DECONVOLUTION VERSUS CONVOLUTION A COMPARISON FOR MATERIALS WITH CONCENTRATION GRADIENT

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1 Materials Structure, vol. 7, nuber () 43 DECONVOLUTION VERSUS CONVOLUTION A COMPARISON FOR MATERIALS WITH CONCENTRATION GRADIENT David Raaa Departent o Electronic Structures, Faculty o Matheatics and Physics, Charles University, Ke Karlovu 5, CZ Prague, E-ail: raaa@karlov..cuni.cz Abstract Eiciency o three deconvolution ethods used in X-ray powder diraction analysis is copared or aterials with concentration gradient. The irst deconvolution ethod shown in this coparison is a odiication o the classical Stokes ethod. In two other ethods, the re-convoluted intensities are itted to the easured data using the least-square procedure. The deconvoluted proile is represented either by a linear cobination o haronic unctions or by coeicients describing the decoposition o easured proile into the basis o instruental unctions. The results o deconvolution ethods are urther copared with results o an alternative approach, which is a convolution o the instruental proile with an analytical unction describing the diraction on saples with concentration gradient. In this case, tuneable paraeters o the analytical unction ollowing ro the icrostructure odel are varied to arrive at the best atch between the easured and re-convoluted intensities. Analysing diraction proiles easured on aterials with concentration gradient, the ost reliable inoration on the concentration proile was obtained using a cobination o the above approaches. The direct deconvolution was used to get basic inoration on the shape o concentration proiles. The inal or o the concentration proiles was adusted by reining ree paraeters o the diusion odel. Keywords: X-ray powder diraction, convolution, deconvolution, and concentration gradient. 1. Introduction Decoposition o diraction proiles is a typical task in the diraction proile analysis. The ain proble is to separate the instruental broadening ro the easured diraction pattern in order to obtain the pure physical proile containing inoration on icrostructure o the aterial. This eans that the well-known equation or convolution hx ( ) g ( ygx ) ( ydy ) (1) has to be solved or (y), which represents the shape o the physical proile. Function h (x) describes the easured diraction proile; g (x-y) is the instruental unction (response o the apparatus). The instruental unction covers usually the line broadening due to the spectral width o radiation, angular resolution o the diractoeter optics as well as a variety o aberrations. Typical exaples or aberrations causing the instruental broadening in the conventional Bragg-Brentano geoetry are the saple transparency and the local saple displaceent due to the lat specien. There are two dierent approaches how to treat the experiental diraction pattern aected by instruental broadening. The irst one eploys direct deconvolution. The second one solves Eq. (1) through convolution o physical and instruental proiles, assuing that the physical proile can be approxiated by a unction o a liited nuber o paraeters, which ollow ro the icrostructure odel o the respective aterial. Although the latter approach based on convolutions becoes ore and ore requently used in the last tie (see, e.g., Res. [1-4]), they are still nuerous applications, or which the use o a deconvolution technique is inevitable. The reason is that a proper icrostructure odel cannot be built without an approxiate knowledge o the deconvoluted proiles, an exaple o the being aterials with concentration gradient. Regarding the direct deconvolution o diraction proiles, the excellent coprehensive overview o deconvolution ethods by M. Èeròanský [5] can be recoended to the reader s attention. Aong the large nuber o deconvolution ethods discussed in [5], the ost popular ethods work with the least-square itting o re-convoluted proiles to the easured intensities, see Re. [6] and the reerences therein. The least-square ethod is suitable to solve integral equations o the irst kind, which is also the case o the equation or convolution (1). In this paper, three procedures used or deconvolution o powder diraction patterns are copared. The Stokes ethod [7] working with soothed experiental and instruental data, decoposition o physical proiles into a Fourier series [6] and decoposition o experiental data into a linear cobination o instruental proiles. Materials with concentration gradients were selected or this coparison, as the presence o concentration proiles iplies strongly asyetrical physical broadening, which is not avourable or ost deconvolution ethods. Finally, a icrostructure odel was built or saples with concentration gradient, in which the corresponding physical proile was calculated. Free paraeters o the icrostructure odel were reined to arrive at the best atch between the experiental and the re-convoluted data.. Copeting deconvolution procedures.1. Stokes ethod with Gaussian soothing The original Stokes ethod [7] utilises the well-known property o the Fourier transoration the Fourier trans- Krystalograická spoleènost

2 44 Materials Structure, vol. 7, nuber () oration o convolution o two unctions can be expressed as ultiplication o Fourier transors o these unctions: FT( h) FT( g) FT( ). FT( g) () Thus, the pure physical unction can be obtained ro the easured proile h and ro the instruental proile g using the orula: FT FT( h) FT FT( g ) 1 1 H G Throughout the text, the sybols FT and FT 1 denote the Fourier and the inverse Fourier transoration. The capitals F, G and H denote the Fourier transors. Due to the ill-posed nature o deconvolution, which is caused ainly by the truncation o diraction proile and by the presence o noise in both the experiental and instruental data, the orula (3) does not yield useul results. The solution is oscillating as a rule; the aplitude o the oscillations being coparable with the axiu o the deconvoluted unction. In order to overcoe probles with truncation o diraction proiles, the experiental data ust be pre-processed. In the irst step, the background is subtracted and the issing arginal data are illed by zeros. To avoid probles with the noise, the input data are oten soothed. This eans that typically the high-requency noise is reoved. Such a iltering o input data was used in the odiied Stokes procedure presented in this contribution. Fourier transors o both experiental and instruental proiles were ultiplied by Gaussian unctions: (3) t t F' Hexp Gexp (4) h g This corresponds to the ollowing cobination o the Fourier transors: FT( h) FT( sh ) FT( h sh ) F' FT( ') FT( g ) FT( s ) FT( g s ) The unctions s h and s g are the inverse Fourier transors o the Gaussian unctions ultiplying the unctions H and G in Eq. (4). Further, it holds as a consequence o Eq. (5): g g (5) FT( hs ) FT( ') FT( gs ) FT( ' g s ) (6) h g g Applying inverse Fourier transoration on Eq. (6), we will get a siilar equation or convolutions: 'gsg hsh hs sg (7) The right hand o Eq. (7) was rewritten to arrive at orally sae convolutions on the let and right side. The let and right hands o Eq. (7) are equal i ' s (8) Equation (8) speciies the aount o soothing in the deconvoluted proile. It can siply be shown that the deconvoluted proile is soothed by the Gaussian unction i both g and h are soothed by Gaussian unctions. 1 s s s FT [ FT( s ) FT( s )] h g g t FT 1 exp t exp g 1 g FT exp t FT g 1 t exp h (9) Thereore, the soothing paraeter is given by reciprocal dierence o the paraeters h and g : g h g h (1) The soothing paraeter describes the iltering o the Fourier transor o the deconvoluted proile: t FT( ') F ' F exp (11) To derive the exact or o the soothing unction or,we ust peror the inverse Fourier transoration o Eq. (11): where s FT t ' FT exp 1 1 s, (1) t t 1 exp exp 1 x exp 4 (13) It ollows ro Eq. (13) that the soothing unction is Gaussian in or. The area below the unction s is equal to unity, x sdx exp dx 1, (14) 4 i the inverse Fourier transoration is deined by the orula: FT ( F ) F ( t )exp( itx) dt 1 1 (15) In such a case, the soothing has no eect on the total integrated intensity o the deconvoluted proile. Krystalograická spoleènost

3 Materials Structure, vol. 7, nuber () 45 Equation (1) has several iportant consequences. The deconvoluted proile is autoatically soothed i the experiental data and the instruental proile are soothed. For correct soothing, h ust be less than g, which eans that the experiental unction h ust be iltered to lower requencies than the instruental unction g. I h and g have approxiately the sae value, is very high; thus, the physical unction is not soothed at all. I g is uch larger than h, the soothing o is extree it is the sae as the soothing o h. The coputer code or deconvolution routine based on the odiied Stokes ethod was written in the MATLAB environent. Fourier transors were calculated using the internal MATLAB unction or the ast Fourier transoration and the inverse Fourier transoration... Decoposition o experiental proiles using Fourier expansion The second procedure tested in this paper is the decoposition o diraction proiles using expansion o the physical unctions into the Fourier series. This approach is especially advantageous i the subsequent ethod o data reduction works with the Fourier coeicients, or instance i the Warren-Averbach analysis is applied. The coputing routine used in the test is copletely based on the technique, published by Sánches-Bao and Cubrera [6]. The only exception is that we took the instruental proile in its easured or, instead o approxiating it by an expansion into the Herite polynoials like in [6]. Eploying the expansion o the pure physical proile into a Fourier series, unction takes the or: ( x) C cos( x) S cos( x) with 1 (16) The sybol in Eq. (16) deines the interval, in which the experiental proile was easured. Regarding the equation or convolution (1), the experiental proile can be written as hx ( ) gx ( y) C cos( y) S sin( y) dy 1 C g( x y)cos( y) dy 1 S g( x y)sin( y) dy (17) Equation (17) represents convolutions o the instruental unction g with the basis o haronic unctions: h( x) C [ gcos( y)] S [ gsin( y)] 1 (18) Within the deconvolution procedure, Eq. (18) is solved or coeicients C and S in the least-square sense. Application o this deconvolution routine iplies also an autoatic soothing o deconvoluted data. The aount o soothing is deterined by a nuber o haronic unctions ( in Eq. (16)-(18)). The lower, the higher is the degree o soothing. The related coputer code was written in the MATLAB environent. The convolutions in Eq. (18) were calculated with the aid o the ast Fourier transoration and the inverse ast Fourier transoration (see Eq. ()). The least-square reineent o the coeicients C and S is perored successively or increasing nuber o haronic unctions,. Because o the large expected asyetry o the deconvoluted proiles in saples with concentration gradient, the haronic unctions up to the 4 th order were tested. The best set o C and S was selected according to the lowest su o residuals between the original data h and the re-convoluted data h obtained ro the back convolution: N S [ h'( x) h( x)] [ gh] in n1 N n1 (19) The deconvoluted proile is calculated ro Eq. (16). The suation in Eq. (19) is perored over all experiental data..3. Experiental proile regarded as a linear cobination o instruental proiles The last deconvolution procedure copared here solves the equation or convolution (1), rewritten into the or used or discrete data: h n n g () In the atrix representation, Eq. () takes the or: g g 1 g g 3 1 h1 g1 g g 1 g h g g1 g g 1 h 3 3 g3 g g1 g 4 h4 (1) Consequently, the Dirac -unction corresponds to an identity atrix or the instruental proile and to a single one ebedded in a zero atrix or the physical proile. However, Eq. () represents an ininite syste o linear equations. Moreover, the syste o equations is underestiated, as the data contain noise (rando errors). Thereore, an assuption ust be done to reduce the nuber o equations (the nuber o paraeters) in the linear cobination (). In contrast to the deconvolution procedures eploying the Fourier analysis, we can assue here that the deconvoluted intensities are equal to zero outside a certain range. This assuption reduces substantially the nuber o coluns in the G atrix in Eq. (1). It eans that the syste o linear equations () becoes overestiated. Thus, Krystalograická spoleènost

4 46 Materials Structure, vol. 7, nuber () it can be solved using the least-square ethod. However, it is still necessary to ilter the noise in the deconvoluted data. For iltering, the Fourier soothing or the Golay-Savitzky ethod [8] were alternatively applied. The Fourier soothing eans a convolution o the deconvoluted proile with a Gaussian unction like in the odiied Stokes ethod described above. The Golay-Savitzky ethod represents the direct soothing in the reciprocal space..4. Reineent o ree paraeters o the diusion odel A siple diusion odel was used to describe the concentration proiles in the saples. One-diensional diusion with concentration-independent diusion coeicients in crystallites having the sae size was assued. In that case, the concentration proile takes the or o the error unction (see, e.g. [9]), c erc( x/ Dt), () where x eans the distance in individual crystallites (in individual diusion couples), D the diusion coeicient and t the diusion tie. The doinant actor odiying the shape o concentration proiles and thereore the shape o the corresponding diraction proiles is the ratio o the total length o diusion couples to the square root o Dt. For the concentration ranges investigated here, all particular coponents (TiN, TiC, Ti (C, N), (Ti, Mo) C, (Ti, Mo)(C, N) and MoC 1 x ) crystallise with the ace-centred cubic structure. Their lattice paraeters ollow the Vegard rule quite well [1]. Thus, the dependence o the interplanar spacing on the concentration is represented by the linear unction: d d d1 c (3) In Eq. (3), d represents the interplanar spacing or c =, d 1 the change in interplanar spacing with the concentration o in-diusing or out-diusing species. Neglecting the acroscopic concentration depth proile in the saple, the inluence o absorption o radiation on the shape o diraction proiles can be oitted. Then the diracted intensities depend linearly on the dierential volue o aterial having a certain concentration: dx dx I(sin ) V d(sin ) d( / d) (4) The last part o Eq. (4) ollows ro the Bragg equation. In a urther step, additional line broadening due to the sall size o crystallites ust be taken into account. Thus, the diraction proiles calculated according to Eq. (4) were convoluted with the Cauchy unction: w y ( x w ) (5) Its area is noralised to unity. The Cauchy unction should be exchanged by the Gaussian unction i the additional physical broadening is caused by icro-stress rather than by the sall size o crystallites. In Eq. (5), diraction angle, x, is calculated in the units sin ; the sae units are used also or the width o the Cauchy unction, w. The above odel is characterised by ive ree paraeters (K = (Dt), d, d 1, w and a scale actor noralising the calculated intensities to the experiental ones). Note, that a physically true odel should include two paraeters ore: the iniu and the axiu concentration o the in-diusing species. However, the iniu and the axiu concentrations would strongly correlate with the paraeter o the diusion odel, K, and with the paraeters d and d 1. Thereore, the paraeters related to the interplanar spacing should not be reined in a true odel. They should be taken ro an independent experient. On the contrary, it can be shown that variation in the iniu and axiu concentrations has a siilar inluence on the shape o the resulting diraction proile like the changes in d and d 1. Upon reineent, the linear least-squares ethod was iteratively applied to the irst-order Taylor expansion o the calculated diraction proile: calc eas calc I S I I p in (6) p n The increents to the reined paraeters, p, ollows ro the solution o the atrix equation: (7) n I p calc i I p calc eas calc I pi ( I I ) p All convolutions were calculated nuerically using the ast Fourier transoration and the inverse ast Fourier transoration (according to Eq. ()). 3. Experiental details X-ray diraction experients were perored with a conventional Bragg-Brentano diractoeter (XRD-7, Seiert/ Freiberger Präzisionsechanik). The nickel-iltered radiation o copper anode with a divergence o 1 was transitted by a Soller colliator in priary bea and diracted on the saple. The optics o the diracted bea consisted o a secondary Soller colliator and the receiving slit o.15. The selected angular range was scanned repeatedly with the step size o.1 in. The total counting tie exceeded 6 seconds per step. To arrive at a good experiental resolution, relections in the high-angle region were selected or the easureent. The instruental broadening was easured on the standard saple o cubic LaB 6 oered by NIST. The instruental proile easured on the diraction line (33) is shown in Fig. 1. The easureent was carried out in the sae diraction geoetry and with the sae instruental paraeters, which were used or saples with concentration gradient. For the standard, the counting tie was n calc i Krystalograická spoleènost

5 Materials Structure, vol. 7, nuber () 47 Intensity (counts/s) Diraction angle ( o ) seconds per step. To avoid probles with dierent angular separation o spectral lines in the K 1 /K doublet due to dierent diraction angles o the respective aterials, the data were irst converted into the equidistant scale in sin. The ollowing saples were used to test the deconvolution procedures cubic TiC 1 x, hexagonal Mo C, ternary ixture (Ti x Mo 1 x ) C and quaternary ixture (Ti x Mo 1 x ) (C y N 1 y ). According to the cheical analysis, TiC 1 x contained weight percent o carbon, this corresponds to the stoichioetric ratio TiC.96. The analysed coposition o the MoC 1 x powder was MoC.51 (5.7 weight percent o carbon). The ternary (Ti.85 Mo.15 ) C was produced by hot pressing TiC.96, MoC.51 and graphite in argon atosphere at C. Aterward, it was annealed in argon or 16 days at 145C. The last three saples with the noinal coposition (Ti.75 Mo.5 )(C.75 N.5 ) were prepared by hot pressing the ixture o 5 ole o (Ti.5 Mo.5 ) C and 5 ole o Ti (C.5 N.5 ) in argon atosphere at C. Subsequently, two o these three saples were annealed or 48 and 19 h, respectively, at 145C inone bar Ar. 4. Results and discussion LaB 6,33 Figure 1. Instruental broadening easured on the relection (33) o cubic LaB Coparison o deconvolution procedures Diraction angle ( o ) Figure. Experiental data (open circles) and re-convoluted intensities (solid line) easured with the hot-pressed quaternary (Ti.75 Mo.5 )(C.7 N.3 ) (Figure at the top). At the botto: physical proiles obtained ro the experiental data by using the Stokes ethod, the Fourier expansion and the linear cobination o instruental proiles. All deconvolution procedures outlined above yielded very siilar results or the aority o experiental data. Nevertheless, i we copare only the dierences between the experiental and the re-convoluted data, the ost successul deconvolution routine was that based on the linear cobination o instruental proiles (Section.3.). Furtherore, the ain dierence between individual procedures ollows ro the aount o soothing, which is autoatically included in the respective deconvolution routine. The quality o deconvoluted patterns (and the atch between the re-convoluted and the experiental intensities) depends on the shape o physical proile. I the physical broadening is uch larger than the instruental one, also the deconvolution procedures with high aount o soothing (e.g., the decoposition using Fourier expansion with a sall nuber o Fourier coeicients) yield excellent results. This can be illustrated on the hot-pressed non-annealed quaternary (Ti.75 Mo.5 )(C.7 N.3 ). Ater deconvolution, three distinct intensity axia are clearly visible at 1.49, 13.9 and (Fig. ). These peaks indicate the presence o two (Ti, Mo) (C, N) phases, which are usually denoted as Ti-rich (Ti, Mo) (C, N) (here with the lattice paraeter a = 4.91 Å) and Mo-rich (Ti, Mo) (C, N) (here with a = Å), and the presence o non-stoichioetric cubic MoC.74 (a = Å) [11]. On the other hand, the Stokes deconvolution with Gaussian soothing (Section.1.) was the ost successul routine in extree cases, when the shape o the experiental proile approached the shape o the instruental proile. Generally, this ethod yields the best results i the pure physical proile is very narrow. Such a diraction proile was easured with the TiC powder (Fig. 3). Only two deconvolution procedures (the Stokes ethod and the linear cobination o instruental proiles) are copared here. The reason is that the decoposition o the diraction proile into a series o haronic unctions did not yield a convergent solution up to the 4 th order o Fourier coeicients. Linear cobination o instruental proiles yielded apparently uch soother deconvoluted proile than the Stokes ethod. However, the level o soothing or the linear cobination was too high, which aected the shape Krystalograická spoleènost

6 48 Materials Structure, vol. 7, nuber 1 () Diraction angle ( o ) Figure 3. Figure at the top: experiental data (open circles) and re-convoluted intensities (solid lines) easured with TiC.96. Figure at the botto: deconvoluted proiles obtained using the Stokes ethod and the linear cobination o instruental proiles. The dierences between the re-convoluted proiles and the experiental data are shown in the inset (solid line is or the Stokes ethod, dashed line or the linear cobination o instruental eects). o the re-convoluted proile. The re-convoluted peak was broader than ro the Stokes ethod having a lower axiu (see inset to Fig. 3). Fro the point o view o the coputer tie consuption, the Stokes ethod is the astest one, ollowed by the Fourier decoposition. The slowest ethod is the decoposition o experiental proiles into the linear cobination o instruental proiles, as large atrix equations are solved upon this approach. In particular cases, the ost eicient technique is the concentration proile itting (Section.4.), which works with convolutions o the physical proile calculated using equations () (4) with the Cauchy unction (5) and with the instruental proile. Results o this ethod are reported in the next Section. 4.. Results o the concentration proile itting The concentration proile itting can be regarded as a ethod, which yields physically reliable inoration on the real coposition and the real structure o the saple. However, its use is liited to the cases, in which an appropriate diusion odel can be created. Eploying this technique, it is useul i the concentration proile itting succeeds a deconvolution ethod in order to get undaental characteristics o the diusion odel. In this test, Distance (rel.u.) (d) Diraction angle ( o ) Figure 4. At the top: diraction proile easured on the (15) relection o hexagonal MoC.51 (open circles) copared with the re-convoluted proile (dashed line) and with the diraction proile obtained ro the concentration proile itting (solid line). Inset: the calculated shape o the concentration proile o carbon. On the concentration scale, eans the iniu carbon concentration, 1 the axiu one. At the botto: physical proiles obtained ro deconvolution by using linear cobination o instruental proiles, Fourier expansion, Stokes ethod and concentration proile itting (d). this approach was irst applied to study the hoogeneity o the hexagonal MoC.51 powder. The shape o the deconvoluted proile (Fig. 4) is characteristic or a presence o a steep concentration gradient in the saple. Bulk o the saple has the lattice paraeters a = (3.18.4) Å and c = ( ) Å, which yield the interplanar spacing o.89 Å or the (15) planes. The above lattice paraeters are the result o Rietveld reineent. The proile asyetry towards saller diraction angles (towards larger interplanar spacing and lattice paraeter) agrees well with the positive slope, which describes the dependence o lattice paraeters in MoC 1-x on the carbon concentration [11]. The concentration proile itting yielded the iniu and the axiu interplanar spacing (or the iniu and the axiu carbon concentration) d in =.8893 Å and d ax =.893 Å. Another saple with a concentration gradient had the average coposition o (Ti.85 Mo.15 ) C. Fro the shape o the diraction proile, it is evident that the concentration proile is uch broader than in the previous case (with respect to the size o the diusion couple), which indicates an approaching diusion process (Fig. 5). The lattice paraeter or the lowest concentration o olybdenu in the saple is equal to Å, which is lower than the lattice Concentration Krystalograická spoleènost

7 Materials Structure, vol. 7, nuber () paraeter o TiC (a = Å). This conirs that increasing aount o olybdenu in the host structure o TiC decreases the lattice paraeter o the ternary ace centred cubic (Ti x,mo 1 x ) C phase. The lattice paraeter corresponding to the highest analysed concentration o olybdenu in the saple is Å. As the concentration proile is related to the concentration o olybdenu, the change o lattice paraeter (change o the interplanar spacing) with concentration is negative, which iplies the physical line broadening toward higher diraction angles. The last two exaples illustrate the inluence o increasing diusion tie on the shape o diraction and concentration proiles. The overall coposition o both saples was (Ti.75 Mo.5 )(C.7 N.3 ), the sae as or the irst saple presented in Section 4.1. The saples were annealed or 48 or 19 h at 145C in one bar Ar. In the irst one, the deconvolution procedures discovered presence o two (Ti, Mo) (C, N) phases and a MoC 1 x phase (Fig. 6). The phase coposition is siilar to that in the non-annealed saple. Ater annealing the saple or 48 h at 145C, the titaniu-rich (Ti, Mo) (C, N) had the lattice paraeter a = Å, the olybdenu-rich (Ti, Mo) (C, N) the lattice paraeter a = Å. The non-stoichioetric cubic MoC 1 x had the lattice paraeter Concentration Distance (rel.u.) (d) Diraction angle ( o ) Figure 5. At the top: the intensity band easured on the (4) line o (Ti.85 Mo.15 ) C (open circles). The re-convoluted proile is indicated by the dashed line; the diraction proile obtained ro the concentration proile itting is indicated by the solid line. The scheatic shape o the concentration proile is shown in the inset. At the botto: physical proiles obtained ro linear cobination o instruental proiles, ro the Fourier expansion, ro the Stokes ethod and ro the concentration proile itting (d) Distance (rel.u.) a = 4.79 Å, which corresponds to the stoichioetric ratio x =.74 within experiental accuracy. The concentration proile itting yielded a siilar shape o the physical proile like the deconvolution routines. However, as a siple diusion odel assuing only presence o a single phase with a concentration gradient was eployed, the re-convoluted diraction proile did not it the experiental data properly. The discrepancy can be also seen on the too wide range o lattice paraeters obtained ro the concentration proile itting. The lattice paraeter or the iniu concentration o titaniu is Å, that or the axiu concentration o Ti Å. The deconvoluted pattern o the latter saple annealed or 19 h (Fig. 7) indicates presence o the residual MoC 1 x with the lattice paraeter 4.77 Å. This value corresponds to a slightly lower carbon content in MoC 1 x, x =.73. The deconvoluted diraction proiles o the quaternary (Ti, Mo) (C, N) build a continuous band starting at the diraction angle o 1.8 and ending at The corresponding liits or the lattice paraeters are 4.98 Å or the olybdenu-rich (Ti, Mo) (C, N) and Concentration (d) Diraction angle ( o ) Figure 6. Diraction proile easured with quaternary (Ti.75 Mo.5 )(C.7 N.3 ) annealed or 48 h at 145C in argon atosphere (open circles). The re-convoluted diraction proile is plotted by dashed line, the diraction proile corresponding to the concentration proile shown in the inset by the solid line. At the botto: physical proiles obtained ro the concentration proile itting, ro the Stokes ethod, ro the Fourier expansion and ro the linear cobination o instruental proiles (d). Krystalograická spoleènost

8 5 Materials Structure, vol. 7, nuber () Distance (rel.u.) Å or the titaniu-rich (Ti, Mo) (C, N). Typically, the physical proile takes a or o a continuous band i ininitesial diracting volues are the sae or all concentrations. This happens i the concentration proile is lat; i.e. i it can be approxiated by a linear unction. This is the case or saturated diusion at longer diusion ties. As in the previous case, ignoring the presence o the residual MoC 1 x in the diusion odel increases the range o lattice paraeters obtained ro the concentration proile itting. The analysed iniu and axiu concentrations were a in = 4.89 Å and a ax = 4.3 Å Diraction angle ( o ) Figure 7. Diraction proile easured with the quaternary (Ti.75 Mo.5 )(C.7 N.3 ) annealed or 19 h at 145C in argon atosphere (open circles). The re-convoluted diraction proile (plotted by dashed line) reconstructs the experiental data quite well. The dierence between the diraction proile calculated or the concentration proile shown in the inset (plotted by solid line) and the experiental data is uch larger, which is due to the siplicity o the structure odel. Figure at the botto shows physical proiles obtained ro the concentration proile itting, ro the Stokes ethod, ro the Fourier expansion and ro the linear cobination o instruental proiles (d). Concentration 1. (d) which has a negligible line broadening due to the crystallite size or icrostrain. Coparison o the deconvolution procedures with concentration proile itting illustrated the powerulness o the concentration proile itting. However, the reliability o results strongly depends on suitability o the icrostructure odel. Thus, a cobination o both techniques shall be recoended. A deconvolution ethod should be applied to get preliinary data, which are necessary to build an appropriate icrostructure odel. For the inal reineent o the odel, the itting o calculated diraction proiles on the experiental data yields ore reliable results than the deconvolution. Acknowledgeents The author would like to thank to Dr. Pablo Wally (Vienna University o Technology, Austria) or supplying the saples. The inancial support o the proect # GAUK-154/99 by the Grant Agency o the Charles University Prague is greatly appreciated. Reerences 1. P. Scardi: A new whole powder pattern itting approach in R.L. Snyder, J. Fiala and H.J. Bunge (ed.), Deect and Microstructure Analysis by Diraction, International Union o Crystallography, Oxord University Press, P. Scardi and M. Leoni, J. Appl. Cryst. 3 (1999) Y.H. Dong and P. Scardi, J. Appl. Cryst. 33 () J.I. Langord, D. Louër and P. Scardi, J. Appl. Cryst. 33 () M. Èeròanský: Restoration and preprocessing o physical proiles ro easured data in R.L. Snyder, J. Fiala and H.J. Bunge (ed.), Deect and Microstructure Analysis by Diraction, International Union o Crystallography, Oxord University Press, F. Sánchez-Bao and F.L. Cubrera, J. Appl. Cryst. 33 () A.R. Stokes, Proc. Phys. Soc. 61 (1948) A. Savitzky and M.J.E. Golay, Anal. Che. 36 (1964) J. Crank: The Matheatics o Diusion, nd ed., Oxord University Press, Oxord, United Kingdo, P. Wally, PhD Thesis, Vienna University o Technology, E. Rudy, Journal o the Less-Coon Metals 33 (1973) Concluding rearks Coparison o three deconvolution procedures has shown that the deconvoluted physical proiles dier only in extree cases, when a high degree o soothing is unavourable. This is the case, or instance, i the pure physical proile is very narrow (approaching the Dirac -distribution), or i one o the edges o the physical proile is very steep. Steep edges o physical proiles are particularly characteristic or aterials with concentration proile, Krystalograická spoleènost

9 Materials Structure, vol. 7, nuber () 51 Krystalograická spoleènost