High-Resolution Strain/Stress Measurements by Three-Axis Neutron Diﬀractometer

materials

Article High-Resolution Strain/Stress Measurements by Three-Axis Neutron Diﬀractometer

Pavol Mikula *, Vasyl Ryukhtin , Jan Šaroun and Pavel Strunz Department of Neutron Physics, Nuclear Physics Institute ASCR, v.v.i., 250 68 Rež,ˇ Czech Republic; [email protected] (V.R.); [email protected] (J.Š.); [email protected] (P.S.) * Correspondence: [email protected]; Tel.: +420-2-6617-3159

Received: 2 November 2020; Accepted: 27 November 2020; Published: 30 November 2020

Abstract: Resolution properties of the unconventional high-resolution neutron diffraction three-axis setup for strain/stress measurements of large bulk polycrystalline samples are presented. Contrary to the conventional two-axis setups, in this case, the strain measurement on a sample situated on the second axis is carried out by rocking the bent perfect crystal (BPC) analyzer situated on the third axis of the diffractometer. Thus, the so-called rocking curve provides the sample diffraction profile. The neutron signal coming from the analyzer is registered by a point detector. This new setup provides a considerably higher resolution (at least by a factor of 5), which however, requires a much longer measurement time. The high-resolution neutron diffraction setting can be effectively used, namely, for bulk gauge volumes up to several cubic centimeters, and for plastic deformation studies on the basis of the analysis of diffraction line profiles, thus providing average values of microstructure characteristics over the irradiated gauge volume.

Keywords: residual stresses; neutron diﬀraction; three axis setting; high resolution; bent crystal monochromator; bent crystal analyzer

1. Introduction Residual stresses are typical phenomena associated, e.g., with the welding of different kinds of structural material. Stresses are generally responsible for the deformation of structures during production and subsequently influencing the behavior of these structures during service [1–3]. However, they also occur when external load or some kind of shape forming is applied to the sample. X-ray diffraction and neutron diffraction are excellent nondestructive techniques for the determination of strain/stress fields in polycrystalline materials. X-ray diffraction, which due to a strong attenuation of X-rays in the material, is limited to sample surface measurements. On the other hand, neutron diffraction, thanks to a low attenuation of neutrons in most materials, is suitable for strain scanning of bulk samples. The conventional neutron diffraction method is based on the precise determination of the relative change in the dhkl-spacing of particularly oriented crystalline grains in the gauge volume [1–3]. In neutron and X-ray diffraction, the angular positions of the diffraction lines are determined by the well-known Bragg condition 2dhkl sin θhkl = λ (θhkl—Bragg angle, λ—the wavelength). The elastic · strain ε is defined as ε = ∆dhkl/d0,hkl. It is in fact the relative change in the lattice spacing with respect to the value d0,hkl corresponding to a strain-free sample element. The strain in the material is a tensor, and for its determination, at least three strain components should be obtained. However, only one component parallel to the scattering vector Q is determined from the measurement of one diffraction line (see the inset in Figure1). Therefore, the strain measurements have to be carried out for three geometrical positions. It is clear that for strain evaluation, the determination of the d0,hkl value also represents a key task [1–3]. The differentiation of the Bragg condition provides a simple formula for the strain ε as ε = cot θhkl ∆θhkl. According to this formula, the strain ε in the material − ·

Materials 2020, 13, 5449; doi:10.3390/ma13235449 www.mdpi.com/journal/materials Materials 2020, 13, x FOR PEER REVIEW 2 of 7

Materials 2020, 13, 5449 2 of 7 the strain ε in the material brings about a change in the scattering angle 2θhkl by Δ(2θhkl) of the position of the diffraction line with respect to the position of the line of the strain-free element. Therefore, from the angular shift in the diffraction line Δθ , the average lattice macrostrain brings about a change in the scattering angle 2θhkl by ∆(2θhkl)hkl of the position of the diffraction line component over the irradiated gauge volume element can be determined. The neutron strain/stress with respect to the position of the line of the strain-free element. Therefore, from the angular shift instrument is a 2-axis diffractometer (see Figure 1) equipped with a position-sensitive detector (PSD), in theusually diffraction used line for ∆theθhkl mapp, theing average of residual lattice strains macrostrain inside polycrystallinecomponent over materials. the irradiated The spatial gauge volumeresolution element mainly can be determined determined. by the The beam neutron defining strain optics/stress is usually instrument of the order is a 2-axisof millimeters. diffractometer The (see Figurediffractometer1) equipped is usually with equipped a position-sensitive with a focusing detector monochromator (PSD), usually (bent perfect used crysta for thel) that mapping has the of residualcurvature strains optimized inside polycrystalline to achieve a maximum materials. luminosity The spatial and resolution resolution mainly with respect determined to a chosen by the beam definingdiffraction optics line [1 is–3 usually]. Together of with the order the focusing of millimeters. monochromator, The di fftheractometer beam optics is system usually includes equipped with aslits focusing before monochromatorand after the sample (bent that perfect determine crystal) the dimensions that has theof the curvature gauge volume optimized element to and, achieve of a maximumcourse, luminosity the position and-sensitive resolution detector with for respect imaging to a the chosen beam di profileffraction diffracted line [ 1 by–3 ]. the Together irradiated with the focusingelement monochromator, [4–6]. The strain/stress the beam scanner optics can systemalso be includesequipped slitswith beforean auxiliary and after machine the sample, e.g., for that thermo-mechanical manipulation with the sample. Then, the resolution of the instrument results determine the dimensions of the gauge volume element and, of course, the position-sensitive detector from the combination of uncertainties coming from the thickness and radius of curvature of the bent for imagingcrystal themonochromator, beam profile the di ffwidthsracted of by the the slits irradiated (in most cases element of 0.5 [–42– 6mm]. The), the strain divergence/stress of scanner the can alsoincoming be equipped and diffracted with anbeam auxiliary by the irradiated machine, element e.g., for, and thermo-mechanical the spatial resolution manipulation of the position with- the sample.sensitiveThen, detector. the Their resolution combinationof the results instrument in a total results uncertainty from usually the combination forming a Gaussian of uncertainties image comingin fromthe detect theor thickness with a FWHM and radius of about of (5 curvature–10) × 10−3 rad of the, which bent can crystal be sufficient monochromator, for the measurement the widths of the slitsof the (in angular most cases shifts of of 0.5–2 diffraction mm), lines the divergence generated by of strains.the incoming However, and such diff racted resolution beam is notby the irradiatedsufficient element, for the and investigation the spatial of resolution the changes of in the the position-sensitive diffraction profile, which detector. would Their, e.g., combination allow for resultsmicrostructure in a total uncertainty characterization usually of forming plastically a Gaussian deformed imagepolycrystalline in the detector samples. with a FWHM of about Encouraged3 by the first recent preliminary results [7], we have decided to study resolution (5–10) 10− rad, which can be sufficient for the measurement of the angular shifts of diffraction lines properties× of a new unconventional high-resolution neutron-diffraction three-axis setup for large generated by strains. However, such resolution is not sufficient for the investigation of the changes bulk polycrystalline samples experimentally with the aim of a possible exploitation in the field of in the distrain/stressffraction profile,measurements. which The would, most e.g., important allow obtained for microstructure results are presented. characterization of plastically deformed polycrystalline samples.

Scattering vector Q Diffracted wave Incident wave vector k f vector k i

d 2 hkl

Diffracting planes

Figure 1. Scheme of the conventional strain scanner. Figure 1. Scheme of the conventional strain scanner. Encouraged by the first recent preliminary results [7], we have decided to study resolution 2. Materials and Methods properties of a new unconventional high-resolution neutron-diffraction three-axis setup for large bulk polycrystallineSeveral years samples ago [8–10] experimentally, the first attempts with were the made aim ofwith a possiblea high-resolution exploitation three-axis in the setting field, of as schematically shown in Figure 2a. Following the drawing shown in Figure 2a (for small widths of strain/stress measurements. The most important obtained results are presented.

2. Materials and Methods Several years ago [8–10], the first attempts were made with a high-resolution three-axis setting, as schematically shown in Figure2a. Following the drawing shown in Figure2a (for small widths of the samples), the best resolution from this setting can be obtained when minimizing dispersion Materials 20202020,, 1313,, 5449x FOR PEER REVIEW 3 of 7 the samples), the best resolution from this setting can be obtained when minimizing dispersion of the of the whole system. When solving the problem in momentum space, it means that the orientation wholeMaterials system. 2020, When13, x FOR solving PEER REVIEW the problem in momentum space, it means that the orientation3 of 7of the ofΔk themomentum∆k momentum elements elements related to related the monochromator to the monochromator and analyzer and analyzershould be should matched be to matched that of tothe that sample,the ofsamples), the and sample, the this best can and resolution be this achieved can from be this achieved by setting proper can by radiibe proper obtained of radiicurvatures when of minimizing curvatures of the dispersion bent of the perfect bent of the perfect crystal crystalmonochromatorwhole monochromator system. andWhen analyzer. andsolving analyzer. the Contrary problem Contrary in to momentum the to the conventional conventional space, it means two two-axis-axis that strain/stress the strain orientation/stress scanner, scanner, of the the didiffractionffractionΔk momentum profilesprofiles elements with the related three-axisthree -toaxis the setup monochromator are obtained and by analyzer rocking should the bent be matched perfect to crystal that of (BPC) the sample, and this can be achieved by proper radii of curvatures of the bent perfect crystal analyzer situatedsituated onon the the third third axis axis of of the the di ffdiffractometerractometer, and, and the neutronthe neutron signal signal is registered is registered by the by point the monochromator and analyzer. Contrary to the conventional two-axis strain/stress scanner, the detectorpoint detector (see Figure (see Figure2)[ 7,8 ].2) By[7,8]. properly By properly adjusting adjusting the curvature the curvature of the analyzer,of the analyzer the three-axis, the three setting-axis diffraction profiles with the three-axis setup are obtained by rocking the bent perfect crystal (BPC) exploits the focusing in both real and momentum space. The resolution of this alternative setting was settinganalyzer exploits situated the focuson theing third in axisboth of real the diffractometer,and momentum and space.the neutron The resolutionsignal is registered of this byalternative the α optimizedsettingpoint was detector by optimized using (see a well-annealedFigure by using 2) [7,8]. a well By- -Feproperlyannealed low-carbon adjusting α-Fe low steel the-carbon rodcurvature (technically steel of rod the analyzer,(technically pure iron th withe three-axispure C iron less with than 1%)C less ofsetting than the diameter exploits1%) of thethe of diameter 8focusing mm (standard in of both 8 mm real sample). (standard and momentum For sample). the experimental space. For Thethe experimentalresolution studies, theof this three-axisstudies alternative, the neutron three - opticsaxis settingneutron diffractometer was optics optimized diffractometer installed by using at a thewell-installedRežˇannealed researchat the α-Fe Řež reactorlow-carbon research LVR-15 steelreactor rod (Research LVR (technically-15 ( CentreResearch pure eironRž,ˇ Centre with Husinec, Řež, CzechHusinec,C less Republic) Czech than 1%) Republic) and of equipped the diameter and withequipped of 8 Si(111)-monochromator mm (standardwith Si(111) sample).-monochromator For and the Ge(311)-analyzer experimental and Ge(311) studies, single-crystal-analyzer the three- single slabs- ofcrystal dimensionsaxis slabs neutron of200 dimensions optics40 diffractometer4 and200 ×20 40 ×installed40 4 and1.3 20 at mm × the 403 Řež×(length 1.3 research mm3 width(length reactor × thickness),LVR-15 width (×Research thickness), respectively, Centre respecti Řež was, used.vely, × × × × × × Thewas Si(111)Husinec,used. The monochromator Czech Si(111) Republic) monochromator providing and equipped the providing with neutron Si(111)-monochromator wavelengththe neutron of wavelength 0.162 and nm Ge(311)-analyzer had of a0.162 fixed nm curvature single-had a fixed with crystal slabs of dimensions 200 × 40 × 4 and 20 × 40 × 1.3 mm3 (length × width × thickness), respectively, thecurvature radius R withM of the about radius 12 m, R andM of the about radius 12 of m curvature, and the of radius the analyzer of curvature was changeable of the analyzer in the range was was used. The Si(111) monochromator providing the neutron wavelength of 0.162 nm had a fixed fromchangeable 3.6 to 36in the m. range Figure from3 shows 3.6 to some 36 m. results Figure of 3 theshows optimization some results procedure of the optimization related to the procedure setting curvature with the radius RM of about 12 m, and the radius of curvature of the analyzer was shown in Figure2a, when searching for a minimum FWHM of the rocking curve, and the optimum relatedchangeable to the setting in the rangeshown from in Figure 3.6 to 36 2a, m. when Figure searching 3 shows some for a results minimum of the FWHM optimization of the procedurerocking curve , α curvatureand relatedthe optimum of to thethe settinganalyzer. curvature shown The of in optimumthe Figure analyzer. 2a, radiiwhen The of searching curvatureoptimum for aradi of minimum thei of Ge(311) curvatureFWHM analyzer of of the the rocking forGe(311)-Fe(110) curve analyzer, and αfor-Fe(211) αand-Fe(110) the reflections optimum and α- curvatureFe(211) have been reflections of the found analyzer. ashaveRA The been= 9 optimum m found and radiiRasA R= ofA3.6 =curvature 9 m,m respectively.and of R theA = Ge(311) 3.6 However,m, analyzerrespectively. it has beenHowever,for found α-Fe(110) it thathas been high-resolutionand αfound-Fe(211) that reflections high determination-resolution have been ofdetermination found the lattice as RA changes= of9 mthe and lattice can RA be= changes achieved3.6 m, respectively. can even be achieved on large irradiatedevenHowever, on large gauge itirradiated has volumes been found gauge (see that Figurevolumes high-resolution2b), (see though Figure determination slightly 2b), though relaxed. of theslightly lattice relaxed. changes can be achieved even on large irradiated gauge volumes (see Figure 2b), though slightly relaxed. BPC monochromator BPC monochromator Roc king BPC BPC monochromator Rocking BPC BPC monochromator (b) Roc kingBPC analyzer  M analyzerRocking BPC  M (a ) (b) analyzer  M (a ) analyzer  M LMS LMS LMS LMS Slit LSA Slit LSA  LSA  LSA          RM R  R RM  M  M   S S Polycrystalline S S Polycrystalline RRAA PolycrystallinePolycrystalline RA RA sample sample PointPoint sample sample Point Point detectordetector detectordetector

FigureFigure 2. ThreeThree-axis 2. Three-axis-axis diffractometer di ﬀdiffractometerractometer settings settings employing bent bent perfect perfect crystal crystal (BPC ( (BPC)BPC) monochromator) monochromator and and analyzer as used in the feasibility studies (RM, RA—radii of curvature, θM, θA—Bragg angles) for analyzer as used used in in the the feasibility feasibility studies studies (R (RMM,, RRAA—radii—radii of of curvature,curvature, θθMM,, θθAA——BraggBragg angles) for vertical (a) and horizontal (b) positions of a polycrystalline sample. This figure has been reprinted vertical ((aa)) andand horizontal horizontal (b ()b positions) positions of of a polycrystalline a polycrystalline sample. sample. This This ﬁgure figure has beenhas been reprinted reprinted from from Ref. [7]. Ref.from [ 7R].ef. [7].

RA = 3.6 m Si(111)+-Fe(110)+Ge(311) Si(111)+-Fe(211)+Ge(311) 4000 RA = 3.6 m R = 4.5 m -Fe(110) -  = 8 mm -Fe(211) -  = 8 mm RA = 3.6A m Si(111)+-Fe(110)+Ge(311) R = R 4.5 = m 3.6 m Si(111)+-Fe(211)+Ge(311) 8000 4000 A A R = 6 m -Fe(110) -  = 8 mm -Fe(211) -  = 8 mm RA = 4.5A m R = R 6 m= 4.5 m 8000 A A R = 9 m 3000 R = 9 m R = 6A m A R = 6 m 6000 A A R = 18 m R = 18 m R = 9A m 3000 A R = 9 m A A (b) 6000 R = 18 m (a) A 2000 RA = 18 m

Intensity / relative 4000 (b)

Intensity / relative

(a) 2000

Intensity / relative 4000 2000 1000

Intensity / relative

2000 0 1000 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0  / deg  / deg 0   -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0  / deg Figure 3. Analyzer rocking curves for different radii of curvature and for (a /) degα-Fe(110) and (b) αFigure-Fe(211) 3. reflections.Analyzer rocking curves for different radii of curvature and for (a) α-Fe(110) and (b) α- Fe(211) reflections. 3. ResultsFigure 3. Analyzer rocking curves for different radii of curvature and for (a) α-Fe(110) and (b) α- AfterFe(211) the reflections. adjustment of the diffractometer setting for optimum resolution, the first step was measuring the resolution of the experimental setting for different diameters of the well-annealed

Materials 2020, 13, x FOR PEER REVIEW 4 of 7

3. Results Materials 2020, 13, 5449 4 of 7 After the adjustment of the diffractometer setting for optimum resolution, the first step was measuring the resolution of the experimental setting for different diameters of the well-annealed α- α-Fe(110)Fe(110) standard standard samples samples situatedsituated inin thethe verticalvertical position (see Figure Figure 44).). Concerning the the resolution resolution representedrepresented by the by FWHMthe FWHMof the of analyzer the analyzer rocking rocking curves, curves, it can itbe can seen be from seen Figure from 4 Figure that the 4 diameterthat the of thediameter sample of (generally the sample the (generally thickness) the plays thickness) an important plays an role. important In the nextrole. twoIn the steps, next thetwo influence steps, the of theinfluence slit width of on the the slit resolution width on the properties resolution for properties two diameters for two of diameters the well-annealed of the well- sampleannealed of sample 4.9 and 8 mmof 4.9 in theand horizontal 8 mm in the position horizontal was position studied was (see studied Figures (see5 and Figures6). In 5 both and cases,6). In both three cases slit, widths three slit of 5, 10,widths and 20of mm5, 10 were, and used.20 mm This were can used. be useful This can when be comparinguseful when the compar effecting of dithefferent effect widths of different of the irradiatedwidths sample,of the irradiated as the vertical sample, dimension as the vertical of the irradiateddimension sampleof the irradiated does not playsample a principal does not role play with a respectprincipal to the role resolution with respect of the to experimental the resolution setting. of the experimental As expected, setting. it can be As seen expected, from Figuresit can be5 andseen6 thatfrom the irradiatedFigures 5 and width 6 that of the the irradiated sample (represented width of the insample our case (represented by the slit in width) our case plays by the an slit important width) role.plays However, an important for the slitrole. widths However, of 5 and for 10the mm, slit thewidthsFWHM of 5is and also 10 slightly mm, the influenced FWHM is by also the diameterslightly influenced by the diameter of the sample. The inspection of Figures 4–6 reveals that with the of the sample. The inspection of Figures4–6 reveals that with the exception of the slit width of 20 mm, exception of the slit width of 20 mm, in all other cases, the angular resolution was sufficiently high in all other cases, the angular resolution was sufficiently high for observation of not only elastic for observation of not only elastic strains, but also changes in the diffraction profiles for strains, but also changes in the diffraction profiles for microstructural (microstrains, mean grain size) microstructural (microstrains, mean grain size) studies of plastically deformed samples on the basis studies of plastically deformed samples on the basis of diffraction profile analysis [11,12]. Furthermore, of diffraction profile analysis [11,12]. Furthermore, the comparison of Figure 5c with Figure 6c shows thethat comparison in the case of when Figure the5c slit with width Figure is much6c shows larger that than in the the diameter case when of the the sample, slit width the isdiameter much larger itself thanhas the a diameternegligible ofinfluence the sample, on the the FWHM diameter of the itself rocking has curve. a negligible Finally, influence a slightly ondeformed the FWHM α-Fe(110)of the rockinglow-carbon curve. steel Finally, rod aof slightly  = 4.9 mm deformed put in theα-Fe(110) horizontal low-carbon position for steel two rod slit of widthsϕ = 4.9 was mm measured.put in the horizontalThe corresponding position for experimental two slit widths rocking was measured. curves are Theshown corresponding in Figure 7. It experimental should be pointed rocking out curves that arefor shown measur in Figureements7 related. It should to Figures be pointed 4b and out 5, the that same for measurementssample was used, related however, to Figuresafter annealing.4b and5 , theIf same we compare sample wasFigure used, 7a,b however, with Figure after 5a,b, annealing. the effect Ifof we plastic compare deformation Figure7 ona,b the with FWHM Figure of5 a,b,the therocking effect of curve plastic, as deformation well as the diffraction on the FWHM profileof itself the, rockingis evident curve, and could as well be assufficient the diff ractionfor a possible profile itself,evaluation is evident of andthe microstructure could be sufficient parameters for a possible on the basis evaluation of the diffraction of the microstructure profile analysis. parameters Of course, on theif basis the samples of the di areffraction situated profile at the analysis. diffractometer Of course, with a if high the samplesaccuracy, are the situated relative atchange the di inff ractometerthe lattice withconstant a high accuracy,ε = ΔdS/d0 the (strain) relative can changealso be inderived the lattice from constantthe peak εposition= ∆dS/ dshift0 (strain) ΔθA. The can peak also beshift derived ΔθA fromcorresponds the peak position to the change shift in∆θ theA. scattering The peak angle shift Δ∆(2θAθScorresponds) as ΔθA = −Δ(2 toθS the). Then, change with in the the help scattering of the angledifferentiated∆(2θ ) as ∆ formθ = of∆ the(2θ Bragg). Then, condition with the ΔdS help/d0 = of−Δ theθS cot diff θerentiatedS (d0 is the formlattice of spacing the Bragg of the condition virgin S A − S ∆dSsample),/d0 = ∆ θtheS cot averageθS (d0 elasticis the latticestrain values spacing εR of (radial the virgin component) sample), and the ε averageL (longitudinal elastic straincomponent) values − εR (radialwithin component)the irradiated and gaugeεL volume(longitudinal could be component) evaluated. within the irradiated gauge volume could be evaluated.

5000 FWHM = 0.071 + 0.004 deg

4000 No slit 3000 (a)

2000

Intensity / relative 1000

0 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3  / deg A Figure 4. Rocking curves for the annealed α-Fe(110) low-carbon steel rods in vertical position: (a) ϕ = 2, MaterialsFigure 2020, 14.3 ,Rocking x FOR PEER curves REVIEW for the annealed α-Fe(110) low-carbon steel rods in vertical position: (a) =5 of 7 (b) 4.9, and (c) 8 mm. 2, (b) 4.9, and (c) 8 mm.

Figure 5. Rocking curves for the annealed α-Fe(110) low-carbon steel rods of ϕ = 4.9 mm in horizontal Figure 5. Rocking curves for the annealed α-Fe(110) low-carbon steel rods of  = 4.9 mm in horizontal position: (a) 5, (b) 10, and (c) 20 mm Cd slits. position: (a) 5, (b) 10, and (c) 20 mm Cd slits.

Figure 6. Rocking curves for the annealed α-Fe(110) low-carbon steel rod of  = 8 mm in horizontal position: (a) 5, (b) 10, and (c) 20 mm Cd slits.

Figure 7. Rocking curves for the slightly deformed α-Fe(110) low-carbon steel rod of  = 4.9 mm installed in horizontal position for (a) 5 mm slit width and (b) 10 mm slit width.

4. Discussion The obtained results shown in Figures 5–7 prove that the presented diffraction method provides a sufficiently high angular resolution, represented by the FWHM of the diffraction profile, which allows for the possibility of studying some plastic deformation characteristics (root-mean-square microstrains, as well as the effective grain size), by applying shape analysis on the neutron diffraction peak profiles [11,12]. The advantage of this method permits the investigation of large volumes of polycrystalline samples (several cubic centimeters), thus providing average values of microstructure characteristics over the irradiated gauge volume, e.g., as a function of macroscopic strain loaded on the sample by an auxiliary instrument. Further application of this method can be found in strain/stress studies of textured samples with a large size of grains, as the conventional diffraction method requiring small irradiated gauge volume would be problematic. Naturally, it could also be used for strain scanning in the polycrystalline materials similarly to the conventional two-axis strain/stress scanners when evaluating strains in small gauge volumes (few cubic millimeters) from peak position shifts ΔθS. However, for such measurements, this method would not be practical,

Materials 2020, 13, x FOR PEER REVIEW 5 of 7

Figure 5. Rocking curves for the annealed α-Fe(110) low-carbon steel rods of  = 4.9 mm in horizontal position: (a) 5, (b) 10, and (c) 20 mm Cd slits. Figure 5. Rocking curves for the annealed α-Fe(110) low-carbon steel rods of  = 4.9 mm in horizontal Materials 2020position:, 13, 5449 (a) 5, (b) 10, and (c) 20 mm Cd slits. 5 of 7

Figure 6. Rocking curves for the annealed α-Fe(110) low-carbon steel rod of  = 8 mm in horizontal Figure 6. Rocking curves for the annealed α-Fe(110) low-carbon steel rod of ϕ = 8 mm in horizontal position:Figure (6.a) R5,ocking (b) 10 ,curves and (c for) 20 the mm annealed Cd slits. α -Fe(110) low-carbon steel rod of  = 8 mm in horizontal position:position: (a) ( 5,a) (5,b )(b 10,) 10 and, and (c )(c 20) 20 mm mm Cd Cd slits. slits.

Figure 7. Rocking curves for the slightly deformed α-Fe(110) low-carbon steel rod of ϕ = 4.9 mm Figure 7. Rocking curves for the slightly deformed α-Fe(110) low-carbon steel rod of  = 4.9 mm installedFigure in 7. horizontal Rocking curves position for forthe ( aslightly) 5 mm deformed slit width α and-Fe(110) (b) 10 low mm-carbon slit width. steel rod of = 4.9 mm installedinstalled in inhorizontal horizontal position position for for ( a()a )5 5 mm mm slit slit widthwidth and (b) 10 mmmm slit slit width width. . 4. Discussion 4. Discussion4. Discussion The obtained results shown in Figures5–7 prove that the presented di ffraction method provides a sufficientlyTheThe obtained highobtained angular results results resolution, shown shown in in Figures represented Figures 5 5––77 prove prove by the thatFWHM the presentedof the diff diffraction ractiondiffraction profile, method method which provides provides allows fora sufficientlya the sufficiently possibility high high of angular studying angular resolution, someresolution, plastic represented represented deformation byby characteristics the FWHM ofof (root-mean-square thethe diffraction diffraction profile, profile, microstrains, which which asallows wellallows for as for the the epossibilityff possibilityective grain of of studying size),studying by some some applying plastic plastic shape deformation analysis characteristics characteristics on the neutron (root (root di-ffmean-ractionmean-square-square peak microstrains, as well as the effective grain size), by applying shape analysis on the neutron diffraction profilesmicrostrains, [11,12 as]. well The as advantage the effective of grain this size), method by applying permits shape the investigation analysis on the of neutron large volumes diffraction of peak profiles [11,12]. The advantage of this method permits the investigation of large volumes of polycrystallinepeak profiles [11,12]. samples The (several advantage cubic of centimeters), this method thus permits providing the investigation average values of oflarge microstructure volumes of polycrystalline samples (several cubic centimeters), thus providing average values of microstructure characteristicspolycrystalline over samples the irradiated (several cubic gauge centimeters), volume, e.g., thus as a functionproviding of average macroscopic values strain of microstructure loaded on the characteristics over the irradiated gauge volume, e.g., as a function of macroscopic strain loaded on samplecharacteristics by an auxiliary over the instrument.irradiated gauge Further volume application, e.g., as of a thisfunction method of macroscopic can be found strain in strain loaded/stress on the sample by an auxiliary instrument. Further application of this method can be found in the sample by an auxiliary instrument. Further application of this method can be found in studiesstrain/stress of textured studie sampless of textured with a largesamples size with of grains, a large as size the conventionalof grains, as the diff conventionalraction method diffraction requiring strain/stress studies of textured samples with a large size of grains, as the conventional diffraction smallmethod irradiated requiring gauge small volume irradiated would gauge be volume problematic. would Naturally,be problematic. it could Naturally, also be it usedcould for also strain be method requiring small irradiated gauge volume would be problematic. Naturally, it could also be scanningused for in the strain polycrystalline scanning in materials the polycrystalline similarly to materials the conventional similarly totwo-axis the conventional strain/stress two scanners-axis used for strain scanning in the polycrystalline materials similarly to the conventional two-axis whenstrain/stress evaluating scanners strains when in small evaluating gauge strains volumes in small (few cubicgauge millimeters)volumes (few from cubic peak millimeters) position from shifts strain/stress scanners when evaluating strains in small gauge volumes (few cubic millimeters) from ∆θSpeak. However, position for shifts such Δ measurements,θS. However, for this such method measurements would not, this be practical, method would because not the be step-by-step practical, rockingpeak position curve analysis shifts Δ wouldθS. However, make it much for such more measurements time-consuming., this On method the other would hand, not when be using practical the, sample with a width of about 10 mm (or more), the resolution of the conventional two-axis scanner would be so relaxed that it would not allow us to investigate the effect of elastic strains from the shifts in the diffraction profiles nor the effects of plastic deformation on their FWHM and the form. As can be seen in Figure1, the width of the sample gauge volume introduces a decisive uncertainty to the resolution, i.e., to the FWHM of the diffraction profile imaged by the position-sensitive detector.

5. Conclusions The presented high-resolution neutron diffraction method can be successfully used for the strain/stress measurements, namely, on bulk samples exposed to an external thermo-mechanical load, e.g., in a tension/compression rig and in an aging machine. The bulk samples in the form of a rod with a diameter of several millimeters can be investigated in the vertical, as well as horizontal, position. Similarly, the bars of a rectangular form could also be tested. In comparison to the conventional Materials 2020, 13, 5449 6 of 7 two-axis neutron diffraction strain/stress scanners, the three-axis alternative offers a considerably higher ∆d/d resolution. However, the presented method, if used for conventional elastic strain/stress scanning, would be much more time-consuming. On the other hand, when using a large-volume sample accompanied with a considerably higher detector signal, this drawback can be partly eliminated. Nevertheless, the presented method could be useful, namely, at high-flux neutron sources where the experimental results can be obtained within a reasonable measurement time. It can be stated that the presented method using the three-axis neutron diffraction setting can offer further complementary information to that achieved by the other methods commonly used.

Author Contributions: Conceptualization and project administration, P.S. and J.Š.; writing—original draft preparation, P.M.; neutron diffraction measurements, P.M. and V.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: Measurements were carried out in the CANAM NPI CAS Režˇ infrastructure (MŠMT project no. LM2015056). The presented results were also supported in the frame of LM2015074 infrastructure MŠMT project “Experimental nuclear reactors LVR-15 and LR-0”. Furthermore, J. Šaroun and V. Ryukhtin acknowledge support from the Czech Academy of Sciences in the frame of the program “Strategie AV21, No.23” and P. Mikula acknowledges support from ESS participation of the Czech Republic—OP (CZ.02.1.01/0.0/0.0/16_013/0001794). The authors also thank B. Michalcová for significant assistance in measurements and basic data processing. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Noyan, I.C.; Cohen, J.B. Residual Stress: Measurement by Diffraction and Interpretation, 1st ed.; Springer: New York, NY, USA, 1987. 2. Hutchings, M.T.; Krawitz, A.D. Measurement of Residual and Applied Stress Using Neutron Diffraction. In NATO ASI Series; Hutchings, M.T., Krawitz, A.D., Eds.; Kluwer Academic Publisher: Amsterdam, The Netherlands, 1992; Volume 26. 3. Stelmukh, V.; Edwards, L.; Santisteban, J.R.; Ganguly, S.; Fitzpatrick, M.E. Weld stress mapping using neutron and synchrotron X-ray diffraction. Mater. Sci. Forum 2002, 404–407, 599–604. [CrossRef] 4. Mikula, P.; Vrána, M.; Lukáš, P.; Šaroun, J.; Wagner, V. High-resolution neutron powder diffractometry on samples of small dimensions. Mater. Sci. Forum 1996, 228–231, 269–274. [CrossRef] 5. Mikula, P.; Vrána, M.; Lukáš, P.; Šaroun, J.; Strunz, P.; Ullrich, H.J.; Wagner, V. Neutron diffractometer exploiting Bragg diffraction optics—A high resolution strain scanner. In Proceedings of the ICRS-5 Conference, Linköping, Sweden, 16–18 June 1997; Volume 2, pp. 721–725. 6. Seong, B.S.; Em, V.; Mikula, P.; Šaroun, J.; Kang, M.H. Unconventional performance of a highly luminous strain/stress scanner for high resolution studies. Mater. Sci. Forum 2011, 681, 426–430. [CrossRef] 7. Mikula, P.; Šaroun, J.; Ryukhtin, V.; Stammers, J. An alternative neutron diffractometer performance for strain/stress measurements. Powder Diffr. 2020, 35, 185–189. [CrossRef] 8. Vrána, M.; Lukáš, P.; Mikula, P.; Kulda, J. Bragg diffraction optics in high-resolution strain measurements. Nucl. Instrum. Methods Phys. Res. A 1994, 338, 125–131. [CrossRef] 9. Macek, K.; Lukáš, P.; Janovec, J.; Mikula, P.; Strunz, P.; Vrána, M.; Zaffagnini, M. Austenite content and dislocation density in electron beam welds of a stainless maraging steel. Mater. Sci. Eng. A 1996, 208, 131–138. [CrossRef] 10. Hirschi, K.; Ceretti, M.; Lukáš, P.; Ji, N.; Braham, C.; Lodini, A. Microstrain measurement in plastically deformed austenitic steel. Textures Microstruct. 1999, 33, 219–230. [CrossRef] 11. Delhez, R.; de Keijser, T.H.; Mittemeijer, E.J. Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis. Fresenius’ Z. Anal. Chem. 1982, 312, 1–16. [CrossRef] Materials 2020, 13, 5449 7 of 7

12. Davydov, V.; Lukáš, P.; Strunz, P.; Kužel, R. Single-line diﬀraction proﬁle analysis method used for evaluation of microstructural parameters in the plain ferritic steel upon tensile straining. Mater. Sci. Forum 2008, 571–572, 181–188. [CrossRef]

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