Calculation of the instrumental profile function for a powder diffraction beamline used in nanocrystalline material research Luca Rebuffi*a,b, Paolo Scardib Elettra-Sincrotrone Trieste, Strada Statale 14, Km 163.5, 34149 Basovizza, Trieste, Italy; b Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy a
*
[email protected]
ABSTRACT Ray-tracing algorithms are used to simulate the instrumental function of a synchrotron beamline targeted to the advanced characterization of nanocrystalline materials by powder diffraction. The characteristics of the source, a bending magnet in the present case of study, and the optics influence the instrumental profile, which is a key parameter for obtaining information on the nanostructure. We combine the SHADOW simulation with the calculation of powder diffraction profiles from standard materials, into a high-level workflow environment based on the ORANGE software, allowing us to integrate data analysis fitting software with realistic information.
1. INTRODUCTION Synchrotron radiation is one of the main tools in the study of nanomaterials, in particular for powder diffraction, which fully exploits the high brilliance, energy selectivity and focusing conditions now routinely available at many beamlines. However, most facilities have been designed for studying conventional materials, whereas a modern approach to nanomaterials requires a complete control of the diffracted signal, and therefore of the optics and general set-up of the beamline [1]. We combine the SHADOW [2] simulation with the calculation of powder diffraction profile from standard materials, into a high-level workflow environment based on the Open Source ORANGE [3] software, allowing us to integrate data analysis fitting software with realistic information [4,5]. We developed algorithms and software tools capable of reproducing optical elements in a realistic form, so to evaluate the effects of aberrations, with the final purpose of reconstructing and representing the instrumental function of the beamline, with the possibility of investigating the role of each separate element. The results of this work, and in a more general sense the emerging paradigm, could be of interest to many different beamlines currently employed for X-ray spectroscopies. Direct applications will include catalysts, heavily deformed materials and other nanocrystalline systems.
2. INSTRUMENTAL PROFILE FUNCTION: A REALISTIC RAY TRACING APPROACH X-ray Diffraction (XRD) Line Profile Analysis (LPA) is a well-known technique for the study of materials microstructure [6,7]. Peak profiles in the diffraction pattern are modified in shape, intensity and position by microstructural effects, like the shape and size distribution of the crystalline domains (aka crystallites), and/or lattice distortions present in the system (microstrain). Together with these physical sources, the observed profile contains the instrumental contribution, which is the combined effect of the photon source energy and spatial distribution, the optical layout setup and the quality of its elements.
When dealing with nanostructured material, crystalline domain size distribution plays a crucial role, and the Scherrer’s formula [8,9] correlates the measured integral breath 2 with the volume-weighted mean crystallite size D and V
the incoming photon beam wavelength :
2
K
(1)
D cos V
Just considering the inverse proportionality occurring between the observed quantity and the average crystallite size, it comes clear how the error on the crystalline domain size diverges for small values of integral breadth, i.e. when instrumental effects are the main feature. Therefore, the LPA capability of determining the characteristics of nanostructured materials, mostly with large crystalline domains, is strongly affected by the shape and the stability of the Instrumental Profile Function (IPF). Synchrotron radiation seems the most appropriate choice to collect high quality diffraction data, thanks to the high beam brilliance, energy selectivity, and focusing conditions. But even a simple powder geometry can be affected by considerable aberrations [1,10], and the IPF needs to be “well-behaving”, i.e., easily represented in convenient form for data analysis. Several mathematical description of instrumental effects are available in literature, in particular describing the optical origin of the diffracted beam divergence [11,12], and the optical aberrations effects [13-16], used by methods for data analysis like WPPM [17] to build a mathematical parametric representation of the IPF, available for calibration and fitting procedures. In any case, the optical nature of the instrumental contribution to the diffraction pattern suggests a ray-tracing simulation approach for its description, prediction and analysis [18,19]. In the present work we use SHADOW, a well-known software for the simulation of realistic effects on the beam transport through optical elements, as the basis for a realistic ray-tracing simulation of a powder diffraction capillary sample (Debye-Scherrer geometry). The instrumental profile can then be obtained without any direct mathematical representation, describing it in terms of the contribution of every single optical component, from the photon source to the detector
3. A MODERN RAY-TRACING TOOL ORANGE is a python based software, representing active object, i.e. containing data and procedures, as widgets in a desktop, exchanging data. Connecting wires between widgets, as visible in Figure 1, represent the data flux. By calling SHADOW via its python API [20], it was possible to create SHADOW objects representing the different optical elements and photon sources as widgets available to the user (see Figure 2), exchanging a SHADOW object representing the photon beam as the I/O data passing through the wires. With the same architecture some SHADOW Preprocessor utilities displayed by the python API interface, has been integrated as widgets, automatically communicating with the SHADOW objects widgets.
Figure 1 - Ora ange Graphic User U Interface: Integration I of SHADOW S objeccts as widgets and a wires.
Figure 2 - OR RANGE-SHAD DOW integratioon: architecture schema
Using Matplootlib python lib brary [21], several plotting fuunction have beeen introducedd (as dedicated widgets and inntegrated into SHADO OW objects wid dgets) in order to reproduce thhe same level of o visual inform mation given by b the older SH HADOW graphic user interface (see Figure F 3).
Figure 3 - Ex xample of plottiing functionalityy, integrated into a SHADOW W optical elemen nt widget
A special feaature of recursiive and cumulaative simulations has been inntroduced, in orrder to improvve the statisticaal quality of the simullation, avoidin ng the use of a single hugee amount of generated g rayss, but executinng a series off several simulations with w a small number n of geneerated rays, wiith the double advantage of monitoring thee simulation during d its execution annd reducing th he memory alllocation (see Figure 4). Thhis feature is particularly p usseful with the powder diffraction simulation, where the executioon time of the algorithm a is noot linear with thhe number of raays.
Figgure 4 - Recurssive simulation management m
4. POWDER DIFFRACTION RAY-TRACING: DEBYE-SCHERRER GEOMETRY The main effect of the beam divergence (distribution), coming from the optical elements and source characteristics, on a powder diffraction pattern is a peak broadening, showing a dependence on the 2 angle, that has been mathematically described in [11,12], and it is usually represented and parameterized with the Caglioti’s equation [22,23] for the full width half maximum (FWHM) of the instrumental peak profiles, here represented as pseudo-Voigt curves:
FWHM
W V tan
U tan 2
(2)
The typical shape of this effect is shown in Figure 5.
Figure 5 - Instrumental Peak Broadening: MCX beamline at Elettra-Sincrotrone Trieste at 11 keV
A complete instrumental profile function characterization, in Debye-Scherrer geometry, can be found in a recent study made at the MCX beamline at the Italian synchrotron Elettra-Sincrotrone Trieste [24], where the three parameters are obtained analyzing the diffraction pattern from a sample of LaB6 (Lanthanum Hexaboride), a standard reference material produced by NIST [25]. A similar characterization can also be made with Silicon [26]. Thus, a realistic ray-tracing approach starts from the simulation of the interaction of the SHADOW photon beam with a capillary filled by such a standard reference material, generating a diffracted photon beam and prosecuting the raytracing onto the optical elements laying on the path from the sample to the detector. The incident beam is obtained by a SHADOW ray-tracing simulation of the beamline, using its capability of adding realistic features to the optical elements, like reflectivity (both for mirrors and crystals) and slope error. With reference to Figure 6, for each ray incident on the capillary a random point is generated along the path between the entry and the exit points, then the diffracted beam is generated rotating the wave vector around SHADOW X axis, by the nominal Bragg angles corrected for the angular divergence and the energy dispersion of the ray. In order to reproduce the diffraction rings of a powder, the diffracted ray is then rotated around the ideal optical axis, the SHADOW Y axis, by a random angle within a range determined the successive angular acceptance of the optical system driving the signal to the detector.
Figure 6 - SHADOW axis systems
Every diffraction peak is normalized to the most intense one, calculated from the structure factor square modulus and the multiplicity of the reflection. The experimental diffraction pattern is collected by a 2 angle stepped scan of the diffracted signal, which is simulated via a repeated SHADOW ray-tracing of the detector optical system, rotated step by step around the SHADOW X axis. Two possible optical systems are available: a couple of collimating slits between the sample and the detector or an analyzer crystal with an entrance slit. The final pattern can be normalized with the Lorentz-Polarization and Thermal factors, using the following expressions [27-30]:
LP 2
T 2
1 sin sin 1 sin sin
exp 2B
1 Q
1 cos 2 2
bragg
bragg
sin
1 Q cos 2 2 cos 2 2
1 cos 2 2 cos2 2 2
mon
with a system of slits
mon
ana
(3)
with an analyzer crystal
2
(4)
Ih Iv is the degree of polarization (around 0.95 for the Ih Iv synchrotron radiation, Ih e Iv are, respectively, the intensity of the horizontally and vertically polarized radiation parts), mon is the angle between the incident beam and the first monochromator crystal, B is the Debye-Waller coefficient (in the present work, for simplicity, we consider an average B value). An example of a full diffraction pattern simulation is visible in Figure 7. Where
bragg
is the nominal Bragg angle of the reflection, Q
Figuree 7 - LaB6 simu ulated diffractioon pattern, show wn by the XRD capillary widgeet
The simulatiion takes into account severral source of aberrations, a likke the displaceement of the capillary c respecct to the goniometric center, the diisplacement off the slits resppect to the ideeal optical patth and a simpple capillary wobbling w approximatedd model, corresponding to a percent p increasse in diameter. The softwaree allows a bacckground to bee added to thee generated difffraction patterrn, selecting annd/or combiniing three different functions: constan nt value, Chebbyshev polynom mial of the firrst kind up to 6th degree andd exponential decay. d A random noisee of adjustable intensity is geenerated aroundd the selected background b currve. Finally, the simulation can n take into acccount the absoorption of the material, reduucing the initial intensity I0 of each incoming andd diffracted ray y according to the law:
I
,x
Where
I 0 exp e
x
(5)
is the linear absorption a coeffficient at the photon p wavelenngth , calculatted with the xrraylib API [31], is the
material denssity and x the path p of the ray inside the capiillary. The absorptiion effect is allso consideredd calculating thhe source poinnts of the diffrracted rays witth a random generator g based on a prrobability expo onential distribution accordinng to the transm mitted intensityy law (eq. 5), raather than the usual u flat distribution:
P
,x
K exp e
x
(6)
Where K is a normalization n factor. An example of the generatted source poinnts of the diffraacted beam wiith the absorption calculationn activated is shown s in Figure 8, andd the effect on n the peak intennsities is show wn in figure 9. The main effeects of creatingg an apparent capillary c displacementt, and an apparrent smaller Deebye-Waller cooefficient are cllearly visible annd reproducedd. Absorption from f the material of the cappillary is not taken t into acccount, because it acts only on o the intensitty of the diffracted siggnal, as a reducction factor coonsiderable, in first approxim mation, constantt for all the rayys. This approxximation is justified byy the thicknesss of the capillarry wall (aroundd 0.01 mm).
Figure 8 - Generated G source points of the diffracted beam m, on a 0.1 mm m diameter capilllary, with the absorption a calcu ulation activated. Thee ZY plane sectiion refers to thee SHADOW axiis system.
Figure 9 - Comparison betw ween simulated diffraction proofiles of LaB6 att 11 keV, with an nd without the absorption calcculation n resp pect to the centrral peak. activated. The patterns are normalized
5. RESULTS: THE MCX BEAMLINE B Results wherre checked making a compleete simulation of o the MCX beamline. With electron beam m energy of 2 GeV, G the source is a bending magnett with critical energy of 3.2 keV. k The total length of the beamline is arround 36 meterr and the optical layouut is composed d by a first verttically collimatting cylindricaal bendable Pt--coated mirror, followed by a Si(111) double crystal monochrom mator, with a sagitally s bendable second crystal c [32], thhen by a second vertically focusing cylindrical bendable Pt-coaated mirror. Thhe optical systtem, from the sample to the detector at a distance of 0.95 m, is composed byy two horizontaal slits, with addjustable apertuure.
A complete ray-tracing r sim mulation representing a pow wder diffractionn experiment at a MCX is shoown in Figure 10. The experiment for f characterizing the IPF useed a 0.1 mm caapillary filled by b NIST 660a LaB6, L and phooton beam enerrgy of 11 keV.
Figure 10 - Ray-tracing simulation s of a powder diffraction experimen nt at MCX: layoout appearance in the user inteerface
After a carefful optimizatio on of the optiical setup, in order o to let thhe simulation the t most closeely reproduce the real conditions off the experiment, Figure 11 and a 12 show, respectively, r a comparison beetween experim mental diffraction peak and simulatedd one, and betw ween experimeental instrumenntal peak broaddening and sim mulated one. As visible frrom the figure,, the general agreement a is goood, but somee discrepanciess in the shape of the peak annd in the instrumental broadening are present, com ming out of possible differences in the sourcce spatial and divergence d disttribution, and the crysstals diffraction n profiles (in SHADOW cryystals are moddeled as perfect) and by noot considering possible effects of a tw wist of the seco ond sagitally bendable crystaal of the monocchromator.
Figure 11 - Co omparison betw ween experimenttal LaB6 (1,1,0)) peak at 11 KeV V and the simulated one.
Figurre 12 - Compariison between exxperimental insttrumental peak k broadening att 11 KeV and th he simulated onee
As a concludding remark, itt can be notedd the asymmetrry of the expeerimental profille, a weak butt visible featurre on the high-angle taail of the peaak, which is a fingerprint of o absorption. By improvingg the simulatioon to accountt for the mentioned missing m optical effects, we caan expect an even better agreement betweeen model and data; then it would w be possible to tuune the absorp ption parameteers, reproducinng this asymm metric shape, prroviding not only o an estimatte of the density of a measured m samp ple, but also ann upper limit foor this quantity in order to preevent strong abberrations.
6. CO ONCLUSION NS We introducced new softw ware for realisstic ray-tracingg of powder diffraction, to become a toool for simulaating the instrumental effects in pow wder diffraction profiles at synchrotron s beeamlines. As ann off-line tool, it can be adoopted by beamline useers to drive ex xperiment desiggn and samplee preparation according a to thhe beamline layyout and beam m energy, and by beam mline scientists to improve thee performance of existing beaamlines. It cann also become a valid tool to improve quality of dessign of optical components annd beamline laayouts, with a realistic-experi r iment-oriented approach. Several featuures are under development, like the usagee of experimenntal rocking cuurves, bendablee crystals twistt effects, capillary wobbbling effects and a diffractometer eccentricitty effects. Finally, the future f availabillity of the wavvefront propagaation during thhe ray-tracing, coupling SHA ADOW with SR RW [20], will open thee door to the sim mulation of thee coherent diffr fraction.
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