The Convolutional Multiple Whole Profile (CMWP) Fitting Method


Download The Convolutional Multiple Whole Profile (CMWP) Fitting Method


Preview text

crystals
Article
The Convolutional Multiple Whole Profile (CMWP) Fitting Method, a Global Optimization Procedure for Microstructure Determination
Gábor Ribárik 1, Bertalan Jóni 1 and Tamás Ungár 1,2,* 1 Department of Materials Physics, Eötvös Loránd University Budapest, H-1117 Pázmány P. sétány 1/A, Hungary; [email protected] (G.R.); [email protected] (B.J.) 2 Department of Materials, The University of Manchester, Manchester M13 9PL, UK * Correspondence: [email protected]
Received: 19 June 2020; Accepted: 9 July 2020; Published: 17 July 2020
Abstract: The analysis of line broadening in X-ray and neutron diffraction patterns using profile functions constructed on the basis of well-established physical principles and TEM observations of lattice defects has proven to be a powerful tool for characterizing microstructures in crystalline materials. These principles are applied in the convolutional multiple-whole-profile (CMWP) procedure to determine dislocation densities, crystallite size, stacking fault and twin boundary densities, and intergranular strains. The different lattice defect contributions to line broadening are separated by considering the hkl dependence of strain anisotropy, planar defect broadening and peak shifts, and the defect dependent profile shapes. The Levenberg–Marquardt (LM) peak fitting procedure can be used successfully to determine crystal defect types and densities as long as the diffraction patterns are relatively simple. However, in more complicated cases like hexagonal materials or multiple-phase patterns, using the LM procedure alone may cause uncertainties. Here, we extended the CMWP procedure by including a Monte Carlo statistical method where the LM and a Monte Carlo algorithm were combined in an alternating manner. The updated CMWP procedure eliminated uncertainties and provided global optimized parameters of the microstructure in good correlation with electron microscopy methods.
Keywords: X-ray line profile analysis; neutron line profile analysis; CMWP; global optimum; dislocation densities; grain size; planar defects; Monte Carlo method

1. Introduction
Line profile analysis (LPA) of X-ray and neutron diffraction patterns has proven to be a powerful method for quantitative and qualitative characterization of lattice defects in crystalline materials [1–7]. Formally, there are two different approaches for treating diffraction line broadening. The top-down approach uses closed-form profile functions, like Gaussian, Lorentzian, pseudo-Voight, or Pearson-VII, for fitting peak profiles [8–11]. Since these profile functions are of ad hoc mathematical character, it is difficult to establish a sound correlation between specific lattice defects and the parameters of these functions. The bottom-up approach is based on physical profiles developed by using the physical properties of specific lattice defects [2–6,12–15]. Size profile calculation is based on optical principles [16] and the concept of column lengths [17,18]. Size distribution is taken into account, assuming log-normal size distribution [6,12,19]. Strain broadening is described by the Krivoglaz–Wilkens theory of diffractions in dislocated crystals [2,3,20,21]. The Krivoglaz–Wilkens theory has been extended to heterogeneous dislocation distributions [4,22–24], to small dislocation loops in irradiated crystals [25], and to infinitesimal dislocation dipoles in strongly deformed crystals [26]. Line broadening caused

Crystals 2020, 10, 623; doi:10.3390/cryst10070623

www.mdpi.com/journal/crystals

Crystals 2020, 10, 623

2 of 19

by planar defects has been treated theoretically [27–29] and modeled numerically for twinning and staking faults [30–32]. Broadening related to elastic intergranular strains is modeled by taking into account the elastic anisotropy of the materials [25,33,34].
The Levenberg–Marquardt optimization procedure has been used successfully in innumerable cases, cf. [5–7,19,25–35] to determine crystal defect types and densities. In the present work, we developed a more robust procedure where the Levenberg–Marquardt and the Monte Carlo methods are applied consecutively to provide the global optimum values of the physical parameters characterizing microstructures. In a previous short letter-type report [15], the basics of the method were outlined briefly. Here, we provide a detailed and more elaborate description of the procedure and show its power by analyzing the depth dependence of the dislocation density, the planar defect density, and the grain size in a Zr matrix and in ZrH precipitates in a hydrated Zr specimen.

2. Fundamental Principles of the Convolutional Multiple-Whole-Profile (CMWP) Optimization Procedure
The convolutional multiple-whole-profile (CMWP) optimization method is based on physically modeled profile functions of different microstructure elements [12–15]. The two fundamental microstructure elements are size and strain [1]. In diffraction patterns they combine as convolution, and in a particular hkl diffraction peak they are

Ihkl(s) = IhSkl(s) ∗ IhDkl(s),

(1)

where IhSkl(s) and IhDkl(s) are the size and strain profiles. The variable s is

s = K − ghkl

(2)

where ghkl is the fundamental reciprocal lattice vector of the hkl reflection and K is an arbitrary reciprocal space vector. Diffraction peaks are three-dimensional in reciprocal space [35]. In powder diffraction experiments, the intensity distributions are integrated along the surface perpendicular to the diffraction
vector, ghkl, and Equations (1) and (2) reduce to one dimension in the radial direction.

Ihkl(s) = IhSkl(s) ∗ IhDkl(s)

(3)

where s = 2(sinθ − sinθB)/λ, θ and θB are the diffraction angle and the exact Bragg angle of the hkl

peak, λ is the wavelength of radiation and * indicates convolution. The equivalent of Equation (1) in

Fourier space is

Ahkl(L) = AShkl(L)ADhkl(L)

(4)

where L is the Fourier variable. If planar defects, intergranular strains, and instrumental effects become substantial, the above equation is extended:

Ahkl(L) = AShkl(L) ADhkl(L) APhkDl (L) AIhGklS(L) AIhnkslt(L)

(5)

where APhkDl (L), AIhGklS(L), and AIhnkslt(L) are the Fourier transforms of the profile functions of planar defects, intergranular strains, and instrumental effects. The Fourier transforms AShkl(L), ADhkl(L), APhkDl (L), and AIhGklS(L) are modeled using physical properties of these lattice defects [15,25,30–34], whereas AIhnkslt(L) is determined by using measured patterns of defect-free standard materials. The latter can be determined
by measuring the patterns of standard specimens, e.g., of Si, CeO2, diamond, or LaB6 standards. The diffraction pattern is calculated from the inverse Fourier transforms of Ahkl(L):

ICalc(2θ)= hkl I0hklFT−1 AShkl(L) ADhkl(L) APhkDl (L) AIhGklS(L) AIhnkslt(L) 2θ − 2θh0kl

(6)

Crystals 2020, 10, 623

3 of 19

where θh0kl and I0hkl are the exact Bragg position and the intensity of the hkl reflection. The least-squares optimization is made by minimizing the weighted sum of squared residuals (WSSR):

WSSR = i [ICalc(2θi) + BG(2θi)] − IMeas(2θi) 2/w2i (7)

where BG(2θi) is the background intensity at 2θi and wi are weights applied to the ith measured data. The size profile is calculated by taking into account the shape and size distribution of coherently
scattering domains [17]. Assuming log-normal size distribution, the Fourier transform of size profiles can be written as [12,19]

AS(L) =

m3 exp(4.5σ2) er f c

ln(√|L|m)

√ − 1.5 2σ



m3 exp(2σ2)|L|

3



ln(|L|m) √

|L|3

ln(|L|m)

(8)

3

er f c

√−


2σ + 6 er f c

√ 2σ

where L is the Fourier variable, m and σ are the median and variance of the log-normal size distribution density function, and erfc is the complementary error function. Equation (8) is valid for spherical crystallites. The CMWP procedure yields the option to evaluate elliptical flat of long crystallites using the ellipticity parameter e which is the ratio of the long and short axis of rotational ellipsoids. With m and σ, the arithmetic-, area-, and volume-weighted mean crystallite diameters can be calculated [36].

j = m exp(kσ2)

(9)

where k = 0.5, 2.5, and 3.5 for arithmetic-, area-, and volume-weighted means, respectively, and j stands for these averages.
The Fourier coefficients of the strain profile are [1]

ADhkl(L)= exp( − 2π2L2 g2 ε2g,L )

(10)

where ε2g,L is the mean square strain (mss). Krivoglaz showed that strain broadening can only be caused by one-dimensional linear defects [20,21], and calculated ε2g,L for dislocations at small L values:

ε2

ρCb2 D

=

ln

(11)

g,L



L

where ρ, C, and b are the density, the contrast, and the Burgers vector of dislocations, and D is the size
of the crystal. This expression is logarithmically singular with increasing crystal size and is only valid for small L values. Wilkens [2,3] corrected the expression by replacing the crystal size with the effective outer cut-off radius of dislocations, Re, and calculated ε2g,L in the entire L range:

ε2

ρCb2

=

f (η)

(12)

g,L



where η=L/Re. The f (η) function is logarithmic at small L values and hyperbolic at large L values. The explicit form of f (η) is given in Equations (A6) to (A8) in Reference [2] and is shown in Figure 1. The figure shows f (η) as a solid blue line, and the logarithmic and hyperbolic components are shown as dash-dot-dot and dashed lines, respectively.
The profile functions of faulted and twinned crystals consist of several sub-profiles [1]. Typical sub-profiles of the 311 reflection of copper containing 4% intrinsic stacking faults are shown in Figure 2 [30]. The number, positions and breadths of sub-profiles depend on the hkl indices of the fundamental Bragg peaks [1,27–32]. In the CMWP procedure, the shifts and breadths of the sub-profiles are parameterized as a function of the density of specific planar faults or twin boundaries [30,31]. The fractions of the sub-profiles are based on the multiplicities within the hkl

stands for these averages. The Fourier coefficients of the strain profile are [1]

𝐴𝐷ℎ𝑘𝑙(𝐿) = exp( - 22L2g2〈𝑔2,𝐿〉)

(10)

where 〈𝑔2,𝐿〉 is the mean square strain (mss). Krivoglaz showed that strain broadening can only be

Crystals 2020, 10, 623 caused by one-dimensional

linear

defects

[20,21],

and

calculated

〈𝑔2,𝐿 〉

for

dislocations

at

small

4 of 19 L

values:

reflection. The shifts and breadths of the sub-profile𝐶sba2re giv𝐷en by fifth-order polynomials of the planar fault density. The coefficients of the polyn〈om𝑔2,𝐿i〉a=ls, a4long𝑙𝑛w(i𝐿th) the fractions of sub-profiles, are(1l1is)ted in
the mwahteerrieals,’Cs,paencdifibcaprearthaemdeetnesrittya,btlhees.coCnMtraWst,PanudsetshethBeusregetarsbvleescttoor eovf adliuslaotceattihonesd, eanndsitDy iosfthpelanar defectssiz. eTohfethpearcarymsteatle.rTthaibsleexsparreessfiroene itsoleodgiatrbityhmusicearlsly. Bsiansgeudlaornwtihtheoinrectriecaaslincogncsriydsetarlastizoenasn, idt wis aosnslyhown that thvealisdubfo-rpsrmofialel sLavraeluthese. sWuimlkeonfsa[2s,y3m] cmoreretrcitceadl tahnedexapnreasnstiio-nsybmy mrepetlariccianlgLthoerecnrytzsitalnsfizuenwctitohnth[3e0,31]. The thefefoecrteivtiecaoludteerrciuvta-otifof nrasdwiuesroefvdeisrlioficeadtiobnys,nRue,manedriccaallcusilmateudla〈ti𝑔2o,𝐿n〉siunstihnegenthtiereDLIrFaFnagXe: software [32].
The anti-symmetrical component of the profile fu𝐶nbct2ion is produced by interference between two overlaCprypstainlsg20s20u,b10-,rxeflFOeRctPiEoEnRsRiEnVrIEeWciprocal sp〈ac𝑔2e,𝐿.〉 O=ne4of th𝑓(ese) sub-reflections corresponds to t4h(1oef2p1)8arent, wherewahsetrhee=oLth/Reer. Tchoerrfe(s)pfounndctsiotnoitshleogtawriitnhmcriyc sattaslm. Talhl eL svyamluems eatnrdichaylpaenrdbotlhiceaat nlatrig-seyLmvmalueetrsi.cTahleparts of theexspulbiFc-iirgteufflorerem1c.tTioohfnef(sWa)ilrikseengcsiovfurernneciltniaotnEeqd[2u]wa(tbiilotuhnesceu(aArcv6he)).toTothh(eAedr8o)ta-indodRt-edcfaaesnrhebnlinceeeci[sh2t]ahraeanlcodtgeiasrriistzhheomdwicnbpyainrat,Fabisgriuenratedh1et.hTahned an anti-sfyigmumrKeersitvhroyogwlpasazrf(aapm)pareostxeairms[ao2tl7iiod]n. bT[l2uh1ee].slTienhteew, daonasvdhatlchuueervlsoeagiraseritathhlesmohiycppaenrrbadomlhiceytppeearrritbz.oe(ldCicoapcsoyarmigfphuotnnbceytniotcsonuaorrtefessyhtaoocwfkninagsfault and twdaisnhRd-diebonátr-sidiktoy[t1,2aa]n.n)dddaarseheindclliundese,dreisnpethcteivpealyr.ameter tables freely available through the web [30,31].

The profile functions of faulted and twinned crystals consist of several sub-profiles [1]. Typical sub-profiles of the 311 re8flection of copper containing 4% intrinsic stacking faults are shown in Figure 2 [30]. The number, positions and breadths of sub-profiles depend on the hkl indices of the fundamental Bragg pea6ks [1,27–32]. In the CMWP procedure, the shifts and breadths of the sub-
profiles are paramf e(te)rized as a function of the density of specific planar faults or twin boundaries
[30,31]. The fractions of4the sub-profiles are based on the multiplicities within the hkl reflection. The shifts and breadths of the sub-profiles are given by fifth-order polynomials of the planar fault density. The coefficients of the polynomials, along with the fractions of sub-profiles, are listed in the materials’ specific parameter table2s. CMWP uses these tables to evaluate the density of planar defects. The parameter tables are free to edit by users. Based on theoretical considerations, it was shown that the sub-profiles are the sum0 of a symmetrical and an anti-symmetrical Lorentzian function [30,31]. The theoretical derivations were verified by numerical simulations using the DIFFaX software [32]. The anti-symmetrical component of the profile function is produced by interference between two overlapping sub-reflec-t2io0ns in recipro2cal space. One4 of these sub6-reflections co8rresponds to the
parent, whereas the other corresponds to the twin crystal. The symmetrical and the anti-symmetrical
Fpiagrutsreof1.thTehseuWb-irlekfelencstifounnscatiroenco[2r]re(lbaltueedcwuritvhee).acThheotdhoetr-danodt-dcaanshbleincehaisratchteerliozgeadribtyhma ibcrepaadrtt,haasnidn tahneaKnrtiiv-soygmlamz eatpryprpoaxriammaetitoenr [[2271]]..TThheesedtawshocvuarlvueesisatrheeahlsyoppearrbaomlicetpearirzte. d(Caospayfruignhcttiboyn coofustratecskyinogf Rfaiubáltriakn[d12t]w.)in density, and are included in the parameter tables freely available through the web [30,31].

√ FigurFeig2u. rAen2.iAntnuiintitvueitisvcehsecmheamtiactiinc tienrteprrperteattaiotinonooffththeeddiissllooccaattiioonnddiippoolelechcharaarcatcetreprapraamraemteert,eMr,=MRe√=R. e ρ (a) R(aan)dRoamnddomislodcisaltoicoantiodnisdtrisibtruibtuiotino,n,wwhheerere MM  11.. (b(b) )DDisilsolcoactiaotnios nasrraarnrgaendgeind sitnronstgrodnipgoldeipole configcounrafitgiuornatwiohnerwehMere M1.T1h.eTshizeesiozfethofetrheegiroengisoonfs tohfethmeamteartiearliainl i(na)(a(r)e(drecdirccilrec)lea)nadnd(b()b()g(rgaryaycircle) and tchireclneu) manbdetrheofnpumlubsear nodf pmluisnaunsddmisilnoucsatdiiosnlosca(tuiopnas n(udpdaonwd ndoTws)niTns)thine tthweotwreogrieogniosnasraerethtehesame. The rseadmaer.rTohwesreadrearfororwRseaarendforthReebalnudetdheoublbuleedaorurobwlesararroewfos rartehefoarvtheeraagveerdagiselodcisaltoicoantidonisdtaisntcaen.ce.
2.1. P2h.1y.sPichaylsaicnadl aSnedcoSnecdoanrdyarPyaPraamraemtetresrsininththeeCCMMWWP PPrroocceedduurree
The Tahime aoimf tohfethCeMCWMPWPprporcoecdeudurere isis tthhee qquuaalliittaattiivveeaannddquqaunatintatittiavetivcehacrhacatrearicztaetrioiznatoiof n of micromsticrruocsttururcetsuirnescirnyscrtyaslltianlleinme amtaetreirailasl.s.TThhee ccrryyssttaallllooggrraapphhicicstsrutrcutucrteuroef othfethdeiffderiffenetreconmt cpoomnepnotsnents in theinmthaetemriaatlesriiaslsasissuasmsuemdetdo tboebkenkonwownn. .IInn tthhiiss sseennssee,,CCMMWWPPisids idffieffreenret,nbtu, tbcuotmcpolmempelenmtareynttoary to RietvReiledtvmeledthmoedthso. dTsh. eThpeapraamrameteetresrsuuseseddinintthhee ooppttiimmiizzaatitoionnpprorcoecdeudrue raereadreivdidivedidiendtointwtoo tgwroougpsroups
of (i) physical and (ii) secondary ones. Physical parameters include the size distribution parameters, of (i) mphaynsdica, lthaenddis(lioi)casteicoonnpdaararmy eotneress,.PahnydsMica, tlhpeaprlaamnaertderesfeicntcdleundseititehse, siozredfiosrtrsitbacuktiinognfpaualrtasmofeters,
twin boundaries, and the elastic intergranular strain, IGS. The peak positions and peak intensities are also fitted during the optimization procedure; however, these parameters are only used to improve the match between the measured and calculated diffraction patterns.
2.1.1. The Density and the Arrangement Parameter of Dislocations

Crystals 2020, 10, 623

5 of 19

m and σ, the dislocation parameters, ρ and M, the planar defect densities, α or β for stacking faults of twin boundaries, and the elastic intergranular strain, εIGS. The peak positions and peak intensities are
also fitted during the optimization procedure; however, these parameters are only used to improve the match between the measured and calculated diffraction patterns.

2.2. The Density and the Arrangement Parameter of Dislocations
Crystals 2020, 10, x FOR PEER REVIEW

5 of 18

Krivoglaz and coworkers [20,21] showed that the Fourier transform of the strain profile is a
logarithmKicrifvuongcltaizonanodf Kcoawsolroknegrsa[s2D0,2/L1]isshlaorwgeerdtthhaant uthneitFyo, uwrhieerretraDnsifsoarmnuomf btheer sotfraleinngptrhofdiliemisenasions. In Relfoegreanricthesm[i2c0f,2u1n]ctDionwaosf sKugagselsotnegd taos bDe/Lthies glraarginersitzhea.nTuhneitpyr,owblheemrewDithisthaisnudmefbineritioofnloenfgDthis the samedaismienntshioenesl.aIsnticRsetfoerreednceens e[r2g0y,2o1]f dDiswloacsatsiuogngse, swtehdictho bbeecothmeegsrlaoignasriizthe.mTichaellpyrsoibnlgemulawr iitfhththeius pper
definition of D is the same as in the elastic stored energy of dislocations, which becomes integrlaotgiaornithlimmiictailslysesitntgoublaertihf ethceryusptpaelrsiiznete.gIrnattihoen cliamseitoisf seelat stoticbesttohreecdryesntaelrgsiyz,et.hIinstphreocbalseemofiselsaosltviced by settinsgtotrheeduepnperegryi,nttheigs rpartoioblnemlimisitsoblyvetdhebyavseerttaingge dthiestuapnpceeroifndteigsrloatciaotniolinmsi[t3b7y,3t8h]e. aWvielrkaegnesdriestaalnizceed the correloaftiodnislboectawtioenesn t[h37e,3e8la].stWicislktoernesdreenaleirzgedy otfhediscloorcrealtaiotinons abnedtwstereaninthberoealdaestnicingstoorfeddifefrnaecrtgiyonopf eaks,
and redpisllaocceadtioDnsinanthdestKrariinvobgrolazdefonrinmguolfadwiffirthacthioenepffeeacktsiv, aenodurteeprlcaucetd-oDff irnadthieusKorifvdogisllaozcfaotrimonusl,aRwei[t2h,3,39].

Wilketnhse ienftfreoctdivueceoduttehreccuot-nocfefprtadoifu“sreosftrdiicstleodcalytiornasn,dRoem[2”,3d,3is9l]o. cWatiiloknendsisintrtirbouduticoends tahnedccoanlcceupltatoefd the

strain“rfuesntrcitcitoend,lyf (rηa)n, dinomth”edeinsltoicreatLionradnigsteri[b3u].tioTnhseacnodnccaelpcut laastesdumtheessthraaitntfhuencetqiouna,lfn(u)m, inbethreoefnptliures and minusL srtarnagigeh[t3p].aTrahleleclosnccreepwt adsissulomcaetsiothnastatrheereaqnudaolmnulymdbiesrtroibfuptleuds wanidthminincyulsinsdtrearisghotf pRaerraaldleili,swcrhewere the cylinddeirssloccoavtieorntshaerewrhaonldeocmrylsytadl.isStriinbcuetethdewviathluine ocyf lRineddeerpseonfdRseornadthiie, wachtuerael dthiseloccyalitniodnerdsecnosviteyr, tWheilkens introdwuhcoeldetchreysdtiaml. eSninsicoentlheessvpalaureamofetReerdMep=enRdes√oρnttohde eascctruiabledtihselodciastlioocnadtieonnsaitryr,aWngielkmenesntisnt[r3o].dTucheedvalue
the dimensionless parameter M = Re√ to describe the dislocation arrangements [3]. The value of M of M iiss llaarrggee wwhheenntthheeddisilsolocactaitoinosnsaraereunucnocrorerlraetleadteadndanadrraarnrgaendgreadnrdaonmdloymanlydatnhedrtehleatreedlastteradinstfriaeilndsfields are ofalroenogf-rlaonngge-rcahnagreaccthearr. aIcntecro.nItnracsotn, Mtraisst,smMailsl wsmhaelnl owphpeonsiotpe-psoigsinted-sisigloncadtiisoloncsaatiroenisn astrreoinngsctororrneglation close ctooreraeclahtiootnhecrloasnedtotheearcehlaottehderstaranidn tfhieeldrse,ladtuede tsotrsacinreefineilndgs,, adrueeotfoshscorrete-rnainngg,eacrheaorafcstehro.rAt-rnaningteuitive schemchataircaicltleurs.trAantioinntiusitsihvoewscnhienmiantiFcigilulurestr2a,twiohnerise s1h2odwisnloicnatiinonFsigaurreera2n, dwohmerley d12istdriisbluocteadtioinnsFaigreure 2a

and inraansdtormonlyg disptorilbeuctoendfignuFriagtuioren 2ina Fanigduirne 2abs.trIonntghedicpaosleeocfornafnigduormatidonistirnibFuigtiuorne, 2thb.eIsntrtahiencfaiesledosfreach

out mruancdhofmurdthisetritbhuatinonth, tehaevsetrraaignefideildslsorceaatciohnoudtismtauncchef,udrdtihsleocr,t(htahne trheedacviercralegeisdRiseloacnadtiothnedbisltuaencdeo, uble

arrowddtishloce, (athveerraegdecidrcisleloiscaRteioannddtihsetabnlucee)d, owuhbelereaarsroiwn tthhee acvaesreagoef sdtirsoloncgatdioinpodliestcaoncnef)i,gwuhraetrieoans,inthtehestrain

fieldscaarsee sotfrostnrgolnygsdcirpeoenleecdoannfidguRrae tbioenco, mtheessstrhaoinrtefireltdhsanardedsistlrooc.nTglhyesMcreveanleudeainsdinRde ibreeccot mcoersreslhaotriotenr with the avthearangdedisdloci.sTlohceaMtiovnadluisetiasnicned: irect correlation with the average dislocation distance:

MM==RRee√√ρ== 𝑑 𝑅R𝑒 e d𝑑d𝑖𝑠i𝑙s𝑜lo𝑐 c

(13) (13)

RRaannddoomm oorr ssttrroonngg ccoorrrerelaltaetdeddidslioslcoactiaotnioanrraarnrgaenmgeenmtsencatsn cbaen cbaellecdallwedeakweoarkstorronsgtrodnipgoldeipole

characchtaerrasc.teIrns.thInetfihersftircsat scea,seM, Misilsalragregrerththaannuunnity, M ≫ 11, ,wwhehreeraesaisnitnhethsecsoencdoncadsec,aisteis, ictloisectlose to

unityuonriteyvoernevsmenaslmlear:lleMr: ≤M1. 1D. Dueuetotoththeerreecciipprociittyyoofflelennggththscsaclaesleisn icnrycsrtyalstaanldarnedciprerocicparl ospcacl es,pace,

V-5Cr-5Ti

110

1

I/IMAX
0.1

FWHM
30 min at 800 oC M=1.6

4 GPa =37 M=5.8

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

K [1/FWHM]

FigurFeig3u. rIent3e.nIsniteyndsiitsytrdibisuttriobnustioonfsthoef {t1h1e0}{1r1e0fl}ercetfiloenctsioonfsaoVf-5aCVr-5CTir-a5lTloi yaldloeyfodrmefoerdmbeyd hbiyghh-ipgrhe-ssure torsiopnreasts4urGePtoarpsiroenssaut r4eGtoPathperesshseuarre stotrtahine sohfeγar=st3r7ai(nreodf cu= r3v7e()readndcuarfvte)rahnedataifntegrthoe8a0ti0ng◦Cto(b80lu0eCcurve)
with l(obgluaerictuhrmvei)cwinitthenlosgitayristhcmaliecninotremnsaitlyizsecdalteontohremmaliazxeidmtoumtheinmtaexnismituiems ianntedntshiteieFsWanHd Mthe. MFWorHeMd.etails aboutMmoraetedreitaalilpsraobcoeustsimnagtearniadl pthroechesesaint gtraenadtmtheenhtseactatnrebaetmfoeuntnsdcainn bReeffoeurenndcien [R4e0f]e.rence [40].

When the strain fields are of long- or short-range character, the tail regions of diffraction peaks decay faster or slower. Figure 3 shows the intensity distributions of the {110} reflections of a V-5Cr-5Ti [40]

Crystals 2020, 10, 623

6 of 19

When the strain fields are of long- or short-range character, the tail regions of diffraction peaks decay faster or slower. Figure 3 shows the intensity distributions of the {110} reflections of a V-5Cr-5Ti [40] alloy deformed by high-pressure torsion at 4 GPa pressure to the shear strain of γ = 37 (blue curve) and after.
Heating to 800 ◦C (red curve) with a logarithmic intensity scale normalized to the maximum and the FWHM. The dislocation density did not decrease after heat treatment at 800 ◦C for 30 min, but the dislocations rearranged into narrow dipole configurations [40]. This rearrangement of dislocations is a kind of recovery process before recrystallization, when the dislocations annihilate and their density drops to low values. In the as-deformed state (blue curve), the tail of the peak decays much faster than in the heat-treated (red curve) recovery state. In the recovery state, the plus–minus dislocations form close pairs of dipoles for which the efficiently screened strain fields are of short-range character. The change of the M parameter from 5.8 to 1.6 is the quantitative measure of rearrangement of the dislocation distribution during the recovery process. When the strain fields are of long-range character, i.e. M 1, the profile tails will be short and the profiles will have a bell-shaped character. Meanwhile, when the strain fields are of short-range character, the profile tails will be long and the profiles will be more of Lorentzian character. This is one of the reasons why attributing Gaussian or Lorentzian components in a pseudo-Voight profile function to strain or size broadening is physically incorrect.

2.3. The Contrast Factor of Dislocations
The mean square strain is proportional to the contrast, C, of dislocations. In X-ray or neutron diffraction, the contrast has the same physical meaning and origin as in TEM. It depends on the relative orientation between the Burgers and line vectors, b and l, the particular diffraction vector, g, and the elastic constants of the crystal, cijkl: C = C(b,l,g,cijkl) [2,3,19]. In a polycrystal, or if all possible Burgers vectors are randomly populated, C can be averaged either over the permutations of the hkl indices or all possible Burgers vectors [41,42]. In the case of cubic and hexagonal crystals, the average contrast factors, C, can be written as [5,43,44]

C = Ch00 1 − qH2 ,

(14)

Chk.l = Chk.0 1 + q1x + q2x2 ,

(15)

where Ch00 and Chk.0 are the average contrast factors for of the h00 and hk.0 type reflections, H2 = (h2k2 + h2l2 + k2l2)/(h2 + k2 + l2)2, and x = (2/3)(l/ga)2 (a is the basal lattice constant of the hexagonal crystal).
In the CMWP code, Equation (15) is included in a more direct manner as Equation (15a) [12]:

Chk.l = Chk.0 1 + a1H12 + a2H22 ,

(15a)

where H12 = {[h2 + k2 + (h + k)2]/l2}/[h2 + k2 + (h + k)2 + (3/2)(a/c)2l2], H22 = l4/[h2 + k2 + (h + k)2 + (3/2)(a/c)2l2], a1 = q1 and a2 = q2 + q1/[(2/3)(c/a)2]. The mss in Equation (12) consists of ρ, C, and b in the product. Therefore, only ρ and the hkl dependent q or q1 and q2 can be determined independently via the CMWP procedure. The scaling factors Ch00 and Chk.0 can only be refined after the q or q1 and q2 values have been determined. The same is true for the Burgers vector values of , , and slip modes in hcp materials [15,45]. The effect of changing dislocation density on the elastic constants
and the Ch00 and Chk.0 scaling constants was investigated in References [46,47]. In Reference [47], it was shown that the effects could change the absolute values of dislocation densities up to 40%.

2.4. Determination of Slip Modes in hcp Crystals
Once the a1 and a2 parameters are determined, the true dislocation densities, ρtrue, and the
, , and slip mode fractions in hcp materials can be determined [15,45]. A simpler and more

Crystals 2020, 10, 623

7 of 19

straightforward method than in References [15,45] is briefly described here. The mss can be written as the sum of the partial mss values, ε2i,g,L , related to the different slip modes:

ε2g,L = ε2i,g,L = 41π ρiCib2i f (η) , (16)

i

i

where i stands for the different slip modes. We can assume that the strain function f (η) is global for all hk.l peaks:
ρ ∗ Ce f f b∗2 = 41π ρiCib2i , (17)
i
where ρ* and Ce f f are the dislocation density and the contrast factor values given by the CMWP procedure. Here, Ce f f = C*hk.0 (1 + a1H12 + a2H22), where C*hk.0 is the input value of the scaling factor and q1* and q2* are the contrast factor parameters given by the CMWP procedure. The fractions, ξ a , ξ c and ξ c+a of the partial mss values are

ξi = ρiCib2i / ρ ∗ Ce f f b∗2) ,

(18)

where i ξi = 1. The partial dislocation densities will be

ρi = ξiρ ∗ Ce f f b∗2 /(Cib2i .

(19)

Since Equations (5) must hold for all hk.l values, ξi can be obtained using the least-squares method. The CMWP procedure allows the evaluation of diffraction patterns consisting of more than one
phase. In such cases, the number of physical parameters increases. In the simplest case of a cubic
single-phase material with a spherical crystallite shape and no planar defects, the number of physical parameters will be five: ρ, M, and q for the strain profile and m and σ for the size profile. In an hcp specimen, containing two different phases, the number of physical parameters becomes twelve: ρ, M, m, σ, and q1, q2, and ρ1, M1, m1, σ1, and q11, q12, where the upper index ’1’ is for the second phase. (Note that the major phase is not labeled.) Since line profile analysis evaluates only the shape and broadening of diffraction peaks, the peak positions and maxima are treated in CMWP as secondary
parameters. The peak shifts caused by faulting or twinning are taken into account by the shifts of the
sub-profiles included in the parameter files for planar defects [30,31].

2.5. Algorithms Used for Solving Equation (7)
Equation (7) requires a non-linear problem with correlations between the different parameters to be solved. The possibilities are (a) nonlinear least-squares algorithms [48], (b) direct search methods [49], and (c) statistical methods [50]. The most frequently used nonlinear least-squares algorithms are the Newton method, the steepest descent or conjugate gradient method, and the LM [51,52] procedure. The nonlinear least-squares algorithms find the local minimum of the WSSR in the parameter space, which does not necessarily coincide with the global minimum of the WSSR. Statistical methods [37,39] can be used to find the global minimum of the WSSR in the parameter space. In order to find the global minimum in the CMWP procedure, we applied a special MC statistical algorithm and the LM nonlinear least-squares algorithm in a multiple successive process. The MC method only optimizes the physical parameters. Based on physical considerations, each parameter is restricted between a minimum and maximum value which cannot be bypassed; however, these can be edited by the user. In consecutive iterations, the new parameter values are searched in the proximity of the previous ones:

ain ∈ ain−1 + ∆in, ain−1 − ∆in

(20)

Crystals 2020, 10, 623

8 of 19

where i is the parameter index and n is the current iteration number. ∆in is defined as

 

∆i0,

n ≤ n0

Crystals 2020, 10, x FOR PEER R∆EVi IE=W

n−n0
∆i (1/4) n0 ,

n  0

n0 < n ≤ n1 .

8(2o1f )18

shown

in

Figure

4.

In

the

 special

MC

alg∆oimriint,hm

developed

fno1r


we

applied

a

bias

towards

the

prevailing parameter values in the consecutive iterations. The consecutive parameter values follow a

cubiTchpervobalaubeiloitfy∆finuniscttihoensvasm. e𝑥d𝑛𝑖 ugreinngertahteedfirbsyt na0riatnerdaotmionnsu. mDbuerirnggefnuerrtahteorr:iterations, ∆in decreases

exponentially to ∆imin.

Once

∆in

reaches𝑖∗∆imin,𝑖

when
𝑖

n

reaches

3

𝑖

n1,

the

∆in

values

remain

constant.

For example, if ∆i /i = 50, the value𝑎o𝑛f =∆i 𝑛i(s2r𝑥e𝑛a−ch1e)d a+ft𝑎e𝑛r−a1bout 4000 iterations, as shown(i2n2)

0 min

min

Fwighuerere4.𝑥I𝑛𝑖nt[h0e,1s]p. eTchiaelsMtaCr ianlg𝑎o𝑛𝑖r∗ithinmdidceavteeslotpheadt tfhoirsCcoMnWsePc,uwtiveeapvpalluieedhaabsinaos ttoywetabrdeesnthaeccperpevteadiliansga pbaertatmerevtearluveatlhuaens i𝑎n𝑛𝑖 t−h1e. Tchoensbeicausetidvecuibteicraptiroonbsa.biTlihtye fcuonncsteiocnutiisvsehpoawrnamscehteemr vaatilcuaellsyfionllFoiwguarecu4bb.ic probabDiuliteytofutnhcetiboinasvesd. xpinrogbeanbeirlaittyedfubnycatiorann, dthoempnaruammbeetrergespnearcaetoisr:sampled on a finer scale around

preceding regions of

tvhaelupeasr.amHoetwerevspera,cea.sTFhigeucaroine∗nd=4ibt∆iosinhno2fwxoinrs−,act1hcee3pr+etiniasgi athlseonaewgopoadracmhaentecres,of𝑎𝑖saims pling

remote (22)

n−1

𝑛

where

xin



[0, 1].

The

star

in

ain∗

ind𝑎ic𝑖 a=tes{𝑎th𝑛𝑖∗a, t
𝑛

this𝑊co𝑆n𝑆s𝑅e𝑛cu
not

yet

been

accepted

as(2a3)

better value than ain−1. The biased cubic pr𝑎o𝑛𝑖b−a1b, ility𝑊fu𝑆n𝑆c𝑅ti𝑛on 𝑊is 𝑆s𝑆h𝑅o𝑛w−n1 schematically in Figure 4b.

(a) ai +i (b)

0.4 0=0.4

n-1 n-1

i

ai n

n i = (1/4)(n-n0)/n0

0.2

n0

ai

n-1

0.0 0

n0 2000

4000

i = n

n1

6000 8000
n

10000

ai -i n-1 n-1 0.0

0.5

X 1.0

Figure 4. (a) Half-breadth of the parameter interval, ∆in𝑖, as a function of the number of iterations, n. UFpigtourtehe4.fi(ras)t Hnoaslft-ebprsea(hdothrizoofntthael rpeadralimnee)t,e∆r inin𝑖 sttearyvsalc,ons𝑛t,aanst.aBfeutnwcetieonnthofe tshteepnsunmo banerdonf1i,t∆erina𝑖 tdioecnasy, sn. exUppotnoetnhtieaflilryst(bnlousetelpinse()h. oBreiyzoonntdalnr1e,d∆lini𝑖nseta),ysc𝑛onstsatyans tco(hnosrtaiznotn. Btaeltwgreeeenn tlhinees)t.e(pbs) nSocahnedmna1t,icd𝑛radweicnagys oef xtphoenpeanrtaiamlleyte(brluvaelluinees)a. inBevyso. ntdhne1r, an𝑛dosmtaygsecnoenrsattaendt (nhuomrizboenr,taxling, rienenthleinbei)a. s(ebd) Sccuhbeimc aptriocbdarbaiwliitnyg fuonf ctthioenpuasreadmientetrhevasplueecsia𝑎l 𝑛M𝑖 Cvsa. ltghoerirtahnmd.om generated number, 𝑥𝑛𝑖 , in the biased cubic probability
function used in the special MC algorithm.

Due to the biased probability function, the parameter space is sampled on a finer scale around precedTinhge verarlouress.oHf tohweepvaerra, maseFteigruvraelu4bessharoewdse, ttheremreiniseadlsionategromosdocfhapn%cefroafcstiaomnps loinf gthreemWoStSeRre[g5i3o]n.sIf oaf tphaerptiacuralamreWteSr SsRpavcae.luTehies cloarngdeirtiothnafnorthaeccbeepsttinvgaltuhee, nWeSwSpRabersta,mfoeutenrds,uanintiisl the actual iteration, but
within a certain p% fraction of WSSRbest, i.e.,

WaiSS=R>WaSin∗S,Rbest ANWDSSWRSnS
(2(234) )

n

ain−1,

WSSRn WSSRn−1

then the parameter values corresponding to this WSSR value are stored in an array. At the end of the

fittinTghepreorcreodrsuroef,tthhee plaarrgaemsteatnerdvthaleusems aalrleesdteotfeermacihnpedarianmteetremr svaolfupe%inftrhaectairornasyowf itlhl beeWcoSnSsRid[e5r3e]d.

Ifasa pthaertpicluulsarmWinSuSsRavbaslouleuties learrrgoerrs tohfanthteherebspesetctvivaleuep,aWramSSeRtebresst,wfoiuthndthuengtilvethne pa%ctunaulmitbereart.ioTnh,e

bruetlawtivtheinerarocresrtairne pc%alcfuralactieodn aonfdWlSisStRedbesta,cic.eo.r,dingly. The convergence criteria of the special MC

procedure are that (i) the number of steps reaches a user-defined value and (ii) that the 1 + p% statistics reaches at leasWt NSpS, Rw>heWreSNSRp ibsesat nAeNdDitaWblSeSmRi
WSSR was not larger than the last WSSR value. The systematic testing of the combined LM + MC thaelgnotrhitehpmarsahmoewteerdvtahluatesnc0o=rr2e0s0p0o,npdi=ng3.t5o%thaisndWSNSpR= v1a0l0uegaarveesetoxrceedlleinntanpaatrtrearyn. fAitttitnhgesenanddofgtohoed fiptthinygsipcarlopceadraumree,ttehrevlaalrugeessteavnedn itfhtehsemdaifllferastctoifoneapcahttpearrnasmweeterre vcoalmuepliincattheeda. rOrauyr wexipllebrieecnocnessihdoewreedd atshtahte tphleusM-mCinupsroacbesdoulurete oeurrtolirnseodf tahbeorveespiesctiidveeapllayrasmueitteedrs iwnitthhethecagsieveonf pl%inenupmrobfeirl.e Tahnearlyelsaitsi.veIn earrgorreseamreenctalwcuitlhattehdeacnodplyisritgedhtaccoconrdditiniognlys.oTfhCeMcoWnvPe,rtgheenscoeucrrciteercioadoef othfethspiseMciaCl MisCacpcreoscseibdluerienatrhee

software package, which is free to use for academia purposes.

2.3. Organization of the Combined LM and MC Algorithms
The total number of parameters can become quite large when more than ten diffraction peaks and more than one phase comprise the diffraction pattern. In such cases, the LM procedure can get

Crystals 2020, 10, 623

9 of 19

that (i) the number of steps reaches a user-defined value and (ii) that the 1 + p% statistics reaches at least Np, where Np is an editable minimum number of MC iterations in which the WSSR was not larger than the last WSSR value. The systematic testing of the combined LM + MC algorithm showed that n0 = 2000, p = 3.5% and Np = 100 gave excellent pattern fittings and good physical parameter values even if the diffraction patterns were complicated. Our experience showed that the MC procedure outlined above is ideally suited in the case of line profile analysis. In agreement with the copyright conditions of CMWP, the source code of this MC is accessible in the software package, which is free to use for academia purposes.
2.6. Organization of the Combined LM and MC Algorithms
The total number of parameters can become quite large when more than ten diffraction peaks and more than one phase comprise the diffraction pattern. In such cases, the LM procedure can get frustrated and the optimization procedure can get stuck. This problem is overcome by sectioning the LM procedure. One full cycle of the optimization procedure consists of the following sections: (1) MC optimization, (2) LM optimization of only the peak positions, (3) LM optimization of only the peak heights, (4) LM optimization of only the physical parameters, (5) LM optimization of all parameters including the physical parameters, the peak positions, and peak heights, and (6) readjustment of the BG. The cycle of six steps is repeated until convergence is reached.
2.7. Systematic Comparison of the Performance of the LM and MC Procedures
Though the LM optimization procedure is a well-established and widely used analytical method, in nonlinear problems, the parameter space can have more than a single minimum and LM can get stuck in one which is not the absolute minimum. The implementation of the MC procedure combined with LM aimed to find the global minimum in the parameter space. A systematic analysis was carried out to check the performance of both the LM procedure alone and the combined alternative with LM plus MC. The most critical parameters were the dislocation density concomitant with the dipole character number, i.e., ρ and M. One of the Zircaloy-4 X-ray diffraction patterns containing about 10% δ-hydride, discussed in detail in the next paragraph, was evaluated by varying the starting values of ρ over 5 orders of magnitude, from 10−3 × 1014 m−2 to 100 × 1014 m−2. The results are shown in Figure 5. CMWP is governed by optimizing the parameters for obtaining the smallest weighted sum of squared residuals (WSSR). The WSSR values are shown in Figure 5a vs. the starting values of ρ when LM alone (open blue triangles) and LM combined with MC (open red circles) were applied. The figure shows that LM alone could not find the minimum of WSSR, especially when the initial ρ values were larger than the optimum, which was about 0.9(± 0.1) × 1014 m−2. The combined application of LM + MC always found the optimum WSSR whatever the initial values of ρ were. The dislocation density and the dipole character number, ρ and M, are shown vs. the initial ρ values in Figure 5b. The figure shows that both ρ and M varied up to about two orders of magnitude when only LM was applied: 0.7 ≤ ρ ≤ 100 × 1014 m−2, 0.4 ≤ M ≤ 60. With the combination of LM + MC, the optima of both ρ and M were found within reasonable error margins: 0.7 ≤ ρ ≤ 1.13 × 1014 m−2, 10 ≤ M ≤ 60. The relatively large variation of M does not mean that it could not be determined. Whenever the value of M is of the order of 10 or larger, it only means that the dipole character of dislocations is very weak. Or, in other words, the dislocation arrangement is “random” [2,3].

Figure 5b. The figure shows that both  and M varied up to about two orders of magnitude when only LM was applied: 0.7   1001014 m−2, 0.4  M  60. With the combination of LM+MC, the optima of both  and M were found within reasonable error margins: 0.7    1.13  1014 m−2, 10  M  60. The relatively large variation of M does not mean that it could not be determined. Whenever the Cvraylsutaelso20f2M0, 1i0s,o62f3the order of 10 or larger, it only means that the dipole character of dislocations i1s0voef r1y9 weak. Or, in other words, the dislocation arrangement is “random” [2,3].

1000 Zr + ~10% Hydride (a) Zr + ~10% -Hydride (b)

LM

100

LM+MC

final

WSSR

100

final

final LM

10

 LM+MC

final

Mfinal LM

Mfinal LM+MC

1

[1014 m-2], M 

10

1E-3 0.01 0.1

1

10

100

 [1014 m-2] initial

1E-3 0.01 0.1

1

10

100

 [1014 m-2] initial

Figure 5. Systematic investigation of the LM and MC optimization procedures. (a) The WSSR vs. the

iFniigtiuarlev5a.luSyesotef mthaetidcisinlovceasttiiognatdioennsoiftyt,hρeinLiMtial,awndheMnCLMoptailmonizea(toiopnenprbolcueedturriaens.g(lae)s)TahnedWthSeSRLMvs.atnhde

initial value of the dislocation den MC (open red circles) optimization

spitryo,ce𝜌dinuitrieasl,

when LM alone (open blue triangles) and the LM and were used. (b) The final dislocation density and dipole

cMhCara(ocpteernpraerdacmirectleers)voaplutiems,izρafitniaolnanpdroMcefidnuarl,evssw. ethree uinsietdia.l(,bv)aTluheesf,inρainlitdiailslaoncdatMioninidtiealn, sinityloagnadridthipmoilce

scchaalreasc, tcearlcpualraatmedetuesrivnaglueietsh,e𝜌rftinhael LaMndal𝑀ofninea(l,ovpse.nthbeluineittiraial,nvgalleuseasn, d𝜌ignriteiaelnantdria𝑀nginlietisa)l,oirntlhoegaLrMithamnidc

MscCaleosp, tciamlcizualatitoend pursoincegdeuitrheesr(othpeenLMredalcoirncele(sopanend bblluacektrdiaontsg)l.eTshaenvdegrtrieceanl btrliaacnkglliense)soirntdhiecaLteMerarnodr

bMaCrs.oAptlilmcuizravteisoninptrhoecetwduorfiesgu(orepsenarreeodncliyrctloesguainddebthlaeckeydeo. ts). The vertical black lines indicate error

bars. All curves in the two figures are only to guide the eye.

2.8. Extension of CMWP for Handling Satellites or Diffuse Scattering

2.5. Extension of CMWP for Handling Satellites or Diffuse Scattering Guinier–Preston (GP) zones [54,55], small dislocation loops [56,57], or diffuse scattering of solute

atomGs [u5i8n]iecra–nPprerostdounc(eGsPa)tezlolintes o[5r4d,5iff5]u,ssemsaclaltdteisrlioncgaptieoanklsoaorposu[n5d6,5o7r]b, eolrodwifftuhesefuscnadttaemrienngtoalf Bsorlaugtge

pateoamkss. [5T8y]pciacnalpsraotdeullciteesaitnelltihteslorwdeirffiunsteenscsaittyteriannggpeseaokfsdairfforuancdtioonr bpeelaokws tohfeafupnrdoatomne-nirtraaldBiraategdg

Zircaloy-2 specimen are shown in Figure 6a. The vertical arrows point to the satellite peaks produced

either by vacancy- (on the left side of the peaks) or interstitial-type dislocation loops (on the right

side of the 00.2 reflection). The interstitial-type satellite on the 10.1 peak cannot be seen because it is

blending in with the broader main peak. The strain profile given by Equations (10)–(12) relies strongly

on the tail regions of Bragg peaks. Satellites or diffuse scattering peaks, not related directly to the

strain profiles, may distort the physical parameter values defining the true strain profiles and the true

strain parameters. The option to handle satellites or diffuse scattering peaks is also implemented in the

CMWP procedure [25]. Irradiation-induced dislocation loops evolve from knocked out vacancies and

interstitials in irradiated materials [56,57,59–62]. Larson and Young [61] and Mason et al. [62] showed

that irradiation-induced dislocation loops have a very wide size distribution, ranging from a few tenths

to a few hundred nanometers. The strain fields of small dislocation loops overlap and strengthen each

other inside the loop regions. As a consequence, within the strained regions of the small vacancy-

or small interstitial-type loops, the lattice constants increase or decrease relative to the matrix lattice

constants, respectively [61,63]. The strained regions around small loops produce satellite peaks around

the fundamental Bragg reflections in proton-, neutron-, or ion-irradiated materials [25,57,62]. Based

on the theoretical description of satellites and diffuse scattering of point defect clusters and small

dislocation loops [61,63,64], we developed a procedure to handle satellites in the CMWP procedure [25].

Taking into account the small size and wide size distribution of the strained volumes of small loops,
the satellite profiles, IhSkAlT(s + ∆s), were modeled as size profiles according to Equation (8), where ∆s is the shift of the satellite peaks relative to the main hkl reflection. In References [64–66], it was

shown that although the strained volumes are coherent with the matrix, the intensities scattered by

the strained volumes are incoherent; therefore, the intensities of the main and the satellite peaks

are additive. The satellite intensities are usually a few percent of the intensities of the main peaks;

therefore, the optimization of the main diffraction pattern and the satellite peaks can be done in separate

steps. The two steps can be repeated subsequently until the procedure converges. Figure 6b,c shows

zoomed parts of the measured (open circles) and CMWP-calculated (red lines) diffraction patterns of a Zircaloy-2 specimen neutron irradiated to the fluence of 13.1 × 1025 n/m2 at about 300(± 25) ◦C [58,67].

Preparing to load PDF file. please wait...

0 of 0
100%
The Convolutional Multiple Whole Profile (CMWP) Fitting Method