Quantitative powder diffraction using a (2+3) surface diffractometer and an area detector
Giuseppe Abbondanza, Alfred Larsson, Francesco Carlá, Edvin Lundgren, Gary S. Harlow
aa r X i v : . [ phy s i c s . a pp - ph ] F e b Quantitative powder diffraction using a (2+3) surface diffractometer andan area detector
Giuseppe Abbondanza , Alfred Larsson , Francesco Carl´a , Edvin Lundgren , and Gary S.Harlow Division of Synchrotron Radiation Research, Lund University, 221 00 Lund, Sweden NanoLund, Lund University, 211 00 Lund, Sweden Department of Chemistry, University of Copenhagen, 2100 Copenhagen, Denmark Diamond Light Source, Didcot OX11 0DE, United Kingdom * Corresponding authors: [email protected], [email protected]
Abstract
X-ray diffractometers primarily designed for surfacex-ray diffraction are often used to measure the diffrac-tion from powders, textured materials, and fiber-texture samples in so-called 2 θ scans. There are, how-ever, very few examples where the measured intensityis directly used, such as for Rietveld refinement, asis common with other powder diffraction data. Al-though the underlying physics is known, convertingthe data is time-consuming and the appropriate cor-rections are dispersed across several publications, of-ten not with powder diffraction in mind. In this pa-per we present the angle calculations and correctionfactors required to calculate meaningful intensitiesfor 2 θ scans with a (2+3)-type diffractometer andan area detector at the I07 beamline, Diamond LightSource. We also discuss some of the limitations withrespect to texture and instrumental resolution, andwhat kind of information one can hope to obtain. Since the early studies by Debye & Scherrer (1916),powder x-ray diffraction (PXRD) has become a well-established characterization technique. It has provedto be a fundamental tool for phase identification andstructure determination of materials. Quantitativeanalyses of PXRD data enables access to informa-tion such as size, strain and stress of the crystallites,the number of different phases in multi-phases ma-terials, atomic and unit lattice parameters. PXRDdata quality improves significantly when synchrotronx-ray beams are employed, which provide a: highphoton flux, enhanced collimation, tunable energies,and a superior angular resolution. In synchrotronsurface x-ray diffraction (SXRD), an experimental
Spin the specimen during the measurementRietveld refinement Use equation (5) for angle calculations.Correct intensity as in equation (36).Using equations (39) and (40). E x pe r i m en t a l D a t a t r ea t m en t A na l ys i s Peak profile analysis(IRF, size and strain)More crystallites will satisfy the Bragg condition.A good powder will show uniform diffraction rings.
Figure 1: Flowchart outlining the steps and the rel-evant equations involved in the investigation of pow-dered samples from the experiment to the determi-nation of the structure.setup composed of a (2+3) diffractometer, an areadetector is often used. The (2+3) diffractometer waspresented by Vlieg (1997) and its combination withan area detector was explored by Schlep¨utz et al. (2005). Although this setup was originally designedfor SXRD, it can be used in PXRD and in grazing-1ncidence x-ray diffraction (GIXRD) by rotating thedetector about the diffractometer center in longi-tudinal and equatorial 2 θ scans, across the Debye-Scherrer cones. In this scenario, the major benefit ofthe (2+3) diffractometer is the control over the ori-entation of the sample and of the detector (and thusof the scattering vector) which enables convenient in-vestigations of specimen textures and preferred ori-entations.The phenomena involved in PXRD are well-knownand have been extensively studied in the last century.A significant step forward in the analysis of PXRDdata was the Rietveld method (Rietveld, 1966; Ri-etveld, 1967; Young, 1993). Such refinement is usu-ally performed on PXRD patterns where the inten-sity is plotted as a function of 2 θ . Therefore, with the(2+3) setup at the I07 beamline (Nicklin et al. , 2016),it is necessary to integrate the two-dimensional datacollected by the area detector into a one-dimensionalpattern. Furthermore, a series of intensity correc-tions should be applied to the measured intensities,to obtain the structure factors that depend on theunderlying crystallography of the sample.In this work, we present the calculations and thecorrection factors needed to extract quantitative in-formation from 2 θ scans with (2+3) diffractometersand area detectors. The calculations are part ofa process intended for the characterization of crys-talline materials, as illustrated by the flowchart inFig. 1. We assessed the validity of the calculationsand of the correction factors by refining a LaB refer-ence sample. Furthermore, we calculated the instru-mental resolution function (IRF) of the setup andcompared the integrated intensities collected at dif-ferent experimental geometries, namely capillary intransmission, grazing-incidence and Bragg-Brentano. The experimental work was conducted at the beam-line I07, a hard x-ray (8-30 keV) high resolutiondiffraction beamline at Diamond Light Source, UK(Nicklin et al. , 2016). The x-ray beam had an en-ergy of 20 keV and a size of 100 µ m vertically and200 µ m horizontally. To record the powder diffrac-tion intensities, a Huber (2+3) diffractometer (Vlieg,1998) and a Pilatus 100K detector were used.A schematic diagram of the Huber (2+3) diffrac-tometer is shown in Fig. 2. The coordinate frame ofreference depends on whether the sample geometryis horizontal (blue) or vertical (red). In both modesof operation the sample is mounted on a hexapod(Micos), which allows scanning of the sample trans-lations and rotations for the initial alignment. The z xy Horizontal geometry Vertical geometryz xy (cid:1) v (cid:0) h φ h φ v ɣ δν Figure 2: Schematic of a (2+3) diffractometer. Thearrows point towards the positive direction of rota-tion in the case of horizontal (blue) or vertical (red)geometry and for the detector (black).grazing-incidence angle of the synchrotron beam ontothe sample surface and the azimuth are given by therotation of α h and φ h in the horizontal geometry, andby α v and φ v in the vertical geometry.The three detector circles are fully independent ofthe two sample circles and allow for radial scans inthe horizontal plane ( γ ) or out of the horizontal plane( δ ), and rotation of the detector around its surfacenormal ( ν ).The Pilatus 100K area detector consists of an arraymodule of 487 ×
195 pixels with size 172 µ m × µ m, resulting in an active area of 83.8 × (Kraft et al. , 2009). In this section we present the coordinate transforma-tions to a assign a diffraction angle 2 θ and an az-imuthal angle χ to every pixel of an area detector,given the distance R of the detector from the centerof the diffractometer and the nominal detector angles γ and δ . The angles and the wavevectors involved inthe calculations are schematically shown in Fig. 3.The following derivation is based on the reciprocalspace coordinate calculations given by Schlep¨utz etal. (2011).These calculations are valid for the horizontal ge-ometry and those for the vertical geometry are givenin §3.2. In both the geometries, the laboratory frameof reference has its zero-coordinates on the diffrac-tometer center and the y-axis points in the directionof the synchrotron beam. Therefore, the wavevectorof the incoming x-ray beam is2 = k , (1)where k = 2 π/λ is the magnitude of the wavevectorand λ is the wavelength of the x-ray beam. Similarly,a generic scattered wavevector such as the one in Fig.3 (a) can be represented as k ′ = k cos δ sin γ cos δ cos γ sin δ . (2)In the assumption that the scattering is elastic (i.e., | k | = | k ′ | = k ), the angle 2 θ can be found using thedefinition of scalar product as follows: k · k ′ = | k || k ′ | cos 2 θ, (3) k · k ′ = k cos δ cos γ, (4)2 θ = arccos(cos δ cos γ ) . (5)The azimuthal χ angle can be calculated as χ = arctan( tan δ tan γ ) . (6)Therefore, when δ and γ are known for each pixelof the detector, it is possible to assign a 2 θ and a χ value to each pixel. To this aim, we need to calculatethe exact position of the detector center.We define the detector center as the point that isimpinged by the direct beam when the detector is inthe zero position (i.e., when γ = δ = ν = 0, as shownin figure 2). In this position, the coordinates of thedetector center are given by calculating the “centerof mass” of an image recorded while the direct beamimpinges the detector: c = 1 S X i,j I ( i, j ) p ( i, j ) , (7)where ( i, j ) are the detector pixel coordinates, c isthe vector pointing to the mass center, S is the sumof all intensities detected in the image, I ( i, j ) and p ( i, j ) are the intensity and the vector describing theposition of the ( i, j ) pixel.Let us denote the coordinates of the detector cen-ter found in equation (7) with ( c x , c z ). A pixel withcoordinates ( i, j ) has an offset (∆ x, ∆ z ) from the de-tector center given by:∆ x = ( c x − i ) w x , (8)∆ z = ( c z − j ) w z , (9) Δ x Δ z dR z yx Δ r ( cx, cz ) (i, j ) (0, 0)j i k k' γ2θ ab δ Figure 3: (a) Schematic representation of the angles γ , δ , 2 θ and χ , subtended by the incoming beamwavevector k and a generic scattering wavevector k’ and (b) diagram diagram showing the offsets ∆ x and∆ z from the detector center of a generic ( i, j ) pixel.The vector d (i.e., the distance of the pixel from thediffractometer center) subtends the angles δ p and γ p .where w x and w z are the width of the pixel alongthe x and z directions, respectively (for a Pilatus100K detector w x = w z = 172 µm ). Fig. 3 (b) illus-trates the typical offsets in pixel position describedabove. Here we assign the (0,0) coordinates to theupper left detector pixel. When the detector is atthe zero position, the position of a generic ( i, j ) pixelin the laboratory coordinates is simply x p y p z p = ∆ xR ∆ z . (10)For non-zero detector angles, the new ( x p , y p , z p )coordinates of the ( i, j ) pixel are found by rotatingthe vector (∆ x, R, ∆ z ) by γ , ν and δ around the z, yand x axes, respectively. Therefore, the pixel positionin the laboratory frame of reference becomes x p y p z p = Γ × N × ∆ × ∆ xR ∆ z , (11)where Γ, N and ∆ are the matrices describing therotations around the z, y and x axes, defined as fol-3ows: Γ = R z ( γ ) = cos γ − sin γ γ cos γ
00 0 1 , (12) N = R y ( ν ) = cos ν ν − sin ν ν , (13)∆ = R x ( δ ) = δ − sin δ δ cos δ . (14)Note that the result can be affected by the order ofmultiplication since the matrix product is not com-mutative. We followed the convention where the or-der of the rotation is R z , R y and R x .The angle values for a generic pixel γ p and δ p canbe calculated as follows: γ p = arctan( z p y p ) , (15) δ p = arcsin( x p d ) , (16)where d is the distance of the pixel from the centerof the diffractometer given by d = q ∆ x + R + ∆ z . (17)To demonstrate the use of the calculations above,equations (15), (16), (5) and (6) were used to producethe angle maps in Fig. 4 for the case when γ =30°and δ =20°, assuming a detector distance of 897 mm,which is the distance at which the data were collectedin this work. Apertures and slits are used in PXRD to control thebeam size, divergence and angular resolution. High-density metals are often used in the fabrication ofslits due to their low x-ray transmission. When usingguard slits, the effective angles ( γ p , δ p ) are defined bythe aperture of the slits, as shown in Fig. 5. The ν rotation axis is always perpendicular to the apertureplane for every nominal detector angle ( γ , δ ), i.e.the slit aperture rotates with the detector. In thisassumption, the coordinates of the aperture in thelaboratory coordinate system are given by x s y s z s = Γ × ∆ × R s , (18) Figure 4: 2D plots of the angle values assigned toeach pixel of a Pilatus 100K detector positioned at γ =30°, δ =20° and R=897 mm. The plots show γ p (a), δ p (b), 2 θ (c) and azimuthal χ p (d).where R s is the slit distance from the diffractome-ter center. The effective detector angles are now de-pendent on the distance between the pixel and theslit and are given by d d,s = [( x d − x s ) + ( y d − y s ) + ( z d − z s ) ] , (19) γ p = arctan( z p − z s y p − y s ) , (20) δ p = arcsin( x p − x s d d,s ) . (21) In the vertical geometry the surface normal of thesample and the y-axis lay in the horizontal plane,while the x-axis points upwards. The equations forthe incoming beam wavevector and those for the off-set from the detector center are the same as in the4 z x (xp, yp, zp)RRs(xs, ys, zs)
Figure 5: Diagram showing how the detector anglesare affected by the use of guard slits.horizontal geometry, i.e., equations (1), (8) and (9),while the generic scattering vector is given by k ′ = k sin δ cos δ cos γ cos δ sin γ . (22)Since the y-component has not changed, the equa-tion for the 2 θ angle is the same as in equation (5).The equation for the χ angle (6) is also the same inboth geometries. The rotation matrices, however, aredifferent in this frame of reference and are given by γ = R x ( γ ) = γ − sin γ γ cos γ (23) N = R y ( ν ) = cos ν ν − sin ν ν (24)∆ = R z ( δ ) = cos δ sin δ − sin δ cos δ
00 0 1 (25)As in the case of horizontal geometry, equation (11)applies to the calculation of pixel coordinates.
The intensity of a powder diffraction pattern can beexpressed as I (2 θ ) ∝ Φ M hkl | F hkl (2 θ ) | P ( γ, δ ) L (2 θ ) V (26)where Φ is the incident photon flux, M hkl is themultiplicity (i.e., the number of symmetry-equivalentreflections contributing to a single peak), F hkl is thestructure factor, P (2 θ ) is the polarization factor, L (2 θ ) is the Lorentz factor and V is the sample vol-ume from which the diffracted intensity arises. De-viations from this are often due to the presence of apreferred orientation or crystal texture. The x-ray beam produced by the I07 undulator hasa strong horizontal polarization (Nicklin et al. , 2016)and therefore the scans in the horizontal plane (i.e., γ -scans) are more affected by this factor than thescans out of the horizontal plane (i.e., δ -scans). Forthis reason it is more convenient to express the po-larization factor P as a function of ( γ , δ ) using thefollowing expression that has been presented by sev-eral authors (Vlieg, 1997; Schlep¨utz et al. , 2005) P ( γ, δ ) = p h (1 − cos δ sin γ ) + (1 − p h )(1 − sin δ )(27)The polar plot in Fig. 6 (a) illustrates the variationof the polarization factor during an in-plane ( γ ) scan.Conversely, it will be less significant for an out-ofplane ( δ ) scan, where the polarization factor will befairly constant. A detailed derivation of the Lorentz factor was givenby Buerger (1940), where it is defined to be pro-portional to the time that a reflection stays in theBragg condition. The Lorentz factor depends on thetype of experiment performed and especially on thescanning variable employed to detect reciprocal space(Vlieg, 1997). For instance, in single-crystal diffrac-tion, the integrated intensity is measured by rotatingthe sample over the entire width of a reflection whilethe detector position is fixed, in a so-called rockingscan or Φ-scan. In that case, the measured intensi-ties need to be corrected by a geometrical factor, i.e.the Lorentz factor, that expresses the relative timespent by each point in reciprocal space in the reflect-ing position during the Φ-scan. In PXRD, instead,the detector is rotated while the sample position isstationary, thus there is virtually no access to recip-rocal space volume. However, due to the randomorientation of the crystallites, a powder can be seenas a single crystal that is rotated along the Φ-axisand an axis orthogonal to the Φ rotation.In PXRD, the Lorentz factor L is the productof three terms. The first term is the “Darwinian”single-crystal part (Darwin, 1922), that accounts forchanges in integration volume as a function of 2 θ : L = 1 / sin 2 θ. (28)5 Figure 6: Intensity correction factors calculated ata distance of 897 mm from the diffractometer cen-ter. Polar contours of polarization factor (a) andLorentz factor (b), as a function of the diffractionand the azimuthal angles. Flat detector correctioncalculated for a Pilatus 100K with center pixel coor-dinates ( c i , c j ) = (246 , θ values. Ifwe express the radius of the base of a generic Debye-Scherrer cone as 2 πk sin 2 θ , the fraction recorded bythe detector is k ∆ χ/ πk sin 2 θ and therefore the sec-ond term will be proportional to L = ∆ χ/ sin 2 θ, (29)where ∆ χ is the range of azimuthal χ values ac-cessible by the detector at a given γ and δ and itcan be considered constant (Als-Nielsen & McMor-row, 2011). This effect is evident in Fig. 7, where thediffraction intensity from a borosilicate capillary con-taining NIST LaB SRM 660c is plotted as a functionof 2 θ and χ . Here, the range of accessible χ valuesdecreases significantly with the increase of 2 θ . The third term is proportional to the number of ob-servable lattice points at the same time, and thereforeto the circumference of the base-circle of the Debye-Scherrer cones. If we denote a particular recipro-cal lattice vector with G hkl , this circumference is G hkl sin ( π − θ )= G hkl cos θ . In the assumption thatthe crystallites and thus their reciprocal lattice pointsare homogeneously distributed on the Ewald sphere,this term is proportional to L = cos θ. (30)Note that L is not proportional to the possiblepermutations of (h,k,l) since this is already accountedfor by the multiplicity factor M hkl in equation (26).The Lorentz factor is given by the product L L L as in equation (31), which is rearranged in equation(32): L (2 θ ) = 1sin 2 θ θ cos θ (31) L (2 θ ) = 1sin θ sin 2 θ (32)As shown in Fig. 6 (b), the Lorentz factor has arather quick decay as a function of 2 θ , regardless ofthe azimuthal angle χ . It is perhaps the most signif-icant intensity correction that needs to be applied toexperimental data, especially at small 2 θ . In addition to the polarization and the Lorentz factor,the decrease of the subtended solid angle for the dif-ferent pixels has to be taken into account. Two con-tributions normally describe this change. The firstone is due to the fact that pixels away from the de-tector center are also further away from the diffrac-tometer center, as depicted in Fig. 8. To accountfor this change in distance the measured intensitiesshould be multiplied by C d = d R . (33) C d is greater than unity for every pixel of the de-tector except the detector center C, where d = R .The second contribution is due to the non-normalincidence of the scattered beam owing to the flat de-tector surface and is given by Schlep¨utz et al. (2011) C i = 1cos (arctan ∆ r/R ) , (34)where ∆ r = √ ∆ x + ∆ z .For a better visualization of these correction fac-tors, the product C d × C i is plotted in Fig. 6 (c)6igure 7: Diffracted intensity as a function of 2 θ and χ originating from a NIST LaB SRM 660c encapsulatedin a 1 mm diameter borosilicate capillary. The data was collected in the Debye-Scherrer geometry by scanningthe detector in the horizontal plane ( γ -scan).for the same detector distance R. The correction be-comes more significant close to the edges of the de-tector image, where it shows a maximum change ofabout 0.35 % from the detector center. The signifi-cance of this correction increases as the detector dis-tance decreases or as the detector size increases. CP Figure 8: The flat surface of the Pilatus 100K (bluerectangle) intersects a sphere centred in the diffrac-tometer (orange frame) in the point C (the detectorcentre). A generic diffracted beam would impingethe point C normally and subtend a non-normal an-gle with any other generic point P.
The sample volume that gives rise to diffracted in-tensity depends on the geometry of the experimentand is often a function of the scattering angle 2 θ , theincidence angle α and the angle between the samplesurface and the diffracted beam γ = 2 θ − α . Thediffracting specimen volume is often modelled as anexponential aberration. Table 1 summarizes this cor-rection for some common experimental geometries.As a general remark, the interaction volume is pro- portional to the area of illuminated sample, A , andis inversely proportional to the linear absorption co-efficient, µ . The corrected intensity is given by I corr = I obs C d C i LP V , (35)where I obs is the observed intensity. Analysis of the line-profile shape is useful for the de-termination of crystallite size and strain of a spec-imen. However, the width of a diffraction peak isa function of 2 θ and it depends not only on speci-men properties but also on experimental conditionsand sample size. Furthermore, instrumental featuressuch as monochromators, collimating and refocusingoptics, slits, beam divergence and energy bandwidthinfluence the broadening of diffraction peaks (Gozzo et al. , 2006).As the mentioned causes of broadening have ei-ther Gaussian or Lorentzian nature, the most usedfitting function for PXRD peaks are based on Voigtor pseudo-Voigt line shapes. The full width at halfmaximum (FWHM) of the Voigt profile depends onthe widths of the associated Gaussian and Lorentziancomponents Γ G and Γ L (Thompson, Cox, et al. , 1987;Thompson, Reilly, et al. , 1987). The Gaussian widths contain information on the in-strumental resolution function (IRF) and on the sam-7able 1: Volume corrections for some common experimental geometries.Geometry Equation Note ReferencesFlat-platesymmetric(Bragg-Brentano) V SR = A µ - (Cheary et al. , 2004)(Egami & Billinge, 2003)Flat-plateasymmetric(grazing incidence) V AR = Aµ (1 + sinαsinγ ) − - (Toraya et al. , 1993)(James, 1967)Capillary intransmission(Debye-Scherrer) A ( θ ) = A L cos ( θ ) + A B sin ( θ ) A L = 2 I ( z ) − L ( z ) − I ( z ) − L ( z ) z A B = [ I (2 z ) − L (2 z )] /z z = 2 µrI ν = ν th-ordermodified Besselfunction L ν = ν th-ordermodified Struvefunction (Dwiggins, 1972)(Sabine et al. , 1998)ple strain. An analytical description of the IRF wasgiven by Caglioti et al. (1958):Γ r = ( U tan θ + V tan θ + W ) / . (36)Specimen contributions to the Gaussian widths arethe expression of crystal defects, dislocations and de-formation of the unit cells, in what is known as inho-mogeneous strain broadening:Γ s = 4 ǫ tan θ, (37)where the root mean square strain ǫ is a coefficientthat depends on the elastic compliance and the me-chanical properties of the specimen. Since this contri-bution is proportional to tan θ , some authors mergedthe strain contribution into equation (36). For exam-ple, Thompson, Cox, et al. (1987) enclosed the coef-ficient ǫ into the constant V, while Wu et al. (1998)into the constant U. In this work we adopt the so-lution presented by Thompson, Reilly, et al. (1987),where the Gaussian width broadening is expressed asΓ G = (Γ r + Γ s ) / . (38)The Lorentzian widths Γ L take into account thespectral bandwidth of the source and the samplecrystallite size through the following equation (Cox,1991): Γ L = X tan θ + Y / cos θ. (39)The X coefficient depends on the monochromatingoptics and it is in the order of magnitude of 10 − for most of the synchrotron beamlines where Si(111)crystals are employed. The dependence of the band-width term on tan θ can be derived by differentiat-ing Bragg’s law. The second term in equation (39)is the Scherrer crystallite size contribution. Here, Y = Kλ/D where K is a dimensionless shape fac-tor (generally close to unity), λ is the wavelength ofthe x-ray radiation, and D is the crystallite size. Oneshould remember that the use of equation (39) relatesto the size of the coherent diffraction domains ratherthan the size of the crystallites per se (Scherrer, 1912;Patterson, 1939; Hargreaves, 2016).Equation (39) can be rearranged asΓ L cos θ = X sin θ + Y. (40)Plotting Γ L cos θ against sin θ in equation (40)would produce a line where the intercept dependson the crystallite size while the slope depends on thebeam spectral bandwidth. Such a plot is known as aWilliamson-Hall plot (Williamson & Hall, 1953) andan example of such analysis is given in §6.1. The beam footprint on the sample causes a broad-ening that affects the instrumental resolution. Thisis especially true for grazing-incidence geometries,where the beam spills over the sample and illumi-nates it over its whole length. A depiction of thisgeometric effect is shown in Fig. 9. Assuming thatevery volume element illuminated by the beam scat-ters, all intensity occurring at the diffraction angle2 θ is spread out into a radial range β on the detec-tor. The angular spread for in-plane scans (Fig. 9 (a)and for out-of-plane scans (Fig. 9 (b) are respectivelygiven by β i = 2 arctan( w sin 2 θ R ) , (41) β o = 2 arctan( w sin(2 θ − α )2 R ) , (42)8here w is the sample width and R is the sample-to-detector distance.Fig. 9 (c) shows a simulated intensity profile of anAu(311) peak at 20 keV (2 θ = 29.1996°), for differentsample widths, assuming a scan out-of-plane and agrazing-incidence angle of 0.1°. The simulated peakswere calculated as Voigt profiles where the Lorentzianand Gaussian components were determined by theIRF of the I07 beamline at Diamond Light Source(see §6.1). The β o broadening was calculated usingequation 42 and summed to the Gaussian componentof the Voigt profile as follows:Γ G = (Γ r + β o ) / . (43)A more insightful way to account for the beamfootprint is to determine the peak profile change in-duced by specimen absorption. Equations (41) and(42) work on the assumption that the intensity of theincoming beam does not change significantly throughthe whole sample width w . This is not always truesince the intensity decays exponentially as describedby Beer-Lambert’s law (Swinehart, 1962). Therefore,a change in diffraction peak profile due to absorptioncan be modelled as an exponential function. Such ef-fects are well-described for several diffraction geome-tries in the work of Rowles & Buckley (2017). Oncethe transmission profile function has been modelled,it can be convoluted with a Voigt or pseudo-Voigtprofile in a refining algorithm. This approach notonly works for correcting the beam footprint effect,but also accounts for possible peak asymmetries. Rietveld refinement is a well-established analysismethod for PXRD data and it is widely employedin the characterisation of polycrystalline materials(Rietveld, 1966; Rietveld, 1967; Young, 1993). Thismethod consists of fitting the experimental data witha calculated intensity profile which is based on thestructural parameters of the material. A non-linearleast-squares algorithm (or other optimization strat-egy) finds the parameters for a theoretical profile thatbest matches the experimental intensities. The Ri-etveld method can be used to find unit cell param-eters, phase quantities, crystallite size and strain,atomic coordinates and texture. Furthermore, itis possible to model texture and preferred orienta-tions for example using spherical harmonics (Whit-field, 2009), although this is outside the scope of thepresent article.The quality of the data and having a good start-ing model are what mainly determines the success ofthe refinement. For a good PXRD measurement thex-ray beam size should be comparatively larger than (cid:2) i θ θ yxw θ θɑβ o (cid:3) z ab R Figure 9: Schematic depiction of the geometricbroadening due to the beam footprint at grazing-incidence: top view of a γ -scan (a) and side viewof a δ -scan(b). Simulation of an Au(311) peak at 20keV, at a grazing angle of 0.1°, scanned out-of-plane,for increasing sample widths (c).the crystallite size. In this way, the statistical signifi-cance of the detected intensity is maximised due to alarger interaction volume, which leads to the forma-tion of homogeneously continuous diffraction rings.9n common PXRD setups the sample is spun to fa-cilitate the measurement of uniform rings and this isalso possible with (2+3)-type diffractometers by ro-tating the φ h in the horizontal geometry or φ v in thevertical geometry.To perform Rietveld analysis, software like Full-Prof (Rodriguez-Carvajal, 1993), GSAS-II (Toby &Von Dreele, 2013) and DIFFRAC. SUITE TOPAS(Coelho et al. , 2011) has been developed. In oursomewhat unconventional case, where the data hasbeen manually corrected by Lorentz and polarizationfactors, e.g. by using equation (35), it is possibleto disable FullProf from applying further instrumen-tal corrections by selecting Lorentz Polarization notperformed as the diffraction geometry. However, wecould not determine how to disable the instrumen-tal corrections in GSAS-II without modifying thePython code making such corrections. We did nottest if this was possible in TOPAS. reference The LaB powder is a standard reference materialcommonly used in powder diffraction for the cali-bration of diffraction line positions and shapes. Asample of this powder was encapsulated in a borosil-icate capillary of 0.5 mm radius, mounted vertically(i.e., with the capillary axis parallel to the z-axis)and measured in a series of γ radial scans in a rangeof 5° to 60°. In order to measure continuous powderrings, the capillary was rotated by 0.5° along the z-axis (i.e., increasing φ h in Fig. 2) between each scan.A total of 135 γ -scans were combined, each one at adifferent φ h .The angle calculations presented in §3 were usedto plot intensity against 2 θ in the 1-dimensional pat-tern shown in Fig. 10 and to correct the data byLorentz-polarization factor, flat detector effects andinteraction volume from a capillary in transmissiongeometry (see Table 1). Therefore the corrected dataare proportional to the squared module of the struc-ture factors and the multiplicity of each diffractionpeak. Since the LaB standard has very well-definedunit cell parameters, the position of the diffractionpeaks were used to calibrate the sample-to-detectordistance and the nominal position of the γ and δ mo-tors.The good agreement of the calculated model to theobserved data is indicated by fairly low residuals (seeFig. 10). This shows that the setup and data pro-cessing employed in this work can be used not only toinvestigate lattice parameters and phases, but also torecord meaningful intensities proportional to struc-ture factors and symmetry multiplicity. Deviationsfrom this proportionality are a clear sign of texture and preferred orientation. The pattern in Fig. 10 was used to determine the IRFof the I07 beamline at Diamond Light Source. Fur-thermore, a different LaB powder sample (Sigma-Aldrich, grain size 10 µ m), prepared by spin coatingon a Si substrate, were measured in grazing-incidenceand in Bragg-Brentano geometry. As discussed in §5.1, the peak widths contain in-strumental as well as specimen-related information.Although software like FullProf and GSAS-II havebuilt-in options to refine such parameters, a preciseknowledge of the IRF is required to obtain signif-icant information. In this work, we processed thedata from the NIST LaB standard described in theprevious section to calculate the IRF.All the peaks in Fig. 10 were fitted one by onewith a Voigt profile. The FWHM reported in Fig.11 (a) shows a broadening that has both Gaussianand Lorentzian contributions. The Gaussian andLorentzian FWHMs of the peak widths were ex-tracted and reported in Fig. 11 in the form of aCaglioti function (a) and a Williamson-Hall plot (b).The dominance of Gaussian component in the broad-ening can be explained by the large average crystal-lite size i.e., above 1 µ m as certified by NIST. Dueto such a large crystallite size, the Scherrer contribu-tion to the Lorentzian broadening is negligible andoutside the limits imposed by coherent scattering do-mains (Miranda & Sasaki, 2018). Furthermore, sincethe unit cell of LaB is not known to show inhomo-geneous strain features, we can assume that Γ s inequation (38) is zero and that the Gaussian broad-ening in Fig. 11 (b) is only due to instrumental con-tributions. Therefore, fitting the Gaussian widthswith the Caglioti function will produce a triplet ofU, V and W values which describe the IRF. It shouldbe noted that the result of the fitting is affected bythe choice of the units expressing the angle. In thiswork, the Gaussian FWHMs were expressed in de-grees. Furthermore, the Williamson-Hall plot in Fig.11 (a), based on equation (40), provides the X andY values which are characteristics of the x-ray beambandwidth and the specimen grain size, respectively.All these fitting parameters are reported in Table 2with their respective standard errors.10igure 10: Rietveld refinement of a NIST LaB SRM 660c PXRD pattern, measured with a 20 keV x-raybeam, performed with Fullprof using the pseudo-Voigt line shape. Before the refinement, the data wascorrected by: Lorentz and polarization factors, flat-detector corrections and interaction volume of x-rayswith a capillary. The background was subtracted by fitting a Chebychev polynomial (16 coefficients).Table 2: Fitting parameters of the Caglioti function(U, V, W) and of the Williamson-Hall plot (X, Y),with their respective standard errors.Value Standard errorU 2 . × − . × − V 1 . × − . × − W 5 . × − . × − X 7 . × − . × − Y 1 . × − . × − A sample of LaB spin-coated on a Si(100) sub-strate was measured in Bragg-Brentano and grazing-incidence geometries, using a beam energy of 20 keV.The sample was prepared by mixing 500 mg of a LaB powder (Sigma-Aldrich, nominal grain size = 10 µm )with 10 mg of ethyl cellulose (Sigma-Aldrich, 48.0-49.5 % (w/w) ethoxyl basis) as a binding agent, in 2mL of ethanol. The mixture was spin-coated on a 6mm square of Si(100) at 800 rpm. Fig. 12 shows theLaB (211) peak a for the different geometries em-ployed in this work, namely Debye-Scherrer (takenfrom the capillary data in §5.3), Bragg-Brentano andGIXRD at fixed grazing-incidence of 0.1° and 0.2°.The plots exhibit a shift in the peak position fromthe theoretical diffraction angle, 2 θ calc , which can beexplained by refraction of light in the LaB film cov-ering the Si substrate. For small incidence angles anda thin film with material constant δ , Lim et al. (1987) Figure 11: Peak broadening analysis of a LaB stan-dard reference material. The total FWHMs are plot-ted together with the Gaussian FWHMs, which arefitted by a Caglioti function [see equation (36)] in (a),while the Lorentzian FWHMs, multiplied by cos θ ,are fitted by a Williamson-Hall plot in (b).11able 3: Experimental and theoretical ∆2 θ due torefraction in a LaB film.Incidence angle α =0.2° α =0.1°Experimental shift (°) 0.03841 0.07142Theoretical shift (°) 0.03488 0.06909Relative error (%) 9.19 3.27modelled this shift as follows:∆2 θ = δ sin 2 θ calc × (2 + sin α sin 2 θ calc + sin 2 θ calc sin α ) . (44)The positions of the peaks in Fig. 12 for α h =0.1°and 0.2° were calculated in terms of center of massand used to calculate the experimental peak shift bysubtracting 2 θ calc =21.0467° for LaB (211). The ex-perimental shift is reported against the theoreticalshift calculated using equation (44) in Table 3. Thediscrepancy between the two set of values can be ex-plained by density inhomogeneities in the LaB film,which also contains ethyl cellulose for the practicalpurpose of consolidating the film. Furthermore, theLaB used for this layered sample is not a standardand it could have a slightly varied lattice parameterfrom what reported in literature after the dissolutionin ethanol. Although this kind of sample is not rec-ommended for calibration of XRD setups, it couldbe used as a reference sample to determine instru-mental broadening at grazing-incidence. In order tocalibrate the detector distance and the motor posi-tions accurately, a well-calibrated standard is alwaysrecommended. The angle calculations presented in this work describehow to convert data measured by a (2+3) diffrac-tometer equipped with an area detector into a 1-dimensional XRD pattern, more familiar to the chem-istry and material science communities. We also col-lected some dispersed knowledge on the phenomenacontributing to the peak widths and on the intensitycorrections. Since the calculations and the correc-tions are energy-independent, they can be applied todata collected at beamlines operating with hard x-rays (8-30 keV) as well as with high energies (40-150keV). The Python code used in this work for such cor-rections and angle calculations is available on Github( github.com/giuseppe-abbondanza/pyLjus ).Quantitative PXRD using surface diffractometersis not often done. This work should however facilitate Figure 12: LaB (211) peak for different experi-mental geometries: Debye-Scherrer, Bragg-Brentano,GIXRD out-of-plane at α =0.1° and 0.2°. The shiftdecreases with an increasing incidence angle and canbe explained by refraction inside the LaB film.quantitative studies using such instruments, includ-ing for example in situ studies under grazing inci-dence conditions.Furthermore, the calculations presented in thiswork can be modified to apply to diffractometers withother geometries (e.g., (2+2), 4-circle and 6-circle)and they can even apply to setups where the detec-tor is not mounted on a motor and therefore has afixed position, facing the direct beam. In this config-uration, one can assume that effectively γ = δ =0°. Acknowledgements
Measurements were performed on the I07 beamlineat the Diamond Light Source. This work was fi-nancially supported by the Swedish Research Councilthrough the R¨ontgen-˚Angstr¨om Cluster “In-situ HighEnergy X-ray Diffraction from Electrochemical Inter-faces (HEXCHEM)” (Project no. 2015-06092) andproject grant ”Understanding and Functionalizationof Nano Porous Anodic Oxides” (project no. 2018-03434) by the Swedish Research Council. G. Abbon-danza acknowledges financial support from NanoL-und under grant p05-2017. We wish to thank R. Fe-lici for valuable discussions.
References
Als-Nielsen, Jens & Des McMorrow (2011).
Elementsof modern X-ray physics . John Wiley & Sons.12uerger, M. J. (Nov. 1940). eng.
Proceedings of theNational Academy of Sciences of the United Statesof America , pp. 637–642.Caglioti, G., A. Paoletti, & F.P. Ricci (1958). NuclearInstruments , pp. 223–228.Cheary, R. W., A. A. Coelho, & J. P. Cline (Feb.2004). eng. Journal of research of the National In-stitute of Standards and Technology , pp. 1–25.Coelho, A. A. et al. (2011).
Powder Diffraction ,S22–S25.Cox, D.E. (1991). “Powder diffraction”. Handbook onSynchrotron Radiation . Ed. by G.S. Brown & D.D.Moncton. North-Holland. Chap. 5.Darwin, C.G. (1922).
The London, Edinburgh, andDublin Philosophical Magazine and Journal of Sci-ence , pp. 800–829.Debye, P & P Scherrer (1916). Nachr. Kgl. Ges. Wiss.G¨ottingen, Math. Physik. Kl.
Dwiggins Jnr, C. W. (Mar. 1972).
Acta Crystallo-graphica Section A , pp. 219–220.Egami, Takeshi & Simon JL Billinge (2003). Under-neath the Bragg peaks: structural analysis of com-plex materials . Elsevier.Gozzo, Fabia et al. (June 2006).
Journal of AppliedCrystallography , pp. 347–357.Hargreaves, J. S. J. (2016). Catalysis, Structure &Reactivity , pp. 33–37.James, R.W. (1967). The Optical Principles of theDiffraction of X-Rays . G. Bell and Sons Ltd., Lon-don.Kraft, P. et al. (2009).
IEEE Transactions on NuclearScience , pp. 758–764.Lim, G. et al. (July 1987). Journal of Materials Re-search , pp. 471–477.Miranda, M. A. R. & J. M. Sasaki (Jan. 2018). ActaCrystallographica Section A , pp. 54–65.Nicklin, Chris et al. (Sept. 2016). Journal of Syn-chrotron Radiation , pp. 1245–1253.Patterson, A. L. (Nov. 1939). Phys. Rev. (10),pp. 978–982.Rietveld, H. M. (Apr. 1966). Acta Crystallographica , pp. 508–513.– (Jan. 1967). Acta Crystallographica , pp. 151–152.Rodriguez-Carvajal, Juan (1993). Physica B ,pp. 55–69.Rowles, Matthew R. & Craig E. Buckley (Feb. 2017).
Journal of Applied Crystallography , pp. 240–251.Sabine, T. M. et al. (Feb. 1998). Journal of AppliedCrystallography , pp. 47–51.Scherrer, P. (1912). “Bestimmung der inneren Struk-tur und der Gr¨oße von Kolloidteilchen mittelsR¨ontgenstrahlen”. Kolloidchemie Ein Lehrbuch . Berlin, Heidelberg: Springer Berlin Heidelberg,pp. 387–409.Schlep¨utz, C. M. et al. (July 2005).
Acta Crystallo-graphica Section A , pp. 418–425.Schlep¨utz, C. M. et al. (Feb. 2011). Journal of AppliedCrystallography , pp. 73–83.Swinehart, Donald F (1962). Journal of chemical ed-ucation , p. 333.Thompson, P., D. E. Cox, & J. B. Hastings (Apr.1987). Journal of Applied Crystallography ,pp. 79–83.Thompson, P., J.J. Reilly, & J.M. Hastings (1987). Journal of the Less Common Metals , pp. 105–114.Toby, Brian H. & Robert B. Von Dreele (Apr. 2013).