A critical analysis of the X-ray diffraction intensities ...

$: A critical analysis of the X-ray diffraction intensities ...$
Rameshwari Naorema , Anshul Guptab , Sukriti Mantric , Gurjyot Sethid , K. V. ManiKrishnae ,Raj Palaf , Kantesh Balanib , Anandh Subramaniamb

aDepartment of Physics, Indian Institute of Technology Kanpur, Kanpur, India.bDepartment of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, India.cSchool of Materials Science and Engineering, University of New South Wales, Sydney, NSW, Australia.dDepartment of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A. eMaterials Science Di-vision, Bhabha Atomic Research Center (BARC), Trombay, Mumbai, India.eMaterials Science Division, Bhabha Atomic Research Center (BARC), Trombay, Mumbai, India.fDepartment of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, India.

A critical analysis of the X-ray diffractionintensities in concentrated multicomponent alloys

The decrease in the X-ray diffraction Bragg peak intensityfrom concentrated multicomponent alloys (CMA), has beenmodeled in literature akin to the effect of temperature. In thecurrent work, experiments and computations are used to com-prehend the effect of atomic disorder in CMA on the Braggpeaks of powder diffraction patterns. Ni–Co–Fe–Cr–Mn andCu–Ni–Co–Fe–V have been used as model systems for thestudy. It is proved that the intensity decrease is not insignifi-cant, but is not anomalous either. A recipe is evolved to com-pare the Bragg peak intensities across the alloys of a CMA. Itis demonstrated that it is incorrect to model the effect of anincrease in atomic disorder in a CMA, as a temperature effect.A ‘good measure’ of lattice distortion is identified and furtherit is established that full width half maximum is a good mea-sure of the bond length distortion. It is demonstrated that thetrue strain due to bond length distortion is of significantlylower magnitude than that given by a priori measures of lat-tice strain. In the scheme of categorization of defects in crys-tals, it is argued that CMA is a separate class (as distinct fromtype-I and type-II defects); which should be construed as adefected solid, rather than a defect in a solid.

Keywords: X-ray diffraction; Lattice strain; Concentratedmulticomponent alloys.

1. Introduction

The current work pertains to X-ray diffraction (XRD) studyof concentrated multicomponent alloys (CMA). Excellenttexts have summarized the progress in XRD techniques,which span more than half a century [1–2]. The concept ofCMA, though of comparatively recent origin, has also beencovered in comprehensive textbooks [4, 5].

In the last decade, intense studies have been carried outon CMA [4, 6]. In many of these alloys, one or more disor-dered solid solutions (DSS) typically with a bcc or fccstructure, are entropically stabilized in competition withpossible intermetallic compounds and hence these alloyshave been termed high entropy alloys (HEA) [5]. This is ty-pically achieved by the mixing of five or more elements inequimolar proportions in the alloy [7–9]. The strain in the

lattice, arising from atomic size differences between theelements is usually characterized by DRmax or d (or dr) pa-rameters [10, 11]:

DRmax ¼Maxðri � �rÞ

�rð1Þ

d ¼X

n

i¼1

Ci 1�ri

�r

� �2" #1=

2

ð2Þ

where, �r ¼P

n

i¼1Ci ri, ri is the radius of i

th element in the al-

loy, n is the total number of elements in the alloy, Ci is theatomic fraction of the ith element in the alloy. Owen et al.[12] have used a product of d and �r i. e. (�uDOwen ¼ d � �r) asa measure of the lattice distortion. The ‘intrinsic lattice dis-tortion’ (�uDYeh) is an important parameter, which has beenused in conjunction with the Debye–Waller factor, for thecomputation of the intensity of the Bragg peaks [13]:

�uDYeh ¼X

n

i¼1

dieff � d

� �2

" #1=2

ð3Þ

where, �d is the average lattice parameter and

dieff ¼

P

n

i

Cjð1þDVij

ViÞ1=3

di. The parameters Cj, di, Vi and

DVij are: the atomic fraction, the lattice constant of the ith

element, the unit cell volume of the ith element and the dif-ference in the unit cell volumes between the ith and the jth

elements, respectively. It is to be noted that the intensity re-ferred to above is the area under the Bragg peak (integratedintensity) above the baseline [14].

Feng andWidom [15] have used variance as a measure oflattice distortion given by the following formula.

rD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1n

X

i

ðri2Þ �1n

X

i

ri

!2v

u

u

t ð4Þ

On closer inspection, this parameter (rD) is found identicalto �uDOwen. Another parameter of recent origin, which has

R. Naorem et al.: A critical analysis of the X-ray diffraction intensities in concentrated multicomponent alloys

Int. J. Mater. Res. (formerly Z. Metallkd.) 110 (2019) 5 393

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been used as a measure of the lattice distortion, is the intera-tomic spacing matrix (Sd) [16].

X-ray scattering from solid solutions have been studied formore than sixty years [1, 17]. One of the important contribu-tions is due to Huang [18], wherein the effect of the forma-tion of dilute solid solutions on the XRD intensities was com-puted. The assertions made in the work of Huang [18], whichinclude the \weakening of the maxima" and the \presence ofdiffuse maxima", have subsequently been experimentallyconfirmed [19–21]. The effect of temperature, atomic disor-der, strain, crystallite size and crystalline defects on theXRD pattern has been comprehensively described [2, 3, 22].Considerable attention has been given to the effect of thesefactors on the diffuse scattering present between the Braggpeaks [23–25]. Defects have been categorized into twoclasses, based on the displacement they create [2]: (I) havingr�2 (or faster) decay and (II) r�

3=2 (or slower) decay. Only de-fects of type-II show broadening of the Bragg peaks [2]. Forexample, isolated point defects belong to the first class, whiledislocations belong to the second class.

Yeh et al. [13] have considered the effect of an increasein the number of elements (in the Cu–Ni–Al–Co–Cr–Fe–Sisystem) on the XRD powder pattern intensities. They havemodeled the effect of atomic disorder, with concomitant in-crease in lattice strain, akin to a thermal effect according tothe equation [3, 26]:

MT ¼ 8p2hu2iTsinhk

� �2

¼6h2T

mkH2M

uðxÞ þx

4

h i sinhk

� �2

ð5Þ

where, hu2iT is mean square displacement of atoms, h isPlanck's constant, T is the temperature at which the experi-ment has been conducted, m is the atomic mass, k is theBoltzmann constant, u is the Debye function, HMis thecharacteristic temperature and x is HM=T (h & k have theirusual meaning according to Bragg's equation). This leadsto a decrease in the intensity of the peaks by the factoreð�2MT Þ (the Debye–Waller temperature factor, IT). It is tobe noted that, HMis technically different from the Debyetemperature (HD), which is used in the context of the Debyetheory of specific heat; however, the difference between thevalues of HM and HD is not usually large [3]. Additionally,Calamiotou et al. [27] have used the following equation toaccount for the contributions to lattice distortion arisingfrom thermal and non-thermal effects.

8p2Uiso ¼6h2T

mkH2M

uðxÞ þx

4

h i

þ d2 ð6Þ

where, 8p2Uiso is the isotropic atomic displacement param-eter (ADP). The first term in Eq. (6) is due to thermal ef-fects and the second term can be considered to arise fromstatic displacements. In a subsequent work of rather recentorigin, Yeh [28] has attributed the diminished intensity ofthe Bragg peaks in the Ni–Co–Fe–Cr–Mn system to latticedistortion, which supposedly results in a high magnitude ofdiffuse scattering.

The intensities observed in the powder diffraction pat-terns obtained from CMA arise from two contributions[29]: (i) from the Bragg peaks and (ii) diffuse scattering be-tween the peaks. An important work of recent origin in the

context of diffraction from CMA is due to Owen et al. [12].Using neutron diffraction and allied computations, they ad-dressed the question: \are the lattices in high entropy alloysseverely distorted?". They used total scattering data to un-derstand the local atomic environments, via pair distributionfunctions (PDF). It is to be noted that their work did not fo-cus on the broadening of Bragg peaks. Important observa-tions/conclusions from their work include: (i) Bragg peaksdid not show \pronounced dampening", (ii) the widths ofthe PDF peaks of HEA were not \disproportionately larger"than that of simpler alloys, (iii) thermal vibrations in HEAwere the highest amongst the alloys studied, (iv) there is noindication of anomalously large local lattice strain in HEAand (v) the breadth of the PDF peak is dominated by thermaleffects rather than static displacements due to atomic disor-der. It is important to note that Owen et al. [12] envisagedatomic disorder within the Uiso term in the Rietveld refine-ment process (i. e. akin to the effect of temperature and asconsidered by Yeh et al. [13]). Further, it is not clear if thecomputations performed using an \average atom" [12] cancapture the essential features of an actual alloy. The problemof anomalous decrease in the intensities of Bragg peaks hasalso been addressed by Guo et al. [30]. They have high-lighted the issue of fluorescence and further have stated that:\Nevertheless, the reason why the (111) peak, and this peakalone showed an intensity decrease with an increasing N, re-mains unknown and needs further study" [30]. The recentworks of Oh et al. [31], Okamoto et al. [32] and Song et al.[33] are also noteworthy. Using density functional theory(DFT) and extended X-ray absorption fine structure(EXAFS), Oh et al. [31] have shown that the magnitude ofthe average distortions is small, but the mean deviation ofthe distribution of distortion values is large. Okamoto et al.[32] have used XRD and first principle calculations to ar-rive at the atomic displacements in a CMA. Using ‘super-cells’ for computations, Song et al. [33] have shown thatthe \local lattice distortion of the refractory HEAs is muchmore significant than that of the 3d HEAs". They high-lighted the limitations of using empirical atomic radii.

The addition of alloying element(s), leading to the forma-tion of a CMA, has the following major effects [26]: (i)change in lattice parameter leading to a shift in peak position;(ii) change in the average atomic scattering factor leading to achange in the intensity of peaks; (iii) change in the solidustemperature; (iv) change in the level of absorption. Point (iv)is of importance, as for a given radiation some elements maycause fluorescence, thus leading to a reduced Bragg peak in-tensity along with a concomitant increase in background in-tensity. Additionally, the elastic constants will change on in-creasing the number of elements in an alloy.

In general, an increase in the temperature causes four im-portant effects on the XRD powder patterns [1, 26]: (i) peakshift due to lattice parameter increase, (ii) decrease in the in-tensity of the Bragg peaks (given by the Debye–Waller tem-perature factor), (iii) increase in the diffuse background withincrease in (sinh/k) and (iv) diffuse scattering ‘tail’ aroundthe Bragg peaks. Standard formulations found in textbooks[1, 26] consider the effect of thermal vibrations as a decreasein the intensity of the Bragg peaks, via a decrease in peakheight, without an increase in peak width (usually character-ized by the full width half maximum (FWHM)). Advancedtheoretical treatments of the subject can also be found inthe literature [3, 34]. The difference in the nature of the dif-


394 Int. J. Mater. Res. (formerly Z. Metallkd.) 110 (2019) 5

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fuse scattering between static (due to atomic disorder) anddynamic (due to thermal vibrations) disorder has beenclearly delineated in the literature [23, 24].The atomic ‘disorder’ in CMA/DSS arises because of

three reasons: (i) probabilistic occupation of lattice sitesaccording to the stoichiometry (henceforth referred to as‘elemental disorder’ [35]), (ii) displacement of atoms fromlattice sites due to atomic size differences (henceforth re-ferred to as positional disorder), and (iii) thermal atomicvibrations. Point (i) increases the configurational entropyof the system and hence is expected to play a role in thestabilization of a DSS. The static and dynamic aspects ofdisorder are best understood by considering two scenarios:the heating of pure Ni from 0 K (case-a) versus alloying itwith Co at 0 K (case-b). In case-a, the atoms vibrate abouttheir mean positions (i. e. the reference lattice is still intactand a time averaged picture will reveal ‘smeared out’atoms in lattice sites), while in case-b the ‘lattice’ is per-manently distorted. The atomic size effect in an alloy onthe XRD pattern has been investigated for a considerablelength of time [3, 36–38]. Guinier [39] has arrived at ananalytical expression, wherein terms related to positionaldisorder, Laue scattering and peak asymmetry have beendelineated. Using an illustrative example in one dimen-sion, Guinier showed that peak broadening occurs due tothe formation of a disordered alloy. Important literatureexists on the effect of alloying on the diffuse scattering in-tensity between the Bragg peaks [24, 25, 36]. It is impor-tant to note at this juncture that the size difference betweenthe elements constituting a CMA is usually small.

The current work aims at the following tasks with regardto (powder) XRD patterns from CMA.(1) Comprehend the effect of atomic disorder on the

FWHM and intensity of the Bragg peaks.(2) Show that it is incorrect to model the effect of atomic

disorder as a change in temperature.(3) Establish that the primary reasons for the decrease in

intensity, on the formation of the Cu–Ni–Al–Co–Cr–Fe–Si multicomponent alloys ([13]), are due to a de-crease in the average atomic scattering factor andchange in the average absorption factor and not dueto atomic disorder.

(4) Show that the decrease in intensity of the Bragg peakscannot be correlated with the number of elements.

(5) Study the effect of change in solidus temperatureacross alloys on the Bragg intensities.

(6) Evolve a methodology to compare XRD peaks acrossalloys and to isolate the effects of lattice strain.

(7) Identify an a priori parameter for lattice strain, whichcorrelates with the broadening of Bragg peaks.

(8) Demonstrate that such a priori parameters are notgood measures of bond length distortion and that thebond distortion in single phase CMA is of signifi-cantly lower magnitude than that hitherto anticipated.

(9) Prove that FWHM of the Bragg peak is a good mea-sure of strain arising due to bond length distortion.

(10) Establish that the atomic disorder in CMA constitutesa defect forming a separate class; wherein, the defectis of non-local nature and permeates the whole crystal.

It is to be noted that much of the literature has focused ondiffuse scattering from dilute solid solutions, while theemphasis of the current work is on the Bragg peaksand concentrated alloys. Another point to be noted is

that, even equimolar binary solid solutions at room tem-perature are rare and hence the occurrence of disorderedequimolar solid solutions with two to five elements is ex-tremely rare.

To achieve the abovementioned tasks, model computa-tions are performed in addition to experimental investiga-tions. The Ni–Co–Fe–Cr–Mn (labelled as system-SA) andCu–Ni–Co–Fe–V (system-SB) alloy systems are chosen asmodel systems for the current study, as these alloys form asingle phase fcc solid solution [40, 41]. In other respects theSA and SB systems are of diverse nature, as will become evi-dent in the discussion of the results. Additionally, the alloysinvestigated by Yeh et al. [13] (Cu–Ni–Al–Co–Cr–Fe–Si,system-SY) are also synthesized to perform a comparativestudy and to unequivocally answer the question: \is there ananomalous decrease in XRD peak intensities on the forma-tion of CMA?". The atomic radius difference between theelements is small in both the systems (SA & SB), which istypical for most of the single phase HEA systems. This im-plies that the positional disorder in the CMAs is small andso is expected to be its effect on the XRD peak profile.

In the context of the formation of CMA, the atomic radiilisted in the literature are often used for the computation ofa priori measures of ‘lattice strain’. In this context the fol-lowing important points are to be noted.(1) The atomic radius of an element in an alloy is expected

to be dependent on the local environment, although thiseffect is not significant [32, 42]

(2) Multiple listings of atomic radii exist in the literature(e. g. the Goldschmidt radius [43, 44], radii based on em-pirical formulations [45, 46]).

The issue of the type of atomic radii listing to be used inalloys for the computation of distortion, has also been ad-dressed [32, 33]. In metallic systems, the Goldschmidt'slisting of radii seems to serve well [47]. In the currentwork the Goldschmidt radius (for 12-coordination) hasbeen used for computations, which is expected to be thelogical choice for alloys with the fcc structure. The terms‘lattice strain’ and ‘lattice distortion’ have been inter-changeably used to essentially depict atomic displace-ments from a perfect lattice. It is important to note thatthe strain computed using a priori parameters (e. g. �uDOwen& DRmax), may not be a good measure of the bond lengthdistortion in CMA [33]. This implies that, the bond lengthdistortion has to be compared with the a priori measuresof lattice strain. The lattice strain as measured by parame-ters such as d & DRmax is dimensionless, while lattice dis-tortion as measured by parameters such as �uDOwen & �r hasdimensions of length. The terms CMA and DSS [6] havebeen used as synonyms in some places in the current work;however, it should be noted that the acronym DSS refers tothe formation of a single phase disordered solid solution inthe present context.

It is expected that this work will suitably augment theclassic works of Huang, Guinier, Krivoglaz and Warrenand the contemporary works of Owen et al. [12], Oh et al.[31], Okamoto et al. [32] and Song et al. [33]. In this con-text, it is important to note that XRD facilities are easily ac-cessible as compared to that for neutron diffraction. Thefollowing effects are not included in the current work andform the scope for future work:(i) multiple scattering effects,(ii) effect of short range order (SRO),



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(iii) dependence of the atomic radii on the environment, and(iv) atomic disorder on the diffuse scattering.To keep the contents of the manuscript ‘tangible’, only im-portant parts of the formalism and results are presented inthe main document and ‘subsidiary’ information has beenmoved to the supplemental material section [the hyperlinkto the electronic supplemental material (ESM) is given inthe data accessibility statement]. Subsequent to a criticalanalysis of the results from the literature, a systematic at-tempt is made via computations and experiments to estab-lish that CMA warrant special consideration and need tobe contrasted with dilute solid solutions.

2. Experimental details

Equimolar alloys belonging to SA, SB and SY systemswere synthesized by vacuum arc melting, followed by an-nealing for 24 hrs at 800 8C for achieving good composi-tional homogeneity. The number of element(s) in the alloysis designated by a numeral after the system (e. g. SA1 ispure Ni, SA2 is the Ni–Co binary, SA3 is Ni–Co–Fe tern-ary, SA4 is Ni–Co–Fe–Cr quaternary, etc.). To minimizetexture and microsegregation which could be present in theas-cast samples, the specimens were: (i) filed to micron sizedparticles and (ii) the filed particles/powder was annealed at800 8C for 48 hrs under vacuum (in quartz tube with Ti get-ter). The importance of removal of texture has been high-lighted by Guo et al. [30]. This is to remove microstrain in-duced in the material because of the filing operation. Insome of the literature pertaining to HEA, it is not clear if tex-ture effects have been adequately accounted for (e.g. [13,28]). The heat treatment process additionally avoids peakbroadening effects due to dislocations and grain size. Theas-cast bulk samples were characterized for microstructurein a scanning electron microscope (SEM) and the composi-tions were determined using energy dispersive X-ray analy-sis (EDX). A Bruker D8 Focus (dwell time of 250 s step–1,step size 0.028, Cr-Ka radiation) and PANalytical Empyrean(dwell time of 250 s step–1, step size 0.028, Cu-Ka radiation)were used for the XRD studies. The solidus temperatures ofthe samples were determined by performing differentialscanning calorimetry (DSC) using a Perkin Elmer (STA8000) instrument (heating rate of 10 K min–1, temperaturerange of 1200–1600 8C).Multiple parameters and methods have been used to

quantify/analyze peak broadening in XRD. These include:FWHM, variance, integral breath (b) methods, Fouriermethods and double Voigt methods [2, 14, 48–50]. Thebook by Mittemeijer and Welzel [14] gives an excellentsummary of the methods. In the current work, FWHM isused as a measure of peak broadening given its simplicity,albeit the fact that integral breath has been recommendedby some investigators [51]. The experimental peak broad-ening (FWHMexperimental) is determined from a Gaussianprofile fit. Other functions have been used in the literature[49] and functions like the Lorentzian and the pseudo-Voigt differ essentially in the ‘tail region’ of the profile.The Gaussian profile has been used in the current work tomaintain consistency with the computational peak profilefitting, wherein it is a natural choice [52]. A comparisonof the different methods of peak fitting for the cases inthe current study can be found in the supplementary mate-rial. The readers may consult Chapter 4 of Ref. [14] for an

illustrative description of the peak fitting methods. TheKa2 peak is subtracted before ‘peak fitting’ using Matchsoftware [53]. This standard method is based on Rachin-ger‘s algorithm [54], which uses Fourier transform of theXRD peak profile with the Stokes’s deconvolution proce-dure [55], to distribute the statistical error and to isolatethe Ka1 peak. The FWHM of the first elements in the re-spective systems (FWHMelement) has been used as refer-ence to isolate the broadening purely due to atomic levelstrain (FWHMatomic strain). This is achieved by using:FWHM2

atomic strain¼ FWHM2experimental � FWHM2

element.This procedure effectively ‘subtracts’ the broadeningfrom instrumental and other sources, which are not dueto atomic disorder/strain. Despite the fact that extremecare was taken to minimize the effect of experimentalvariables including the level of the sample surface, itsflatness, etc., it should be noted that the intensities ob-tained are subject to errors inherent in the method. Thisfurther emphasizes the use of FWHM as a definitive pa-rameter in the context of CMA. To make a comparisonof Bragg data, not only across systems, but between ex-perimental and computational routes, the most intense111 peak has been used.

3. Computational methodology

Computations were performed to understand the effect ofthe following on the XRD Bragg peak: pure positional dis-order (case-i), pure elemental disorder (case-ii) and com-bined positional and elemental disorder (case-iii).

A c · c · c cubic cluster with fcc structure was generatedby placing elements on perfect lattice sites. The cluster wasbuilt of ‘c’ fcc unit cells in each orthogonal direction (with-in the supercell approximation). Models of equimolar mul-ti-component alloys were constructed by progressively in-creasing the number of elements from one to five in thiscluster. As the cluster size was increased (by increasing‘c’), the broadening due to system size (also known as crys-tal truncation rods, CTR [56]) decreased. The largest clustersize used in the current computations was with c = 11. In thedisordered fcc alloys the 100 peak is absent, while the 111and 200 peaks are present (h, k, l unmixed). An importantpoint to be noted is that exact stoichiometry cannot beachieved for all compositions for the clusters considered.However, the deviation from the desired composition isnegligible, especially in a large sized 11 · 11 · 11 cluster.For example in the 5 · 5 · 5 cluster there are 500 atoms(N = 500) and for the ABC ternary alloy the closest to equi-molar stoichiometry that can be obtained is A166B167C167. Inthe abovementioned approach, a comparison across sys-tems is made between clusters of the same size, which is ad-vantageous as compared to an approach where supercellsize is varied with stoichiometry (e.g. [33]).

To evaluate the effect of ‘pure’ positional disorder on theXRD peak intensities (case-i), a model system was createdwith Ni atoms on a perfect lattice. Further, the atoms weredisplaced with respect to their mean positions by a magni-tude ‘u’. The direction of ‘u’ was randomly chosen, whilethe magnitude was kept fixed for a particular computation.In this approach the magnitude of ‘u’ gives a direct controlon the degree of distortion, as will become evident in thediscussion of the results. The intensity and FWHM of aBragg peak (111 peak) was tracked as a function of the



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magnitude of the displacement (u). To emphasize, in thishypothetical construct there exists pure positional disorder,without the influence of the atomic scattering factor of al-loying elements added.

To study the effect of atomic scattering factor of the ele-ments on the intensities of the Bragg peaks (case-ii), atomsof different elements were introduced on a perfect lattice.This was achieved by randomly placing elements on anaverage lattice (i. e. the lattice parameter was computedusing the radius of an ‘average atom’). This is a case of pureelemental disorder without positional disorder. The numberof elements was increased from two to five.

To generate clusters with both positional and elementaldisorder (case-iii), atoms according to a prescribed compo-sition were first introduced into a perfect average lattice(as in case-ii) and they were further allowed to relax toequilibrium positions using a molecular dynamics (MD) si-mulation as described later in this section. For each of theabove-mentioned cases (i–iii), five random clusters werecreated and XRD patterns were computed according to theapproach described below. The broadening due to crystal-lite size was subtracted by using the computed FWHM forthe pure element of the series. This is akin to the procedurefor the experimental data, wherein a Gaussian profile wasassumed for the Bragg peaks (additional details can befound in the supplemental material). The scatter in the com-puted FWHM for the five random configurations is shownas an ‘error bar’ in the results presented.

For the clusters considered above (cases-i, ii & iii), inten-sity-2h plots were computed using the equation [3]:

I ¼X

p

X

q

fp fq exp2 p ik

� �

~s�~s0ð Þ �~rpq

ð7Þ

where, fp & fq are the atomic scattering factors of the pth andqth atoms; ~s & ~s0 are unit vectors in the direction of dif-fracted and primary beams respectively and ~rpq is the dis-placement vector between the pth and qth atoms. It is note-worthy that: (i) for crystallites in random orientations thisequation simplifies to the Debye scattering equation [57]and (ii) this is the fundamental equation used for the deriva-tion of other formulae, wherein the intensity is split intothose arising from fundamental reflections, short range or-der (diffuse intensity), size effect modulation, etc. Thesplitting process typically involves approximations and isnecessitated by the computational effort involved for largecrystallites. In the current work, Eq. (7) was used withoutsimplifications and hence no approximations are involved.A Matlab [58] code is written to evaluate Eq. (7) for thec · c · c cluster. It is to be noted that this methodology hasthe advantage that each type of atom is treated explicitly,unlike the ‘average atom models’, which are sometimesused in the literature. As mentioned earlier, a Gaussian pro-file fit for the intensity was assumed, though other functionshave been used in the literature [14, 59].

The effect of the atomic scattering factor, the absorptionfactor and the ‘temperature factor’ on the peak intensity isconsidered in Section 3 of the ESM. The effect of the multi-plicity factor is taken ‘out of the picture’ by comparing in-tensities of a given peak (the 111 peak).

As mentioned before, subsequent to placing atoms of twoor more types on a perfect lattice, the system was allowed to

equilibrate to a ‘relaxed’ configuration (local energy mini-mum) in an MD simulation. Energy minimization was fol-lowed by equilibration under isobaric (P = 1 bar) and isother-mal conditions (T = 300 K), using the LAMMPS software[60]. Binary EAM pair potentials were used in the computa-tions and were obtained either from the NIST repository orby mixing the pure component‘s pair potentials [61]. Themethod was validated using reference data (details can befound in ESM). The conjugate gradient algorithm was usedfor the minimization of the energy, with force tolerances forthe relaxation as 10–6 eV (energy) and 10–6 eV Å–1 (force)[62]. It is to be noted that the use of EAM potentials, facili-tates the computation of large supercells, in contrast to den-sity functional theory based simulations (e. g. [33]), whereinthe supercell size is constrained by the computational time.The average displacement of the atoms on the formation

of the CMA (designated as �uAC), can be computed fromthe position vector of the atoms in the perfect lattice(~ro) and that from the MD simulation (~ri) using:

�uAC ¼ 1N

P

N

i¼1~ri �~roð Þ. The parameter �uAC is a measure of

the strain, arising from the displacement of the atoms from aperfect lattice. The symbol �uAC has been chosen for this a pos-teriori measure of strain, over Dd ([33]); to avoid confusionwhich may arise due to the use of ‘d’ in Bragg's equation. Inthe special case of pure positional disorder (case-i discussedearlier), �uAC is the same as ‘u’. To reduce the effect of a finitecluster size, the values of FWHM and �uAC computed for var-ious cluster sizes, are extrapolated to an ‘infinite’ size.

4. Lessons from a critical analysis of powderdiffraction patterns in theCu–Ni–Al–Co–Cr–Fe–Si system

The Cu–Ni–Al–Co–Cr–Fe–Si alloy system (SY) investi-gated by Yeh et al. [13] has the following inherent complex-ities: (i) the presence of more than one phase for alloys be-yond SY2; (ii) the formation of a compound in alloys SY5,SY6 and SY7 (with a total of three phases); (iii) the massabsorption coefficient (l/q) for the elements used in the al-loy varies drastically for Cu-Ka radiation, and (iv) changein the average atomic scattering factor (favg ¼

P

n

i¼1Cifi, where

Ci is the mole fraction & fi is the atomic scattering factor of

the ith element) across the alloys in the system. The differencein mass absorption coefficients for the elements leads to avariation of l=q (SY1 to SY7). The mass absorption coeffi-cient for the alloys is computed as:

l

q

� �

alloy

¼X

n

i¼1

l

q

� �

i

wi ð8Þ

where, for the ith element, wi is the weight fraction, li is thelinear absorption coefficient and qi is the density. The soli-dus temperature will change across the alloys in the systemwith concomitant effects on the thermal diffuse scatteringand the intensity of the Bragg peaks. Formation of multiplephases in alloys beyond SY2 implies that the volume (orweight) fraction of material giving rise to a peak (say the(111) peak from the fcc phase) decreases, thus making a di-rect comparison of intensities across alloys difficult. Theabove-mentioned aspects render a fruitful comparison of



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XRD data across systems (SY1 to SY7), to isolate the ef-fects of lattice strain, an arduous task.

The intensity of a given peak across alloys is: (i) directlyproportional to the square of the average atomic scatteringfactor (f 2avg), and (ii) inversely proportional to the absorption

factor 1=2lalloy� �

for the alloy. This integrated factor

f 2avg=2lalloy� �

111(designated as ISA) represents the intensity

variation due to scattering factors and absorption coefficientsacross the alloys. As stated before, the melting point (Tm) orthe solidus temperature (Ts) of the system changes acrossthe alloys of the SY system. If the intensity of a given peak(say the 111 peak) is compared across samples, this aspecthas to be accounted for. Other factors being equal, the ampli-tude of the vibrations scale with the homologous temperature(T/Tm) and hence change in solidus temperature due to thechange in composition, is akin to heating or cooling the sam-ple (depending on the decrease or increase in the solidus tem-perature, respectively). There exists no precise way of incor-porating this effect on the intensity of the Bragg peaks (asmaterial properties, especially the elastic properties changeacross systems) and further integrating this term with that re-sulting from the inclusion of the atomic scattering factor andabsorption factor (ISA). However, the change in solidus tem-perature can be approximately accounted for as follows:(i) a pure element is considered as the starting material,(ii) the change in solidus temperature is tracked with an in-crease in the number of elements, (iii) this is treated as heat-ing or cooling of the elemental metal (depending on decreaseor increase in solidus temperature), (iv) this effect on the in-tensity of the Bragg peaks is modeled as a change in ‘T’ inEq. (4) (resulting in a new factor IT), and (v) ISA and IT aremultiplied to obtain a new integrated factor ISAT. It is impor-tant to note that this factor IT should be conceived akin to the‘real’ temperature factor, unlike the atomic disorder modeledas a temperature factor. In an alloy series with increasing n,the alloy with n-1 elements can be considered as the refer-ence for the alloy with n elements; but, in the current workthe pure element (n = 1) is considered as the reference. Thisis necessitated by the fact that HM data is not available formost of the multi-component alloys in the literature.The various parameters related to the SY system ((i)

atomic scattering factor (f), (ii) mass absorption coeffi-cient (l/q), (iii) solidus temperature (Ts), and (iv) atomicradius are listed in Table 2 of ESM. The following pointscan be noted from the table: (i) the average atomic scatter-ing factor (favg) changes across the alloys with a drastic de-crease from SY2 to SY3; (ii) the mass absorption coeffi-cient is a function of the radiation used (Cu-Ka, Cr-Ka,etc.); (iii) the solidus temperature (Ts) has a bandwidth ofmore than 200 8C. These variations result in a variation inthe intensity of 111 Bragg peak across the alloys (Fig. 1a).The percentage variations in intensity (minimum to maxi-mum) due to these factors are: (i) 66% in IS, (ii) 307% inIA and (iii) 0.13% in IT. Hence, the variation in Ts acrossthe alloys has the least effect, while the absorption factorplays the biggest role (using Cu-Ka radiation).

A comparison of the mass absorption coefficients forthree different radiations (Cu-Ka, Cr-Ka, Mo-Ka), acrossthe alloys in the SY system is shown in Fig. 1b. It can be

seen from the plots that the effect of the variation of (l/q)across alloys could have been avoided by using Cr-Ka oreven Mo-Ka radiation. Finally, to compare intensitiesacross multiple radiations, the value of ISAT obtained is mul-tiplied by another factor (designated I1) to obtain Inet [63].Details regarding the same can be found in the ESM. Thefactor I1 is dependent on two parameters: (i) the wavelength(k) of the radiation used, and (ii) the critical excitation vol-tage (Vk). Figure 2a shows the plot of Inet across the alloysin the SY system for Cu-Ka, Mo-Ka and Cr-Ka radiations,assuming that a single phase fcc alloy is obtained; which isclearly not the actual case. It is seen from the figure that,the variation in the intensity of the 111 peak across the al-loys in the SY system is small for Mo-Ka and Cr-Ka radia-tions, as compared to Cu-Ka radiation. Given that a low in-tensity is obtained with Mo Ka radiation, it is preferable touse Cr-Ka for investigations involving the comparison ofthe intensities across systems. The experimental results ofthe current investigation (SY system) for Cu and Cr-Ka ra-diations are shown in Fig. 2b ((i) & (ii)). The intensities ofthe 111 peaks have been normalized with respect to the in-

(a)

(b)

Fig. 1. (a) Plot of the computed 111 Bragg peak intensity across the al-loys in the SY system, due to variations in the: (i) atomic scattering fac-tor (IS ¼ f 2avg), (ii) absorption factor IA ¼ 1= 2lalloyð Þ

h i

for Cu-Ka radia-tion and (iii) solidus temperature (which leads to the temperaturefactor, IT). (b) Plot of the mass absorption coefficient l=qð Þalloy

h i

forthree different radiations (Cu-Ka, Mo-Ka and Cr-Ka), for the SY sys-tem.



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tensity for the pure element. The data extracted from thework of Yeh et al. (Figure 2 in [13]) has been overlaid onthe figure for comparison (Fig. 2b (iii)). To address the is-sue of the change in phase fraction of the fcc phase as wetraverse across the alloys in the system, the intensity hasbeen (approximately) corrected taking into account this as-pect (Fig. 2b (iv) for Cu-Ka & Fig. 2b (v) for Cr-Ka). Thetrends observed in Fig. 2a, corroborate well with the experi-mental investigations for the SY system after the correctionfor the phase fraction, using Cu and Cr-Ka radiations(Fig. 2b). This further strengthens the argument in favourof the use of Cr-Ka radiation for the investigation of alloysacross the SY system. Corresponding results and other de-tails for the SY system can be found in Section 5.1 of theESM. It is curious to note that the variation in I111net with n

using Cu-Ka radiation, displays a visual trend similar to thatresulting from treating lattice distortion as a thermal effect(as in Fig. 5, Yeh et al. [13]).

It is important to note that, the distortion factor used byYeh et al. [13] (�uDYeh) is different from that used by Owenet al. [12]. The use of �uDYeh as a measure of lattice distortionresults in a severe ‘damping’ of the Bragg peaks on the for-mation of CMA. Further, the temperature corresponding tothe value of �uDYeh used by Yeh et al. is 5985 K for the SY7alloy (using Eq. (4)), which ‘seems rather unreasonable’.The corresponding temperature using �uDOwen is 520 K. Thevalidity of the method of averaging of the intensity obtainedfrom the bcc and fcc structures used by Yeh et al. [13], isalso not clear.

The remaining important factor for comparison of inten-sities across alloys is that arising from atomic disorder,which leads to a decrease in the intensity of the Braggpeaks. This aspect is considered in the next section for thesingle phase alloys systems SA and SB.

In summary, the above analyses lead us to the following‘recipe’ for a fruitful and meaningful comparison of peakintensities across alloys (in CMA). (1) an alloy systemwhich forms a single phase CMA must be chosen. (2) a ra-diation that shows minimum fluorescence (absorption) for

(a)

(b)

Fig. 2. Intensity plots in the SY system. (a) Plot of I111net (computed)across the alloys for Cu-Ka, Mo-Ka and Cr-Ka radiations. (b) Experi-mentally obtained 111 peak intensities (current investigation) for Cu-Ka and Cr-Ka radiations ((i) & (ii)). The data extracted from Fig. 2 inthe work of Yeh et al. [13] have been overlaid for comparison (iii).The plot of the intensities corrected for the phase fraction of the fccstructure are labelled (iv) for Cu-Ka and (v) for Cr-Ka radiations.

Fig. 4. Experimental XRD patterns obtained from the alloys in the SAsystem using Cr-Ka radiation. The inset to the figure shows the com-puted XRD pattern in the vicinity of the 111 Bragg peak.

Fig. 3. The case of pure positional disorder in Ni. (a) Plot of FWHM asa function of �uAC. (b) Variation in the intensity with �uAC.



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the elements in the alloy system (e. g. Mo-Ka for the SYsystem) is preferred and if this is not possible, a radiationmust be chosen, wherein the variation in absorption factoris small across the alloys (e. g. Cr-Ka). (3) The results mustbe corrected/normalized for variation of: (a) average atomicscattering factor, (b) absorption factor and (c) solidus tem-perature across alloys. Once the above-mentioned factorsare taken into account, any remaining variation in the inten-sity of the Bragg peaks across the alloys is purely due to theeffect of the strain arising from atomic disorder. Needless topoint out, the sinh/k dependence of the atomic scatteringfactor, temperature factor and the Lorentz-polarization fac-tor have to be considered.

5. Results and discussion (SA and SB systems)

The results obtained from computations performed for thecase of pure positional and pure elemental disorder are con-sidered briefly first. Additional details regarding the samecan be found later in the manuscript and in Section 4 ofESM. The results for the case of pure positional disorder(in the absence of elemental disorder) are shown in Fig. 3,

which shows the plot of FWHM and intensity as a functionof �uAC. The following important conclusions can be drawnfrom the figure. (i) Pure positional disorder which perme-ates the entire crystal, leads to a broadening of the Braggpeaks. (ii) The magnitude of broadening is approximatelylinear with the distortion (�uAC); (iii) The intensity monoto-nically decreases with distortion. Pure elemental disorder(without positional disorder) does not lead to peak broaden-ing; however, there is an intensity change due to change inthe average atomic scattering factor (favg). These modelcomputations form the reference point for easy analyses ofthe other results in the current work.

Figure 4 shows the experimental XRD patterns obtainedfrom the alloys of the SA system using Cr-Ka radiation. It isclearly seen that there is no anomalous decrease in the inten-sity of the peaks, on increasing the number of elements in thealloy. XRD patterns obtained using Cu-Ka radiation for theSA system and from the SB system (Cu & Cr-Ka radiations)can be found in Section 5.2 of the ESM. The inset to Fig. 4shows the computed XRD pattern in the vicinity of the 111Bragg peak. The peak shifts are due to a change in the aver-age lattice parameter. The intensity of the peaks depends onmany factors as discussed before and is considered again la-ter in the manuscript. It is to be noted that the peaks are sym-metric, within the resolution of the experiments and compu-tations; which is in contrast to that observed for defectssuch as edge dislocations and stacking faults [64].

The results related to the 111 peak from SA and SB sys-tems are presented in this section, while those for the 200peak can be found in ESM (Sections 5 & 6). The FWHMfor the pure elements of the two series (i. e. SA1 and SB1)for the 111 peak are: 0.0498 (SA1, Cr-Ka) and 0.0538(SB1, Cr-Ka). The variation of the experimentally (usingCr-Ka) and computationally determined FWHM for the111 peak, with DRmax and �uDOwen for the SA system, is shownin Fig. 5. The inset to Fig. 5a shows the variation in FWHMwith the number of elements in the SA system. As men-tioned in Section 2, the value of the FWHM of the Braggpeaks is representative of the lattice distortion. For a mea-sure of lattice distortion, the use of FWHM of the Braggpeaks is preferred over the use of FWHM of the PDF (asmeasured from total scattering [12]). This helps in the de-coupling of the contributions from static (due to atomic dis-order) and dynamic disorder (due to temperature), as theBragg peak width is not affected by thermal effects. How-ever, it should be noted that the Bragg data is related to theaverage structure, unlike the total scattering data [12]. Theerror bar associated with the computational plot corre-sponds to five different configurations used in the study.Two important points can be noted from the figure: (i) thepeak broadening observed (FWHM) is significant (in therange of *0.078–0.098), especially in contrast to that ex-pected due to purely elemental disorder (Fig. 3) and (ii) areasonable match exists between the experimental andcomputational plots, which include the trendlines. Thebroadening of the Bragg peaks was envisaged by Guinier[1] for the case of a linear equiatomic disordered binary al-loy. The value of broadening observed is comparable to thatobtained for cold worked metals (e. g. peak broadening of*0.088 is reported for the 111 peak in cold worked Al[63]). The correspondence between the experimental andcomputational results can be considered as good, as withan increase in the number of elements, the computational

(a)

(b)

Fig. 5. Plot of FWHM (experimental and computational) versus (a)DRmax and (b) �uDOwen for the SA system (111 peak). The inset to (a)shows the variation of FWHM with the number of elements in the SAsystem. The inset to (b) shows the variation of FWHM with �uAC.



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result is expected to yield poorer approximation. This is dueto the number of binaries involved and the methodologyused to obtain the binary potentials.

The corresponding plots for the SB system and plots ofFWHM with other measures of distortion (like d & Ravg)are shown in the ESM. A monotonic increase in FWHMwith �uDOwen is observed in both SA and SB systems. The var-iation of FWHM with DRmax is monotonic for the SA sys-tem, but not for the SB system. It is to be noted that thecomputed values capture the combined effect of elementaland positional disorder. The parameter �uDOwen is a measureof the distortion with reference to an ‘average lattice’, re-presented as a ‘root mean square quantity’ and seems to bethe best amongst the a priori measures of lattice distortion(with d as the second best).

The inset to Fig. 5b, shows a plot of the variation of theFWHM in the SA alloy series with �uAC, which shows a goodcorrespondence with the plots in the main figure. This re-veals that, an a priori computed parameter like �uDOwen, corre-lates well with FWHM. It is to be noted that, �uAC is a param-eter from the computations and experimental data pointscorresponding to this parameter have been overlaid in theinset to Fig. 5b. The computational model slightly over-estimates the values FWHM and the distortions. The term‘correlates’ in the current context implies a monotonic in-crease. Important aspects regarding the use of parametersof lattice strain like �uDOwen and �uAC are discussed later inthe manuscript.

The variation in the intensity of the 111 peak with thenumber of elements in the SA system (experimental), usingCu-Ka and Cr-Ka radiations, is shown in Fig. 6a ((i) and(ii)). The variation in Inet with the number of elements inthe system is shown in Fig. 6b. Figure 6a ((iii) and (iv))shows the plots of (i) and (ii) corrected for the variation inInet. Thus, the intensity variation seen in the plots (iii) and(iv) arises purely due to lattice strain. It can be concludedthat the intensity decrease due to the formation of CMA,which is about 10–20%, is not insignificant. However, thiscannot be considered as anomalous either. The plot of in-tensities, which have been corrected for the variation in Inet,

with �uDOwen and DRmax are shown in Fig. 6c and d. A mono-tonic decrease in the intensities with these parameters is ob-served. The corresponding plots for the SB system areshown in Fig. 7. Two important differences to be noted be-tween Fig. 6 and Fig. 7 are: (i) in the SB system the latticedistortion (�uDOwen) does not increase monotonically with thenumber of elements and (ii) the intensity variation is not amonotonic function of DRmax. An important point to benoted is the increase in the intensity of the 111 Bragg peakbetween SB3 and SB4 (as seen clearly in Fig. 7c). This im-plies that the decrease in the observed intensity cannot becorrelated with the number of elements in an alloy. Anotherpoint to be emphasized is that the decrease in the intensitywith �uDOwen is monotonic for both the SA and SB systems,while this is not the case with DRmax in the SB system. Thisreinforces the confidence placed on �uDOwen as a measure oflattice distortion.

Figure 8a shows the variation in the computed 111 Braggpeak intensity, with the number of elements, in the SA sys-tem. The variation arising purely due to lattice distortion(plot-(iii)) is obtained from the total magnitude (plot-(i))by subtracting the value obtained due to elemental disorderon the perfect lattice (plot-(ii)). Figure 8b shows the varia-tion in computed intensity, due to the component arisingfrom the lattice distortion, as a function of �uAC. Similar tothe experimental observations, a monotonic decrease inthe intensity is observed with lattice distortion (�uAC).

A study of the plots of FWHM of 111 peaks with �uDOwen or�uAC reveals that the lattice strain as measured by these pa-rameters, has a progressively diminishing effect on theBragg peak broadening. This, on the one hand, effectively‘avoids’ the occurrence of ‘anomalous effects’ with an in-creasing strain; but, on the other hand, raises the question:\Are these measures of lattice strain truly representative ofthe bond length distortion?". A true measure of latticestrain, which is responsible for the strain energy in the crys-tal, should be computed from the actual bond length distor-tion. Bond length distortion in an fcc crystal can be defined

as: DD ¼ 12N

X

N

i¼1

X

12

j

Dij � D0ij

�

�

�

�

�

�

� �

; where, Dij is the bond

Fig. 6. Variation in the intensity of the 111 peak. (a) Plot of experi-mental data with the number of elements in the SA system, using Cu-Ka (i) and Cr-Ka (ii) radiations. Inset (b) Plot of Inet with the numberof elements for three radiations (Mo-Ka, Cu-Ka & Cr-Ka). The plots(i) and (ii) corrected for the variation in Inet are shown in (iii) & (iv) of(a). Inset (c) Plot with �uDOwen for Cu-Ka & Cr-Ka radiations. Inset (d)Plot with DRmax for Cu-Ka & Cr-Ka radiations.

Fig. 7. Variation in the intensity of the 111 peak. (a) Plot of experi-mental data with the number of elements in the SB system, using Cu-Ka (i) and Cr-Ka (ii) radiations. Inset (b) Plot of Inet with the numberof elements for three radiations (Mo-Ka, Cu-Ka & Cr-Ka). The plots(i) and (ii) corrected for the variation in Inet are shown in (iii) & (iv) of(a). Inset (c) Plot with �uDOwen for Cu-Ka & Cr-Ka radiations. Inset (d)Plot with DRmax for Cu-Ka & Cr-Ka radiations.



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length between the ith atom and its nearest neighbour the jth

atom, D0ij is the equilibrium bond length (ri + rj) between the

atoms and N is the number of atoms. The corresponding strain

can be computed as: eDD

¼ 12N

P

N

i¼1

P

12

j

Dij�D0ij

D0ij

�

�

�

�

�

�

�

�

� �

.

Figure 9a and b pertains to the case of pure positionaldisorder in nickel (computed), which was considered ear-lier. The main plot shows the variation of FWHM withDD and the inset shows the plot of DD with �uAC. Figu-re 9c–e pertains to the computed results from the SA sys-tem. Figure 9c shows the variation of FWHM with bonddistortion (DD). Insets to the figure show the variation ofDD with �uDOwen (Fig. 9d) and DD with �uAC (Fig. 9e). As di-dactically utilized before, experimentally obtained FWHM

data are studied as a variation of computationally obtainedDD. The following observations and conclusions can bedrawn from this figure, analyzed in conjunction withFig. 3. (i) For both Ni and the SA system; the computedFWHM scales approximately linearly with DD and the ‘sa-

turation’ observed in the FWHM–�uDOwen plot is absent. Thisunambiguously establishes the fact that FWHM is the truemeasure of bond distortion. (ii) The value of DD ‘satu-rates’ with increasing �uDOwen and explains the reason be-hind the nature of the FWHM–�uDOwen plots. The DD–�uACplot does not show this saturation behaviour; but is not lin-ear either. This further demonstrates that, a priori mea-sures like �uDOwen or d, or even an a posteriori measure like�uAC of the strain, do not accurately represent the bond dis-tortion in CMA. (iii) In the ‘special case’ of pure unre-laxed positional disorder, �uAC is commensurate with truelattice distortion (Fig. 9c), as observed from the approxi-mately linear trendline. Thus, the approximately linearnature of the variation of FWHM with �uAC (Fig. 3), is ex-plained. The corresponding plots for the SB system, high-lighting the importance of DD, can be found in the ESM(Section 6.1).The percentage increase in the lattice distortion be-

tween SA2 and SA5, as computed by various measures isas follows (the value of these parameters for SA5 is in-cluded in the brackets) (i) 100% for �uDOwen (0.018 Å), (ii)226% for DRmax (0.026), (iii) 114% for d (0.015), (iv)39% for �uAC (0.021 Å) and (v) 33% for DD (0.012 Å).The corresponding values for the SB system can be foundin Section 6.1 of the ESM. This calculation, in conjunc-tion with the results presented earlier, implies that the in-crease in bond length distortions on increasing the numberof elements in a CMA is of significantly lower magnitudethan that measured by a priori measures of lattice strain.The bond distortion, which is a true measure of strain inthe alloy and which contributes to the strain energy, isnot large as previously anticipated in the standard litera-ture. This implies that the strain energy penalty to thecrystal is lower and explains the relative ease of the for-mation of DSS in the presence of multiple elements. Toemphasize, entropy has to overcome the effects of a lowermagnitude of ‘true strain’ in the crystal and hence it is notsurprising that DSS form in CMA. The availability ofatoms of multiple sizes helps in this regard (as illustratedin the ESM Section 7.1). In this context it is important tonote that large lattice strains will lead to the formation ofa compound [11].

Fig. 9. The case of pure positional disorder in nickel: (a) the variation of FWHM with DD, and inset (b) plot of DD with �uAC. Results from the SAsystem: (c) the variation of FWHM with bond distortion (DD), inset (d) plot of DD with �uDOwen and inset (e) DD with �uAC.

Fig. 8. (a) The variation in the computed 111 Bragg peak intensitywith the number of elements in the SA system: (i) due to pure elemen-tal disorder, (ii) due to a combination of elemental and positional disor-der and (iii) Due to that arising from lattice distortion alone. Inset (b)The variation in computed intensity, due to the component arising fromlattice distortion, as a function of �uAC.



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The importance of the source of atomic radii, used for thecalculation of lattice strain parameters, was highlighted inthe introduction (Section 1). A few points are worthy of dis-cussion here. Owen et al. [12] have concluded that the in-crease in the PDF peak width cannot be correlated with thevalue of d. In our analysis, this stems from the use of bccCN-8 radii for Cr (& perhaps hcp radius for Co). Using thefcc CN-12 listing of Goldschmidt radii [43, 44], the valueof d increases in the following order for the alloys investi-gated by Owen et al. [12]: Ni-20Cr (d = 0.00322) < Ni-25Cr (d = 0.00348) < Ni-33Cr (d = 0.00378) < Ni-37.5Co-25Cr (d = 0.00992) < NiCoFeCrMn (d = 0.01481). Thiswould imply an increase in the PDF width with the value ofd (and �uDOwen). Further, this would necessitate an alterationin their conclusions [12] and this has been one of the motiva-tions of the current study. It is to be noted that the penternaryalloy used in the investigations of Owen et al. [12] is the onelabelled as SA5 in the current study. Song et al. [33] haveused elemental [65] and Pearson‘s 12-CN radii listing [66],for the computation of atomic size mismatch (d). They haveshown that the value of d increases in the following order:CoFeNi (d = 0.00957) < CoCrFeNiMn (d = 0.01056) <CoCrFeNi (d = 0.01181). These values of d have been calcu-lated from Pearson’s listing and if Goldshmidt's listing of ra-dii (12-CN) [43, 44] is used, the following order emerges:CoFeNi (d = 0.00769) < CoCrFeNi (d = 0.01055) < CoCrFe-NiMn (d = 0.01481). This implies that the d parameter corre-lates well with lattice distortion and the observations ofSong et al. [33] regarding the ability of d to predict local lat-tice distortions, have to be modified.

In the categorization of defects based on the decay of thedisplacement fields [2], it was noted that defects belongingto the type-II category show peak broadening. Thermal dis-order is a type-I defect in the Scheme. As discussed before,in the case of thermal disorder the reference lattice is ‘intact’,which is in sharp contrast to the case of CMA. From the re-sults presented, it is clear that the atomic disorder present inCMA has characteristics of both type-I & II defects (i. e. de-crease in intensity is akin to thermal disorder and peak broad-ening is similar to that caused by a type-II defect like disloca-tion). Further, the Bragg peaks from CMA are symmetric,unlike those for certain type-II defects (e. g. edge disloca-tions and stacking faults [64]); which result in peak asymme-try. Thus, the disorder in CMA should be categorized under aseparate class. In CMA, a defect is not localized in a perfectcrystalline matrix, but the disorder permeates the whole crys-tal. That is, the atomic disorder is not a defect in a crystal, butthe crystal is ‘defected’, wherein the defect is never infinite-simally small in any part of the crystal.

6. Summary and conclusions

Two systems (Ni–Co–Fe–Cr–Mn (SA) & Cu–Ni–Co–Fe–V(SB)) were computationally and experimentally investi-gated to evaluate the effect of atomic disorder on the Braggpeak, in a powder XRD pattern from CMA. The formationof DSS is observed in the SA and SB systems. The Cu–Ni–Al–Co–Cr–Fe–Si (SY) system was used as a referencefor the studies. Computations were performed using thefundamental equation for diffraction, to understand the ef-fect of lattice distortion on the FWHM of the XRD peaks.The following conclusions can be drawn from the study.

1) The intensity variation observed in the SY alloy sys-tem is primarily due to the variation in the atomic scatteringfactor and the absorption coefficient and not due to atomicdisorder. Based on the lessons learnt from the alloy system,a ‘recipe’ is developed to compare XRD intensities acrossalloy systems; such that the effect of lattice strains on theXRD peaks can be isolated. In the analysis the effect ofchange in the solidus temperature across alloys is included.This is a temperature factor in the ‘true’ sense, unlike latticedistortion modeled as a thermal effect.

2) (i) Pure positional disorder (in the absence of elemen-tal disorder) leads to an increase in the peak width(FWHM), along with a decrease in the intensity.

(ii) Pure elemental disorder (without positional disorder)does not lead to peak broadening; however, there is an in-tensity change due to a change in the average atomic scat-tering factor (favg).

3) (i) The FWHM of a given XRD peak increases with thestrain in the lattice, which arises due to the formation of aCMA. The lattice distortion parameter (�uDOwen) seems to bestcorrelate with the values of FWHM, as compared to param-eters like DRmax, d & Ravg.

(ii) The atomic diameter as listed by Goldschmidt (for12-coordination) gives the best results for the computationof �uDOwen.

(iii) Given that the peak broadening observed is an im-portant signature of CMA, it is incorrect to model the effectof lattice distortion as a ‘temperature factor’.

4) (i) On the formation of a CMA, the intensity of theBragg peaks decreases with lattice strain; but this aspectcannot be correlated with the number of elements, whichdecides the configurational entropy.

(ii) The decrease in intensity is not insignificant; how-ever, this cannot be termed as anomalous either. In fact,for some combinations of alloys and radiation used (e.g.with Cr-Ka radiation for alloys SB3 & SB4), the intensityobserved may actually increase with an increase in numberof elements in a CMA, which essentially implies the predo-minance of other factors over strain.

5) (i) If a priori measures such as �uDOwen or DRmax, or aposteriori measures like �uAC, are used as parameters of lat-tice strain, it may be construed that the lattice strain has aprogressively diminishing effect on the FWHM and the in-tensities of the Bragg peaks.

(ii) The bond length distortion, which can be directly re-lated to the strain energy cost to the crystal, can be per-ceived as a ‘true measure’ of lattice strain. It is establishedthat FWHM, which is an experimentally and computation-ally accessible quantity, is a measure of true lattice strainarising from bond length distortion. This can thus be usedto separate the static and dynamic components of disorder.

(iii) It is demonstrated that the bond length distortion inCMA is of significantly lower magnitude than that givenby a priori measures of strain; which effectively ‘avoids’the occurrence of ‘anomalous effects’ with an increasingnumber of elements.

6) In CMA the entire crystal can be construed as a de-fected solid, rather than a defect in a solid (i. e. the atomicdisorder permeates the whole crystal and is of non-localnature). The disorder in CMA should be categorized as aseparate class; as distinct from type-I (decay * r�2) andtype-II (decay * r�

3=2) defects.



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It is expected that the lessons learnt and conclusions drawncan be considered as general, which are applicable to otherconcentrated single phase systems. However, further investi-gations on diverse systems, especially with non-equimolarcompositions, are needed to confirm the assertions.

Ethics statement

The consent of all the authors was sought before submission of themanuscript.

Data Accessibility Statement

The data related to the article are contained in the figures in the maindocument and in the Electronic Supplemental Material (ESM)[http://home.iitk.ac.in/*anandh/Archive/ESM_X-Ray_Diffraction_Intensities_in_Concentrated_Multicomponent_Alloys.pdf].

Competing Interests Statement

The authors declare no competing interests.

Acknowledgments

The authors thank Gyanendra Pratap Singh (I.I.T. Kanpur) for helpwith computation of solidus temperature of alloys of the SY systemusing CALPHAD. Prof. S. Anantha Ramakrishna (I.I.T. Kanpur) andProf. Madhav Ranganathan are kindly acknowledged for discussions.Rupesh Chafle had kindly helped with the computation of bond lengthdistortion.

Author Contributions

Rameshwari Naorem and Sukriti Mantri performed the computationsof the X-ray diffraction patterns. Anshul Gupta carried out the experi-mental investigations. Gurjyot Sethi, K.V. Mani Krishna and Raj Palacomputed the relaxed coordinates using MD simulations. Kantesh Ba-lani and Anandh Subramaniam designed the Scheme of investigationsand prepared the manuscript. The current affiliations of the authorshave been cited. The work was carried out at I.I.T. Kanpur.

Funding

This research work did not receive any specific grant from fundingagencies in the public, commercial, or not-for-profit sectors.

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(Received June 26, 2018; accepted October 29, 2018; on-line since March 25, 2019)

Correspondence address

Prof. Dr. Anandh SubramaniamDepartment of Materials Science and EngineeringIndian Institute of TechnologyKanpur- 208016 U.P.INDIATel: +91 512 259 7215e-mail: [email protected] page: home.iitk.ac.in/*anandh

BibliographyDOI 10.3139/146.111762Int. J. Mater. Res. (formerly Z. Metallkd.)110 (2019) 5; page 393–405# Carl Hanser Verlag GmbH & Co. KGISSN 1862-5282



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A critical analysis of the X-ray diffraction intensities ...

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