Rietveld refinement of energy-dispersive synchrotron measurements

Daniel Apel*, I, Manuela KlausI, Christoph GenzelI and Davor BalzarII

I Helmholtz-Zentrum Berlin fur Materialien und Energie, Albert-Einstein-Straße 15, 12489 Berlin, GermanyII University of Denver, Dep. Physics and Astronomy, 2112 E Wesley Ave, 80208 Denver, Colorado, USA

Received July 13, 2011; accepted October 4, 2011

Energy-dispersive diffraction / Synchrotron radiation /Rietveld method / Line-profile analysis /Crystal microstructure

Abstract. In the past two decades the energy-dispersivediffraction (EDD) method has become a powerful tool inmany fields of materials research such as residual stress,texture, and crystal structure analysis, because of its favor-able ratio of a comparatively low experimental effort inform of a simple and fixed instrumental setup to a highinformation content included in the measured diffractionpatterns. However, mainly due to the rather poor instru-mental resolution only little work has been done so far toapply the well-established methods of diffraction line pro-file analysis to the EDD data. In the paper, a Rietveldprogram is introduced that allows for particle size andstrain broadening analysis by refining the whole EDDpattern. With the examples of synchrotron measurementsperformed on the materials science beamline EDDI atBESSY II on instrumental standard as well as samplesexhibiting size and/or strain broadened diffraction lines, itis demonstrated that the generalized Thompson, Cox &Hastings approach (TCH) using pseudo-Voigt functionsfor describing the diffraction line profiles yields sound andreliable results on the materials microstructure. For a firstproof of the theoretical assumptions this Rietveld programis based on, the Pawley approach was used to extract thepeak intensities obtained from powder samples affected bymicrostructural broadening. An excellent agreement withthe results of the size-strain round robin was obtained. Fu-ture enhancements of the program code which aim at itsapplication to full residual stress and microstructure analy-sis in the near surface zone and also in the material vol-ume of polycrystalline materials are discussed.

Introduction

In their famous experiment from 1912, the pioneers ofX-ray diffraction Laue, Friedrich and Knipping used poly-chromatic X-rays to demonstrate that the periodic atomic

structure of a single crystal space lattice generates well-defined diffraction patterns when it is illuminated by short-wave X-ray light. Therefore, the first X-ray diffraction ex-periment about 100 year ago at the same time was the firstenergy-selective experiment because any lattice plane (hkl)filters out the wavelength lhkl ¼ hc=Ehkl that fulfillsBragg’s law for the angle 90� � qhkl between its normaland the incoming beam. The reason why it took morethan 50 years since that time before the white beam dif-fraction technique was applied to polycrystalline materialsindependently in 1968 by Giessen & Gordon [1] andBuras et al. [2] is due to the fact that the detection ofEDD patterns (i.e. diffracted intensity as a function of thephoton energy) requires appropriate solid state detectorsystems with efficient energy resolution, which were notavailable in the early years of X-ray diffraction.

The main advantages of the EDD method compared toangle-dispersive diffraction (ADD) are the simple experi-mental setup and the fact that complete diffraction patternswith a multitude of diffraction lines Ehkl are obtained with-in short measuring times under fixed but arbitrary diffrac-tion angles 2q. The bibliography on the first ten years ofEDD compiled by Laine & Lahteenmaki [3] revealed thepotential and the great variety of applications of this meth-od in materials research. In view of the characteristics ofEDD mentioned above it is not astonishing that most ofthese activities focused on such investigations that requirea large number of diffraction lines like crystal structuredetermination [4], texture [5] and quantitative phase analy-sis [6, 7], or on measurements using large and heavy sam-ple environments like high pressure cells, where the angu-lar range for observing the diffracted signal is stronglyconfined [8]. Already at a very early stage of EDD a num-ber of publications dealt with the question of the instru-mental resolution [9, 10], which had been consideredessential for the application of this method to crystal struc-ture determination and refinement [11–14].

Whereas most of the progress made in the first decadeafter the introduction of the EDD method was achieved byemploying conventional X-ray sources, the modern 3rd

generation high-brilliance synchrotron facilities availabletoday provide much better possibilities for advanced whitebeam diffraction experiments. The high photon flux, thenegligible small angular beam divergence, as well as thewide energy spectrum extending from a few eV up to a

934 Z. Kristallogr. 226 (2011) 934–943 / DOI 10.1524/zkri.2011.1436

# by Oldenbourg Wissenschaftsverlag, Munchen

* Correspondence author (e-mail:daniel.apel@helmholtz-berlin.de)

few hundred keV have opened up new prospects for EDDexperiments. For example, concerning X-ray residualstress analysis (XSA), numerous efforts have been madein the past two decades to develop and enhance methodsthat allow for high resolution real space strain [15–17]and stress [18, 19] scanning in the bulk and the near sur-face region of polycrystalline materials, respectively, or,for simultaneous in-situ white beam diffraction [20, 21]and tomography experiments [22]. Other approaches aimat transferring methods that have been designed for depth-resolved XSA in the ADD mode to the EDD. In [23] itwas shown that the so-called Laplace-space methods,which make use of the exponential attenuation of X-raysby matter, can highly benefit from the EDD technique.The lattice strain ehkl obtained for each line Ehkl in thediffraction pattern can be assigned to a different (average)information depth hthkli, which provides additional infor-mation for evaluating residual stress depth profiles.

With the materials science beamline EDDI for EnergyDispersive DIffraction at BESSY II, a new synchrotroninstrument has been developed six years ago, which isespecially dedicated to the analysis of near surface prop-erty gradients in polycrystalline materials [24]. Photon en-ergies up to about 120 keV provided by a 7 T multipolewiggler allow for residual stress gradient analysis in me-chanically surface treated materials [24, 25], thin film sys-tems [26, 27] or high resolution texture depth profiling[28]. However, due to the rather poor intrinsic detectorresolution of DE=E � 10�3 . . . 10�2, until now most ofthe investigations performed in the EDD mode only ex-ploit the absolute shift DEhkl of the diffraction lines or thechanges in the integrated line intensities, DIhkl, in order toevaluate I. kind and averaged II. kind residual stresses andcrystallographic texture, respectively.

Only little work has been done so far in the field ofEDD line profile analysis. Gerward et al. [29] demon-strated that the EDD method can be successfully appliedto analyze crystallite size broadening. The authors investi-gated magnetite powder and used a Gaussian function tofit the line profiles. They separated the size and strain con-tribution to line broadening by plotting FWHM2 as a func-tion of E2, calculating the domain size and microstrain fromthe intercept and slope of a linear plot, respectively. Bycomparing the results for the calculated domain size withresults from ADD and electron microscopy measurementsa very good agreement was found. More than 20 yearslater, Otto [13] first used a pseudo-Voigt function to fit theEDD line profiles. He showed that the peak profiles of thestandard reference material measured at a synchrotronsource were essentially pure Gaussian whereas he foundan increasing Lorentzian contribution to the peak profilesin plastically deformed ductile f.c.c. based materials(Cu3Au and Au). He performed a single-line profile analy-sis to determine the domain size and microstrain.

In contrast to [29, 13], where the analysis was carriedout for each Bragg peak separately, in present work thewhole spectrum is considered. Moreover, our approachfocuses on the adaption of the generally accepted ADDline broadening model in Rietveld refinement to the EDD.Based on comprehensive studies on the influence of var-ious intrinsic and extrinsic parameters on diffraction line

shift and broadening measured with EDD solid state ger-manium detectors [30], a Rietveld program is introducedfor the refinement of EDD data. Since EDD patterns pos-sess high information content in form of a multitude ofdiffraction lines Ehkl including higher order reflections,each of them stemming from different average depths be-low the materials surface, the Rietveld method [31, 32] isparticularly predestinated for EDD data evaluation. Bymeasurements of W and LaB6 standard powder samples,instrumental broadening is demonstrated to be acceptablefor the application of the EDD method to the analysis ofsize and strain broadening.

With the examples of size-broadened CeO2 powder,which had been extensively characterized within the scopeof a round robin test [33] and of ball-milled W––Ni pow-der mixtures showing both size and strain broadening [34],the correctness and capability of the developed formalismis confirmed. The program in its present form is consid-ered as a first important step on the way to a comprehen-sive software package provided in user operation at theEDDI beamline. In its final stage of development it shallallow for whole powder pattern refinement in order to de-termine elastic (macro) strain and stress tensors [35, 36] aswell as microstructural size and strain broadening [37] in-cluding the depth dependence of these material properties.

Model for energy-dispersive Rietveld refinement

A Rietveld program for the evaluation of the EDD datameasured at the EDDI beamline was developed by thefirst author. The program was coded in Matlab R2009b.The principle of the Rietveld method is to minimize theweighted sum of squared residuals (WSS) of the observedintensities yobs

i and the calculated intensities ycalci of the

entire diffraction pattern using a variety of refinable para-meters [1]. The minimization of the WSS is performedaccording to the non-linear least-squares-method using the“trust-region-reflective” algorithm [38, 39] implemented inMatlab. The intensities ycalc

i of the EDD spectrum are cal-culated according to the following formulae:

ycalci ¼ S

PNk¼1

WðEkÞ AkðEÞ

� PkðE; q0ÞMkjFkj2 l3kGkðEi � EkÞ þ ybkg

i : ð1Þ

Where S is the scale factor, W(Ek) is a factor that correctsfor the wiggler spectrum of the 7 T multipole wiggler,Ak(E) is the energy dependent absorption correction,Pk(E, q0) is the polarization factor, Mk is the multiplicity ofthe k-th Bragg reflection, Fk is the structure factor, Gk isthe peak profile function and yi

bkg is the background inten-sity.

In contrast to ADD the 2q angle is constant throughoutthe measurement of the EDD spectrum, thus the Lorentzfactor is a constant and is absorbed in the scale factor S.The polarization factor Pk(E, q0) depends on the polariza-tion of the incident beam P(E) [40]. Due to the horizontalpolarization of the incident synchrotron beam the factorPk(E, q0) is approximately unity for diffraction experimentsperformed in the vertical scattering plane [40]. The absorp-

Rietveld refinement of EDD synchrotron measurements 935

tion correction factor Ak(E) corrects the calculated intensi-ties for the effects of sample absorption and air absorption.As the photon energy changes from reflection to reflectionthe attenuation factor is calculated separately for each re-flection.

The instrumental resolution of EDD experiments de-pends on the intrinsic resolution of the detector as well ason the beam divergence Dq defined by the diffraction geo-metry and setup [13]:

DE

E¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG0

E

� �2

þ 5:55 � e � FE

þ ðcot q DqÞ2s

; ð2Þ

where G0 is electronic noise contribution, F the Fano fac-tor, e: the energy for creating an n�/nþ pair. The early ap-proaches to analyze line broadening by means of EDDusing conventional laboratory X-ray sources suffered fromboth the poor detector resolution and the large beam diver-gence. Therefore, the instrumental line profiles were notonly broad and of Gaussian shape (detector based), butshowed in many cases also more or less pronouncedasymmetry. As a result of the poor resolution, the sample-related (physical) broadening was masked by the instru-mental broadening in most cases, as it gave a dominantcontribution to line broadening.

At the present day, the detector resolution has not im-proved significantly, but the availability of modern syn-chrotron radiation and advances in instrumentation faci-litates a significantly better geometrical resolution whichresults in a much narrower instrumental line profile. More-over, a pure Gaussian function is not suitable anymore tofit the observed line profile but rather some combinationof Gauss and Lorentz functions has to be used. An over-view of the available functions for fitting diffraction linescan be found, for instance, in [41, 42]. Nowadays, themost widely used in Rietveld refinement program, basedon ADD techniques, is the generalized Thompson, Cox &Hastings (TCH) [43] pseudo-Voigt model for diffractionline profiles:

G2G ¼ P=cos2 qþ U tan2 qþ V tan qþW ; ð3Þ

GL ¼ X=cos qþ Y tan qþ Z ; ð4Þ

where G is the full width at half maximum (FWHM) ofthe line profile, P, U, V, W, X, Y and Z are refinable para-meters and G and L denote Gauss and Lorentz profiles,respectively. The parameters X and P will relate to sizebroadening and Y and U to strain broadening. The refinedparameters then can be used to calculate the volume-aver-aged domain size DV according to the well known Scherrerequation [44] and the maximum (upper limit) of strain eaccording to the Stokes–Wilson approximation [45]:

bSð2qÞ ¼ K � lDV cos q

; ð5Þ

bDð2qÞ ¼ 4e tan q ; ð6Þ

where S denotes size, K is the Scherrer constant (close tounity), l is the wavelength, DV the volume-averaged do-main size, D denotes distortion and e the “maximum”strain. bS and bD are the integral breadths of the size- andstrain broadened line profiles, respectively.

Since the TCH pseudo-Voigt model is the most com-mon and established profile function used to determine theaverage domain size and microstrain by means of Rietveldrefinement we have chosen this model to be the basis ofour line broadening analysis method for the refinement ofthe EDD line profiles. Bragg’s law for EDD is given by

Ehkl ½keV� ¼ 6:199

sin q0

1

dhkl ½�A�; ð7Þ

where Ehkl is the X-ray photon energy of the diffractionline, dhkl the interplanar spacing and q0 the (fixed) Braggangle. Using Eq. (7) the formulas (5) and (6) can be trans-formed from the angular scale to the energy scale [29]:

bS ¼K � 6:199 ½keV �A�

DV sin q0; ð8Þ

bDðEÞ ¼ 2e E : ð9Þ

It can be seen from Eq. (8) that the size broadening termbS is independent of energy whereas the strain broadeningterm bD(E) in Eq. (9) depends linearly on the energy. As aresult, the size and strain contributions to the line broad-ening are easily separated.

A simple linear relationship between the FWHM ofEDD measurements GED and the energy, which describesthe experimental findings in most cases with sufficient ac-curacy, is obtained from Eq. (2) by multiplying both sideswith the energy E and Taylor expansion up to the firstorder term. Following [46, 47] one has

GED ¼ U � E þW ; ð10Þ

where U and W are refinable parameters. Based on Eq. (10)and taking advantage of the difference in energy depen-dence of the contributions from size and strain to EDDline broadening (Eqs. (8) and (9)) the size (P, X) and thestrain (U, Y) contributions to the FWHM of the Gauss andLorentz profiles from the TCH pseudo-Voigt model in theEqs. (3) and (4) can be written as:

G2G ¼ Pþ U � E2 ; ð11Þ

GL ¼ X þ Y � E ; ð12Þ

where G is the full width at half maximum (FWHM) ofthe line profile, P, U, X and Y are refinable parameters andG and L denote Gauss and Lorentz profiles, respectively.Eqs. (11) and (12) can be amended by a term dependenton the first power and square root of energy, respectively.These two terms would allow for small corrections to theEqs. (11) and (12), if necessary, and should be refinedonly for the standard sample (i.e. LaB6 or W in thiswork). The parameters X and P will relate to size broad-ening and Y and U to strain broadening. Therefore, to ob-tain the physical contribution to the broadening, the fourparameters in Eqs. (11) and (12) need to be refined. Be-fore estimating physical broadening of a sample under in-vestigation (sam), these refined values have to be correctedfor instrumental effects, which are determined by refine-ment of line profiles of the “standard” sample (stand). Wecan write [33]:

Geff ¼ Gsam � Gstand ; ð13Þ

936 D. Apel, M. Klaus, C. Genzel et al.

where G stands for P, X, U and Y. It should be noted thatEq. (13) is a formal representation of the relationshipswhich exist between the FWHM in the convolution/decon-volution of Gauss (quadratic addition) and Lorentz pro-files (linear addition), respectively. The effective value(eff) depicts the pure physically broadened profile para-meters. As the parameters in Eqs. (11) and (12) areFWHM’s, they should be converted to integral breadths ofsize-broadened and strain-broadened profiles before calcu-lating associated domain size and strain values. Conversionfactors are [48]:

bL

GL¼ p

2;

bG

GG¼ 1

2

ffiffiffiffiffiffiffiffip

ln 2

r; ð14Þ

where GL and GG are calculated from the effective para-meters determined using (13). Then the Lorentz and Gaussintegral breadths are combined for both size and strainparts according to the relation [48]:

bi ¼ ðbGÞiexp ð�kÞ

1� erf ðkÞ ; k ¼ bLffiffiffipp

bG

; ð15Þ

where i stands for S and D. Only now can bS and bD berelated to the corresponding values of DV and e, accordingto Eqs. (8) and (9).

In addition to the Bragg peaks in the EDD spectrum,fluorescence lines and escape peaks appear. Escape peaksfrom the Ge detector are produced when the photon en-ergy is above the K-edge energy of Ge, EK ¼ 11.104 keV.For each Bragg peak at energy EB exceeding EK, twoescape peaks are possible and they can occur at ener-gies Eesca ¼ EB � 9.876 keV and Eescb ¼ EB � 10.983 keV.Both, the escape peaks and fluorescence lines are fittedwith a Gaussian function, whereas the FWHM of both isdescribed with a linear function according to Eq. (10).

Experimental

Instrumental characterization

Synchrotron X-ray experiments were carried out at thematerials science beamline for Energy Dispersive Diffrac-tion (EDDI) of Helmholtz-Zentrum Berlin, situated at theBerlin synchrotron radiation facility BESSY II. For a de-tailed description see [49, 24]. The photon source of theEDDI beamline is a superconducting 7 T multipole wig-gler [50]. Figure 1 shows its energy spectrum as recordedby a Low Energy Germanium (LEGe) detector at a verylow ring current of 90 pA and then extrapolated to300 mA.

With regard to Rietveld refinement of the EDD pat-terns, the calculated intensities ycalc

i need to be correctedfor the Wiggler spectrum, as it is considered in Eq. (1) bythe factor W(Ek). As can be seen in Fig. 1, the Wigglerspectrum changes when using a filter in the primary beam.Therefore, a correction function defined specifically forthe filter used during the experiment needs to be appliedto correct the calculated intensities. The appropriate func-tion can be chosen in the Rietveld program.

The beamline makes direct use of the white synchrotronbeam as it is provided by the wiggler. Therefore, the only

optical elements are an absorber mask and several slit sys-tems at different positions in the beamline, which areneeded to define a well-defined beam cross-section. Fordetailed information see [49]. The last slit system (S2) po-sitioned on the primary beam side is set to 0.5 � 0.5 mm2

throughout the experiment. The diffracted beam is definedby a double slit system (S3 and S4) and the equatorial slitaperture was set to 50 mm throughout the experiment. Forexperiments to determine the influence of the equatorialslit aperture on the line width, the aperture size was variedfrom 30 mm to 240 mm. The results show that for an aper-ture size up to 100 mm no significant broadening of thediffraction lines occurs. Studies regarding the variation ofthe axial slit aperture showed that the influence on bothline broadening and line-shift is negligible for the geome-trical diffraction setup used in the experiments.

Due to the rather simple setup of the beamline the in-strument resolution mainly depends on the intrinsic detec-tor resolution [51] and the broadening from the total beamdivergence, which is the sum of the divergence in the pri-mary and in the diffracted beam. For the determination ofthe instrument resolution the standard reference material

Rietveld refinement of EDD synchrotron measurements 937

Fig. 1. Energy spectrum of the 7 T multipole wiggler (photon fluxthrough a pinhole of 1� 1 mm2 30 m behind the source), scaled to aring current of 300 mA.

Fig. 2. EDD pattern of the standard reference material LaB6

SRM660b, measured at 2q ¼ 12�. The counting time was 30 s.

LaB6 SRM660b (NIST) and a W powder sample weremeasured. The EDD patterns are shown in Figs. 2 and 3.

To validate the assumptions made in Eq. (9), that thefull width at half maximum (FWHM) depends linearly onthe energy E, diffraction patterns for the two powder sam-ples were recorded at different 2q0 angles. The diffractionlines were then fitted with a standard pseudo-Voigt func-tion to determine the FWHM. The obtained values of theFWHM plotted as a function of energy E are shown inFigs. 4 and 5.

It can be seen that the FWHM shows the predictedlinear dependency on energy regardless of which scatter-ing angle is chosen for the measurement. However, it canbe seen that the slope and intercept of the linear functionchanges slightly with increasing scattering angle. There-fore, it is advised to determine the instrumental resolutionfunction separately for each scattering angle used duringthe measurements.

Consequently, the proposed implementation of theTCH pseudo-Voigt model is tested on the measured dataof the two samples under consideration. As the crystalstructure determination is not the focus of this investiga-tion and relative intensities strongly depend on the correc-tions for the wiggler spectrum, we have chosen the Paw-ley method [52] for the extraction of intensities in most ofthe examples presented in this paper. To demonstrate thedifferences between Rietveld and Pawley refinement, theFigs. 6 and 7a show the results obtained for the applica-tion of both methods to W powder.

Rietveld refinement (Fig. 6) which is based on fixedintensity ratios defined by the structure factors Fhkl reveals(at least for some reflections) significant discrepancies be-tween measured and calculated intensities. The influenceof the wiggler spectrum on the quality of the results isclearly to be seen by comparing the refinements in Fig. 6aand b, respectively. Both EDD patterns where recordedunder different storage ring conditions and refined usingthe wiggler spectrum shown in Fig. 1 (black curve). Satis-factory results, however, were only obtained for the pat-tern in Fig. 6a, whereas large (and obviously systematic)deviations were observed in Fig. 6b. The reason for thisfinding is not yet clear so far and will be the subject of

further investigations, but it may be assumed that the wig-gler spectrum is incorrectly described in case (b).

Pawley refinement (Fig. 7a), on the other hand, treatsthe peak intensities as free parameters and therefore,yields a much better agreement for the same EDD patternas shown in Fig. 6b in this respect. For that reason thePawley approach was used to refine all the EDD patternsshown in the following examples. The result of the refine-ment for the LaB6 SRM660b powder is shown in Fig. 7b.

From Fig. 7 it can be seen that the line profiles of bothsamples can be well described by the modified TCH pseu-

938 D. Apel, M. Klaus, C. Genzel et al.

Fig. 3. EDD pattern of W powder, measured at 2q ¼ 12�. The count-ing time was 600 s.

Fig. 4. Plot of the FWHM of the diffraction lines of LaB6 SRM660bas a function of energy E recorded at various 2q angles.

do-Voigt model. Figure 8 plots the instrumental resolution,Dd/d, calculated from the FWHM of the two powder sam-ples under consideration, as obtained by the refinement ofthe line profile parameters. Although the resolution isdominated by the strong intrinsic detector resolution, LaB6

SRM660b yields narrower line profiles than the W pow-der, as expected. Therefore, we report all domain size andstrain values relative to LaB6 SRM660b.

Application to size and strain broadenedline profiles

In order to confirm a validity of modeling energy depen-dence of FWHM of line profiles with the Eqs. (11) and(12), we chose two samples: (i) Ceria powder that wasused in the Size-Strain Round Robin [33], which exhibitspredominantly size-broadening effects and (ii) W-Ni ball-milled powder analyzed earlier [34], which shows bothsize- and strain-related broadening. The former sample wasmeasured on a variety of different geometries/sources andgives an opportunity to compare absolute values of coher-ently domain size and strain, as obtained from the EDDImeasurements, additional to just considering the quality offit.

Elemental powders of Ni having a purity of 99.8 wt%and a particle size smaller than 50 mm and W having apurity of 99.95 wt% and a particle size of 44–77 mm weremixed with a composition of 15 at% W. The powder mix-ture was milled in a planetary ball mill (Fritsch Pulveri-sette P6) using a WC-Co vessel (93.8 wt% WC, 6 wt%Co) with WC-Co milling balls of the same purity. Themilling time varied between 0.5 h and 150 h. For the ex-periments described in this paper a powder mixture milledfor 0.5 h and 150 h were chosen.

Rietveld refinement of EDD synchrotron measurements 939

Fig. 5. Plot of the FWHM of the diffraction lines of W as a functionof energy E recorded at various 2q angles.

a�

b�Fig. 6. Rietveld refinement of two EDD patterns of W powder(2q ¼ 12�) taking into account the actual intensity ratios between theindividual reflections. See text for details.

Results and discussion

The refined diffraction pattern of ceria Round-Robin sam-ple collected at 2q ¼ 10� with the counting time of 600 sis shown in Fig. 9. The coherent domain size and strainvalues were calculated for patterns collected at 8� and 10�.

The obtained value for the coherent domain size averagedat 226(31) A, which is in very good agreement with theresults reported in [33]. The strain parameters refined tozero in both cases (8� and 10�).

The measured EDD line profiles of the W––Ni powdermixture samples milled for 0.5 h and 150 h are shown inFig. 10. It can be seen that the milling process lead to abroadening of the line profiles. Moreover, it can be seenthat the peak maximum intensity of the Ni peaks de-creased during the milling process, as a consequence ofthe broadening. Peculiarly, the maximum intensity of theW peaks increased after 150 h of milling, which was alsoobserved during ADD experiments, as reported in [34].

The refinement result for the sample milled for 0.5 h,taken at 2q ¼ 10�, is shown in Fig. 11.

For the sample milled for 150 h, a slight shift of the Nipeaks to lower energies can be seen (see Fig. 10). Theshift of the peak positions of Ni indicates that the latticeparameter of Ni increased during milling, which can beattributed to structural distortions induced by the diffusionof W atoms into the Ni matrix (see [34]), which lead tothe formation of a Ni(W) solid solution. It was shown

940 D. Apel, M. Klaus, C. Genzel et al.

Fig. 7. Refined EDD patterns (Pawley method) of (a) W powder(same spectrum as shown in Fig. 6b) and (b) LaB6 SRM660b, meas-ured at 2q ¼ 12�.

Fig. 8. Resolution Dd/d as a function of interplanar spacing d for theEDDI instrument, as calculated from the FWHM obtained by refinement(Pawley method) of the LaB6 and W powder, measured at 2q ¼ 8�.

Fig. 9. Refined EDD pattern (Pawley method) of the ceria samplemeasured at 2q ¼ 10�. The counting time was 600 s.

Fig. 10. The measured EDD patterns of the W-Ni powder samplemilled for 0.5 h and 150 h, respectively (2q ¼ 10�). The inset showsthe observed peak shift of Ni to lower energy values due to formationof the Ni(W) solid solution during milling.

a�

b�

elsewhere [34] that the observed asymmetry of the peakscan be attributed to compositional variations of W insidethe Ni matrix. The asymmetry can then be regarded as theconsequence of the overlap of two peaks, one attributableto a Ni(W) solid solution with a lower W content, andtherefore with a smaller lattice parameter, and the otherattributable to a Ni(W) solid solution with a higher Wcontent, and therefore with a larger lattice parameter. Toaccount for the asymmetry, two Ni phases, differing inlattice parameter, were refined. The result of the refine-ment for the sample milled for 150 h, measured at2q ¼ 10�, is shown in Fig. 12.

The calculated values for domain size and strain forboth samples measured at 2q ¼ 10� and 16� are listed inTable 1. Additionally, the results from [34] are also shownin Table 1 for comparison. Here, LaB6 SRM660b wasused as instrumental standard. For the sample milled for0.5 h, the strain parameters of W diffraction lines refinedto zero (for 2q ¼ 10� and 16�). Therefore, either no or asmall amount of strain, not detectable with our instrument,is introduced into the W particles, which is expected con-sidering the short milling time and the hardness of W.This is in good agreement with the small strain of 3 � 10�5

measured in [34]. The refinement of the size parameters Pand X yielded an average domain size of 362(184) A and325(89) A for 2q ¼ 10� and 16�, respectively. For thesample milled for 150 h, the strain parameters refined to avalue of 3.7(0.9) � 10�3 and 3.3(0.7) � 10�3, which is ingood agreement with the result from [34], where a strainof 1.3 � 10�3 was evaluated. The EDD refinement of the

size parameters P and X yielded an average domain sizeof 605(335) A and 315(68) A. Interestingly, the domainsize of the W particles milled for 150 h measured at2q ¼ 10�, 605(335) A, is larger than the one for the sam-ple milled for 0.5 h. In contrast, for the sample measuredat 2q ¼ 16� a small decrease in domain size is observed,which is in agreement to the observed decrease in domainsize from about 540 A to 160 A in the ADD experiments.

The size parameters of Ni milled for 0.5 h and 150 h,measured at 2q ¼ 10�, refined to zero, in effect yieldingan infinite domain size. However, the strain parametersrefined to non-zero values and we obtained a value for thestrain of 2.3(0.9) � 10�3 for the sample milled for 0.5 h.For the sample milled for 150 h the strain was calculated to4.7(0.3) � 10�3. This observed increase in strain is in agree-ment with the ADD experiments (see Table 1). The refine-ment of the size parameters for the samples milled for 0.5 hand 150 h measured at 2q ¼ 16� yielded an average do-main size of 606(238) A and 313(74) A. The strain para-meters yielded values of 1.9(0.9) � 10�3 and 4.4(0.9) � 10�3.The observed decrease in domain size is in qualitativeagreement to the observed decrease from 340 A to 140 Ain the ADD experiments.

It can be seen that, in the case of the W-Ni powdermixture, the larger 2q angle in the EDD experiments givesmore reliable results for the domain size, whereas thestrain values are similar. Hence, future work should eluci-date if there is a dependency of the instrumental resolution(especially the size-related parameters P and X) on thescattering angle.

Rietveld refinement of EDD synchrotron measurements 941

Fig. 11. Refined EDD pattern (Pawley method) of the W––Ni powdersample milled for 0.5 h, measured at 2q ¼ 10�. The counting timewas 600 s.

Table 1. Calculated values for domain size and microstrain for the W––Ni powder samples milled for 0.5 h and 150 h, measured at 2q ¼ 10� and16� and the results from the ADD experiments.

Data 2q ¼ 10� 2q ¼ 16� single-line analysis ADDdomain size [A] microstrain domain size [A] microstrain domain size [A] microstrain

W 0.5 h milled 362(184) �(U, Y ¼ 0) 325(89) �(U, Y ¼ 0) 540 3 � 10�5

W 150 h milled 605(335) 3.7(0.9) � 10�3 315(68) 3.3(0.7) � 10�3 160 1.3 � 10�3

Ni 0.5 h milled �(P, X ¼ 0) 2.3(0.9) � 10�3 606(238) 1.9(0.9) � 10�3 340 7 � 10�4

Ni 150 h milled �(P ¼ 0) 4.7(0.3) � 10�3 313(74) 4.4(0.7) � 10�3 140 2.4 � 10�3

Fig. 12. Refined EDD pattern (Pawley method) of the W––Ni powdersample milled for 150 h, measured at 2q ¼ 10�. The counting timewas 600 s.

Since the W––Ni powder sample is rather complex con-sidering the observed increase of maximum intensity of Wpeaks and the asymmetry due to the variation in composi-tion of the formed Ni(W) solid solution during milling,the values for domain size and strain are rather challen-ging to accurately evaluate. Moreover, it is noted that rela-tively large standard uncertainties of obtained results likelystem from counting statistics and large instrumental broad-ening, relative to the milling-induced broadening. How-ever, the obtained values for strain show good agreement tothe ones obtained in [34] using the single-line method inADD. It was shown in [42] that single-line methods cangive unreliable and significantly different values of do-main size, whereas strain values vary less among differentanalytical approaches used.

Conclusions and outlook for the future

The experimental results presented in this work confirmthat the EDD method can be successfully applied to X-rayline profile analysis in order to determine microstructuralparameters. By measuring samples with negligible physi-cal broadening (i.e. LaB6) under various geometrical con-ditions, intrinsic detector broadening was shown to give amajor contribution to instrumental line broadening, result-ing in almost pure Gaussian profiles. However, as demon-strated by means of powder samples with well-definedparticle size and strain broadening, the resolution of theEDD synchrotron setup used for the experiments wasfound to be sufficient for separating instrumental and phy-sical broadening contributions. Moreover, distinct changesin the line profile shape from pure Gaussian to Lorentzianwere observed, which allowed application of the pseudo-Voigt function for profile fitting. In order to study micro-structural parameters, a Rietveld-refinement program wasdeveloped based on the Matlab platform. The Thomson,Cox, and Hastings [43] model for line profiles wasexploited and adapted for EDD. The examples consideredin this paper were selected with the aim to demonstratethat the approach is in principle suitable to evaluate EDDdata with regard to line broadening. For this reason, allmeasurements were performed on powder samples, whichare assumed to be free of macro residual stress and crys-tallographic texture.

The findings obtained under different diffraction condi-tions were compared and assessed in the light of resultsobtained by means of well-established ADD line profileanalysis. In most cases a good qualitative and even semi-quantitative agreement was found, but there are also someunanswered questions which require further work on boththe experimental and the evaluation side. So the EDDrefinement of a data set obtained for a rather complextwo-phase material system showing both size- and strainbroadening measured at a larger 2q angle yields evenmore reliable results than a data set which was recorded atsmaller 2q. First indications of an improvement of the in-strumental resolution with increasing scattering angle,which would support this finding, were observed (Figs. 4and 5) and thus, further efforts in this direction will benecessary. Furthermore, in order to allow for quantitative

phase and structure analysis, progress in EDD data refine-ment on the absolute scale (i.e. including the intensity ra-tios) has to be achieved, which requires an improved un-derstanding of all parameters that affect the primary andthe diffracted photon spectrum, respectively.

Further enhancements concerning the application ofEDD Rietveld refinement should aim at the analysis of(macro) residual stress distributions within polycrystallinematerials. Considering for example the near surface zoneof mechanically treated technical parts or thin film sys-tems, macro residual stress fields which may considerablyvary within the penetration depth of the X-rays and/or tex-ture (gradients) have to be taken into account in the eva-luation procedure. Since each diffraction line Ehkl includesinformation from a different average information depthhthkli, parameters which describe the depth dependence ofthe residual stresses and (at a later stage) also of texture,have to be added to the refinement model. Future enhance-ments of the approach introduced in this paper for process-ing the EDD data will aim at the simultaneous refinementof complete sets of diffraction patterns measured in var-ious modes of X-ray residual stress analysis (see, e.g. [23,24]), which includes also depth profile analysis performedin transmission geometry using small gauge volumes.

Acknowledgments. The authors would like to thank the reviewers fortheir valuable comments and suggestions on the manuscript.

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