research papers Journal of Treatment of grazing-incidence small-angle X-ray Applied Crystallography scattering data taken above the critical angle ISSN 0021-8898 A. Martorana,a,b* A Longo,b F. d'Acapito,c C. Maurizio,d E. Cattaruzzae and Received 7 August 2000 F. Gonellae Accepted 16 January 2001 aDipartimento di Chimica Inorganica, UniversitaÁ di Palermo, Viale delle Scienze, I-90128 Palermo, Italy, bICTPN-CNR, via Ugo La Malfa, 153, I-90146 Palermo, Italy, cINFM and European Synchrotron Radiation Facility, GILDA-CRG, BP 220, F-38043 Grenoble, France, dINFM and Dipartimento di Fisica, UniversitaÁ di Padova, via Marzolo, 8, I-35131, Padova, Italy, and eINFM and Dipartimento di Chimica Fisica, UniversitaÁ di Venezia, Dorsoduro 2137, I-30123 Venezia, Italy. Correspondence e-mail: nino@ictpn.pa.cnr.it The equations taking into account refraction at the sample surface in grazing- incidence small-angle X-ray scattering (GISAXS) when the angle between the incoming beam and the sample surface is slightly larger than the critical angle are derived and discussed. It is demonstrated that the refraction of both the incoming and the scattered beam at the sample surface affects the GISAXS pattern and that, when a planar bidimensional detector perpendicular to the incoming beam is used, the effect depends on the azimuthal detector angle. The smearing of the pattern depending on the size of the illuminated sample area in grazing incidence is estimated by simulations with Cauchy functions of different widths. The possibility of integrating the recorded intensities over a suitable azimuthal angular range and then of making the correction for refraction is also analysed, employing simulations involving the intensity function of mono- # 2001 International Union of Crystallography disperse interacting hard spheres. As a case study, the refraction correction is Printed in Great Britain ± all rights reserved applied to the investigation of a Cu±Ni implant on silica glass. 1. Introduction allows the investigation of anisotropic implants (Babonneau et al., 1999). On the other hand, the scattered beams recorded at Grazing-incidence small-angle X-ray scattering (GISAXS) the same 2 but at various azimuthal ' angles strike the (Levine et al., 1989) is a powerful tool for the structural sample surface with different inclination, thus requiring characterization of thin (micrometre scale) super®cial layers suitable correction for refraction. In the literature, the containing nanosized particles. Samples can be obtained by correction is given only on the plane containing the incoming ion implantation (Babonneau et al., 1999; d'Acapito et al., beam and the surface normal of the sample, that is, at ' = 0 1998; Cattaruzza et al., 2000), sol±gel synthesis (Kutsch et al., (Kutsch et al., 1997). The equations allowing one to obtain the 1997) or vapour deposition (Naudon & ThiaudieÁre, 1997). internal angles (2 0, '0) from the detector angles (2 , ') are Investigation of particles deposited on the surface is usually derived in the next section of this paper. performed at the critical angle c; however, when dealing with Despite the small vertical size of the incident beam, its buried nanostructures, the working angle must be slightly projection on the sample surface is very long when working higher than c to permit a controlled limited beam penetration in grazing incidence (typically a few centimetres), so that the in the matrix. In the latter geometry, the refracted beam acts as limits of the illuminated sample region are actually deter- the effective primary beam of the SAXS experiment and the mined by the overall sample length. The smearing of the scattered radiation is likewise refracted on leaving the sample. recorded pattern arising from the sample size increases with Thus, correction is required in order to retrieve the actual the scattering angle (becoming noteworthy in the wide-angle scattering angle 2 0 from the measured 2 between the region) and decreases with the distance of the sample from incoming beam and the outgoing X-rays. the detector element. An estimate of the extent that The use of an experimental setup consisting of a planar refraction and sample smearing can affect a pattern detector perpendicular to the incident beam assures better recorded under grazing-angle geometry has been achieved counting statistics, with respect to linear devices, and also by simulations ranging from small to wide X-ray 152 A. Martorana et al. Grazing-incidence small-angle X-ray scattering J. Appl. Cryst. (2001). 34, 152±156 research papers scattering angles and involving Cauchy functions of different widths. In principle, the equations taking into account the refrac- tion effect should be applied to every (2 , ') pair of detector angles. However, it is shown by simulations involving the scattering intensity of monodisperse interacting hard spheres (Guinier & Fournet, 1955) that for isotropic implants a simpli®ed procedure is allowed, consisting ®rst of azimuthal integration of the recorded intensities and then correction for refraction according to an average ' angle. An application to the actual case of a Cu±Ni-implanted layer on silica glass is also described. 2. Refraction correction The correction for refraction at the sample surface of both the incident and the scattered beams is based on the scheme drawn in Fig. 1. The angles (2 , ') are measured at the detector, whereas (2 0, '0), the actual scattering angles, are referred to a virtual detector placed within the implanted layer and perpendicular to the refracted incoming beam. The equations for the determination of (2 0, '0) are based on Snell's law and on three-dimensional trigonometry: cos 0 cos = 1 ; 1 Figure 1 Scheme of directions and angles for refraction correction. n sin cos sin 2 cos ' sin cos 2 ; 2 surf is the unit vector normal to the sample surface; ninc is the unit vector in the direction of the incoming beam; nir is the unit vector in the direction of the cos 0 cos = 1 ; 3 refracted incoming beam; nsc is the unit vector in the direction of the scattered beam; nsr is the unit vector in the direction of the refracted cos cos 2 sin sin = cos cos ; 4 scattered beam; is the angle between the planes (nsurf, ninc) and (nsurf, nsc); 2 is the angle between ninc and nsr, measured at the detector; 2 0 is the scattering angle between n cos 2 0 cos 0 cos 0 cos sin 0 sin 0 ; 5 ir and nsc; and 0 are the angles of the incoming beam at the sample surface before and after refraction, respectively (notice that the angle /2 + is evaluated between the cos '0 sin 0 sin 0 cos 2 0 = cos 0 sin 2 0 ; 6 positive directions of nsurf and ninc, so that the incoming beam actually strikes the sample travelling from left to right in the ®gure); 0 and are where (1 ) is the real part of the refractive index. the angles of the scattered beam at the sample surface before and after In Fig. 2, the values of Q/Q [Q = (4 sin )/ , Q0 = refraction, respectively; ' is the azimuthal detector angle between the (4 sin 0)/ , Q = Q Q0] are drawn as a function of the planes (nsurf, ninc) and (ninc, nsr); '0 is the azimuthal scattering angle detector angle 2 , for the , and values ( = 0.25 , = 7.7 between the planes (nsurf, nir) and (nir, nsc). The angles of the drawing are not to scale, but the relations between them are in agreement with 10 6, = 1.49 AÊ) of a previously reported experiment equations (1)±(6): 0 < 0 < , 0 < 0 < , 2 0 < 2 , ' < '0. (Cattaruzza et al., 2000). From inspection of Fig. 2, the 2 correction is seen to be slightly dependent on ' and more allowing the estimation of the correction for absorption of the effective at the smaller 2 values; it is negative (that is, 2 0 = 2 transmitted intensity without the approximation '0 ' ' | 2 |) and decreases in absolute value with 2 , whereas the (Kutsch et al., 1997). correction concerning the azimuthal angle ' is positive, decreasing with 2 and increasing with '. This behaviour can be understood qualitatively by inspection of Fig. 1. The 3. Smearing effects and simulations correction for refraction of the scattered beam can be Equations (5) and (6) are valid under the assumption that the regarded, on the detector plate, as a shift parallel to n illuminated sample has negligible size, so that unique (2 , ') surf of the intensity recorded at a given pixel. On a ®xed scattering angles can be de®ned at each detector pixel. Whereas the ring (that is, at constant 2 ) the shift at ' = 0 concerns only 2 , condition on ' is, to a fairly good approximation, ful®lled by whereas at increasing ' values it tends to become tangential to the narrow cross section of the incoming beam used for the scattering ring, thus giving rise to increasing | '| and grazing-incidence experiments, the 2 values may be subject to decreasing | 2 | values. an uncertainty increasing with 2 and decreasing with the From equations (1)±(6), the path p(2 , ') within the sample distance of the detector element from the sample. A scheme of of a scattered beam generated at depth z and detected at the experimental setup is shown in Fig. 3. (2 , ') can be easily calculated as A normalized Cauchy function, p 2 ; ' z= sin 0 ; 7 I0 2 0 2 =f 2 2 2 0 2 00 2g; 8 J. Appl. Cryst. (2001). 34, 152±156 A. Martorana et al. Grazing-incidence small-angle X-ray scattering 153 research papers is calculated by Gaussian quadrature. The integration range is relative to an illuminated sample region of width L = 10 mm. According to Fig. 3, the glancing angle 2 (x) for the element x is given by 2 x a tan y= D x ; 10 where y = Dtan(2 ), 2 2 (0) and D = 500 mm; 2 0(x) in equation (9) is then obtained from 2 (x) by equation (5). The approximation introduced in equation (10) by ignoring the angle of grazing incidence of the incoming beam with the sample surface is negligible. At ®rst glance, the smeared pro®les obtained by equation (9) are indistinguishable from the unsmeared ones for 2 < 3 , whatever the value of , whereas the Cauchy functions, in particular the sharpest ones, are severely distorted in the wide-angle region. Figure 2 Because of the faint dependence of 2 on ', an effective Q/Q, as a function of the detector angle 2 , for the GISAXS procedure seems to be, for suitable ' intervals and isotropic parameters: = 0.25 , = 7.7 10 6, = 1.49 AÊ. The different curves, implants, ®rst to integrate the two-dimensional detector going downwards from the topmost, are relative to ' = 0, 10, 20, 30 and 40 . intensity values over ' and then to perform the correction on 2 . Possible smearing effects are estimated in the following was used to estimate the in¯uence of the size of the illumi- simulation. The function nated area on the uncertainty of 2 . This function, although not related to any actual SAXS application, can be subjected I0 Q0 2 Q0R = 1 8V0=V1 2Q0R ; 11 to rigid translation and, therefore, retaining a constant shape, proportional to the scattering intensity of monodispersed can account for the mere smearing effects at different glancing interacting hard spheres (Guinier & Fournet 1955), is taken angles. The grazing incidence patterns were calculated for the into account; R is the particle radius, V0 its volume, V1 the above-reported , and parameters, for ' = 0 and for several volume per particle, and 2 00 values ranging from the small- to the wide-angle region. Different values of , varying from 0.25 to 1 , were used to t 3 sin t t cos t =t3: 12 investigate the dependence of the smearing effects on the degree of sharpness of the scattering pro®le. The smearing integral (9) is calculated as a function of '; a The smearing integral, further integration on ' corresponds to summing the `observed' intensity values on pixels equidistant from the trace of the direct beam on the detector. In Fig. 4, two sets of RL=2 I simulated data are drawn, relative to R1 = 30 and R2 = 60 AÊ, sm 2 dx I 2 0 x ; 9 L=2 respectively. The other parameters are (8V0/V1) = 4 for both the calculated patterns, L = 10 mm, D = 500 mm and 'min = 10 , 'max = 26.5 ; the choice of 'min, 'max depends, as will be discussed on the next section, on actual experimental limita- tions. Furthermore, a ®tting to the `data' of the unsmeared intensity has been performed, taking into account the refrac- tion effects for 'av = 18 and re®ning an overall scale factor, the particle radius R and the packing factor (8V0/V1). From inspection of Fig. 4, it is evident that only on a logarithmic intensity scale can some difference be appreciated between the simulated data and the ®tted pattern; the broader minima of the `data' arise mainly from the dependence on ' of the 2 correction. The values of the optimized parameters, R1 = 30 AÊ and (8V0/V1)1 = 4.0 for the ®rst ®tting run, R2 = 60 AÊ and (8V0/V1)2 = 3.9 for the second, are very close to those of the simulated data. Therefore, it is possible to conclude that one can ®rst integrate the rough intensity values in suitable ' Figure 3 Scheme of the GISAXS experimental setup. D is the distance from the intervals and then correct for refraction according to an centre of the illuminated sample to the detector; y is the distance of the average ' value. As the data have been simulated with a pixel from the trace of the incoming beam. The small angle (some sample-to-detector distance that is de®nitely smaller than tenths of a degree, in grazing-incidence geometry) between the incoming beam and the sample surface is approximated as = 0 , so that in the actual experimental con®gurations, the good ®ttings of drawing the sample surface is parallel to the incoming beam. unsmeared functions demonstrate a fortiori that smearing 154 A. Martorana et al. Grazing-incidence small-angle X-ray scattering J. Appl. Cryst. (2001). 34, 152±156 research papers angles, taking into account that recording at ' < 10 is hindered by the beam stop, while intensities at ' > 26.5 are affected by the Yoneda peak (Yoneda, 1963; Babonneau et al., 1999). The correction for absorption does not produce appreciable effects, owing to the very small thickness of the implanted layer (about 1000 AÊ). In Fig. 5, a comparison of the integrated scattered intensity with and without refraction correction is presented (Figs. 5a and 5b, respectively); ®ts to the experimental scattering curves are also shown. The model of interacting spherical clusters used for the ®tting procedure is based on the local mono- disperse approximation [LMA, which assumes complete correlation between particle size and position within the sample (Pedersen, 1994)], on a Weibull-like cluster size Figure 4 distribution and on a Perkus±Yevick structure factor; all these GISAXS intensities of interacting monodispersed hard spheres [equation hypotheses are valid for these composites (Cattaruzza et al., (11)]. Details around a minimum for the pro®les corresponding to hard- sphere radii R = 60 AÊ and R = 30 AÊ, respectively, are shown. Exp30 and 2000). Exp60 (heavy lines) are the simulated data from equation (11) taking into Fitting parameters are reported in Table 1; the correction account sample smearing [equation (9)], refraction and ' integration for refraction essentially affects the parameters that are from ' = 10 to ' = 26.5 . The dashed lines (Fit30 and Fit60) are relative involved in the structure factor and are therefore related to to ®tted pro®les calculated without sample smearing and taking into account the refraction correction at 'av = 18 . the position of the maximum of the scattering curve. In particular, the large decrease in the hard-sphere volume effects caused by the size of the illuminated sample area can fraction HS (Pedersen, 1994) is evident; this is caused by the be neglected in the analysis of GISAXS data. shift of the GISAXS pattern towards smaller scattering angles, which results in the estimation of larger distances between the interacting objects. The difference between 4. A case study HS = 0.24 reported in Table 1 and HS = 0.28 previously assessed Equations (5) and (6) give the correction 2 as a function of (Cattaruzza et al., 2000) can be ascribed to the better signal-to- the experimental parameters (incidence angle , index of noise ratio achieved by use of the ' integration. As the refraction 1 , incoming wavelength ), and of the (2 , ') refraction correction involves a roughly rigid translation of the angles. To study how the correction for double refraction GISAXS pattern, it is not surprising that the form factor is not affects experimental scattering data, we report here as a case concerned very much; indeed, the average radius of the clus- study the analysis of a silica slide sample containing Cu±Ni ters is nearly constant in ®tting runs (a) and (b). alloy nanoclusters in a layer, of thickness 100 nm, below the glass surface. The sample was prepared by double implanta- tion of Cu+ and Ni+ ions in silica; details of preparation 5. Conclusions parameters, transmission electron micrographs and proce- The correction for refraction of the incoming beam and of the dures for recording the GISAXS pattern of the sample scattered beam at the sample surface in GISAXS patterns has considered here are reported elsewhere (Cattaruzza et al., 2000). While all the angles used for the correction are set out by the scattering geometry with an accuracy that depends on the experimental apparatus, the correction for the index of refraction 1 is calculated from the experimental deter- mination of the critical angle c of the composite system. c is calculated from the re¯ectivity curve as the angle corre- sponding to the half maximum of re¯ected intensity. The accuracy of its determination depends on the system, as the presence of Kiessig fringes could introduce an uncertainty in the determination. In the present case, the error was found to be around 5%; thus negligible in the following analysis. In the previously reported study (Cattaruzza et al., 2000), only the scattering data relative to the smallest available ' angle were analysed. According to the simulations reported in the previous section, the experimental intensities are now Figure 5 radially integrated in the range 10 ' 26.5 and then Scattered intensity of Cu+ + Ni+ implanted silica obtained by radial integration of the two-dimensional scattering pattern (a) with or (b) corrected for refraction according to 'av = 18 . The integration without correction for double refraction of the X-ray beam. Best ®tting range is limited to the ' values available for all the relevant 2 curves (within the LMA approximation) are also reported. J. Appl. Cryst. (2001). 34, 152±156 A. Martorana et al. Grazing-incidence small-angle X-ray scattering 155 research papers Table 1 Furthermore, it has been demonstrated that, whereas for Fitting parameters within the LMA of the scattering curve (a) after and anisotropic implants the correction given by equations (5) and (b) before correction for double refraction of the X-ray beam impinging (6) should be applied for each (2 , ') pair of angles, for on the sample. isotropic implants a simpli®ed procedure can be exploited in HS is the hard-sphere volume fraction in the structure factor; a and b limited ' ranges, consisting of azimuthal integration of the are the parameters that de®ne the Weibull function W(R) = (a/b)(R/b)a 1 exp[ (R/b)a]. recorded intensities followed by correction for refraction according to an average ' angle. Experimental Mean Mean scattering cluster intercluster curve a b (AÊ) radius (AÊ) distance (AÊ) HS (a) 2.48 21.5 19.1 87 0.24 (b) 2.35 21.5 21.0 68 0.39 References Acapito, F. d', ThiaudieÁre, D., Zontone, F. & Regnard, J. R. (1998). Mater. Sci. Forum, 278±281, 891±896. been derived under the assumption of a point-like sample and Babonneau, D., Naudon, A., ThiaudieÁre, D. & Lequien, S. (1999). J. for an experimental setup consisting of a planar detector Appl. Cryst. 32, 226±233. perpendicular to the incoming beam. It has been demon- Cattaruzza, E., d'Acapito, F., Gonella, F., Longo, A., Martorana, A., strated by the simulations reported in x3 that the smearing Mattei, G., Maurizio, C. & ThiaudieÁre, D. (2000). J. Appl. Cryst. 33, effects arising from the ®nite size of the illuminated sample are 740±743. Guinier, A. & Fournet, G. (1955). Small-Angle Scattering of X-rays. negligible in the small-angle scattering region. 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