RaDMaX online: a web-based program for the determination of strain and damage profiles in irradiated crystals using X-ray diffraction
RRaDMaX online : a web-based program for the determination of strain and damage profiles inirradiated crystals using X-ray diffractionA. Boulle a and V. Mergnac ba Institut de Recherche sur les Céramiques, CNRS UMR 7315, Centre Européen de la Céramique,Limoges, France b Université de Limoges, Direction des Systèmes d’Information, Limoges, France.
Abstract
RaDMaX online is a major update to the previously published
RaDMaX (radiation damage inmaterials analysed with X-ray diffraction) software [Souilah, Boulle & Debelle,
J. Appl. Cryst. (2016) , 311-316]. This program features a user friendly interface that allows to retrieve strainand disorder depth-profiles in irradiated crystals from the simulation of X-ray diffraction datarecorded in symmetrical θ/2θ mode. As compared to its predecessor, RaDMaX online has beenentirely rewritten in order to be able to run within a simple web browser, therefore avoiding thenecessity to install any programming environment on the users’ computers. The
RaDMaX online web-application is written in
Python and developed within a
Jupyter notebook implementinggraphical widgets and interactive plots.
RaDMaX online is free and open source (CeCILL license)and can be accessed on the internet at: https://aboulle.github.io/RaDMaX-online/.
1. Introduction
Together with Transmission Electron Microscopy, Raman spectroscopy and Rutherford back-scattering spectroscopy in channelling mode (RBS/C), X-ray diffraction (XRD) plays a crucial rolein the characterization of irradiated materials (Zhang et al. , 2015). XRD distinguishes itself by thefact that it is sensitive to both lattice strain and atomic disorder, and in that it allows to derive thesequantities on a an absolute scale. For instance, whereas Raman scattering and RBS/C quantifydisorder through ad hoc parameters such as the broadening of the Raman lines or the fraction ofdisplaced atoms determined from the the back-scattering yield, XRD in principle allows todetermine the actual atomic displacement probability distribution (Boulle & Debelle, 2016), albeitthis comes with an increased difficulty in the data analysis as compared to others techniques.The analysis of XRD data does not necessarily have to be complex; for instance the determinationof the maximum strain in the irradiated region can be easily obtained by measuring the position ofthe peak emanating from the strained region (Debelle & Declémy, 2010) and an estimation of thestrained depth can be obtained by analysing the evolution of the width of the interference fringes1Sousbie et al. , 2006). However in order to benefit from the full potential offered by XRD, namelythe retrieval of depth-resolved strain and disorder profiles, the simulation of the experimental datais mandatory (Zaumseil et al. , 1987; Klappe & Fewster, 1994; Milita & Servidori, 1995; Emoto etal. , 2009; Boulle & Debelle, 2010; Rieutord et al. , 2013).In 2016, this led to the development of the computer program
RaDMaX [Radiation Damage inMaterials analysed with X-ray diffraction (Souilah et al. , 2016)] which, contrarily to most XRDsimulation software packages, is solely dedicated to the simulation of XRD data recorded fromirradiated crystals. This particular focus allows to significantly simplify the user interface, henceallowing scientists even with only little crystallographic knowledge to analyse their data.
RaDMaX is written in Python which makes it compatible with all major computer operating systems(Windows, Mac OS and Linux). The downside of this multi-platform capability is that it requiresthe installation of a full scientific Python software distribution, which can be a disincentive forsome users to adopt the software. This can happen, for example, when the users don’t haveadministrator privileges on their computer, or when they are not familiar with setting up acomputing environment. For this reason we engaged in developing a web-based version of
RaDMaX that entirely runs within a web browser and, therefore, doesn’t require the installation ofany software, while keeping the possibility for scientists to load and analyse their own data anddownload complete simulation results.This software,
RaDMaX online , described in the present article (as well as the corresponding sourcecode files), can be accessed at the following address: https://aboulle.github.io/RaDMaX-online/. Ascompared to the previous version of the program (Souilah et al. , 2016), the code has almost entirelybeen rewritten in order be able to be run on a web-hosted
Jupyter server that the users can access via a simple html page. In section 2 we provide details regarding the implementation of the program(scientific background and the libraries and technologies used), whereas in section 3 we describethe user interface of the program, particular emphasis being laid on the new features that have beendeveloped since the previous release the
RaDMaX program.
2. Background and implementation
In this section we shall briefly give recall the scientific foundations on which
RaDMaX online relieson; complete descriptions are provided in our previous articles (Boulle & Debelle, 2010; Souilah etal. , 2016). We will also outline the computing libraries that have been used, although we are notgoing to provide the details of the algorithms used since the source code is freely available online asfree software (CeCILL licence). 2 .1 Scientific background In RaDMaX online the diffracted X-ray intensity is computed within the framework of thedynamical theory of diffraction from distorted crystals as first theorized by S. Takagi and D. Taupin(Taupin, 1964; Takagi, 1969) which is the state of the art approach for the analysis of irradiatedcrystals using XRD (Klappe & Fewster, 1994; Milita & Servidori, 1995). More specifically, wemake use of the recursive solutions to the Takagi-Taupin equations provided by Bartels et al. (Bartels et al. , 1986). Within this approach, the irradiated region is divided into N sub-layers inwhich the composition, strain and disorder are assumed to be constant (typical values for N arewithin the 50-150 range). Consistently with earlier studies, the disorder is quantified through theDebye-Waller factor (Speriosu, 1981; Milita & Servidori, 1995; Zaumseil et al. , 1987) : (1)where Q is the scattering vector ( where θ is half the scattering angle and λ is the x-raywavelength), δ u ( z ) corresponds to the random atomic displacement vector at the depth z below thesurface, and the average <...> at depth z is performed in ( x , y ) planes parallel to the crystal surface.The Debye-Waller factor lowers the coherently diffracted intensity; for a perfect crystal, DW = 1,whereas for heavily disordered or amorphous materials, DW → 0. The strain has its usual meaningwhich, in the case of the θ/2θ scan recorded in the direction normal to the surface, correspond to thethird diagonal element of the strain tensor e zz : (2)where d z ( z ) is the inter-planar spacing of the diffracting crystallographic plane family measured atdepth z , and the superscript th refers to the theoretical value.Starting from the unirradiated region of the crystal (which corresponds to the scattering from aperfect single crystal), the total diffracted amplitude is constructed iteratively by combining theamplitudes diffracted and transmitted at each sub-layer interface, up to the surface. Determining thedepth-resolved strain and disorder profiles therefore consists in determining the values of the strainand disorder within each sub-layer, which is hardly feasible without further constraints put on thesystem. In RaDMaX online , we make used of cubic B-spline functions to model the strain anddisorder depth-profiles which allows to reduce the number of fitting parameters by a factor ~ 10(Boulle & Debelle, 2010). Using cubic B-splines, both the strain and the disorder have thefollowing form: (3)where N w S,D is the number of B-splines used to describe the strain (‘S’) and disorder (‘D’) profiles(typical values are in the 5-15 range), w i S,D are the weights to be determined in the fitting procedure3nd B i ,3 ( z ) is the third-degree basis function (Boulle et al. , 2003). With this model, the number offitting parameters is reduced from 2 × N to 2 × N w . NumPy library (https://numpy.org/) which introducesvectorized operations, i.e. operations implicitly working on all components of a vector (Oliphant,2007; van der Walt et al. , 2011), and the
SciPy xrayutilities python package (Kriegner et al. , 2013) which, amongother things, allows to read crystallographic information files ( cif ) which contain the description ofthe structure of the crystal (Hall et al. , 1991). It is here worth mentioning that cif
Jupyter notebook (https://jupyter.org/), a free andopen-source interactive programming environment that runs within a web browser and that allowsto gather the source code, the computational output (including plots, figures, etc. ) as well as richtext within a single document (Perkel, 2018). Interactive widgets ( i.e. sliders, editable text boxes,selection boxes, etc.) have been implemented using the ipywidgets library(https://github.com/jupyter-widgets/ipywidgets ) . Central to
RaDMaX online is the existence ofinteractive strain and disorder depth-profiles that can be graphically manipulated by the user so asto match the computed XRD curve with experimental data. This feature is brought to
Jupyter usingthe bqplot library (https://github.com/bloomberg/bqplot). In short, bqplot is a plotting library whereevery attribute of a plot can be programmatically modified and which allows for high level of userinteractivity (data selection, data manipulation, etc. ).Finally, although
RaDMaX online can be run locally on every user’s computer by downloading thesource code, its intended primary operating mode is to run as a web-hosted application. In such acase, for obvious security reasons, the users should not be able to execute arbitrary code on the webserver. Since recently this is feasible thanks to the voilà library (https://github.com/voila-dashboards/voila). When the URL of the
Jupyter notebook is called, voilà launches the kernel forthat notebook, runs all the cells and populates the notebook model with the outputs. Afterexecution, all the source code cells are hidden and the notebook is converted to html, includingJavaScript widgets which establish the connection to the kernel, and this page is served to the user.The corresponding result for
RaDMaX online is given in Fig. 1.4egarding web hosting,
RaDMaX online
RaDMaX online, the sources for the Docker deployment are free to use andcan be downloaded from: https://github.com/aboulle/RaDMaX-docker.
3. Program description and usage
As shown in Fig. 1,
RaDMaX online is divided is 4 different areas:1. An “upload” button allows the user to send their data to the server (in a two columns [2θintensity] tab- or space-separated ascii format). It should be noted that the data is entirelystored in RAM and no data is saved on the server. By default
RaDMaX online is loaded witha default data set corresponding to an irradiated (001)-oriented ZrO single crystal2. The different tabs (“Experiment”, “Material”, “Strain/disorder”, “Film parameters” and“Fitting parameters”) allow to set all parameters used in the XRD calculations and in thefitting procedure.3. The experimental (filled circles) and calculated (continuous line) XRD curves are shown inthe right-hand side plot, whereas the left-hand side plots are the strain and disorder depth-profiles. These plots are interactive and can be manipulated with the mouse via the controlpoints (filled circles) so as to match the calculated curve with to experimental data.5 Fig. 1. RaDMaX online’s web interface. . Three buttons at the bottom allow to launch or cancel a least-squares refinement of the strainand disorder profiles (using the solution provided by the user as a starting point) as well asto download the simulation results.
As the name suggests, this tab allows to set the parameters related to the XRD experiment (Fig. 1).The first line (“Resolution”, Full-width at half-maximum (“FWHM”) and “Shape”) correspond toparameters describing the angular resolution of the diffractometer. The exact angular resolutionfunction is tightly linked to the type of optics that are used to condition the incident beam. A fastand efficient way to quantify the resolution is to use the the XRD peak from the (unirradiated)single crystal as a reference and to adjust the parameters of an arbitrarily chosen bell-shapedfunction so as to optimally describe the corresponding peak. In
RaDMaX online several function areavailable: Gaussian, Lorentzian, pseudo-Voigt ( i.e. a linear combination of a Gaussian and aLorentzian function) and split ( i.e. asymmetric) pseudo-Voigt. The width of the function is given bythe FWHM parameter, whereas the “shape” parameter correspond to the relative amount ofGaussian (shape = 0) and Lorentzian (shape = 1) in the pseudo-Voigt function. Fig. 2 shows thecalculated diffracted intensity around the 004 reflection of an irradiated (001)-oriented ZrO singlecrystal. The corresponding strain depth-profile is given in Fig. 2a (the Debye-Waller is constant andequal to 1). Increasing the FWHM obviously increases the width of the Bragg peak emanating fromthe virgin region and lowers the fringes contrast (Fig. 2b), whereas increasing the shape parameterenhances the tails of the Bragg peak emanating from the virgin region and also decreases the fringescontrast (Fig. 2d).This set of parameters might also be used to describe the scattering from imperfect single crystals.This is the reason why the possibility of asymmetric functions has also been implemented (a recentexample can be found in (Boulle et al. , 2019)). This is achieved by selecting the split pseudo-Voigtfunction and by providing two, comma-separated values, for either or both the FWHM and shapevalues. These values correspond to left- and right-hand side parameters of the asymmetric function.The effect of a low-angle side asymmetry is shown in Fig. 2c. It can be observed that, incomparison with Fig. 2d, the difference in the diffraction emanating from the irradiated part is notobvious. Conversely, the asymmetry is clearly observed for the Bragg peak for the virgin area.The second line allow to fix the x-ray wavelength, the background and a possible 2 θ offset.Whereas the two former are rather self-explanatory, the latter correspond to an overall shift betweenthe computed and the experimental curve. This offset can be due to a misalignment of thediffractometer or to the fact the material analysed differs from the cif file used for the calculation,resulting in a shift between the Bragg peak of the computed curve and the observed peak. The 2 θ θ; however, forsufficiently small angular ranges (~1° or less) the difference in lattice parameters can be safelycorrected by applying a uniform 2 θ shift. The “Material tab”, Fig. 3a, allows to select the material from a drop-down menu and to set the hkl values of the reflection. Selecting a material from the list computes the structure factor and latticespacing for the corresponding hkl values, using cif files that are located within the ‘structure’ folderof the sources of
RaDMaX online . Specific cif files can be added on demand or, when using
RaDMaX in offline mode, added manually in the corresponding folder.The “Sample type” drop-down menu allows to select the geometry of the irradiated crystal. Possible7
Fig. 2. (a) strain depth-profile used for all calculation in thepresent article. Evolution of the 004 reflection of a (001)-oriented ZrO single crystal with (b) increasing FWHM ofthe resolution function (FHWM = 0.005°, 0.01°, 0.02° and0.04°), (c) increasing asymmetry (FWHM left = 0.005°,0.01°, 0.02° and 0.04°, FWHM right = 0.005°), (d)increasing shape parameter (shape = 0 [Gaussian], 0.1, 0.5and 1 [Lorentzian]). Fig. 3.
The different tabs of
RaDMaX online’s interface. hoices are “Single crystal”, “Thick film”, “Thick film + substrate” and “Thin film”. Thesegeometries are explicated below.Fig 4a, represents the case of a single crystal and the red curve correspond to the strain or disorderprofile. The strain and disorder profiles are determined by the energy loss of the incident projectileswithin the target material and, in general, exhibit a smooth transition between the irradiated and thenon-irradiated regions. This smooth transition is enabled by selecting the “Smooth” profile in the“Strain/disorder” tab (see section 3.3). This option is selected by default. The irradiated thickness isreferred to as t , whereas the thickness of the single crystal is referred to as T . For a sufficiently thicksingle crystal T → ∞. The corresponding XRD curve using the same strain depth-profile as in Fig.2a is shown in Fig. 5a (black curve).For a thick film, T is not infinite but larger than the irradiated thickness ( T > t ). The corresponding T value can be given under the “Film parameters tab”, Fig. 3c. Fig 5a (red curve) shows the XRDcurve in the case of 500 nm thick ZrO film with 200 nm irradiated region. If a substrate isdiffracting in the recorded angular region its nature and the corresponding hkl can be selected underthe same tab (Fig. 3c).Fig. 4c is a special case of a thin film where the irradiated thickness exactly correspond to the filmthickness, t = T . This situation is computed when the “Thin film” geometry is selected whilekeeping the “Smooth” strain profile (the same situation can be computed by selecting “Thick film”and setting t = T ). The corresponding XRD data is given by the blue curve in Fig. 5a: the domelocated at the 2 θ position corresponding to the perfect crystal is due to the narrow, low strain,region in Fig. 2a. Although not impossible, this situation seems rather far-fetched; a more realisticexample where the film thickness is actually thinner than the expected projectile path within thematerial ( T < t ) is represented in Fig. 4d. This situation can be computed using the “Thin film”8 Fig. 4.
The different sample geometries supported by
RaDMaX online . T is the thickness of the film or thesubstrate and t is the depth of the damaged region. Theshaded region schematizes the damaged region of thecrystal. The red curve illustrates the depth-dependence ofthe damage (strain or disorder). The damage depth profile isdetermined by the energy loss of the incident projectileswithin the crystal and therefore smoothly decrease to 0(after one or more maxima) (a-c), except when the filmthickness is smaller than the total projectile path length, inwhich case the damage is abruptly truncated (d). ption combined with the “Abrupt” strain profile under the “Strain/disorder” tab. The calculatedXRD curve is shown as the green line in Fig. 5a, where a perfect thin film signal is easilyrecognized. This tab allows the set all parameters related to the strain and disorder profiles (Fig. 3b). The“Strain/DW depth-profile” allows to select either the “Smooth” or the “Abrupt” B-spline functions,as discussed in section 3.2 above. The “Number of control points” correspond to the number ofinteractive points in the strain and in disorder profiles; from a numerical point of view, thisparameter correspond to the N w constant used in Eq. 3. This value should be adjusted to thecomplexity of the strain/disorder profile as already discussed in (Boulle & Debelle, 2010; Souilah et al. , 2016) . Low values (say, < 10) are well adapted to rather simple profiles, whereas complexprofiles (multimodal or exhibit tiny features) may require larger values (> 10). It should be borne inmind that the number of fitting parameter is equal to 2 N w ; hence, increasing N w value might yield tomore unstable least-squares fitting procedures.The “Damaged depth”, as discussed in section 3.3, is the depth required for the strain or thedisorder to converge to their bulk values (0 and 1, respectively). Fig. 5b shows the evolution of theXRD curve when the damaged depth takes the following values: 50 (black), 100 (red), 200 (blue)and 400 nm (green). Obviously, decreasing the damaged depth leads to broader profiles and morewidely spaced interference fringes. This feature can be used to estimated the damage depth from the9 Fig. 5. (a) calculated XRD curves corresponding togeometries depicted in Fig. 4. Single crystal (black), thickfilm (red), thin film smooth (blue) and thin film abrupt(green). (b) calculated curves for increasing damageddepths: 50 nm (black), 100 nm (red), 200 nm (blue) and400 nm (green). (c) calculated XRD curves for anincreasing numbers of sub-layers used in the computation:10 (black), 30 (red), 60 (blue) and 100 (green) sub-layers.(d) zoom on the 73.2 – 73.4° region.
RD signal.The “Number of data points” text field correspond to the number of sub-layers used in thecalculation of the XRD intensity (the parameter N mentioned in section 2.1). This parameter is notstraightforward to determine. Fig. 5c and 5d show the influence of this parameters when it takes thefollowing values: 10 (black), 30 (red), 60 (blue) and 100 (green). It can be noticed that no visibledifference can be observed between the 60 and 100 sub-layers cases. A rule of thumb is that 10 to20 sub-layers should be used for the tiniest feature in the strain/disorder profile. In the case of thestrain depth-profile show in Fig. 2a, the abrupt decrease of strain ranges from 120 to 180 nm, that isa 60 nm thickness, which, with above criterion gives a sub-layer thickness of 3 to 6 nm, which inturn gives a total number of sub-layers ranging between 33 to 66 in perfect agreement with theobservation of Fig. 5c,d. Increasing the value of N increases the depth resolution but increases thecomputation time; conversely decreasing this value accelerates the computation at the expense ofthe depth resolution.Finally, two sliders can be used to apply a global scale factor to the strain and the Debye-Wallervalues, where the latter is constrained to lie within the [0,1] interval. This last section of the user interface allows choose the least-squares fitting algorithm and thepermitted variation range of the fitted parameters. The default algorithm is the “trust reflectivealgorithm” as implemented in
SciPy (Branch et al. , 1999). Regarding the bounds set on theparameters, it is recommended to keep the Debye-Waller factor within its range of physicallymeaningful values, i.e. the [0,1] interval. The bounds on the strain are dependent on the problemand should be determined by the user. It should be noted that the limits can be ignored by selectingthe “Least squares (no bounds)” fitting algorithm. This implements the more usual Levenberg-Marquardt algorithm.Finally, contrarily to previous versions of
RaDMaX, this version does not include the morecomputationally demanding simulated annealing fitting procedure in order not to saturate theresources of the web server.
4. Alternate working modes
As mentioned earlier
RaDMaX online can be accessed on the web. The main address is:https://radmax.unilim.fr/. If, for any reason, the main server is not available, an alternate solutionconsist in running the
RaDMaX notebook on the
Binder web service (https://mybinder.org/). Inshort, this service allows to connect to a software repository (like github, gitlab, etc. ) and to run thecontent within a
Jupyter server that is build on demand, including all the necessary dependencies.10he drawback is that building the image of the custom
Jupyter server might take some times (up toa few minutes) and that, being a free service, the images can not be kept indefinitely on Binder andare therefore deleted after a given period of inactivity. Nonetheless, this service remains a veryconvenient way to share scientific notebooks and even complete web applications like
RaDMaXonline . The corresponding binder link is the following:https://mybinder.org/v2/gh/aboulle/RaDMaX-online/master?urlpath=voila/render/RaDMaX.ipynb Advanced users might want to customize
RaDMaX to fit their own needs. This is feasible bydownloading the source code from https://aboulle.github.io/RaDMaX-online/ and running theRaDMaX.ipynb file in their own
Jupyter
Jupyter ,as well as
SciPy and
NumPy are already included. The only remaining dependencies to be installedare ipywidgets, bqplot and xrayutilities. The last one can only be installed with the Python packagemanager. The following should be typed in a command-line interface: pip install xrayutilities
The others can be installed either via pip or via anaconda’s package manager, i.e. pip install ipywidgets bqplot voila or conda install -c conda-forge ipywidgets bqplot voila If the Python environment has been installed without using Anaconda, the
NumPy , SciPy and
Jupyter packages should be installed as well. There are many ways to do that depending on theoperating system; the following command should work in any environment: pip install numpy scipy jupyter
5. Concluding remarks
RaDMaX online is a free software dedicated to the determination of strain and disorder depth-profiles in irradiated crystals by means of XRD. It is a web-based program that doesn’t require anyinstallation on the users’ computers.
RaDMaX online is entirely based on free and open-sourcetechnologies. The program as well as the source code can be accessed at the following address:https://aboulle.github.io/RaDMaX-online/ The web application can be directly accessed at: https://radmax.unilim.fr/
Acknowledgements
AB is grateful to the NEEDS program (MeSINII project) for partial support of this work.11 eferences
Bartels, W. J., Hornstra, J. & Lobeek, D. J. W. (1986).
Acta Crystallogr. A . , 539–545.Boulle, A., Chartier, A., Crocombette, J.-P., Jourdan, T., Pellegrino, S. & Debelle, A. (2019). Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At. , 143–150.Boulle, A. & Debelle, A. (2010).
J Appl Cryst . , 1046.Boulle, A. & Debelle, A. (2016). Phys Rev Lett . , 245501.Boulle, A. & Kieffer, J. (2019). J. Appl. Crystallogr. , 882–897.Boulle, A., Masson, O., Guinebretière, R. & Dauger, A. (2003). J. Appl. Crystallogr. , 1424–1431.Branch, M. Ann., Coleman, T. F. & Li, Yuying. (1999). SIAM J. Sci. Comput. , 1–23.Debelle, A. & Declémy, A. (2010). Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At. , 1460–1465.Emoto, T., Ghatak, J., Satyam, P. V. & Akimoto, K. (2009).
J. Appl. Phys. , 043516.Hall, S. R., Allen, F. H. & Brown, I. D. (1991).
Acta Crystallogr. A . , 655–685.Klappe, J. G. E. & Fewster, P. F. (1994). J. Appl. Crystallogr. , 103–110.Kriegner, D., Wintersberger, E. & Stangl, J. (2013). J. Appl. Crystallogr. , 1162–1170.Milita, S. & Servidori, M. (1995). J. Appl. Crystallogr. , 666–672.Oliphant, T. E. (2007). Comput. Sci. Eng. , 10–20.Perkel, J. M. (2018). Nature . , 145–146.Rieutord, F., Mazen, F., Reboh, S., Penot, J. D., Bilteanu, L., Crocombette, J. P., Vales, V., Holy, V. & Capello, L. (2013). J. Appl. Phys. , 153511.Souilah, M., Boulle, A. & Debelle, A. (2016).
J Appl Cryst . , 311.Sousbie, N., Capello, L., Eymery, J., Rieutord, F. & Lagahe, C. (2006). J Appl Phys . ,.Speriosu, V. S. (1981). J. Appl. Phys. , 6094–6103.Takagi, S. (1969). J. Phys. Soc. Jpn. , 1239–1253.Taupin, D. (1964). Bull Soc Fr. Minér Crist . , 469.van der Walt, S., Colbert, S. C. & Varoquaux, G. (2011). Comput. Sci. Eng. , 22–30.Zaumseil, P., Winter, U., Cembali, F., Servidori, M. & Sourer, Z. (1987). Phys. Status Solidi A . , 95–104.Zhang, Y., Debelle, A., Boulle, A., Kluth, P. & Tuomisto, F. (2015). Curr Opin Solid State Mater Sci . , 19. 12 igure captions Fig. 1:
RaDMaX online ’s web interface.Fig. 2: (a) strain depth-profile used for all calculation in the present article. Evolution of the 004reflection of a (001)-oriented ZrO single crystal with (b) increasing FWHM of the resolutionfunction (FHWM = 0.005°, 0.01°, 0.02° and 0.04°), (c) increasing asymmetry (FWHM left =0.005°, 0.01°, 0.02° and 0.04°, FWHM right = 0.005°), (d) increasing shape parameter (shape = 0[Gaussian], 0.1, 0.5 and 1 [Lorentzian]).Fig. 3: The different tabs of RaDMaX online’s interface.Fig. 4: The different sample geometries supported by
RaDMaX online . T is the thickness of the filmor the substrate and t is the depth of the damaged region. The shaded region schematizes thedamaged region of the crystal. The red curve illustrates the depth-dependence of the damage (strainor disorder). The damage depth profile is determined by the energy loss of the incident projectileswithin the crystal and therefore smoothly decrease to 0 (after one or more maxima) (a-c), exceptwhen the film thickness is smaller than the total projectile path length, in which case the damage isabruptly truncated (d).Fig. 5: (a) calculated XRD curves corresponding to geometries depicted in Fig. 4. Single crystal(black), thick film (red), thin film smooth (blue) and thin film abrupt (green). (b) calculated curvesfor increasing damaged depths: 50 nm (black), 100 nm (red), 200 nm (blue) and 400 nm (green). (c)calculated XRD curves for an increasing numbers of sub-layers used in the computation: 10 (black),30 (red), 60 (blue) and 100 (green) sub-layers. (d) zoom on the 73.2 – 73.4° region.13ig. 1 RaDMaX online ’s web interface. 14ig. 2(a) strain depth-profile used for all calculation in the present article. Evolution of the 004 reflection of a (001)-oriented ZrO single crystal with (b) increasing FWHM of the resolution function (FHWM = 0.005°, 0.01°, 0.02° and 0.04°), (c) increasing asymmetry (FWHM left = 0.005°, 0.01°, 0.02° and 0.04°, FWHM right = 0.005°), (d) increasing shape parameter (shape = 0 [Gaussian], 0.1, 0.5 and 1 [Lorentzian]). 15ig. 3The different tabs of RaDMaX online’s interface.16ig. 4The different sample geometries supported by
RaDMaX online . T is the thickness of the film or thesubstrate and tt