¥ ë j==0 j 1 û

m ¥ =-å Cgxyjydy()cos()w + Consequently, the Dirac d-function corresponds to an iden- j ò-¥ 0 j=0 tity matrix for the instrumental profile and to a single “one” m ¥ embedded in a zero matrix for the physical profile. How- +-å Sgxy()sin()jydyw j ò-¥ 0 ever, Eq. (20) represents an infinite system of linear equa- = j 1 tions. Moreover, the system of equations is under- (17) estimated, as the data contain noise (random errors). There- fore, an assumption must be done to reduce the number of Equation (17) represents convolutions of the instrumental equations (the number of parameters) in the linear combi- function g with the basis of harmonic functions: nation (20).

m m In contrast to the deconvolution procedures employing =*ww +* hx( )åå Cgj [ cos( jy0 )] Sgj [ sin( jy0 )] (18) the Fourier analysis, we can assume here that the j==0 j 1 deconvoluted intensities are equal to zero outside a certain range. This assumption reduces substantially the number of Within the deconvolution procedure, Eq. (18) is solved for columns in the G matrix in Eq. (21). It means that the sys- coefficients Cj and Sj in the least-square sense. Application tem of linear equations (20) becomes overestimated. Thus,

Ó Krystalografická spoleènost 46 Materials Structure, vol. 7, number 2 (2000) it can be solved using the least-square method. However, it physical broadening is caused by micro-stress rather than is still necessary to filter the noise in the deconvoluted data. by the small size of crystallites. In Eq. (25), diffraction an- For filtering, the Fourier smoothing or the Golay-Savitzky gle, x, is calculated in the units sin q; the same units are method [8] were alternatively applied. The Fourier used also for the width of the Cauchy function, w. The smoothing means a convolution of the deconvoluted pro- above model is characterised by five free parameters (K = file with a Gaussian function like in the modified Stokes Ö(Dt), d0, d1, w and a scale factor normalising the calcu- method described above. The Golay-Savitzky method rep- lated intensities to the experimental ones). Note, that a resents the direct smoothing in the reciprocal space. physically true model should include two parameters more: the minimum and the maximum concentration of the in-dif- 2.4. Refinement of free parameters of the fusing species. However, the minimum and the maximum diffusion model concentrations would strongly correlate with the parameter of the diffusion model, K, and with the parameters d0 and A simple diffusion model was used to describe the concen- d1. Therefore, the parameters related to the interplanar tration profiles in the samples. One-dimensional diffusion spacing should not be refined in a true model. They should with concentration-independent diffusion coefficients in be taken from an independent experiment. On the contrary, crystallites having the same size was assumed. In that case, it can be shown that variation in the minimum and maxi- the concentration profile takes the form of the error func- mum concentrations has a similar influence on the shape of tion (see, e.g. [9]), the resulting diffraction profile like the changes in d0 and d1. cxDt=erfc( / ), (22) Upon refinement, the linear least-squares method was iteratively applied to the first-order Taylor expansion of the where x means the distance in individual crystallites (in in- calculated diffraction profile: dividual diffusion couples), D the diffusion coefficient and 2 é calc ù t the diffusion time. The dominant factor modifying the æ ¶I ö SII=-+å ê measç calc å Dp ÷ú = min (26) shape of concentration profiles and therefore the shape of ç 0 ¶ j ÷ n ëê è j p j øûú the corresponding diffraction profiles is the ratio of the total length of diffusion couples to the square root of Dt. For D the concentration ranges investigated here, all particular The increments to the refined parameters, pj, follows components (TiN, TiC, Ti (C, N), (Ti, Mo) C, (Ti, Mo)(C, from the solution of the matrix equation: N) and MoC1–x) crystallise with the face-centred cubic structure. Their lattice parameters follow the Vegard rule ¶I calc ¶I calc ¶I calc åå×=DpII()meas - calc quite well [10]. Thus, the dependence of the interplanar i 0 n ¶p ¶p n ¶p spacing on the concentration is represented by the linear i j i function: (27)

dd=+ dc (23) All convolutions were calculated numerically using the fast 01 Fourier transformation and the inverse fast Fourier transformation (according to Eq. (2)). In Eq. (23), d0 represents the interplanar spacing for c =0, d1 the change in interplanar spacing with the concentration of in-diffusing or out-diffusing species. Neglecting the 3. Experimental details macroscopic concentration depth profile in the sample, the influence of absorption of radiation on the shape of diffrac- X-ray diffraction experiments were performed with a con- tion profiles can be omitted. Then the diffracted intensities ventional Bragg-Brentano diffractometer (XRD-7, Seifert/ depend linearly on the differential volume of material hav- Freiberger Präzisionsmechanik). The nickel-filtered radia- ing a certain concentration: tion of copper anode with a divergence of 1° was transmit- ted by a Soller collimator in primary beam and diffracted dx dx IV(sinq ) µ= = (24) on the sample. The optics of the diffracted beam consisted d(sinql )dd ( /2 ) of a secondary Soller collimator and the receiving slit of 0.15 mm. The selected angular range was scanned repeat- The last part of Eq. (24) follows from the Bragg equation. edly with the step size of 0.01° in 2q. The total counting In a further step, additional line broadening due to the small time exceeded 60 seconds per step. To arrive at a good ex- size of crystallites must be taken into account. Thus, the perimental resolution, reflections in the high-angle region diffraction profiles calculated according to Eq. (24) were were selected for the measurement. convoluted with the Cauchy function: The instrumental broadening was measured on the standard sample of cubic LaB6 offered by NIST. The in- w strumental profile measured on the diffraction line (332) is y = (25) p()xw22+ shown in Fig. 1. The measurement was carried out in the same diffraction geometry and with the same instrumental Its area is normalised to unity. The Cauchy function should parameters, which were used for samples with concentra- be exchanged by the Gaussian function if the additional tion gradient. For the standard, the counting time was 20

Ó Krystalografická spoleènost Materials Structure, vol. 7, number 2 (2000) 47

3000 LaB ,332 6 2500 120 2000 100

1500 80 60 1000 40 Intensity (counts/20s) 500 Intensity (cps) 20 0 119 120 121 122 123 0 Diffraction angle (o2Q) 120 121 122 123 124 125 126 400 Figure 1. Instrumental broadening measured on the reflection 350 (332) of cubic LaB6. 300 250 (c) seconds per step. To avoid problems with different angular 200 a a 150 separation of spectral lines in the K 1/K 2 doublet due to (b) different diffraction angles of the respective materials, the Intensity (cps) 100 data were first converted into the equidistant scale in sin q. 50 (a) The following samples were used to test the decon- 0 120 121 122 123 124 125 126 volution procedures – cubic TiC1–x, hexagonal Mo2C, ter- o nary mixture (Tix Mo1–x) C and quaternary mixture (Tix Diffraction angle ( 2Q) Mo1–x)(Cy N1–y). According to the chemical analysis, TiC1–x contained 19.33 weight percent of carbon, this cor- Figure 2. Experimental data (open circles) and re-convoluted in- responds to the stoichiometric ratio TiC0.96. The analysed tensities (solid line) measured with the hot-pressed quaternary composition of the MoC1–x powder was MoC0.51 (5.72 (Ti0.75 Mo0.25)(C0.7 N0.3) (Figure at the top). At the bottom: physi- weight percent of carbon). The ternary (Ti0.85 Mo0.15) C was cal profiles obtained from the experimental data by using the produced by hot pressing TiC0.96, MoC0.51 and graphite in Stokes method (a), the Fourier expansion (b) and the linear com- argon atmosphere at 2000°C. Afterward, it was annealed bination of instrumental profiles (c). in argon for 16 days at 1450°C. The last three samples with sion with a small number of Fourier coefficients) yield ex- the nominal composition (Ti0.75 Mo0.25)(C0.75 N0.25) were cellent results. This can be illustrated on the hot-pressed prepared by hot pressing the mixture of 50 mole of (Ti0.5 non-annealed quaternary (Ti0.75 Mo0.25)(C0.7 N0.3). After Mo0.5) C and 50 mole of Ti (C0.5 N0.5) in argon atmosphere at 2000°C. Subsequently, two of these three samples were deconvolution, three distinct intensity maxima are clearly ° ° ° annealed for 48 and 192 h, respectively, at 1450°Cinone visible at 122.49 , 123.09 and 123.75 (Fig. 2). These bar Ar. peaks indicate the presence of two (Ti, Mo) (C, N) phases, which are usually denoted as Ti-rich a’–(Ti, Mo) (C, N) (here with the lattice parameter a = 4.2921 Å) and Mo-rich 4. Results and discussion a”–(Ti, Mo) (C, N) (here with a = 4.3044 Å), and the presence of non-stoichiometric cubic MoC0.74 (a = 4.2788 Å) 4.1. Comparison of deconvolution procedures [11]. On the other hand, the Stokes deconvolution with All deconvolution procedures outlined above yielded very Gaussian smoothing (Section 2.1.) was the most successful similar results for the majority of experimental data. Nev- routine in extreme cases, when the shape of the experimen- ertheless, if we compare only the differences between the tal profile approached the shape of the instrumental profile. experimental and the re-convoluted data, the most success- Generally, this method yields the best results if the pure ful deconvolution routine was that based on the linear com- physical profile is very narrow. Such a diffraction profile bination of instrumental profiles (Section 2.3.). was measured with the TiC powder (Fig. 3). Only two Furthermore, the main difference between individual pro- deconvolution procedures (the Stokes method and the lin- cedures follows from the amount of smoothing, which is ear combination of instrumental profiles) are compared automatically included in the respective deconvolution here. The reason is that the decomposition of the diffraction routine. The quality of deconvoluted patterns (and the profile into a series of harmonic functions did not yield a match between the re-convoluted and the experimental in- convergent solution up to the 40th order of Fourier coeffi- tensities) depends on the shape of physical profile. If the cients. Linear combination of instrumental profiles yielded physical broadening is much larger than the instrumental apparently much smoother deconvoluted profile than the one, also the deconvolution procedures with high amount Stokes method. However, the level of smoothing for the of smoothing (e.g., the decomposition using Fourier expan- linear combination was too high, which affected the shape

Ó Krystalografická spoleènost 48 Materials Structure, vol. 7, number 1 (2000)

250 250 120 1.0 200 0.8 200 150 100 0.6 100 0.4

50 80 0.2 150 Concentration

0 0.0 121.2 121.3 121.4 121.5 121.6 60 0.0 0.2 0.4 0.6 0.8 1.0 100 Distance (rel.u.) 40 Intensity (cps) Intensity (cps) 50 20

0 0 120.5 121.0 121.5 122.0 122.5 123.0 123.5 119.0 119.5 120.0 120.5 121.0 121.5 122.0

1200 500

900 400

300 600 (d) (b) 200 (c) 300 Intensity (cps)

Intensity (cps) (b) (a) 100 0 (a) 0 120.5 121.0 121.5 122.0 122.5 123.0 123.5 119.0 119.5 120.0 120.5 121.0 121.5 122.0 o Diffraction angle ( 2Q) Diffraction angle (o2q) Figure 3. Figure at the top: experimental data (open circles) and Figure 4. At the top: diffraction profile measured on the (105) re- re-convoluted intensities (solid lines) measured with TiC0.96. flection of hexagonal MoC0.51 (open circles) compared with the Figure at the bottom: deconvoluted profiles obtained using the re-convoluted profile (dashed line) and with the diffraction pro- Stokes method (a) and the linear combination of instrumental file obtained from the concentration profile fitting (solid line). profiles (b). The differences between the re-convoluted profiles Inset: the calculated shape of the concentration profile of carbon. On the concentration scale, “0” means the minimum carbon con- and the experimental data are shown in the inset (solid line is for centration, “1” the maximum one. At the bottom: physical pro- the Stokes method, dashed line for the linear combination of in- files obtained from deconvolution by using linear combination of strumental efffects). instrumental profiles (a), Fourier expansion (b), Stokes method (c) and concentration profile fitting (d). of the re-convoluted profile. The re-convoluted peak was broader than from the Stokes method having a lower maximum (see inset to Fig. 3). this approach was first applied to study the homogeneity of From the point of view of the computer time consump- the hexagonal MoC0.51 powder. The shape of the tion, the Stokes method is the fastest one, followed by the deconvoluted profile (Fig. 4) is characteristic for a pres- Fourier decomposition. The slowest method is the decom- ence of a steep concentration gradient in the sample. Bulk position of experimental profiles into the linear combina- of the sample has the lattice parameters a = (3.01028 ± tion of instrumental profiles, as large matrix equations are 0.00004) Å and c = (4.73569 ± 0.00004) Å, which yield the solved upon this approach. In particular cases, the most ef- interplanar spacing of 0.8902 Å for the (105) planes. The ficient technique is the concentration profile fitting (Sec- above lattice parameters are the result of Rietveld refine- tion 2.4.), which works with convolutions of the physical ment. The profile asymmetry towards smaller diffraction profile calculated using equations (22) – (24) with the angles (towards larger interplanar spacing and lattice pa- Cauchy function (25) and with the instrumental profile. Re- rameter) agrees well with the positive slope, which de- sults of this method are reported in the next Section. scribes the dependence of lattice parameters in MoC1-x on the carbon concentration [11]. The concentration profile 4.2. Results of the concentration profile fitting fitting yielded the minimum and the maximum interplanar spacing (for the minimum and the maximum carbon con- The concentration profile fitting can be regarded as a centration) dmin = 0.8893 Å and dmax = 0.8903 Å. method, which yields physically reliable information on Another sample with a concentration gradient had the the real composition and the real structure of the sample. average composition of (Ti0.85 Mo0.15) C. From the shape of However, its use is limited to the cases, in which an appro- the diffraction profile, it is evident that the concentration priate diffusion model can be created. Employing this tech- profile is much broader than in the previous case (with re- nique, it is useful if the concentration profile fitting spect to the size of the diffusion couple), which indicates an succeeds a deconvolution method in order to get funda- approaching diffusion process (Fig. 5). The lattice parame- mental characteristics of the diffusion model. In this test, ter for the lowest concentration of molybdenum in the sample is equal to 4.3192 Å, which is lower than the lattice

Ó Krystalografická spoleènost Materials Structure, vol. 7, number 2 (2000) 49

100 100 1.0 1.0 0.8 90 0.8 80 0.6 80 0.6 0.4 70 0.4

Concentration 0.2 60 0.2 60 Concentration 0.0 0.0 50 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Distance (rel.u.) 40 Distance (rel.u.) 40 30 Intensity (cps) 20 Intensity (cps) 20 10 0 0 120 121 122 123 124 125 121 122 123 124 125 126 127 180 240 160 200 140 120 160 100 80 120 (d) 60 (d) 80 (c) Intensity (cps) 40 (c) Intensity (cps) 40 (b) 20 (b) (a) (a) 0 0 121 122 123 124 125 126 127 120 121 122 123 124 125 o q Diffraction angle (o2q) Diffraction angle ( 2 ) Figure 5. At the top: the intensity band measured on the (422) line of (Ti0.85 Mo0.15) C (open circles). The re-convoluted profile Figure 6. Diffraction profile measured with quaternary (Ti0.75 is indicated by the dashed line; the diffraction profile obtained Mo0.25)(C0.7 N0.3) annealed for 48 h at 1450°C in argon atmo- from the concentration profile fitting is indicated by the solid sphere (open circles). The re-convoluted diffraction profile is line. The schematic shape of the concentration profile is shown in plotted by dashed line, the diffraction profile corresponding to the the inset. At the bottom: physical profiles obtained from linear concentration profile shown in the inset by the solid line. At the combination of instrumental profiles (a), from the Fourier expansion (b), from the Stokes method (c) and from the concentration bottom: physical profiles obtained from the concentration profile profile fitting (d). fitting (a), from the Stokes method (b), from the Fourier expansion (c) and from the linear combination of instrumental profiles (d). parameter of TiC (a = 4.3278 Å). This confirms that increasing amount of molybdenum in the host structure of TiC decreases the lattice parameter of the ternary face cen- a = 4.2790 Å, which corresponds to the stoichiometric ratio tred cubic (Ti ,Mo ) C phase. The lattice parameter cor- x 1–x x = 0.74 within experimental accuracy. The concentration responding to the highest analysed concentration of profile fitting yielded a similar shape of the physical profile molybdenum in the sample is 4.2996 Å. As the concentra- like the deconvolution routines. However, as a simple dif- tion profile is related to the concentration of molybdenum, fusion model assuming only presence of a single phase the change of lattice parameter (change of the interplanar with a concentration gradient was employed, the re-convo- spacing) with concentration is negative, which implies the luted diffraction profile did not fit the experimental data physical line broadening toward higher diffraction angles. properly. The discrepancy can be also seen on the too wide The last two examples illustrate the influence of in- range of lattice parameters obtained from the concentration creasing diffusion time on the shape of diffraction and con- profile fitting. The lattice parameter for the minimum concentration profiles. The “overall” composition of both centration of titanium is 4.3018 Å, that for the maximum samples was (Ti Mo )(C N ), the same as for the 0.75 0.25 0.7 0.3 concentration of Ti 4.2761 Å. first sample presented in Section 4.1. The samples were an- The deconvoluted pattern of the latter sample annealed nealed for 48 or 192 h at 1450°C in one bar Ar. In the first for 192 h (Fig. 7) indicates presence of the residual MoC1–x one, the deconvolution procedures discovered presence of with the lattice parameter 4.2770 Å. This value corre- two (Ti, Mo) (C, N) phases and a MoC1–x phase (Fig. 6). sponds to a slightly lower carbon content in MoC1–x, x = The phase composition is similar to that in the non-an- 0.73. The deconvoluted diffraction profiles of the quater- nealed sample. After annealing the sample for 48 h at nary (Ti, Mo) (C, N) build a continuous band starting at the ° a 1450 C, the titanium-rich ’–(Ti, Mo) (C, N) had the lat- diffraction angle of 122.80° and ending at 123.26°. The a tice parameter a = 4.2897 Å, the molybdenum-rich ”–(Ti, corresponding limits for the lattice parameters are 4.2980 Mo) (C, N) the lattice parameter a = 4.2997 Å. The Å for the molybdenum-rich a”–(Ti, Mo) (C, N) and 4.2887 non-stoichiometric cubic MoC1–x had the lattice parameter

Ó Krystalografická spoleènost 50 Materials Structure, vol. 7, number 2 (2000)

which has a negligible line broadening due to the crystallite size or microstrain. Comparison of the deconvolution procedures with

1.00 concentration profile fitting illustrated the powerfulness of 100 0.95 the concentration profile fitting. However, the reliability of 0.90 results strongly depends on suitability of the microstructure 80 0.85 0.80 model. Thus, a combination of both techniques shall be rec- Concentration 60 0.75 ommended. A deconvolution method should be applied to 0.0 0.2 0.4 0.6 0.8 1.0 Distance (rel.u.) get preliminary data, which are necessary to build an ap- 40 propriate microstructure model. For the final refinement of

Intensity (cps) 20 the model, the fitting of calculated diffraction profiles on the experimental data yields more reliable results than the 0 deconvolution. 121 122 123 124 125 126 200 Acknowledgements

160 The author would like to thank to Dr. Pablo Wally (Vienna University of Technology, Austria) for supplying the sam- 120 (d) ples. The financial support of the project # GAUK-154/99 (c) 80 by the Grant Agency of the Charles University Prague is greatly appreciated. Intensity (cps) (b) 40 (a) 0 References 121 122 123 124 125 126 in o Q 1. P. Scardi: A new whole powder pattern fitting approach Diffraction angle ( 2 ) R.L. Snyder, J. Fiala and H.J. Bunge (ed.), Defect and Microstructure Analysis by Diffraction, International Union Figure 7. Diffraction profile measured with the quaternary (Ti0.75 of Crystallography, Oxford University Press, 1999. ° Mo0.25)(C0.7 N0.3) annealed for 192 h at 1450 C in argon atmo- 2. P. Scardi and M. Leoni, J. Appl. Cryst. 32 (1999) 671. sphere (open circles). The re-convoluted diffraction profile (plot- J. Appl. Cryst. ted by dashed line) reconstructs the experimental data quite well. 3. Y.H. Dong and P. Scardi, 33 (2000) 184. The difference between the diffraction profile calculated for the 4. J.I. Langford, D. Louër and P. Scardi, J. Appl. Cryst. 33 concentration profile shown in the inset (plotted by solid line) and (2000) 964. the experimental data is much larger, which is due to the simplic- 5. M. Èeròanský: Restoration and preprocessing of physical ity of the structure model. Figure at the bottom shows physical profiles from measured data in R.L. Snyder, J. Fiala and profiles obtained from the concentration profile fitting (a), from H.J. Bunge (ed.), Defect and Microstructure Analysis by the Stokes method (b), from the Fourier expansion (c) and from Diffraction, International Union of Crystallography, Oxford the linear combination of instrumental profiles (d). University Press, 1999. 6. F. Sánchez-Bajo and F.L. Cumbrera, J. Appl. Cryst. 33 Å for the titanium-rich a’–(Ti, Mo) (C, N). Typically, the (2000) 259. physical profile takes a form of a continuous band if infini- 7. A.R. Stokes, Proc. Phys. Soc. 61 (1948) 382. tesimal diffracting volumes are the same for all concentra- 8. A. Savitzky and M.J.E. Golay, Anal. Chem. 36 (1964) tions. This happens if the concentration profile is flat; i.e. if 1627. it can be approximated by a linear function. This is the case The Mathematics of Diffusion nd for “saturated” diffusion at longer diffusion times. As in the 9. J. Crank: ,2 ed., Oxford University Press, Oxford, United Kingdom, 1987. previous case, ignoring the presence of the residual MoC1–x in the diffusion model increases the range of lattice param- 10. P. Wally, PhD Thesis, Vienna University of Technology, eters obtained from the concentration profile fitting. The 1995. analysed minimum and maximum concentrations were amin 11. E. Rudy, Journal of the Less-Common Metals 33 (1973) = 4.2829 Å and amax = 4.3020 Å. 43.

5. Concluding remarks

Comparison of three deconvolution procedures has shown that the deconvoluted physical profiles differ only in extreme cases, when a high degree of smoothing is unfavour- able. This is the case, for instance, if the pure physical profile is very narrow (approaching the Dirac d-distribu- tion), or if one of the edges of the physical profile is very steep. Steep edges of physical profiles are particularly characteristic for materials with concentration profile,

Ó Krystalografická spoleènost Materials Structure, vol. 7, number 2 (2000) 51

Ó Krystalografická spoleènost