I. INTRODUCTION
The principle of the neutron diffraction method consists in the precise determination of the dhkl spacing of particularly oriented crystal planes (hkl). In neutron and X-ray diffraction, the angular positions of the diffraction maxima are determined by the well-known Bragg condition 2dhkl ⋅sin θhkl = λ (θhkl – Bragg angle, λ – the neutron wavelength). It offers a unique nondestructive technique for the determination of the strain/stress fields. When defining the elastic strain ε as ε = Δd/d 0,hkl, it means that it is, in fact, a relative change of the lattice spacing dhkl with respect to the strain-free value of the lattice spacing d 0,hkl. The strain within a material is a tensor, and in a single diffraction measurement of strain, one obtains just the vector component parallel to the scattering vector Q which is perpendicular to the reflecting set of planes (see Figure 1). For other strain components, other geometrical positions of the sample need to be chosen. It follows from the definition of the strain ε that the knowledge of the d 0,hkl value is a crucial task (Noyan and Cohen, Reference Noyan and Cohen1987; Hutchings and Krawitz, Reference Hutchings and Krawitz1992; Stelmukh et al., Reference Stelmukh, Edwards, Santisteban, Ganguly and Fitzpatrick2002). After the differentiation of the Bragg condition, we arrive at a simple formula ε = −cot θhkl⋅Δθhkl. This formula indicates that strain ε gives rise to a change in the scattering angle 2θhkl resulting in an angular shift Δ(2θhkl) of the scattering angle (for particular reflecting planes illuminated by a fixed wavelength). Therefore, from the shift in the Bragg angle Δθhkl (relative to that of the stress-free material), it is possible to determine the average lattice macrostrain over the irradiated gauge volume. After obtaining three strain components from the measurement on, e.g., a steel sample, the conversion of strains to stresses is carried out by means of the relation
$$\sigma_x=\frac{E_{hkl}}{(1-2\nu_{hkl})(1+\nu_{hkl})}[(1-\nu_{hkl})\varepsilon^{hkl}_{x}+\nu_{hkl}(\varepsilon^{hkl}_{y}+\varepsilon^{hkl}_{z})]$$
where 
$\varepsilon _{x\comma y\comma z}^{hkl}$ is the x,y,z components of the lattice strain measured at the hkl crystal lattice planes, Ehkl and νhkl are the diffraction elastic Young modulus and diffraction Poisson ratio, respectively. The neutron strain/stress scanner, which is in fact a powder diffractometer equipped with a position sensitive detector (PSD), evaluates the variations of lattice spacing within a sample. The required spatial resolution is usually of the order of millimeters and is determined by the dimensions of the gauge volume. The scanner is usually equipped with a bent perfect crystal (BPC) monochromator and optimized with respect to luminosity and resolution in a limited range of scattering angles 2θ S (Noyan and Cohen, Reference Noyan and Cohen1987; Hutchings and Krawitz, Reference Hutchings and Krawitz1992; Stelmukh et al., Reference Stelmukh, Edwards, Santisteban, Ganguly and Fitzpatrick2002). The instrument can be also equipped with an external loading machine as, e.g., with a tension/compression rig (see Figure 2).
Figure 1. Schematic sketch of the Bragg diffraction geometry.
Figure 2. Conventional strain scanner with a tension/compression rig.
II. HIGH-RESOLUTION THREE-AXIS DIFFRACTION SETTING
Conventional two-axis neutron scanners usually use, for strain determination, beam optics elements, namely focusing and optimally bent monochromator, a system of slits before and after the sample creating a gauge volume element (irradiated by the beam coming from the monochromator) and a PSD imaging the diffraction profile coming from the irradiated gauge volume (Mikula et al., Reference Mikula, Vrána, Lukáš, Šaroun and Wagner1996, Reference Mikula, Vrána, Lukáš, Šaroun, Strunz, Ullrich and Wagner1997; Seong et al., Reference Seong, Em, Mikula, Šaroun and Kang2011). The resolution of the conventional scanner is thus determined mainly by the thickness and curvature of the monochromator, monochromator take-off angle, the widths of the slits (usually in the order of 1–3 mm), the divergence of the beam diffracted by the gauge volume, and the spatial resolution of PSD. The combination of all these features results in an uncertainty (FWHM of the diffraction peak profile image as recorded on the PSD) of about roughly 10−3 rad which appears still sufficiently small for the measurement of Δ(2θhkl) angular shifts brought about by strains. However, the mentioned resolution is not sufficient for measurements of changes of diffraction line profiles imaged by PSD which would permit the carrying out of microstrain and grain size studies for plastically deformed polycrystalline samples. Rather a long time ago, first attempts with a high-resolution three-axis setting as schematically shown in Figure 3(a) were tested (Vrána et al., Reference Vrána, Lukáš, Mikula and Kulda1994; Macek et al., Reference Macek, Lukáš, Janovec, Mikula, Strunz, Vrána and Zaffagnini1996; Hirschi et al., Reference Hirschi, Ceretti, Lukáš, Ji, Braham and Lodini1999). Following the sketch displayed in Figure 3(a) (for small widths of the samples), a maximum resolution from this arrangement can be achieved for minimal dispersion of the whole system. When treating it in momentum space, this means that the orientation of the Δk domains related to the monochromator and the analyzer are matched to that of the sample. For L MS/(R M⋅sin θ M)≠1 and L SA/(R A⋅sin θ A)≠1, a general form for focusing in momentum space (not dependent on α 1 and α 2), which minimizes the dispersion between all elements, can be derived as (Vrána et al., Reference Vrána, Lukáš, Mikula and Kulda1994)
$$\eqalign{2{\rm tan}\,\theta _{\rm S} & = \displaystyle{{{\rm tan}\,\theta _{\rm M}} \over {1-L_{{\rm MS}}/\lpar R_{\rm M}\cdot {\rm sin}\,\theta _{\rm M}\rpar }} \cr & + \displaystyle{{{\rm tan}\,\theta _{\rm A}} \over {1-L_{{\rm SA}}/\lpar R_{\rm A}\cdot {\rm sin}\,\theta _{\rm A}\rpar }}}.$$
Figure 3. Three axis diffractometer settings employing the BPC monochromator and the analyzer as used in the feasibility studies (R M, R A – radii of curvature, θ M, θ A – Bragg angles) for vertical (a) and horizontal (b) positions of a polycrystalline sample of the cylindrical form.
When fulfilling the condition (2), a maximum peak intensity and a minimum FWHM of the analyzer rocking curve can be expected. However, it should be pointed out that in some cases, it is difficult to fulfill it. Of course, this high-resolution performance can be used for macro-strain scanning, but due to the step-by-step analysis with the analyzer, the measurement would be impractical as a result of rather low detector signal. However, due to its high resolution, it can be used for plastic strain studies on the basis of diffraction profile analysis as, e.g., in deformation studies on samples under the external thermo-mechanical load (Vrána et al., Reference Vrána, Lukáš, Mikula and Kulda1994; Macek et al., Reference Macek, Lukáš, Janovec, Mikula, Strunz, Vrána and Zaffagnini1996; Hirschi et al., Reference Hirschi, Ceretti, Lukáš, Ji, Braham and Lodini1999). Recently, it has been found that the setting provides a sufficiently high resolution, though slightly relaxed, when wider slits (e.g. of the width of about 10 mm) or wider samples are used as schematically shown in Figure 3(b). If the same step-by-step analysis were to be applied, the principal drawback, being that of the time taken for such a measurement, would be smaller because the detector signal could be much higher due to the larger horizontal dimension of the sample. It is well known that the vertical direction has a negligible influence on the resolution. The experimental tests were carried out on the three-axis neutron diffractometer installed at the Řež research reactor LVR-15. Si(111)-monochromator and Ge(311)-analyzer single crystals had the dimensions of 200 × 40 × 4 mm3 and 20 × 40 × 1.3 mm3 (length × width × thickness), respectively. The monochromator Si(111) set for take-off angle of 2θ M = 29.95° and providing the neutron wavelength of 0.162 nm had a fixed curvature with the radius R M of about 12 m. The Ge(311) analyzer with the nominal diffraction angle of θ A = 28.39° had a changeable radius of curvature R A in the range from 3.6 to 36 m. Figure 4 demonstrates the difference in the resolution represented by FWHM of the analyzer rocking curve for two positions (vertical and horizontal) of the α-Fe(110) standard sample of the diameter of 5.1 mm. The radius of curvature of the analyzer was R A = 9 m when the dispersion of the whole setting was minimized. In this case, only one Cd slit having a window of 2 × 1 cm2 (height × width) was used. In the next step, a plastically deformed sample of the same material (Grade 08G2S GOST 1050) was tested. The virgin sample was submitted to shear deformation (23%) as well as to drawing deformation (23.2%). The effect of deformation on the rocking curves is documented in the results shown in Figure 5. Inspection of Figures 4 and 5 reveals that for both positions of the sample, it is possible to detect not only the effect of the elastic strains (macrostrains) resulting in a shift of the rocking curve Δθ A but also the effect of plastic deformation (microstrains) resulting in a change of the diffraction profile. Peak shifts for the deformed sample in vertical (Δθ AV = 0.075 ± 0.003°) and horizontal (Δθ AH = 0.013 ± 0.003°) positions are different because different strain components are measured. However, peak broadening ω = ((FWHM def)2−(FWHM virg)2)1/2 is caused by microstrains. Then, by the application of the diffraction profile analysis (Delhez et al., Reference Delhez, de Keijser and Mittemeijer1982; Davydov et al., Reference Davydov, Lukáš, Strunz and Kužel2008), information about the change of the dislocation density and the mean grain size can be obtained. However, it is clear that the method is still based on the time-consuming step-by-step analysis. Therefore, the possibility has been searched how to avoid it and to exploit somehow a position sensitive detector.
Figure 4. Analyzer rocking curves for the virgin α-Fe(110) sample situated on the second axis of the instrument in the vertical (a) and horizontal position (b).
Figure 5. Analyzer rocking curves for the deformed α-Fe(110) sample in the vertical (a) and horizontal position (b).
III. THREE-AXIS ALTERNATIVE EMPLOYING PSD
It has been found that for each point of Δθ A in the vicinity of the peak position of the rocking curve, the diffraction profile as imaged by the PSD linearly shifts in correspondence with the value of the change of the scattering angle 2(Δθ A) (see Figure 6). In fact, the shift of the diffraction image is connected just with the value of 2(Δθ A). The range of linearity depends on FWHM A of the rocking curve. Figure 7 shows an example of this linear dependence which can be used for the calibration of Δθ A against the center position of the diffraction image in the PSD which can be applied in the vicinity of the peak position of the analyzer rocking curve (Δθ A = 0°). It comes from the fact that the strain in the sample changes the scattering angle 2θ S by a value Δ(2θ S). The corresponding bunch of neutrons impinges the analyzer at some deviation Δθ A. As Δθ A is much smaller than FWHM A of the rocking curve, the analyzer fixed at the nominal Bragg angle still diffracts the bunch (possibly with a slightly less intensity) at the deviation of the scattering angle of 2(Δθ A). Naturally, it results in a shift of the PSD image of the diffraction profile with respect to the strain-free position (Δθ A = 0°). As usual, the strains bring about a smaller effect than the Δθ A-span of the calibration function; therefore, it can be used for macrostrain scanning as, e.g., in the samples subjected to a thermo-mechanical load. Furthermore, it has been found that the strain resolution derived from the calibration function depends on the curvature of the analyzer as well as the width of the slit for the incident beam on the sample laid in the horizontal position. These properties are documented in Figures 8 and 9 for the radius of curvature R A = 3.6 m and the widths of the slit at 10 and 5 mm, respectively. However, as can be seen, it has practically no effect on the determination of the elastic strain observed in the deformed sample (the difference in the peak position is within the experimental error). Nevertheless, the change of the line inclination due to the plastic deformation deserves further studies. For evaluation of the obtained data in our case, it is possible to use simple formulae Δθ A ≈ −Δ(2θ S) for large values of R A and Δθ A ≈ −Δ(2θ S)⋅(1 + 1/(R A⋅sin θ A)) for smaller values of R A from which the change of the lattice spacing of the sample can be determined by means of the relation Δθ S = −(Δd S/d 0,S)⋅tan θ S.
Figure 6. Three-axis diffractometer alternative with the fixed position of the analyzer in combination with the PSD detector for vertical (a) and horizontal (b) positions of the polycrystalline sample.
Figure 7. Linear calibration function as taken for the virgin sample laid horizontally. The width of the Cd slit was 10 mm and the bending radius R A = 9 m.
Figure 8. Linear calibration functions as taken for the virgin sample (a) and the deformed one (b) both laid horizontally. The width of the Cd slit was 10 mm and the bending radius R A = 3.6 m.
Figure 9. Linear calibration functions as taken for the virgin sample (a) and the deformed one (b) both laid horizontally. The width of the Cd slit was 5 mm and the bending radius R A = 3.6 m.
IV. SUMMARY
The feasibility of using the high-resolution three-axis diffractometer performance in some special cases of strain/stress analysis of bulk samples of the width of 5–10 mm is presented. This measurement alternative can be utilized, namely, in the case of samples subjected to thermo-mechanical load when the position of the sample is fixed. Otherwise, for the comparison of different samples, attention should be paid to keeping the central point of the irradiated gauge volume in the same position with a high accuracy to avoid an additional error to Δ(2θ S) as well as to Δθ A. We hope that the presented neutron diffraction settings can offer an additional support to complement the information achieved by using the other conventional characterization methodologies.
ACKNOWLEDGEMENTS
Measurements were carried out at the CANAM infrastructure of the NPI CAS Řež supported through MŠMT project no. LM2015056. The presented results were also supported in the frame of the following MŠMT projects: LM2015048 and the infrastructural one – “Experimental nuclear reactors LVR-15 and LR-0.” We thank Ms. B. Michalcová for a significant help in the measurements and basic elaboration of the data.
