research papers
Measurement of Xray diffractionline broadening induced by elastic mechanical grain interaction
^{a}Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals Research), Heisenbergstrasse 3, D70569 Stuttgart, Germany, and ^{b}University of Stuttgart, Institute for Materials Science, Heisenbergstrasse 3, D70569 Stuttgart, Germany
^{*}Correspondence email: e.j.mittemeijer@is.mpg.de
Various graininteraction models have been proposed in the literature to describe the stress and strain behavior of individual grains within a massive aggregate. Diffraction lines exhibit a response to the occurrence of a strain distribution in the diffracting crystallites, selected by the direction of the diffraction vector with respect to the specimen frame of reference, by correspondingly induced diffractionline broadening. This work provides a report of synchrotron diffraction investigations dedicated to the measurement of the experimentally observable diffractionline broadening induced by external elastic loading of various polycrystalline specimens. The experimentally obtained broadening data have been compared with those calculated adopting various graininteraction models. Although such graininteraction models have been proven to accurately predict the average (Xray) diffraction measured lattice strain, as derived from the diffractionpeak position, the present results have demonstrated that the extent of the diffractionline broadening due to grain interactions, as calculated by employing these graininteraction models, is much smaller than the experimentally determined broadening. The obtained results have vast implications for diffractionline broadening analysis and the understanding of the elastic behavior of massive polycrystals.
Keywords: Xray diffractionline broadening; grain interaction; strain.
1. Introduction
The use of diffraction techniques to study lattice strain is not a new strategy (e.g. see Greenough, 1952, and references therein). Elastic loading of a polycrystal results in a strain distribution within the material. The elastically intrinsically anisotropic grains in a massive body cannot deform freely to comply with an imposed state of mechanical stress; instead they must adapt their mechanical response to their surroundings. The understanding of grain interaction in polycrystalline materials is a major problem, of importance in order to understand how a polycrystalline material responds in its individual grains to loads. A wide range of elastic graininteraction models have been developed to describe the strain within the individual crystals of a massive polycrystalline specimen subjected to an external load (Voigt, 1910; Reuss, 1929; Neerfeld, 1942; Hill, 1952; Eshelby, 1957; Kröner, 1958; Vook & Witt, 1965; Witt & Vook, 1968; van Leeuwen et al., 1999; Welzel & Mittemeijer, 2003; Welzel & Fréour, 2007). These models differ not only in their complexity (Reuss and Voigt models can often be solved analytically, whereas the Eshelby–Kröner model can be solved only numerically) but also in their boundary conditions [anisotropic models, such as the Vook–Witt and inverse Vook–Witt models (Welzel & Mittemeijer, 2003), have been found to be valid in predicting the measured strain in thin films under loading, whereas the classical isotropic Neerfeld–Hill model appears appropriate especially for bulk material with spherical, or equiaxed, grains in the specimen]. The more complex the model, the more versatilely applicable it may be. The Eshelby–Kröner model (Eshelby, 1957; Kröner, 1958), for example, can incorporate morphological (grain shape; Koch et al., 2004) texture and crystallographic (orientationdistribution function) texture.
The elastic graininteraction models, as listed above, have been applied to calculate the macroscopic elastic constants of a material or to predict the average lattice strain as measured along the diffraction vector (see §2) in an Xray diffraction (XRD) experiment. Although it is not generally recognized, these models can also be used to calculate the strain variation within an aggregate under loading for the diffracting crystallites, which strain variation results in diffractionline broadening (Sayers, 1984; Funamori et al., 1997; Singh & Balasingh, 2001; Koker et al., 2013).
Diffractionline broadening is induced when grains with (hkl) planes sharing the same normal (parallel to the diffraction vector, as indicated in Fig. 1) experience different (average) strains in the direction of the diffraction vector because of their intrinsically anisotropic elastic behavior. The average strain of a crystallite as a function of its orientation with respect to the specimen frame of reference can be determined according to elastic graininteraction models (Sayers, 1984; Koker et al., 2013). On this basis, the variation of this (average) strain as a function of the angle of rotation, χ, about the diffraction vector, (see §2), can be calculated for a fixed diffraction geometry (i.e. fixed HKL, ψ, φ).
Broadening contributions such as instrumental and finite grain size remain unchanged during elastic loading and unloading experiments. Such sources of broadening are not considering in this work. Here the focus of attention is on the measurement of elastic loadinginduced (reversible) diffractionline broadening. This is a rather unexplored area [exceptions are the experimental observations reported by Funamori et al. (1997) and Singh & Balasingh (2001)], which is at least partly a result of the high resolution necessary to measure such a phenomenon. [Theoretical calculations of elastic loadinginduced diffractionline broadening have been performed by Sayers (1984) and Koker et al. (2013).]
The work presented here consists of diffractionline broadening measurements on polycrystalline specimens of four different metals (Cu, Ni, Nb, W) under two different imposed states of stress (uniaxial and biaxially rotationally symmetric), made using synchrotron radiation. It will be shown that sources of strain variation by elastic grain interactions not captured by the known elastic graininteraction models bring about a substantial part of the observed diffractionline broadening.
2. Theoretical background
2.1. Reference frames and calculations of diffraction averages
Three Cartesian frames of reference, as shown in Fig. 2, are defined:
(1) The crystal frame of reference (): the conventional definition of an orthonormal ), is adopted. A detailed treatment is given by Giacovazzo et al. (2002). For cubic crystal symmetry, the axes chosen coincide with the a, b and c axes of the crystal lattice.
such as the one given by Nye (1957(2) The specimen frame of reference (): the axis is orientated perpendicular to the specimen surface, and the and axes are in the surface plane.
(3) The laboratory frame of reference (): this frame is chosen in such a way that the axis coincides with the diffraction vector in the (Xray) diffraction experiment.
The relative orientation of the laboratory frame of reference with respect to the specimen frame of reference is specified by the angles φ and ψ, where ψ is the inclination angle of the sample surface normal (i.e. the axis) with respect to the diffraction vector (i.e. the axis) and φ denotes the rotation of the sample around the sample surface normal. The angle χ is defined as the rotation of the laboratory frame of reference about the axis (the diffraction vector), where, for = 0°, the frame of reference coincides with the frame of reference. The direction of the diffraction vector is especially important in XRD experiments as this is the direction along which lattice spacing (and thus average lattice strain) is measured.
In the following, a superscript (, or ) is used for indicating the reference frame adopted for the representation of quantities that are orientation specific.
The orientation of each crystallite in the system can be identified by three Euler angles, according to the Bunge convention (Bunge, 1965; Bunge & Roberts, 1969). These angles will be called α, β and γ (Roe & Krigbaum, 1964). See, for example, Koker et al. (2013) for a more indepth discussion. It is usual to associate a set of Euler angles with a vector in the threedimensional orientation (Euler) space G (Bunge, 1982). In this way, each point in the orientation space G represents a possible orientation of the frame of reference with respect to the frame of reference. Texture can be quantified by introducing the orientation distribution function (ODF), , which is a function of the Euler angles, specifying the of crystallites having an orientation in the infinitesimal orientation range around .
In the context of i.e. diffraction averages). A diffraction line contains data on only a subset of the crystallites for which the diffraction planes are perpendicular to the chosen measurement direction. Because only the measurement direction (i.e. the direction of the diffraction vector) is defined, a degree of freedom occurs for the diffracting crystallites: the rotation about the diffraction vector (denoted by the angle χ). For a single HKL diffraction line, the group of diffracting crystallites is selected by specifying the HKL of the reflection and the orientation of the diffraction vector with respect to the specimen reference frame , which can be identified by the angles (φ, ψ). Therefore the subscripts (φ, ψ) and superscripts (HKL) are attached to the corresponding average of a strain tensor (element).
the analyzed volume is generally only a fraction of the volume of the polycrystalline specimen: braces denote volumeweighted averages for diffracting crystallites only (For a fixed diffraction vector (i.e. HKL, φ and ψ have been specified), the average lattice strain and the lattice strain distribution (where the strain varies as a function of rotation χ about the diffraction vector) are related according to
is the representation of the ODF in terms of the measurement parameters and the rotation angle χ. The ODF is now expressed as ; cannot be directly used in equation (1), since the angles χ, φ and ψ are not Euler angles representing a rotation of the system with respect to the system. (They actually provide the rotation of the system with respect to the system .) However, the values of α, β and γ, and thus and thereby at every χ, can be calculated from HKL, χ, φ and ψ. For a more detailed treatment, see Welzel et al. (2003).
In order to compute the strainbroadened diffraction line, both the average lattice strain (directly related to peak position upon loading) and the strain distribution [ at fixed HKL, φ and ψ] are required. The strain distribution can be converted into a frequency function , which describes the fraction of grains in the considered diffraction volume sharing the same diffraction vector and which experience the same (average) strain. Using Bragg's law (for a specified radiation wavelength and the material's strainfree lattice spacing d_{o} for the transition from ∊ to d), the frequency function can be converted to the scale. This transformation results in the straininduced broadening contribution function . The function , when convoluted with the diffraction profile corresponding to unloaded state , yields the `measured' diffraction peak for an elastically loaded polycrystalline aggregate. It is important here to note that incorporates all broadening contributions (such as grain size and instrumental broadening) other than that induced by the external mechanical loading. Upon unloading, the diffraction line relaxes to , as all broadening induced by elastic loading is fully reversible. For a more detailed discussion of such calculations, see Koker et al. (2013).
2.2. Lattice strain versus mechanical stress relationships
Two states of stress in the specimen frame of reference are induced in this study by externally applied loading: biaxially rotationally symmetric and uniaxial. The general expression for the measurable average lattice strain along the diffraction vector obeys
where are the diffraction (Xray) stress factors and describes the loadinginduced state of mechanical stress in the specimen frame of reference. For proof, see Welzel & Mittemeijer (2003). The diffraction stress factors for a quasiisotropic aggregate can be replaced by the diffraction elastic constants (Welzel & Mittemeijer, 2003; Welzel, Ligot et al., 2005). Then, for the case of uniaxial (i.e. for all i,j except ij = 11) and biaxially and rotationally symmetric (; , ) stress states, and considering cubic crystal symmetry, equation (2) simplifies to a linear relationship between the XRD measured average lattice strain and .
(i) Biaxially rotationally symmetric loading:
where S_{1}^{ HKL} and are the (Xray) diffraction elastic constants, is the biaxially applied (and/or residual) stress (), and ψ is the specimen inclination or tilt angle. As this state of stress is rotationally symmetric, is independent of rotation φ about the specimen surface normal.
(ii) Uniaxial loading:
where is the applied uniaxial load along the axis. This state of stress is not symmetric about the surface normal (), and thus depends on the angle of rotation φ about the specimen normal.
2.3. Grain interaction and diffractionline broadening
Graininteraction models can be used to predict the elastic loadinginduced average lattice strain of and the latticestrain variation in polycrystalline aggregates (Koker et al., 2013). The Reuss (1929) model (isotropic grain interaction), in which all grains are in the same state of stress in the specimen frame of reference, overestimates the strain variation in the aggregate and therefore supposedly provides a maximum value for the predicted diffractionline broadening. The Voigt (1910) model (isotropic grain interaction), in which all grains are in the same state of strain in the specimen frame of reference, implies the absence of strain variation in the aggregate and therefore would predict the absence of diffractionline broadening. The (other) isotropic graininteraction models Neerfeld–Hill (Neerfeld, 1942; Hill, 1952) and Eshelby–Kröner (Eshelby, 1957; Kröner, 1958) provide results that are more or less `averages' of the Voigt and Reuss extremes.
Grain interactions within an aggregate can also be anisotropic, implying different types and extents of grain interaction for directions parallel and perpendicular to the specimen surface. For example, the Vook–Witt graininteraction model (Vook & Witt, 1965; Witt & Vook, 1968) assumes that all grains are under the same stress perpendicular to the surface and experience identical strains parallel to the surface. The Eshelby–Kröner model has been shown to converge with the Vook–Witt model for needleshaped grains () (Welzel et al., 2005).
While none of these models showed an ideal match with the data presented in this work, the Reuss (isotropic) and Vook–Witt (anisotropic) models were used here, also recognizing their computational simplicity, for discussion of the observed diffractionline broadenings. [For a comparison between the various graininteraction models for predicting diffractionline broadening, see Koker et al. (2013).]
Texture decreases the effect of latticestrain variation by anisotropic grain interaction on the diffractionline broadening (Koker et al., 2013). Therefore, calculations for the effect of grain interactions on diffractionline broadening for a statistically untextured aggregate result in an overestimation of the expected (graininteractioninduced) diffractionline broadening.
In the presence of texture, the isotropic graininteraction models (Voigt, Reuss, Neerfeld–Hill and Eshelby–Kröner with ) for all H00 and HHH reflections of cubic materials still lead to straight lines in plots of lattice strain versus and predict zero diffractionline broadening. The anisotropic graininteraction models do imply for (also textured) cubic materials that distinct broadening also occurs for the H00 and HHH reflections (Kokeret al., 2013). This is an important result, having direct relevance for the diffractionline broadening observed in this work (§4).
3. Experimental procedures
Synchrotron Xray diffraction stress measurements were conducted at the Max Planck Institute for Intelligent Systems (formerly Metals Research) `surface diffraction beamline' at ANKA, located at the Karlsruhe Institute of Technology (KIT), Germany. All of the measurements discussed in this work were made using radiation with a photon energy of 8.1 keV and a corresponding wavelength of Å, with the exception of the measurements for the uniaxially loaded W dogbone specimens, which were performed using an energy of 10 keV, with Å. Data were collected using a sodium iodide point detector preceded by a set of Soller slits. The counting statistics were kept constant by monitoring the incoming beam current (as opposed to fixed time increments), since the electronbeam current decays as a function of time after each injection.
An overview of the performed experiments is provided by Table 1: materials, loading state, magnitude of loading, measured reflections. The measurable reflections and accessible tilt angles were dictated by the texture and/or the geometry of the loading device mounted on the Eulerian cradle. All diffraction lines discussed in this work were measured in reflection diffraction geometry. It should be noted that the specimens do not plastically deform, as it has been shown experimentally in this work that changes in both diffractionline position and diffractionline width are completely reversible upon unloading (as discussed in §2.1).

3.1. Loading setup
3.1.1. Biaxially rotationally symmetric loading
A heating/cooling chamber (MRI Physikalische Geräte GmbH, Karlsruhe, Germany) was used for in situ XRD measurements. Heating and cooling rates were regulated by the internal PID controller; measurements were made only at fixed temperatures, accurate within ±1 K. Temperature variation within the chamber was found to be negligible by determining from measurements, at different locations at the surface of the film, the strainfree lattice parameter a_{o} (Welzel, Ligot et al., 2005), which depends strongly on temperature for the metals concerned.
Thin films of various metals were sputter deposited on singlecrystal substrates (see §3.2). Single crystals were chosen for the substrates so that elastic grain interactions in the substrate do not occur. The selected substrates have coefficients significantly different from those of the deposited metals. Heating or cooling of the layer/substrate system led to the development of a biaxial symmetric state of stress in the film and the substrate (sub). As the (thick) substrate can be considered as rigid, the thermal misfit is fully accommodated by the metal films. Then the thermal misfit strain in the film is given by
where is the linear coefficient of T, and T_{room} is the room temperature. Using the biaxial elastic modulus of the film M, the corresponding induced stress in the film can be calculated according to
which depends on temperature3.1.2. Uniaxial loading
Socalled `dogbone' specimens were uniaxially loaded through the use of a tensile machine (Kammrath & Weiss GmbH, Dortmund, Germany) mounted on an Eulerian cradle for in situ XRD measurements.
The applied load F and the crosssectional area A in the middle of the dogbone specimen can be used to calculate the stress applied to the specimen,
The corresponding strain can be found using Hooke's law and the elastic (Young's) modulus E of the material. The strain rate was kept constant during loading and unloading of the specimens. During XRD measurements, the distance between the cross heads was kept fixed, i.e. the material was not allowed to creep, and thereby the strain was kept constant.
3.2. Materials
The intrinsic elastic anisotropy parameter (also known as Zener's anisotropy ratio) for a cubic material follows from (Nye, 1957)
where c_{ij} are the components of the singlecrystal elastic stiffness tensor and s_{ij} are the components of the singlecrystal elastic compliance tensor for the polycrystal. represents isotropy, as practically holds for tungsten. The anisotropy parameters for the materials investigated in this work are presented in Table 2. [Gold has been included in this table, even though it is not experimentally investigated here, to facilitate comparison with previous linebroadening studies (Funamori et al., 1997; Koker et al., 2013).]

3.2.1. Biaxially rotationally symmetric loaded specimens
Copper. Copper was sputter deposited to a thickness of 2 µm onto a singlecrystal silicon (100 orientation) wafer [deposition conditions: p_{Ar} = 3 ×10^{3} mbar (1 bar = 10^{5} Pa), P_{Cu} = 200 W]. The conditions of deposition were tuned to produce little to no residual stress in the film at room temperature. The microstructure of the Cu specimen (imaged with a focused ion beam) consists of many equiaxed grains, ranging in size from 0.1 to 2.0 µm. The films possessed a 111 fiber texture (see the 111 polefigure section in Fig. 3).
The film was loaded tensilely by cooling from room temperature (298 K) to a minimum temperature of 153 K. This led to a biaxially rotationally symmetric state of stress with of approximately 280 MPa, as experimentally determined in this work. (Note the occurrence of material stiffening at low temperatures.)
Niobium. Niobium was sputter deposited onto an aluminium singlecrystal substrate (100 orientation) to a thickness of 2 µm (p_{Ar} = 5.8×10^{3} mbar, P_{Nb} = 200 W). After deposition, a tensile residual stress of MPa was measured at room temperature. The film possessed a strong 110 fiber texture. The film was loaded compressively to achieve a state of zero stress by cooling to 153 K. Therefore, the `unloaded' measurements were made in this cooled state; the `loaded' measurements were made at room temperature.
Tungsten. The tungsten film was also sputter deposited to a thickness of 2 µm onto an Al singlecrystal (100 orientatation) substrate (p_{Ar} = 5.8×10^{3} mbar, P_{W} = 200 W). As with the Nb (also bodycentered cubic), a 110 fiber texture was present in the film. The residual stress in the film at room temperature was MPa. Cooling to 153 K led to a measured stress of MPa, i.e. approximately 200 MPa of tensile stress was induced upon cooling.
3.2.2. Uniaxially loaded specimens
Nickel. Dogbone specimens were made from a 3.2 mmthick coldrolled nickel sheet (purchased from GoodFellow, 99.99 at.% purity). The texture that had developed during rolling was neither sharp nor strong. The macroscopic (uniaxial) elastic limit of the Ni sample was measured to be approximately 125 MPa; therefore, the maximum in situ applied uniaxial elastic loading was MPa.
Tungsten. Dogbone specimens were made from a 3 mmthick coldrolled tungsten sheet (purchased from GoodFellow, 99.95 at.% purity). A weak rolling texture was prevalent in the material. The macroscopic (uniaxial) elastic limit was measured to be approximately 300 MPa. This loading limit was measured for material that had a significant residual stress of −600 MPa (determined using XRD). The applied in situ uniaxial loading stress was limited to a maximum of MPa.
3.3. Evaluation of diffraction data
Diffraction lines at multiple inclinations ψ at selected values of φ for a uniaxial state of applied stress and at φ = 0° for a biaxially rotationally symmetric state of stress were measured in the loaded and unloaded states for each of the discussed materials (see Table 1). A Pearson VII profileshape function (Hall, 1977) was used to fit the peaks and thus to extract values for the parameters (i) diffractionline position ( position of the peak maximum) and (ii) integral breadth β, which is the ratio of the maximum peak intensity to the area under the peak. The integral breadth depends less on the precise shape of the peak than on the FWHM. Therefore, the integral breadth was adopted as the width parameter, reflecting the magnitude of the latticestrain variation in the specimen.
The loadinginduced broadening of the diffraction line was defined as the difference in its integral breadth, , of the loaded and unloaded states (see Fig. 4):
In the following, the average lattice strain , the integral breadth β and the loadinginduced broadening have all been plotted as a function of . If no error bars have been indicated on a plot, then they are of the order of the symbol size.
4. Results and discussion
XRD measurements were made of multiple HKL diffraction lines for the pure metal specimens (films and dogbone specimens, as introduced in §3.2) in the loaded and unloaded states. Each specimen was measured before application of the load and after removal of the load to ensure that the applied deformation by loading of the specimen was purely elastic: it was verified that peak position and peak width returned to their original state upon unloading. As an example, the average lattice strain and the integral breadth β, as derived from the diffraction lines measured for a 111 fibertextured Cu thin film, are shown in Figs. 5 and 6, respectively. Similar measured diffractionline broadening results [for , equation (9)] have been obtained for all five specimens (Table 1) and have been plotted in Figs. 7–9. [It should be noted that minor differences in texture magnitude between the calculated texture and the measured one (as shown for the Cu film in Fig. 3), as opposed to differences in texture character, do not significantly influence the elastic response (Welzel et al., 2003).]
Previously, it was thought that the H00 and HHH reflections would not broaden upon elastic loading as the diffraction lattice planes are structurally (and therefore elastically) symmetric with respect to the diffraction vector (Sayers, 1984; Funamori et al., 1997; Singh & Balasingh, 2001). Thus, the isotropic graininteraction models predict that no strain distribution will be induced for the H00 and HHH reflections. However, the local inhomogeneity of the matrix (i.e. the different surroundings for each diffracting grain) will induce a different loading for each of the diffracting grains (also for the H00 and HHH reflections). On the other hand, the anisotropic graininteraction models (while still neglecting the effect of the different surroundings for the diffraction grains) do predict a loadinginduced diffractionline broadening for the H00 and HHH reflections from cubic materials. For example, see the results for the Vook–Witt graininteraction model (open diamonds) in Fig. 7.
It should be noted that the often assumed dependence on microstrain line broadening does not generally hold, and indeed the widths of for example the Nb 110 and 220 reflections do not comply with such behavior [see the discussion by Leineweber & Mittemeijer (2010)].
As the Ni bulk material is significantly less textured than the Cu and Nb films, it is therefore prone to additional diffractionline broadening as a consequence of the local heterogeneity within the aggregate, i.e. the different surrounding for each diffracting grain (much more so than the Cu and Nb films, as was discussed in §2.3).
For the same loading axes and the same magnitude of load, a tensile or a compressive nature of the load has no effect on the latticestrain variation , independent of the type of state of stress (Koker et al., 2013).
Generally, the integral breadth of the only strainbroadened profile is proportional to the square root of the strain variance (Stokes & Wilson, 1944; Wilson, 1962, 1963; Mittemeijer & Welzel, 2008). Let us assume that the strain broadening is small as compared to the instrumental broadening, and a Gaussianshape function is adopted for the instrumental and the only strainbroadened profiles. Then, upon convolution, it follows that the additional broadening in the `loaded' h profile, as compared to the `unloaded' g profile (where ), roughly scales with the strain variance. This, in turn, scales with , with σ as the magnitude of the applied stress (see further below), for isotropic grain interaction (Singh & Balasingh, 2001). However, if a Lorentzian (Cauchy) shape function is adopted for the instrumental and the only strainbroadened profiles, then the additional line broadening in the h profile, as compared to the g profile, would scale with the square root of the strain variance and thus with σ. Indeed, it has often been suggested that a microstrainbroadened line profile has a Gaussian shape (e.g. see Mittemeijer & Welzel, 2008), but this is not generally true and Lorentzian (Cauchy)shaped strainbroadened line profiles have also been observed (e.g. see Vermeulen et al., 1995); for a rigorous discussion on (also the lineprofile shape of) micro (lattice) strain broadening, see Leineweber & Mittemeijer (2010).
Within the restrictions indicated by the above discussion, to facilitate the comparison of experiments with different magnitudes of loading, the parameter is introduced to normalize the loadinginduced broadening with respect to the magnitude of the applied stress (for the same type of state of stress), where the quantity σ is a scalar representing the magnitude of applied principal stress component(s). (Thus, for the biaxially rotationally symmetric stress states, , and for the uniaxial stress states, .) Such results for are shown in Fig. 10 for the biaxially rotationally symmetric applied states of stress and in Fig. 11 for the uniaxially applied states of stress.
The results for (Fig. 10) and for (Fig. 11) clearly exhibit the role of the degree of the intrinsic elastic anisotropy (see Table 2): the degree of loadinginduced broadening, in the case of biaxial loading, increases in the order W Nb Cu and, in the case of uniaxial loading, is larger for Ni than for W. Obviously, the practically intrinsically elastically isotropic W does not exhibit (resolvable) loadinginduced broadening. The adoption of elastic grain interaction as source of the line broadening is in particular also supported by the complete reversibility of the line broadening of the W specimen (which is intrinsically elastically isotropic), whereas this specimen should have shown (remaining) line broadening after unloading if, for example, dislocation line broadening had been induced.
Calculations according to the isotropic Reuss and the anisotropic Vook–Witt graininteraction models, for each of the specimens investigated, were performed. The results can be compared with the experimental data; see Fig. 5 for the average lattice strain (for Cu) and see Figs. 7–9 for the diffractionline broadening data of the Cu and Nb thinfilm specimens and the Ni dogbone specimen, respectively. [To calculate a theoretical value for diffractionline broadening, the straininduced broadening contribution function , as calculated for the experimental conditions according to a given graininteraction model, was convoluted with the measured diffraction peak for the specimen in the `unloaded' state. The breadth of this convoluted, or `loaded', function was then compared with that of the `unloaded' peak to determine the predicted diffractionline broadening . See Koker et al. (2013) for a description of such calculations.]
The predicted amounts of diffractionline broadening according to the isotropic Reuss model and the anisotropic Vook–Witt model are much less than the experimentally observed diffractionline broadening (Figs. 7–9).
The variation in strain throughout the aggregate depends not only on the orientation of an individual grain, as implied by all graininteraction models, but also on the (shape, number/size and) crystallographic orientation, with respect to the specimen frame of reference, of its nearest neighbors, which effects are not considered at all in the graininteraction models. Grains of identical orientation at different locations in the aggregate will not have the same average lattice strain or the same latticestrain distribution within the grains, owing to these local variations in the microstructure. As a consequence the strain variation predicted by the graininteraction models is distinctly smaller than the experimentally determined one (see Figs. 7–9). Yet, the average lattice strain (calculated as a function of peak position for various ψ tilts) is predicted quite well: for example, see Fig. 5.
The abovediscussed observations lead to the conclusion that the current graininteraction models are unable to describe the complete strain distribution within an elastically loading polycrystalline aggregate. Such models are oversimplifications of the grain interactions occurring in reality; however, these simplifications do allow for fairly accurate predictions of the average lattice strain to be made.
The discrepancies between the experimental data and the graininteraction model predictions for the diffractionline broadening can thus be explained by recognizing that several types of strain variation occur within an elastically loaded polycrystalline aggregate: (i) macro, (ii) meso and (iii) microvariations in strain:
(i) Macrovariation in strain is the variation of the average lattice strain, the average taken for the groups of diffracting grains, that occurs upon changing the orientation of the diffraction vector with respect to the specimen frame of reference. This macrovariation in lattice strain is expressed in this article by the variation of (as a function of ).
(ii) Mesovariation in strain is the variation of the average lattice strain, the average now taken per diffracting grain, for the group of diffracting grains sharing a fixed orientation of the diffraction vector. This mesovariation in lattice strain is expressed in this article by the variation of . Two types of mesovariation of lattice strain are distinguished: (a) mesovariation by variable χ (as considered in the graininteraction models) and (b) mesovariation at constant χ due to different local surroundings.
(iii) Microvariation in strain is the variation of lattice strain within an individual grain.
It is essential to recognize that all published graininteraction models, i.e. including those considered in this article, do not take into account the abovedescribed microvariation or the entire magnitude of mesovariation [type (b) is ignored] in lattice strain. Therefore, broadening of the H00 and HHH peaks was observed, although no broadening is predicted by the isotropic graininteraction models and the predicted broadening by the anisotropic graininteraction models is much too small. It has been demonstrated in this work that the diffractionline broadening by grain interaction is substantial and its extent is much larger (a factor of ten) than predicted by any of the published graininteraction models, in contrast with the application of these graininteraction models to macrostress: then they predict diffractionline shift compatible with experimental data.
5. Conclusions
(1) The measured diffractionline broadening induced by elastic loading is much larger than that predicted by elastic graininteraction models as proposed in the literature.
(2) Three sources of latticestrain variation can be identified:
(i) macrovariation in lattice strain: the variation of the average lattice strain for the groups of diffracting crystallites;
(ii) mesovariation in lattice strain: the variation of the average lattice strain per grain for the group of diffracting crystallites;
(iii) microvariation in lattice strain: the variation of strain within a diffracting crystallite.
(3) The graininteraction models presented so far only consider the macrovariation and a part of the mesovariation in lattice strain and ignore the microvariation in lattice strain. In other words, the graininteraction models do not take into account the effects of the different surroundings in the specimen for each of the diffracting grains (of even possibly identical crystallographic orientations with respect to the specimen frame of reference). Hence, the current graininteraction models severely underestimate the diffractionline broadening for general HKL reflections.
(4) Elastic grain interaction induces diffractionline broadening for the H00 and HHH reflections recorded from cubic materials, in contrast with predictions from the isotropic graininteraction models and much larger than predicted by the anisotropic graininteraction models.
(5) The larger the degree of intrinsic elastic anisotropy of a material, the larger the magnitude of strain variation and thus diffractionline broadening.
(6) Texture reduces the overall strain variation in, and thus the diffractionline broadening for, a polycrystalline aggregate owing to the intrinsic elastic anisotropy of the material, as the `range' of orientation variation of the grains within the material is reduced in the presence of texture.
Acknowledgements
We are grateful for the permission for measurements at the Synchrotron Light Source ANKA at the MPIIS beamline for surface diffraction. We thank Mr R. Weigel, Dr M. Mantilla, Dipl.Ing. S. Kurz and Dipl.Ing. J. Stein for assistance during the measurements. The authors also thank Dr F. Theile for assistance with the sputter deposition of the thinfilm specimens and Mrs M. Dudek for assistance with preparation of specimens for Xray diffraction analysis.
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