REVIEW

COPY

NOT FOR D

ISTRIB

UTION

Element sensitive holographic imaging of atomic structure using white x-rays

K. M. Dąbrowski, D. T. Dul, T. P. Roszczynialski and P. Korecki∗Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

White-beam x-ray fluorescence holograms were recorded using a 50 watt x-ray tube for a Cu3Ausingle crystal. Element sensitivity for this model sample was explicitly demonstrated by measuringdistinct holograms using Cu K and Au L x-ray fluorescence. The phase sensitivity of the recordedholograms enabled to obtain projections of the local atomic structure and image the electron densitywithin the unit cell, relative to either Au or Cu atoms. Since the experimental setup does not requireany optics and the angular resolution requirements are loose, application of more powerful laboratoryx-ray sources is straightforward.

PACS numbers: 61.10.-i, 42.40.-i, 78.70.En, 42.30.Rx

Keywords: x-ray holography, local atomic structure, phase-problem, polychromatic x-rays, laboratory x-ray

sources

I. INTRODUCTION

X-ray fluorescence holography1 (XFH) provides themeans for obtaining three-dimensional (3D) images oflocal atomic structure. The main driving force behindits development was the desire to overcome the phaseproblem of standard diffraction methods. An x-ray flu-orescence hologram is formed as a result of the interfer-ence of the incident radiation and x-rays scattered insidethe sample. Therefore, similarly to Gabor holograms,2

x-ray fluorescence holograms contain the amplitude andthe phase information of the scattered radiation. Thisinterference is probed at the atomic sites by measuringx-ray fluoresce emitted by atoms. A 3D image aroundabsorbing atoms can be obtained from holograms usingnumerical reconstruction.3 However, because of a veryweak holographic signal (∼ 10−3 of the average fluores-cence yield) and necessity to record multiple hologramsfor twin-image removal,4,5 experiments usually requireintense synchrotron radiation. In the case of monochro-matic XFH, only a few attempts have been made usingconventional x-ray sources.1,6–9 For review of XFH andrecent application examples see Refs. 10 and 11.

A way around to solve the twin-image problem is to usea white-beam for recording x-ray fluorescence holograms.Transition from monochromatic to polychromatic illumi-nation is accompanied by a qualitative change of the x-ray hologram character.12,13 Because of the short coher-ence length of the white beam, the interference effects areconstrained around forward-scattering. This effect is ex-emplified in Fig. 1(a), which shows spatial modulations ofthe total x-ray field induced by scattering of the incidentpolychromatic plane wave on a single atom. The mainfeature is the specific "shadow" in the forward-direction.The drop in the intensity at the forward scattering di-rection is due to the negative scattering amplitude of x-rays. Figures 1 (b) and (c) show similar calculation for acrystal. In this case, the enhanced forward-scattering re-sults in a localization of the x-ray field minima on dense-packed atomic chains. This effects determines the mainfeatures of the hologram. Since a hologram is measuredby probing the x-ray field at the atomic sites (via flu-

orescence yield emitted from particular elements) whenthe sample is rotated relative to the incident beam di-rection, one can observe signal variations near directionscoinciding with dense-packed directions and planes. Thecalculations from Fig. 1 were performed using the formal-ism of Ref. 14 for a spectrum characteristic of an x-raytube operating at 50 kV as described in the reminder ofthis article.

Until now white-beam x-ray fluorescence hologramswere measured exclusively at synchrotrons. For example,in a recent experiment15 high-resolution projections ofatomic structure around absorbing atoms were observedfor a hard x-ray wiggler radiation with a mean energyabove 60 keV and a spread of almost 40 keV. Initial ex-periments employed total electron yield for measuring thevariation of the x-ray field at atomic sites. Therefore, theterm white-beam x-ray fluorescence holography was firstused in Ref. 16.

In this work we recorded white beam x-ray fluores-cence holograms in a laboratory using a low-power 50watt x-ray tube. Moreover, for the first time we explicitly

(a) (b) ( )c

max

min

1Þ

FIG. 1. (Color online) Spatial modulation of the x-ray fieldintensity due to the interference between incident plane waveand scattered radiation for a white-beam case. (a) Singleatom. (b,c) Crystal. Arrows denote the direction of theincident beam. A white beam x-ray hologram is measuredby probing this modulation at atomic sites (via x-ray flu-orescence) while the crystal is rotated relative to the in-cident beam direction. The "shadowing" effect enables todirectly observe signal variations near directions coincidingwith dense-packed directions and planes and the higher orderdiffraction effects enable 3D imaging.

2

showed element sensitivity of XFH by recording distinctholograms for Au and Cu atoms in the Cu3Au alloy. Theexperiment was possible by exploiting the relative largephotoelectric absorption cross section in the energy rangearound 20 keV (changing with ∼ 1/E3) which compen-sated for the small intensity of an ordinary x-ray tubeas compared to synchrotron radiation. Despite this rela-

tively soft x-ray spectrum, we were able to obtain projec-tions of local atomic structure around Au and Cu atomsand determine the electron density within the unit cellrelatively to those atoms.

II. EXPERIMENT

The scheme of the experimental setup for laboratoryrecording of white-beam x-ray fluorescence holograms ispresented in Fig. 2 (a). A 50 W x-ray tube (50 kV, 1mA, 0.8 Al filter) with a tungsten anode and a spot sizeof 75 µm was used as a source of polychromatic x-rays.A circular aperture with diameter of 8 mm at a distanceof 340 mm from the x-ray tube was used to shape thebeam with approximately 2× 108 photons/s. Originallythis setup was devoted to experiments with polycapillaryoptics.6

A Cu3Au single crystal in the ordered phase with a(001) orientation was used in the experiment. The sam-ple had a diameter of 10 mm and a thickness of 1 mm.The choice of Cu3Au (space group Pm3m, lattice con-stant a = 3.75 Å) was made as it can be seen as a modelsystem in x-ray holography for which computational18

and experimental studies were performed.19 Moreover,for x-ray holograms recorded with a monochromatic 10keV incident beam, significant extinction effects wereobserved.20

The sample was placed at a distance of 610 mm fromthe source and was rotated relative to the incident beamaround two axes (θ, φ). The data were collected on∆φ = 0.15,∆θ = 0.5 mesh in the θ range from 26.5

to 80 as measured from the sample’s normal. For fur-ther analysis the collected data were rebinned in the φdirection to a mesh compatible with the angular reso-lution of the setup determined by the sample’s angulardiameter (< 1). For each sample orientation an x-rayfluorescence spectrum [c.f. Fig. 2(b)] was recorded for 0.5s using an energy resolving silicon drift detector (area of25 mm2, thickness of 500 µm) placed at a distance of 12mm from the sample. The detector was attached to theθ circle and positioned along the surface normal. Thisexperimental geometry enabled to avoid the problem ofsimultaneous recording of Kossel lines which can be ob-served in emission geometry.21 The fluorescence detectionrate was at the level of 105 photons/s. The Cu K andAu L fluorescence yields (YCu and YAu) were computedfrom the spectra and subsequently corrected for dead-time. The total acquisition time was ∼ 90 h. Note, thattimes needed for acquisition of a full multiple-energy dataset at synchrotrons are of the the same order.22

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10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

E [keV]

N(E

) [a

rb.

un

its]

(a)

(d)

(b)

(c)

x-ray tube

0.8 mm Al

8 mm Pbaperture

SDD

q

f

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10 20 30 40 500

2

4

6

8

10

E keV

0

´106

8 10 12 14

8

6

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2

0

10

E keV

´104

Cu K

Au L

FIG. 2. (Color online) (a) Scheme of the experimental setupfor laboratory recording of white-beam x-ray fluorescenceholograms. (b) Spectrum of the emitted x-ray fluorescenceY (E) with visible Cu K and Au L lines. (c) Spectrumof the incident beam N0(E). (d) Effective energy spectraN(E). Small red circles show the spectrum for Cu atomsand large black circles for Au atoms. Dashed and solid linesare Lorentz (E0 = 21.3 keV, ∆E = 11.9 keV) and Gumbel(ε0 = 20.2 keV, ∆ε = 5.9 keV) fits to the data.17

For white-beam XFH, the main parameter influencingthe character of the holograms is the so called effectiveenergy spectrum N(E). Roughly speaking, it is a spec-trum sensed by atoms inside the sample. In previousworks, performed for hard synchrotron radiation, a thinobject approximation was used. In the present work, inorder to take into account the non-negligible escape depthof the fluorescence radiation as compared to penetrationdepth of the incident beam, N(E) has to be defined us-ing formulas taken from x-ray spectroscopy23 i.e. as:N(E) = N0(E)η(E), where N0(E) is the incident beamspectrum, η(E) = µi(E)/[µ(E)/ cos θ + µ(Ei)], µi(E) isthe photoelectric absorption coefficient of atoms of typei, µ(E) is the total absorption coefficient of the sample

3

and Ei is the effective energy of x-ray fluorescence emit-ted by atoms of type i (Au or Cu).

The incident beam spectrum N0(E) was measured di-rectly with the SDD detector by placing it far fromthe source and correcting it by the energetic efficiencycurve. The N0(E) spectrum is shown in Fig. 2(c). η(E)was computed using the procedures from the xraylib

library.24 The resulting effective spectra for Cu and Auatoms, averaged over the sample’s orientation are shownin Fig. 2(d). The spectra are very similar except for thelow energy part where one observes a jump correspond-ing to the Au L3 absorption edge in the Au spectrumand characteristic W emission lines for the Cu spectrum.The area of these lines is small i.e. of the order of 2% ofthe total spectrum.

III. WHITE-BEAM X-RAY FLUORESCENCE

HOLOGRAMS

White-beam x-ray fluorescence holograms recordedfor Au (χAu) and Cu (χCu) atoms are shown in Fig.3. They were extracted from florescence yields mea-sured as the function of the sample orientation rel-ative to the incident beam direction according to:χi(θ, φ) = Yi(θ, φ)/Y

0i (θ, φ)− 1, where Y 0

i (θ, φ) is theslowly varying component of the signal. The backgroundsubtraction was performed using smoothing splines. Theholograms were symmetrized using the Pm3m spacegroup and extended to the full sphere. Areas around the<001> directions were not accessible due to the specificmounting of the detector. For comparison, parts of theexperimental data were replaced by calculated data. Inthe calculation the effective energy spectrum N(E) wasapproximated by the Gumbel function which accountsfor the skewness of the spectrum [c.f. Fig. 2 (d)]. Fordetails of the calculations see Ref. 16.

The most characteristic features of the recorded white-beam holograms are the bands which are centered aroundprojections of dense-packed atomic planes. For a directcomparison Fig. 3(c) shows a purely geometrical projec-tion of the low index atomic planes in the Cu3Au sample.In addition, the data show distinct minima at the cross-ing of these bands which coincide with the dense-packeddirections. For the Cu3Au structure there are essentiallyto kinds of atomic planes. The first type is composed ofboth Au and Cu atoms while the second type containsonly Cu atoms. Hence, one can expect that the holo-grams measured for Au or Cu fluorescence will mainlydiffer near the projection of these planes [c.f. Fig. 3(c)].This effect which is clear in simulated data can be alsodirectly observed in the measured data.

FIG. 3. (Color online) White-beam x-ray fluorescence holo-grams recorded for a Cu3Au crystal. Hologram recorded forAu L fluorescence is shown in (a) and for Cu K in (b). Forcomparison, parts of the experimental data were replaced bycalculated data. Contrast has been enhanced for a better pre-sentation. Both patterns are presented as equal-area projec-tions of the upper part of the hemisphere. (c) Correspondingprojection of dense-packed atomic planes in Cu3Au. Linesaround which the holograms are expected to significantly dif-fer are shown in color. (d) The unit cell of Cu3Au.

IV. RECONSTRUCTION OF ATOMIC

STRUCTURE

In order to retrieve the structural information from themeasured holograms two approaches were used: waveletanalysis16,25 and a linear regression approach.18 The firstmethod is model-free and allows one to obtain projec-tions of the local atomic structure. The linear regres-sion algorithm, which is a generalization of the tomo-graphic approach14 enables 3D imaging of the unit cellbut assumes periodicity and requires the knowledge ofthe unit cell parameters. Moreover, these two methodsrely on different features of the holograms. The wavelettransform uses the information contained around dense-packed atomic directions while the linear regression ap-proach uses the whole visible bands around projectionsof atomic planes.

4

A. Local structure imaging using wavelet analysis

As the wavelet analysis of white-beam x-ray hologramshas been described previously,25 here we will give only abrief description of it. The wavelet analysis takes ad-vantage of unique properties of the holographic signalrecorded for white-beam which can be written as:

χ(k) = −2re

∫

V

ρ(r)

rh(ϑ, r)dV, (1)

where k defines the incident beam direction, r is the po-sition relative to the absorbing atom, ρ(r) is the electrondensity of the sample, re is the classical electron radius,V is the sample volume and ϑ = arccos(k · r). For aLorentzian approximation of the effective spectrum:

h(ϑ, r) ≈ exp(

−∆krϑ2/4)

cos(

k0rϑ2/2

)

, (2)

where k0 is the center and ∆k is the FWHM of N(k). Dueto the short coherence length of a white-beam, all h(ϑ, r)functions are localized around directions in which thescattering atoms are located. They have an almost zeromean and are similar i.e. they differ only in scale and po-sition on the sphere. Note, that these general propertiesare independent of the specific shape of the x-ray spec-trum unless it is sufficiently white. Since h(ϑ, r) func-tions closely resembles wavelets, the spherical wavelettransform26 is a natural way for analysis of white-beamholograms. In particular, a windowed inverse wavelettransform was shown to produce high-quality sphericalprojections of the local atomic structure. It improves thequality of the structural images as compared to a directwavelet transform. However, this happens at the expenseof resolution degradation along the radial direction.

In a recent work16 a compact formula was given forcomputing this transform:

F (k) = Cψ

∫

S2

χ(k′)Ψ(k′, k)dΩ, (3)

where Cψ is a constant and

Ψ(k′, k) = (4)

e−1

4∆krcϑ

2

sin(

krcϑ2

2

)

− e−1

4∆kr0ϑ

2

sin(

kr0ϑ2

2

)

kr0ϑ2,

In this expression ϑ = arccos(k′· k). The inverse win-

dowed wavelet transform is determined by two parame-ters: r0 and rc. In this work r0 was set to the small-est possible value of 2 Å, determined by specific lim-itations of the wavelet transform.16 The rc parameterwas used to adjust the depth of view of the projection∆r ∼ rc(1 + ∆k/k0). Therefore, signals from atoms lo-cated at distances larger then ∆r will be strongly sup-pressed in the projection.

Figure 4 shows projections of the atomic structurearound Au and Cu atoms obtained from holograms pre-sented in Fig. 3 by means of the windowed inverse wavelet

FIG. 4. (Color online) Projections of local atomic struc-ture obtained with the inverse windowed wavelet transform ofwhite-beam fluorescence holograms for Au and Cu absorbingatoms. Images in (a) and (b) were obtained for rc = 2.65 Åand in (c) and (d) for rc = 10 Å. The angular and intensityscales can be directly compared to those from Fig. 3.

transform. The parameters k0 and ∆k were obtainedfrom a fit to the measured spectrum as shown in Fig. 2(d).Images were computed for rc = 2.65 Å (nearest atom dis-tance in the <011> direction) and for rc = 10 Å. Thespots visible in the images coincide with directions ofdense-packed atomic rows, which means that individualatomic chains can be visualized using the wavelet analy-sis of white-beam holograms. For larger rc, spots havinglarger interatomic spacing become visible.

Note, that for the Cu3Au sample the positions of atomsin the local structure around Au and Cu atoms are thesame. These positions are however occupied by differenttypes of atoms. Hence, one can expect that the projec-tion will differ only in intensities. For example in the<011> direction, for an Au absorber, the sequence ofatoms is Cu-Au-Cu. In the unit cell, there are three Cuatoms having different position. Therefore, for an aver-age Cu absorber this sequence is X-Cu-X, where X cor-responds to a site with occupancy 1

3Au and 2

3Cu. Hence,

the nearest neighbour of the Cu absorber has an effec-tively larger scattering amplitude than the nearest neigh-bour of the Au absorber. On the other hand, for the Auabsorber the total number of electrons in this chain is ef-fectively higher than for the Cu absorber. This situationis reflected in the experimental results. For rc = 10 Åthe <011> spots in Au data are significantly stronger ascompared to Cu data. For this relative large rc the totalnumber of electrons in the chain becomes the dominantfactor of the observed intensity. This also explains thevery weak intensity of spots at the <121> and <111> di-rections for Cu data as compared to Au data. However,for rc = 2.65 Å the spots at the <011> directions have

5

comparable intensity in Au and Cu data. Hence for asmall value of rc the dominant contribution to the signalcomes from the nearest atoms. This demonstrates thatwhite-beam fluorescence holograms were recorded withsensitivity enabling to detect atoms in the local struc-ture which produce holographic oscillations at the levelof 10−4 of the total signal. The angular diameters ofvisible spots can be directly converted to spatial resolu-tion in the tangential plane. For example, at the [011]direction for a tangential plane at 2.65 Å the resolutionis 1.4 Å.

B. Reconstruction of electron density with linear

regression

According to Refs. 14 and 27, for a periodic sample,the hologram can be described as a linear superposition ofterms corresponding to different reciprocal space vectorsH:

χ(k) = −

8πreV

∑

H

FHχH(k), (5)

where V is unit cell volume and FH are the structurefactors. A generalization to non-centrosymmetric posi-tions is possible. Again, for a Lorentzian approximationof the effective energy spectrum, a compact form for theχH functions can be obtained. In such a case:14

χH(k) = (6)

H2 + 2k0H · k

W+

∆kH · k

AWln

[

4κ2+(H · k)2

H4

]

and

W = H4 + 4k0H2H · k+ 4κ2

+(H · k)2, (7)

where A = π + 2 arctan(2k0/∆k) andκ+ =

√

k20 + (∆k/2)2. A single intensity bands visible inthe experimental data corresponds to set of all colinearsreciprocal vectors H.

In Ref. 14 a quantitative tomographic algorithm wasproposed for imaging of 3D structure from white-beamholograms. This algorithm, which explicitly takes ad-vantage from the projection-like character of white-beamx-ray holograms is based on measuring of the signal am-plitude at the center of bands where the projection in-terpretation is straightforward. In this work, because ofthe small number of observed bands, a direct applicationof the tomographic algorithm resulted in low-resolutionimages. On the other hand, the softer energy spectrumallowed us to resolve individual χH in a single band (itwas not possible in previous synchrotron experiments).Therefore, the structure factors FH can be obtained bya linear regression procedure.18 Since both the value andsign of FH can be extracted, one is able to reconstructthe sample electron density by a simple Fourier sum:

ρ(r) =1

V

∑

H

FHeiH·r. (8)

FIG. 5. (Color online) Reconstruction of the electron densityfor recorded Au and Cu holograms. Fitted χH basis functionsfor Au (a) and Cu (b) data. (c) and (d) show 3D isosurfaceplots of the reconstructed electron density. (e) and (f) showthe reconstructed electron density at z = 0 and z = a/2 [c.f.Fig. 3(d)] for Au and Cu, respectively.

In the reconstruction procedure 32 basis functions in4 reciprocal space shells were fitted to the experimentaldata. The results of the fitting procedure are shown inFig. 5(a) and Fig. 5(b). The reconstructed holograms cal-culated by means of Eq. (5) and the determined structurefactors well describe the experimental holograms fromFigs. 3(a) and 3(b).

The retrieved structure factors FH were then usedto determine the electron density which is visualized inFig. 5(c-f). The isosurface plots are shown for two dif-ferent values of the reconstructed electron density. Fora better visualization two dimensional cross-sections atz = 0 and z = a/2 are shown.

The analysis of the Au data is straightforward. Thereconstruction shows the electron density ρ(r) inside theunit cell relative to the position of Au atom. The visible

6

strong maxima at the corners correspond to Au atomsand the weaker ones, located at the faces of the unit cell,to Cu atoms. For Cu data, the interpretation is a bitmore complex since there are three non equivalent posi-tions of Cu atoms inside the cell. Therefore data showρ relative to the Cu atoms and averaged over the threepositions. Consequently, the corners of the visualizedunit cells correspond to sites occupied with Cu atomswhile the faces to sites with an occupancy of 2

3Cu and

1

3Au. The nominal (noise limited) resolution of the data

is 1.7 Å.28 However, the precision of atomic position de-termination is an order of magnitude better.

There is a small discrepancy between the reconstructedand expected electron densities. In fact, similar differ-ences are present in the results obtained by means ofthe wavelet analysis. This is partially due to parasiticeffects of secondary fluorescence and total absorption.In future, these effects can be corrected using numeri-cal methods of x-ray fluorescence spectroscopy.23 In ad-dition, the wavelet and linear regression analysis can befurther improved by using a more exact approximationsof the effective spectra.

V. CONCLUSION

When a white x-ray beam illuminates a crystal thediffracted beams give rise to a Laue pattern as discov-ered 100 years ago. Nowadays, the Laue method is mostfrequently used for sample orientation. It analyses all theallowed diffraction beams, except the forward scattered

one (zeroth-order diffraction). In this work we showed,that by using a low power x-ray tube and by probing theinterference of forward scattered white-beam inside thesample one is able to obtain phase sensitive informationabout the structure and obtain element selectivity viafluorescence detection. On the one hand, recording ofwhite beam x-ray fluorescence holograms in laboratorycould be treated just as a curiosity. On the other hand,in the used experimental geometry the angular resolu-tion is mainly limited by the sample size. Therefore, theuse of x-ray sources with larger spots would not result inresolution deterioration. In addition, the presented ex-periment was performed without any x-ray optics neededfor collimation or monochromatization. This means thata standard 60 kV x-ray tube with spot size of 1 mm andpower of 3 kW could be used in experiments. This de-notes almost 100 times more photons on the sample. Byusing rotating anode x-ray tubes the gain in the photonflux could be even much larger. These numbers indicatepossibilities of real structural studies in samples contain-ing 1% of foreign atoms or in thin films or multilayers.

ACKNOWLEDGMENTS

This work was supported by the Polish National Sci-ence Center (Grant No. DEC-2011/01/B/ST3/00506).K.M.D and D.T.D acknowledge the supportfrom the Jagiellonian University’s DSC program(DSC/000692/2012).

∗ pawel.korecki@uj.edu.pl1 M. Tegze and G. Faigel, Nature 380, 49 (1996).2 D. Gabor, Nature 161, 777 (1948).3 J. J. Barton, Phys. Rev. Lett. 61, 1356 (1988).4 J. J. Barton, Phys. Rev. Lett. 67, 3106 (1991).5 T. Gog, P. M. Len, G. Materlik, D. Bahr, C. S. Fadley,

and C. Sanchez-Hanke, Phys. Rev. Lett. 76, 3132 (1996).6 K. M. Dąbrowski and P. Korecki, Nucl. Instrum. Methods

Phys. Res., Sect. B 285, 94 (2012).7 Y. Takahashi, K. Hayashi, and E. Matsubara, Powder

Diffr. 19, 77 (2004).8 M. Tegze, G. Faigel, S. Marchesini, M. Belakhovsky, and

A. I. Chumakov, Phys. Rev. Lett. 82, 4847 (1999).9 S. G. Bompadre, T. W. Petersen, and L. B. Sorensen, Phys.

Rev. Lett. 83, 2741 (1999).10 K. Hayashi, N. Happo, S. Hosokawa, W. Hu, and T. Mat-

sushita, J. Phys.: Condens. Matter 24, 093201 (2012).11 G. Faigel, G. Bortel, C. S. Fadley, A. S. Simionovici, and

M. Tegze, X-Ray Spectrom. 36, 3 (2007).12 P. Korecki and G. Materlik, Phys. Rev. Lett. 86, 2333

(2001).13 P. Korecki, M. Tolkiehn, and D. Novikov, Radiat. Phys.

Chem. 78, S34 (2009).14 P. Korecki, M. Tolkiehn, D. V. Novikov, G. Materlik, and

M. Szymoński, Phys. Rev. B 74, 184116 (2006).

15 P. Korecki, M. Tolkiehn, K. M. Dąbrowski, and D. V.Novikov, J Synchrotron Radiat. 18, 851 (2011).

16 D. T. Dul and P. Korecki, New J. Phys. 14, 113044 (2012).17 The Lorentz function is defined as NL(E) =

(A/2π) ∆E(E−E0)2+(∆E/2)2

where E0 is its mean and ∆E isthe FWHM. The Gumbel function is defined as NG(E) =(A/∆ε)exp [−exp (−(E − ε0)/∆ε)− (E − ε0)/∆ε] whereε0 is its mode and ∆ε determines the spread being∼ 2.57∆ε. In both functions A is the height constant.

18 F. N. Chukhovskii and A. M. Poliakov, Acta Crystallogr.,Sect. A: Found. Crystallogr. 59, 109 (2003).

19 B. Adams, D. V. Novikov, T. Hiort, G. Materlik, andE. Kossel, Phys. Rev. B 57, 7526 (1998).

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21 M. Markó, G. Krexner, J. Schefer, A. Szakál, and L. Cser,New J. Phys. 12, 063036 (2010).

22 S. Hosokawa, N. Happo, and K. Hayashi, Phys. Rev. B 80,134123 (2009).

23 B. Beckhoff, B. Kanngießer, N. Langhoff, R. Wedell, andH. Wolff, Handbook of Practical X-Ray Fluorescence Anal-

ysis (Springer, 2006).24 T. Schoonjans, A. Brunetti, B. Golosio, M. S. del Rio,

V. A. Solé, C. Ferrero, and L. Vincze, Spectrochim. Acta,Part B 66, 776 (2011).

7

25 P. Korecki, D. V. Novikov, and M. Tolkiehn, Phys. Rev. B80, 014119 (2009).

26 J.-P. Antoine, L. Demanet, L. Jacques, and P. Van-dergheynst, Appl. Comput. Harmon. Anal. 13, 177 (2002).

27 P. Korecki, M. Tolkiehn, D. V. Novikov, G. Materlik, andM. Szymoński, Phys. Rev. Lett. 96, 035502 (2006).

28 R. E. Stenkamp and L. H. Jensen, Acta CrystallographicaSection A 40, 251 (1984).