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Precise evaluation of elastic differential scattering cross-sections and their uncertainties in X-ray scattering experiments
Nuclear Instruments and Methods in Physics Research B 187 (2002) 437–446 www.elsevier.com/locate/nimb
Precise evaluation of elastic differential scattering cross-sections and their uncertainties in X-ray scattering experiments M.E. Poletti a,*,1, O.D. Goncßalves a, H. Schechter a, I. Mazzaro b a b
Universidade Federal do Rio de Janeiro, P.O. Box 68528, 21945-970 Rio de Janeiro, Brazil Universidade Federal do Paran a, LOXR, P.O. Box 19091, Curitiba 81531, Paran a, Brazil Received 23 July 2001; received in revised form 26 October 2001
Abstract In this work we propose a method of obtaining elastic scattering differential cross-sections from X-ray scattering experiments with amorphous samples taking into account all background sources and also multiple scattering processes. Background from the ambient and other scattering sources were measured directly. Multiple scattering was calculated through a Monte Carlo code that simulates the actual experiment and may use different theoretical approaches when considering the differential scattering cross-section. The same code, after slight modifications, provides also the attenuation coefficients and the mean scattering angle, which were compared with analytical results. The uncertainty for all experimental quantity was evaluated. As an illustration of the method, the procedure was applied to scattering data obtained for a glandular breast tissue measured inside a cylindrical container. Ó 2002 Published by Elsevier Science B.V. PACS: 87.64Bx; 87.59Ek; 87.66Xa Keywords: X-ray scattering; Elastic scattering differential cross-sections; Amorphous materials
1. Introduction Elastic scattering is a powerful tool not only to get information about electrons from single atoms (inner-shell) but also to determine the spatial
*
Corresponding author. Tel.: +55-21-2562-7470; fax: +5521-2562-7368. E-mail address: [email protected] (M.E. Poletti). 1 Ph.D. Scollarship from Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq).
structure of substances in general. Its application extends from basic physics to technology and medical appliances. Depending on the momentum transfer in the elastic scattering process, it is possible to describe the phenomenon using two different approaches. In the first one, for high momentum transfer, it is possible to use the free atom approximation in which no collective effect is considered. In the second approach, for low momentum transfer, it is necessary to take into account the positions of atoms and molecules in the sample, which is
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 1 1 4 9 - 1
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detectable through structures in the experimental angular distribution of scattered photons (also called ‘‘diffraction pattern’’ or more precisely ‘‘scattering profile’’). The precise limit between these two regions will depend on the sample composition and structure, temperature, the experimental setup geometry being used and on the accuracy of the experiment [1–3]. Although elastic (Rayleigh) cross-sections have usually been measured with radioactive monochromatic gamma-ray sources, it can also be done with X-ray generators. This kind of measurement is particularly effective in obtaining form factors for amorphous solids (plastic), liquids and biological samples, which are of great relevance in medical technology [4–9]. The experimental setup for this kind of measurements is not complex, but on the other hand, it is very difficult to analyze the data since the number of scattered photons detected includes error sources that should be considered in order to obtain precise and reliable elastic differential crosssection. Some of these error sources are: (i) a spread on the experimental scattering angle due to the geometric setup; (ii) scattering other than single scattering in the sample (multiple scattering, fluorescence, scattering from the container, scattering in the air and other background sources) and that must be eliminated either physically or by calculation; (iii) self-attenuation and polarization, that must be taken into account; (iv) normalization of the data since the measurements are not absolute; (v) values of elastic and inelastic scattering cross-sections of isolated atoms that must be known as accurately as possible, since they serve as a base for normalizing the experimental data. Generally, the methods reported in the literature to deal with these corrections can be divided into two categories: (i) analytic and (ii) Monte Carlo simulation. The later one incorporates complete geometry considerations (collimators, samples and beam size) and also all the scattering components. The analytic (or semi-analytic) approach usually considers primary scattering as isotropic, the incident beam as having a uniform profile, the sample as being immersed in the beam and also ignores the contribution of the container to the multi-
ple scattering. Generally attenuation and geometric effects are corrected, but multiple scatter very rarely [10]. Although some few authors have performed these corrections using computer simulation, Monte Carlo techniques are applicable to a wide variety of cases of practical interest [8,11]. They are necessary in cases where analytic corrections are impossible to calculate, for example nonuniform beams (X-ray beams usually have a nonuniform cross-section due to the sources and to shadows cast by slits), samples larger than the beam size, etc. The availability of such Monte Carlo programs permits considerable flexibility in the strategy for an experiment and also provides a quick test for the angular resolution of the system, taking into account the dimensions of the source, detector, sample and collimating system both in the incident and scattering beam. In this work we deal specifically with the analysis of X-ray scattering data including all aspects cited above. As an illustration, a suggested procedure was applied to scattering data obtained for a glandular breast tissue measured inside a cylindrical container. The same procedure is naturally also useful for any similar experiment [9].
2. Experimental aspects The considered experimental setup is present in Fig. 1(a). It is composed of a powder diffractometer operating in transmission mode in order to minimize the attenuation in the sample. The photon source is an X-ray Mo tube (Z ¼ 42, Ka ¼ 17:44 keV, Kb ¼ 19:6 keV) operating at 30 kVp, in order to avoid other harmonics of Mo Ka. The X-ray generator is equipped with a graphite monochromator in the incident beam to select the fluorescence line (Ka), producing a reasonable monochromatic incident beam (E ¼ 17:44 keV with a FWHM ¼ 0:19 keV, Fig. 1(b)). The energy distribution of the incoming beam (Fig. 1(b)) was measured with a silicon crystal analyzer (440). For each crystal position (in relation to the incoming beam) the detector (NaI (Tl)) ‘sees’ one photon energy according to the Bragg law. The beam size was limited by a vertical slit (0:75 8 mm2 ) placed
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Fig. 1. (a) Experimental setup and (b) photon energy spectrum incident on the sample.
in the exit of the monochromator. An additional set of S€ oller slits was placed after and before the samples and still another vertical slit collimator (1:7 8 mm2 ) was placed in front of the detector. The collimators were chosen in order to minimize the beam spread providing that just a small part of the sample be irradiated. Lines linking of incoming beam and the centers of sample and detector as depicted in Fig. 1(a) define the nominal scattering angle. The spectrometer system features an NaI (Tl) detector with the conventional electronic. The detector efficiency at this energy is 65% while the energy resolution is about 5 keV implying that the number of scattered photon includes elastic, inelastic and multiple scattering effects. The number of scattered photons was measured varying the scattering angle between 1.3° and 72° 1 ) with steps of 1=3 degree and (0:02 < x < 0:83 A with counting times around 60 s providing a maximum statistic uncertainty of 2%. For each sample, more than one data set has to be collected in order to test the stability of the system (in the considered experiment we performed three collections for each sample). We shall generally assume that the atomic arrangement of the sample is isotropic, implying that the scattering profile is symmetric around the primary beam. With this assumption it is sufficient, for a given energy, to consider just the one-
dimensional (1D) dependence of the elastic scattering with the scattering angle [10]. The biological sample, a glandular tissue, was measured inside a cylindrical acrylic container with nominal height 15 mm, external diameter 8.2 mm and wall thickness 0.1 mm (3% of the total diameter). The dimensions were chosen in order to provide sufficient single scattering events while minimizing the probability of multiple scattering. The container with the sample was placed on a rotating table performing one revolution in 10 s in order to ensure that all parts of the sample could be equally exposed to the beam. The commonly used geometry is a thin foil target, the reason being that attenuation corrections is easier in this geometry. That is not the most convenient geometry for samples with irregular surfaces, being the cylindrical form more useful. Cylindrical geometry has two advantages. First the attenuation in the sample varies very slowly and second it is possible to minimize surface irregularities by rotating the sample. The disadvantage is that the attenuation correction factor is difficult to calculate. Other sample characteristics that must be known are composition, density and linear attenuation coefficient. The elemental composition was measured using an Elemental Analyzer EA 1110 from CE Instruments (based on thermal conductivity providing the mass proportions of H, C, N and S) giving the following weigh fractions:
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H: 0.093, C: 0.184, N: 0.044, O: 0.678 and S: 0.001. The oxygen amount was obtained assuming that only H, C, N, O and S are present. The density was measured using the usual Buoyancy method (q ¼ 1:04 0:02 g=cm3 ). The linear attenuation coefficient was measured for the sample using the geometry depicted in Fig. 1 with h ¼ 0° and with photons of the same energy as in the scattering experiment, with a well-collimated beam (standard setup of ‘‘narrow beam geometry’’).
the lead sample, and the probability of multiple scattering in the air is very small, the contribution of the lead for additional background can be neglected. Combining these four factors we obtain the counting rate for scattering in the sample alone (N ),
3. Methodology
3.2. Correction for the beam polarization
There are many necessary steps in obtaining the single scattering cross-section. The first one is to subtract from the original data the number of photons originated from every other sources (background). The resulting data have to be corrected for polarization of the incoming beam and multiple scattering. The result is the number of single scattered photons, which can be related to the single scattering cross-section. Each step will be detailed below.
The beam produced by a monochromator crystal is partially polarized [1,2]. The partial polarization of the X-ray beam must then be taken into account because it will affect the subsequent scattering from the sample under study. In this case the collected data must be re-scaled by the factor, Cp , 1 þ jp cos2 h 1 þ cos2 h Cp ¼ ; ð2Þ 1 þ jp 2
3.1. Evaluation of the background components The background depends on the geometry of the experimental setup (collimators, volume of irradiated air, natural background, and scattering in the material used as holder or container) and comes from sources other than the sample. The background can be experimentally subtracted performing a series of measurements for a given scattering angle. In the present work four different acquisitions were made. The first acquisition comprises photons scattered by the samples plus photons scattered by the container plus scattering in the environment (NSþCþG ). The second acquisition (NCþG ) is performed with the empty container. The third (NG ) is the counting rate obtained without the sample and container and provides information about for the environmental scattering and background. The last one, NPb , is obtained with a lead target instead of the sample and provides information about ambient background which does not pass trough the samples. Since the beam hits just a small central region of
N ¼ ðNSþCþG NPb Þ T1 ðNCþG NG Þ T2 ðNG NPb Þ;
ð1Þ
where T1 and T2 are geometric transmission factors for sample and sample plus container, respectively.
where jp is defined as the square of the ratio between the parallel and perpendicular amplitude components of the photon polarization. Because most crystals are non-ideal, jp could not be accurately predicted and this is a limitation for the accuracy of the method. On the other hand this inaccuracy is not expected to be significant when compared to other sources of uncertainty and without loss of generality jp could be obtained from jp ¼ cos2 hm , where hm is the diffraction angle of the monochromator crystal. 3.3. Correction for multiple scattering The number of photons that undergo multiple scattering can be determined analytically only for double-scattering processes and for few specific setup geometries. In general, to evaluate multiple scattering (more than two interactions) it is necessary to use numerical methods. The most popular computational method is the Monte Carlo. In order to check the contribution of multiple scattering, we have written a Monte Carlo code. It is a 3D Monte Carlo code considering a finite size
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source, variable shape of collimators, variable sample geometry (slabs, cylinders, etc.) and a 3D detector. The code was compared with the EGS4 code [14] for chosen cases and a good agreement was achieved. To obtain MS it is necessary to consider some theoretical results for the crosssections as being true. For the total atomic crosssections, the code uses the XCOM data [12]. For the differential inelastic cross-sections it was used or the atomic incoherent scattering function (S) multiplied by the well-known formula of Klein– Nishina [15]. For the elastic scattering cross-section, our code may consider either the atomic form factor (F ) [15] or the molecular form factor [5,9]. The number of single scattered photons detected obtained after all these corrections (N corr ) may then be expressed as N corr ¼ N Cp1 MS ; ð3Þ
dr=dXðhÞ is the differential cross-section comprising inelastic plus elastic; dV ¼ dA dx, dA being the cross-sectional area viewed by the incident beam on the differential volume dV and dx is the voxel size. Considering that the differential scattering cross-section does not vary significantly over the range of scattering angles, comprising the source, target and detector volumes Eq. (4) can be written as dr ðhÞ N corr ðhÞ ¼ N0 nv dX elas Z Z dX eðEÞel1 ðEÞd1 el2 ðEÞd2 dV
where N is the number of detected photons corrected for background, Cp is the correction for polarization of the incoming beam, MS is number of photons that undergo multiple scattered.
ð5Þ
3.4. Determination of experimental differential scattering cross-section Following Kane et al. [13] the total number of scattered photons (single scattered) at a scattering angle h; N corr ðhÞ, is related to the differential scattering cross-section dr=dXðhÞ by an integral over the scatterer volume V and the scattering solid angle X, Z Z dr N corr ðhÞ ¼ ðhÞel2 d2 dV ; dX N0 eel1 d1 nv dX det sample ð4Þ where h is the scattering angle with respect to primary beam, N0 is the incident number of X-ray photons per cm2 , e is the detector efficiency; dX is the differential solid angle viewed by the detector (considering the collimator) and defined by the elemental volume in the sample; l1 and l2 are the attenuation coefficient before and after the scattering; d1 and d2 are the path lengths inside the sample before and after the scattering, respectively; nv is the number of atoms (or molecules) per cm3 ;
det
þ
sample
dr ðhÞ dX inel
Z
Z
dX det
0
eðE0 Þel1 ðEÞd1 el2 ðE Þd2 dV
sample
dr dr 0 ¼ N0 nv eðEÞ ðhÞ GðhÞ þ ðhÞ G ðhÞ ; dX elas dX inel
where dr=dXðhÞelas ðinelÞ corresponds to the mean elastic (or inelastic) cross-section for this range of scattering angles. If either the cross-section is slow varying or the angular acceptance is sufficient small, this quantity will approach the differential crosssection at the mean angle h. E is the energy of the elastic scattered photon and E0 is the energy of the inelastic scattered photon. Since the Compton shift for experimental the angular range is not very large, the detector efficiency was assumed to be the same for both elastic and inelastic scattered photons. GðhÞ and G0 ðhÞ are known as geometric attenuation factor for elastic and inelastic scattering. These factors take into account the attenuation of the incident and scattered beam through the sample volume (known as self-attenuation) and also the distribution of dX due to the geometry (geometric effect). In order to calculate GðhÞ and G0 ðhÞ the Monte Carlo code was modified in order to consider just the geometrical conditions without introducing any other physical constraint when sampling the photon paths during the scattering process. The code generates a normalized distribution (gðhÞ and g0 ðhÞ) of the attenuation correction for each possible scattering angle for a given measuring position (angle) due to finite size of sources, sample and detector. The integration of gðhÞ and g0 ðhÞ
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leads to the desired GðhÞ and G0 ðhÞ. Besides the attenuation correction, the code provides also the angular geometric dispersion and the mean angle for a specific nominal scattering angle. In the present experiment, like in most of high angular resolution experiments, it is required high beam fluency turning impossible to measure the incoming number of photons (N0 ) directly. The absolute efficiency (e) is also difficult to be measured. The alternative, in order to obtain the differential cross-section without knowing these quantities (Eq. (5)), is to perform some normalization with theoretical input in a region of momentum transfer where no interference is expected and the IAM was demonstrated to be valid [10]. Rewriting Eq. (5), N corr ðhÞ dr dr ðhÞelas GðhÞ þ ðhÞinel G0 ðhÞ ; ¼ K 1 nv dX dX ð6Þ 1
where K ¼ N0 eðEÞ. The quantity inside the brackets can be calculated theoretically in the frame of IAM (from here designated as ATheor ) as ! IAM IAM Fm2 dr dr Theor 0 Sm A ¼ NA q G þG ; M dXTh M dXKN ð7Þ 2IAM
Fm
=M ¼
X
ðwi =mi Þfi2 ;
ð7aÞ
ðwi =mi ÞSi ;
ð7bÞ
i
SmIAM =M ¼
X i
where nv was replaced by (NA q=M), NA is the Avogadro’s constant, q the tissue mass density, M its molecular weigh and dr=dXTh and dr=dXKN are the well-known Thomson and Klein–Nishina differential cross-sections, respectively. FmIAM and SmIAM are the molecular form factor and molecular incoherent scattering function, calculated both within the IAM approach. Expressions (7a) and (7b) are useful when the chemical formula of the molecule is not known, which is the case of biological samples. For these samples, the usual elemental analyses give relative weight and atomic mass of each element in the sample (wi and mi ,
respectively). fi and Si are respectively the atomic form factor and the incoherent scattering function of the ith element in the sample. The normalization factor (K) can then be obtained by averaging the ratio between the experimental data (N corr ) and the corresponding theoretical data (ATheor ) for x between xcritical (x value above which there is no more intermolecular interference) and xmax (the maximum x value measured in the experiment), n 1X N corr ðxi Þ 1 xcritical 6 xi 6 xmax : K ¼ ð8Þ n i¼1 ATheor ðxi Þ It should be noticed that since it is not possible to know the exact molecular formula of complex substances like human tissues, also the number of scattering molecules per unit volume (nv ) cannot be achieved. It is then useful to define a new quantity relating both nv and the differential crosssection and given by dr=dXnv ¼ nv dr=dX. Since it is not expected to exist a significant departure between theory and experiment for the inelastic scattering [11], the elastic cross-section times the number of molecules per cm3 , ðnv dr= dXelas Þ may then be obtained immediately by subtracting the theoretical prediction for the inelastic effect from Eq. (7), nv
dr KN corr G0 dr nv ¼ : dXelas G G dXinelas
ð9Þ
From this equation it is also possible to extract the experimental molecular form factors [9].
4. Results and discussion All corrections to be applied to the number of detected photons depend on the apparatus, experimental setup and energy of the incident beam. As an example, these corrections were applied to data obtained using as sample, glandular tissue in a cylindrical container. In the following we discuss the results obtained after each one of the corrections. 4.1. Background Fig. 2 shows the results of measurements performed to obtain the number of photons scattered
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Fig. 2. Relative number of photons scattered by: (1) the container filled with glandular tissue (solid line); (2) the empty container (dashed line); (3) without container in the sample position (dotted line) and (4) detected photons that did not come through the sample, obtained with a thick lead sample (dot-dot-dashed line).
just in the sample. The curve corresponding to the empty container (dashed curve) shows a structure due to the container material (acrylic) and has also contribution from scattering on the air and on the collimating slits (dot curve). The number of scattered photons reaching the detector and passing outside the sample (curve plotted with dot-dotdashed line) is very small (generally less than around 0.6%) when compared with the total number of scattered ones (solid line). For h larger 1 ) all components become than 11° (x ¼ 0:13 A very small, the resulting correction being around 3–5% of the total counting number.
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conditions (including container) and considering (i) possibility of elastic and inelastic scattering taking interference in account for the first one, (ii) possibility of elastic and inelastic scattering without interference and (iii) considering just the inelastic scattering. The results are compared in Figs. 3 and 4. As can be seen from the Fig. 3 the number of photons undergoing multiple scattering is 10% of the total number of photons detected. Around the considered energy it is not recommended to use just multiple inelastic scattering for multiple scattering. Fig. 4 depicts the ratio between each possibility and the most complete simulation (case (i)). Even though the differences between the two first approaches could be as big as 12%, the final results using either method to calculate the multiple scattering correction are different by less than 0.7%. As expected, the square symbols (situation (ii)) are similar (within the uncertainties) to the results obtained in situation (i) for h P 25° 1 ), the region where the interference (x P 0:30 A effects can be neglected. The results obtained considering just inelastic scattering (situation (iii), depicted as circles) are much smaller than situation
4.2. Multiple scattering A preliminary study was performed in order to choose the best way to consider multiple scattering. The first question is how important is the multiple scattering process itself and the second is how important is each effect (elastic with interference, elastic without interference and inelastic scattering) in the composition of multiple scattered. To answer these questions we performed calculations with the Monte Carlo code for glandular tissue, carrying out the complete experimental
Fig. 3. Number of scattered photons obtained for glandular tissue corrected for background and polarization (open circle symbol) compared with the expected number of scattered photons that undergo multiple scattering, calculated considering: (1) inelastic plus elastic scattering and considering interference effects (solid line); (2) inelastic plus elastic scattering without considering interference effects (IAM) (dotted line) and (3) just multiple inelastic scattering (dashed line).
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(i), the difference being larger at low scattering angles, where the elastic–inelastic and elastic– elastic processes are more probable than for larger
angles. The error bars correspond to the statistical uncertainty. Results obtained on other energies and geometries suggest that the number of photons that undergo multiple scattering (MS ) could be invariant with the scattering angle [11,16]. Our results (Fig. 3) show that a significant variation is obtained when elastic scattering is considered.
5. Geometric and attenuation correction
Fig. 4. Ratio between the number of multiple scattered photons calculated in different ways: MSI : inelastic plus elastic scattering and considering interference effects; MSX ð1Þ: inelastic plus elastic scattering without considering interference effects (IAM) (empty square symbols); MSX ð2Þ: just multiple inelastic scattering (dashed line).
The weight functions g and g0 , together with the angular spread, are shown in Fig. 5 obtained by fixing the detector at h ¼ 25° (a) and h ¼ 60° (b). At h ¼ 25° all curves are symmetric. At h ¼ 60° the curves for g and g0 are no long symmetric, but the angular spread is still symmetric around an angle slightly lower than the nominal one (0.13°), a minor shift that occurred at 25° (0.11°). All final results of cross-sections are given as a function of a corrected x, obtained with the mean scattering angle.
Fig. 5. Angular dispersion and weight function for elastic (gðhÞ – square symbols) and inelastic (g0 ðhÞ – circle symbols) single scattering at h ¼ 25° and h ¼ 60°.
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Fig. 6. Attenuation factors for elastic (GðhÞ) and inelastic (G0 ðhÞ) single scattering plotted as a function of scattering angle, obtained by: (1) Monte Carlo simulation (solid and dash line) and (2) analytical calculation ignoring beam profile and geometric effect (dot and dash-dot line).
We have compared our results for G and G0 with a widely used approximation that considers a monochromatic and parallel incoming beam together with a parallel outgoing beam. This approach allows the double-integration in Eq. (6) to be expressed in an analytical form [13], the method being usual for thin target geometry. In the present case, due to the complexity of the cylindrical geometry, a numerical method was used in order to obtain the analytical result for G and G0 (shown in the Fig. 6). These results agree (over the considered angular range) with the correspondent Monte Carlo ones within 3%.
6. Evaluation of the error bars The resulting experimental uncertainties in the measured differential elastic cross-section (vertical error bars) have been estimated by error propagation in Eq. (9) considering statistical uncertainty of counting rate, uncertainty from the Monte Carlo calculations and also from normalization procedures, assuming that all variables are noncorrelated. The statistical errors in N (Eq. (1)) range between 4% and 5%. The statistical uncertainties from simulation (MS and G, the latter including uncertainty in the sample thickness) are
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Fig. 7. Experimental elastic differential cross-section (multiplied by the number of molecules per volume), nv dr=dXelas , as function of momentum transfer, x, together the resulting vertical and horizontal error bar.
between 2% and 3%. The error from the normalization procedure (K) is around 3%. The resulting vertical error bars range between 5% and 7% and are displayed in Fig. 7, together with the obtained elastic differential cross-section times the number of atoms per volume. The horizontal error bar associated with the experimental momentum transfer is given by 2
2 1=2
Dx ¼ Af½ðE=2Þ cosðh=2ÞDh þ ½sinðh=2ÞDE g
;
where E is the photon energy, h is the scattering 1 keV1 ). The angular angle, and A ¼ 1=12:398 (A uncertainty was between 0.4° (at small-angle) and 0.8° (at high-angle) and includes inaccuracy of the mechanical sample positioning (0.02°). The resulting momentum-transfer uncertainties are of Dx ¼ 0:006 (at small-angle) and Dx ¼ 0:01 (at high-angle) which is sufficiently narrow to avoid overlapping of peaks in the experimental amorphous form factor. The results are also displayed in Fig. 7.
7. Conclusions We have presented an accurate method to obtain elastic scattering cross-section for amorphous samples from X-ray scattering experiments
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(scattering profiles) combining a series of complementary measurements and Monte Carlo simulations. Correcting geometric self-attenuation with Monte Carlo methods represents an improvement when compared with the standard analytical methods, which ignores beam profile and geometric effect, resulting in a difference around 3%. The results show that multiple scattering must be considered even in thin samples (for glandular tissue, in the considered geometry and thickness around 1=l, MS contribution could be around 10%). Another conclusion is that the multiple scattering in this energy depends on the scattering angle, but on the other hand, interference due to atoms or molecule positions is not important when considering multiple scattering. The process was applied for other geometries and materials, including thin foil and solid samples. The results should be published soon [9]. The method here described lead to a precise evaluation of elastic cross-section with trustable uncertainties for complex compounds, especially for material of medical interest, e.g. tissues and tissue-equivalent materials, creating new possibilities for material characterization and medical applications.
Acknowledgements This work was supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq), Coordenacß~ ao de Aperfeicßoamento de Pessoal de Nıvel Superior (CAPES) and Fundacß~ ao de Amparo a Pesquisa do Estado do Rio de Janeiro (FAPERJ). We would
like to thank Dr. Michael Farquharson for the careful revision and useful suggestions.
References [1] R.W. James, The Optical Principle of the Diffraction of X-ray, Bell, London, 1962. [2] L.V. Azarrof, Elements of X-Ray Crystallography, McGraw-Hill, USA, 1968. [3] O.D. Goncßalves, C. Cusatis, I. Mazzaro, Phys. Rev. A 48 (1994) 243. [4] A.H. Narten, X-Ray Diffraction Data on Liquid Water in the Temperature Range 4 C-200 C, ORNL Report 4578, 1970. [5] L.R.M. Morin, J. Phys. Chem. Ref. Data 11 (1982) 1091. [6] J. Kosanetzky, B. Knoerr, G. Harding, U. Neitzel, Med. Phys. 14 (1987) 526. [7] A. Tartari, E. Casnati, C. Bonifazzi, C. Baraldi, Phys. Med. Biol. 42 (1997) 2551. [8] D.E. Peplows, K. Verghese, Phys. Med. Biol. 43 (1998) 2431. [9] M.E. Poletti, O.D. Goncßalves, I. Mazzaro, Phys. Med. Biol. 47 (2002) 47. [10] H.P. Klug, L.E. Alexander (Eds.), X-ray Diffraction Procedure for Polycrystalline and Amorphous MaterialWiley, Wiley, New York, 1974. [11] A. Tartari, C. Bonifazzi, C. Baraldi, E. Casnati, Nucl. Instr. and Meth. B 142 (1998) 203. [12] M.J. Berger, J.H. Hubbell, XCOM: Photon Cross Sections on a Personal Computer, NBSTIR 87-3597, National Institute of Standards and Technology, Gaithersburg, MD, USA, 1987. [13] P.P. Kane, L. Kissel, R.H. Pratt, S.C. Roy, Phys. Rep. 140 (1986) 75. [14] W.R. Nelson, H. Hirayama, D.W.O. Rogers, The EGS4 Code System SLAC-265, Stanford Linear Accelerator Center, 1985. [15] J.H. Hubbell, E.A. Veigele, E.A. Briggs, D.T. Brown, D.T. Cromer, R.J. Howerton, J. Phys. Chem. Ref. Data 4 (1975) 471, Erratum, J. Phys. Chem. Ref. Data 6 (1977) 615. [16] D.A. Bradley, D.R. Dance, S.H. Evans, C.H. Jones, Med. Phys. 16 (1989) 851.
Precise evaluation of elastic differential scattering cross-sections and their uncertainties in X-ray scattering experiments M.E. Poletti a,*,1, O.D. Goncßalves a, H. Schechter a, I. Mazzaro b a b
Universidade Federal do Rio de Janeiro, P.O. Box 68528, 21945-970 Rio de Janeiro, Brazil Universidade Federal do Paran a, LOXR, P.O. Box 19091, Curitiba 81531, Paran a, Brazil Received 23 July 2001; received in revised form 26 October 2001
Abstract In this work we propose a method of obtaining elastic scattering differential cross-sections from X-ray scattering experiments with amorphous samples taking into account all background sources and also multiple scattering processes. Background from the ambient and other scattering sources were measured directly. Multiple scattering was calculated through a Monte Carlo code that simulates the actual experiment and may use different theoretical approaches when considering the differential scattering cross-section. The same code, after slight modifications, provides also the attenuation coefficients and the mean scattering angle, which were compared with analytical results. The uncertainty for all experimental quantity was evaluated. As an illustration of the method, the procedure was applied to scattering data obtained for a glandular breast tissue measured inside a cylindrical container. Ó 2002 Published by Elsevier Science B.V. PACS: 87.64Bx; 87.59Ek; 87.66Xa Keywords: X-ray scattering; Elastic scattering differential cross-sections; Amorphous materials
1. Introduction Elastic scattering is a powerful tool not only to get information about electrons from single atoms (inner-shell) but also to determine the spatial
*
Corresponding author. Tel.: +55-21-2562-7470; fax: +5521-2562-7368. E-mail address: [email protected] (M.E. Poletti). 1 Ph.D. Scollarship from Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq).
structure of substances in general. Its application extends from basic physics to technology and medical appliances. Depending on the momentum transfer in the elastic scattering process, it is possible to describe the phenomenon using two different approaches. In the first one, for high momentum transfer, it is possible to use the free atom approximation in which no collective effect is considered. In the second approach, for low momentum transfer, it is necessary to take into account the positions of atoms and molecules in the sample, which is
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detectable through structures in the experimental angular distribution of scattered photons (also called ‘‘diffraction pattern’’ or more precisely ‘‘scattering profile’’). The precise limit between these two regions will depend on the sample composition and structure, temperature, the experimental setup geometry being used and on the accuracy of the experiment [1–3]. Although elastic (Rayleigh) cross-sections have usually been measured with radioactive monochromatic gamma-ray sources, it can also be done with X-ray generators. This kind of measurement is particularly effective in obtaining form factors for amorphous solids (plastic), liquids and biological samples, which are of great relevance in medical technology [4–9]. The experimental setup for this kind of measurements is not complex, but on the other hand, it is very difficult to analyze the data since the number of scattered photons detected includes error sources that should be considered in order to obtain precise and reliable elastic differential crosssection. Some of these error sources are: (i) a spread on the experimental scattering angle due to the geometric setup; (ii) scattering other than single scattering in the sample (multiple scattering, fluorescence, scattering from the container, scattering in the air and other background sources) and that must be eliminated either physically or by calculation; (iii) self-attenuation and polarization, that must be taken into account; (iv) normalization of the data since the measurements are not absolute; (v) values of elastic and inelastic scattering cross-sections of isolated atoms that must be known as accurately as possible, since they serve as a base for normalizing the experimental data. Generally, the methods reported in the literature to deal with these corrections can be divided into two categories: (i) analytic and (ii) Monte Carlo simulation. The later one incorporates complete geometry considerations (collimators, samples and beam size) and also all the scattering components. The analytic (or semi-analytic) approach usually considers primary scattering as isotropic, the incident beam as having a uniform profile, the sample as being immersed in the beam and also ignores the contribution of the container to the multi-
ple scattering. Generally attenuation and geometric effects are corrected, but multiple scatter very rarely [10]. Although some few authors have performed these corrections using computer simulation, Monte Carlo techniques are applicable to a wide variety of cases of practical interest [8,11]. They are necessary in cases where analytic corrections are impossible to calculate, for example nonuniform beams (X-ray beams usually have a nonuniform cross-section due to the sources and to shadows cast by slits), samples larger than the beam size, etc. The availability of such Monte Carlo programs permits considerable flexibility in the strategy for an experiment and also provides a quick test for the angular resolution of the system, taking into account the dimensions of the source, detector, sample and collimating system both in the incident and scattering beam. In this work we deal specifically with the analysis of X-ray scattering data including all aspects cited above. As an illustration, a suggested procedure was applied to scattering data obtained for a glandular breast tissue measured inside a cylindrical container. The same procedure is naturally also useful for any similar experiment [9].
2. Experimental aspects The considered experimental setup is present in Fig. 1(a). It is composed of a powder diffractometer operating in transmission mode in order to minimize the attenuation in the sample. The photon source is an X-ray Mo tube (Z ¼ 42, Ka ¼ 17:44 keV, Kb ¼ 19:6 keV) operating at 30 kVp, in order to avoid other harmonics of Mo Ka. The X-ray generator is equipped with a graphite monochromator in the incident beam to select the fluorescence line (Ka), producing a reasonable monochromatic incident beam (E ¼ 17:44 keV with a FWHM ¼ 0:19 keV, Fig. 1(b)). The energy distribution of the incoming beam (Fig. 1(b)) was measured with a silicon crystal analyzer (440). For each crystal position (in relation to the incoming beam) the detector (NaI (Tl)) ‘sees’ one photon energy according to the Bragg law. The beam size was limited by a vertical slit (0:75 8 mm2 ) placed
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Fig. 1. (a) Experimental setup and (b) photon energy spectrum incident on the sample.
in the exit of the monochromator. An additional set of S€ oller slits was placed after and before the samples and still another vertical slit collimator (1:7 8 mm2 ) was placed in front of the detector. The collimators were chosen in order to minimize the beam spread providing that just a small part of the sample be irradiated. Lines linking of incoming beam and the centers of sample and detector as depicted in Fig. 1(a) define the nominal scattering angle. The spectrometer system features an NaI (Tl) detector with the conventional electronic. The detector efficiency at this energy is 65% while the energy resolution is about 5 keV implying that the number of scattered photon includes elastic, inelastic and multiple scattering effects. The number of scattered photons was measured varying the scattering angle between 1.3° and 72° 1 ) with steps of 1=3 degree and (0:02 < x < 0:83 A with counting times around 60 s providing a maximum statistic uncertainty of 2%. For each sample, more than one data set has to be collected in order to test the stability of the system (in the considered experiment we performed three collections for each sample). We shall generally assume that the atomic arrangement of the sample is isotropic, implying that the scattering profile is symmetric around the primary beam. With this assumption it is sufficient, for a given energy, to consider just the one-
dimensional (1D) dependence of the elastic scattering with the scattering angle [10]. The biological sample, a glandular tissue, was measured inside a cylindrical acrylic container with nominal height 15 mm, external diameter 8.2 mm and wall thickness 0.1 mm (3% of the total diameter). The dimensions were chosen in order to provide sufficient single scattering events while minimizing the probability of multiple scattering. The container with the sample was placed on a rotating table performing one revolution in 10 s in order to ensure that all parts of the sample could be equally exposed to the beam. The commonly used geometry is a thin foil target, the reason being that attenuation corrections is easier in this geometry. That is not the most convenient geometry for samples with irregular surfaces, being the cylindrical form more useful. Cylindrical geometry has two advantages. First the attenuation in the sample varies very slowly and second it is possible to minimize surface irregularities by rotating the sample. The disadvantage is that the attenuation correction factor is difficult to calculate. Other sample characteristics that must be known are composition, density and linear attenuation coefficient. The elemental composition was measured using an Elemental Analyzer EA 1110 from CE Instruments (based on thermal conductivity providing the mass proportions of H, C, N and S) giving the following weigh fractions:
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H: 0.093, C: 0.184, N: 0.044, O: 0.678 and S: 0.001. The oxygen amount was obtained assuming that only H, C, N, O and S are present. The density was measured using the usual Buoyancy method (q ¼ 1:04 0:02 g=cm3 ). The linear attenuation coefficient was measured for the sample using the geometry depicted in Fig. 1 with h ¼ 0° and with photons of the same energy as in the scattering experiment, with a well-collimated beam (standard setup of ‘‘narrow beam geometry’’).
the lead sample, and the probability of multiple scattering in the air is very small, the contribution of the lead for additional background can be neglected. Combining these four factors we obtain the counting rate for scattering in the sample alone (N ),
3. Methodology
3.2. Correction for the beam polarization
There are many necessary steps in obtaining the single scattering cross-section. The first one is to subtract from the original data the number of photons originated from every other sources (background). The resulting data have to be corrected for polarization of the incoming beam and multiple scattering. The result is the number of single scattered photons, which can be related to the single scattering cross-section. Each step will be detailed below.
The beam produced by a monochromator crystal is partially polarized [1,2]. The partial polarization of the X-ray beam must then be taken into account because it will affect the subsequent scattering from the sample under study. In this case the collected data must be re-scaled by the factor, Cp , 1 þ jp cos2 h 1 þ cos2 h Cp ¼ ; ð2Þ 1 þ jp 2
3.1. Evaluation of the background components The background depends on the geometry of the experimental setup (collimators, volume of irradiated air, natural background, and scattering in the material used as holder or container) and comes from sources other than the sample. The background can be experimentally subtracted performing a series of measurements for a given scattering angle. In the present work four different acquisitions were made. The first acquisition comprises photons scattered by the samples plus photons scattered by the container plus scattering in the environment (NSþCþG ). The second acquisition (NCþG ) is performed with the empty container. The third (NG ) is the counting rate obtained without the sample and container and provides information about for the environmental scattering and background. The last one, NPb , is obtained with a lead target instead of the sample and provides information about ambient background which does not pass trough the samples. Since the beam hits just a small central region of
N ¼ ðNSþCþG NPb Þ T1 ðNCþG NG Þ T2 ðNG NPb Þ;
ð1Þ
where T1 and T2 are geometric transmission factors for sample and sample plus container, respectively.
where jp is defined as the square of the ratio between the parallel and perpendicular amplitude components of the photon polarization. Because most crystals are non-ideal, jp could not be accurately predicted and this is a limitation for the accuracy of the method. On the other hand this inaccuracy is not expected to be significant when compared to other sources of uncertainty and without loss of generality jp could be obtained from jp ¼ cos2 hm , where hm is the diffraction angle of the monochromator crystal. 3.3. Correction for multiple scattering The number of photons that undergo multiple scattering can be determined analytically only for double-scattering processes and for few specific setup geometries. In general, to evaluate multiple scattering (more than two interactions) it is necessary to use numerical methods. The most popular computational method is the Monte Carlo. In order to check the contribution of multiple scattering, we have written a Monte Carlo code. It is a 3D Monte Carlo code considering a finite size
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source, variable shape of collimators, variable sample geometry (slabs, cylinders, etc.) and a 3D detector. The code was compared with the EGS4 code [14] for chosen cases and a good agreement was achieved. To obtain MS it is necessary to consider some theoretical results for the crosssections as being true. For the total atomic crosssections, the code uses the XCOM data [12]. For the differential inelastic cross-sections it was used or the atomic incoherent scattering function (S) multiplied by the well-known formula of Klein– Nishina [15]. For the elastic scattering cross-section, our code may consider either the atomic form factor (F ) [15] or the molecular form factor [5,9]. The number of single scattered photons detected obtained after all these corrections (N corr ) may then be expressed as N corr ¼ N Cp1 MS ; ð3Þ
dr=dXðhÞ is the differential cross-section comprising inelastic plus elastic; dV ¼ dA dx, dA being the cross-sectional area viewed by the incident beam on the differential volume dV and dx is the voxel size. Considering that the differential scattering cross-section does not vary significantly over the range of scattering angles, comprising the source, target and detector volumes Eq. (4) can be written as dr ðhÞ N corr ðhÞ ¼ N0 nv dX elas Z Z dX eðEÞel1 ðEÞd1 el2 ðEÞd2 dV
where N is the number of detected photons corrected for background, Cp is the correction for polarization of the incoming beam, MS is number of photons that undergo multiple scattered.
ð5Þ
3.4. Determination of experimental differential scattering cross-section Following Kane et al. [13] the total number of scattered photons (single scattered) at a scattering angle h; N corr ðhÞ, is related to the differential scattering cross-section dr=dXðhÞ by an integral over the scatterer volume V and the scattering solid angle X, Z Z dr N corr ðhÞ ¼ ðhÞel2 d2 dV ; dX N0 eel1 d1 nv dX det sample ð4Þ where h is the scattering angle with respect to primary beam, N0 is the incident number of X-ray photons per cm2 , e is the detector efficiency; dX is the differential solid angle viewed by the detector (considering the collimator) and defined by the elemental volume in the sample; l1 and l2 are the attenuation coefficient before and after the scattering; d1 and d2 are the path lengths inside the sample before and after the scattering, respectively; nv is the number of atoms (or molecules) per cm3 ;
det
þ
sample
dr ðhÞ dX inel
Z
Z
dX det
0
eðE0 Þel1 ðEÞd1 el2 ðE Þd2 dV
sample
dr dr 0 ¼ N0 nv eðEÞ ðhÞ GðhÞ þ ðhÞ G ðhÞ ; dX elas dX inel
where dr=dXðhÞelas ðinelÞ corresponds to the mean elastic (or inelastic) cross-section for this range of scattering angles. If either the cross-section is slow varying or the angular acceptance is sufficient small, this quantity will approach the differential crosssection at the mean angle h. E is the energy of the elastic scattered photon and E0 is the energy of the inelastic scattered photon. Since the Compton shift for experimental the angular range is not very large, the detector efficiency was assumed to be the same for both elastic and inelastic scattered photons. GðhÞ and G0 ðhÞ are known as geometric attenuation factor for elastic and inelastic scattering. These factors take into account the attenuation of the incident and scattered beam through the sample volume (known as self-attenuation) and also the distribution of dX due to the geometry (geometric effect). In order to calculate GðhÞ and G0 ðhÞ the Monte Carlo code was modified in order to consider just the geometrical conditions without introducing any other physical constraint when sampling the photon paths during the scattering process. The code generates a normalized distribution (gðhÞ and g0 ðhÞ) of the attenuation correction for each possible scattering angle for a given measuring position (angle) due to finite size of sources, sample and detector. The integration of gðhÞ and g0 ðhÞ
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leads to the desired GðhÞ and G0 ðhÞ. Besides the attenuation correction, the code provides also the angular geometric dispersion and the mean angle for a specific nominal scattering angle. In the present experiment, like in most of high angular resolution experiments, it is required high beam fluency turning impossible to measure the incoming number of photons (N0 ) directly. The absolute efficiency (e) is also difficult to be measured. The alternative, in order to obtain the differential cross-section without knowing these quantities (Eq. (5)), is to perform some normalization with theoretical input in a region of momentum transfer where no interference is expected and the IAM was demonstrated to be valid [10]. Rewriting Eq. (5), N corr ðhÞ dr dr ðhÞelas GðhÞ þ ðhÞinel G0 ðhÞ ; ¼ K 1 nv dX dX ð6Þ 1
where K ¼ N0 eðEÞ. The quantity inside the brackets can be calculated theoretically in the frame of IAM (from here designated as ATheor ) as ! IAM IAM Fm2 dr dr Theor 0 Sm A ¼ NA q G þG ; M dXTh M dXKN ð7Þ 2IAM
Fm
=M ¼
X
ðwi =mi Þfi2 ;
ð7aÞ
ðwi =mi ÞSi ;
ð7bÞ
i
SmIAM =M ¼
X i
where nv was replaced by (NA q=M), NA is the Avogadro’s constant, q the tissue mass density, M its molecular weigh and dr=dXTh and dr=dXKN are the well-known Thomson and Klein–Nishina differential cross-sections, respectively. FmIAM and SmIAM are the molecular form factor and molecular incoherent scattering function, calculated both within the IAM approach. Expressions (7a) and (7b) are useful when the chemical formula of the molecule is not known, which is the case of biological samples. For these samples, the usual elemental analyses give relative weight and atomic mass of each element in the sample (wi and mi ,
respectively). fi and Si are respectively the atomic form factor and the incoherent scattering function of the ith element in the sample. The normalization factor (K) can then be obtained by averaging the ratio between the experimental data (N corr ) and the corresponding theoretical data (ATheor ) for x between xcritical (x value above which there is no more intermolecular interference) and xmax (the maximum x value measured in the experiment), n 1X N corr ðxi Þ 1 xcritical 6 xi 6 xmax : K ¼ ð8Þ n i¼1 ATheor ðxi Þ It should be noticed that since it is not possible to know the exact molecular formula of complex substances like human tissues, also the number of scattering molecules per unit volume (nv ) cannot be achieved. It is then useful to define a new quantity relating both nv and the differential crosssection and given by dr=dXnv ¼ nv dr=dX. Since it is not expected to exist a significant departure between theory and experiment for the inelastic scattering [11], the elastic cross-section times the number of molecules per cm3 , ðnv dr= dXelas Þ may then be obtained immediately by subtracting the theoretical prediction for the inelastic effect from Eq. (7), nv
dr KN corr G0 dr nv ¼ : dXelas G G dXinelas
ð9Þ
From this equation it is also possible to extract the experimental molecular form factors [9].
4. Results and discussion All corrections to be applied to the number of detected photons depend on the apparatus, experimental setup and energy of the incident beam. As an example, these corrections were applied to data obtained using as sample, glandular tissue in a cylindrical container. In the following we discuss the results obtained after each one of the corrections. 4.1. Background Fig. 2 shows the results of measurements performed to obtain the number of photons scattered
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Fig. 2. Relative number of photons scattered by: (1) the container filled with glandular tissue (solid line); (2) the empty container (dashed line); (3) without container in the sample position (dotted line) and (4) detected photons that did not come through the sample, obtained with a thick lead sample (dot-dot-dashed line).
just in the sample. The curve corresponding to the empty container (dashed curve) shows a structure due to the container material (acrylic) and has also contribution from scattering on the air and on the collimating slits (dot curve). The number of scattered photons reaching the detector and passing outside the sample (curve plotted with dot-dotdashed line) is very small (generally less than around 0.6%) when compared with the total number of scattered ones (solid line). For h larger 1 ) all components become than 11° (x ¼ 0:13 A very small, the resulting correction being around 3–5% of the total counting number.
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conditions (including container) and considering (i) possibility of elastic and inelastic scattering taking interference in account for the first one, (ii) possibility of elastic and inelastic scattering without interference and (iii) considering just the inelastic scattering. The results are compared in Figs. 3 and 4. As can be seen from the Fig. 3 the number of photons undergoing multiple scattering is 10% of the total number of photons detected. Around the considered energy it is not recommended to use just multiple inelastic scattering for multiple scattering. Fig. 4 depicts the ratio between each possibility and the most complete simulation (case (i)). Even though the differences between the two first approaches could be as big as 12%, the final results using either method to calculate the multiple scattering correction are different by less than 0.7%. As expected, the square symbols (situation (ii)) are similar (within the uncertainties) to the results obtained in situation (i) for h P 25° 1 ), the region where the interference (x P 0:30 A effects can be neglected. The results obtained considering just inelastic scattering (situation (iii), depicted as circles) are much smaller than situation
4.2. Multiple scattering A preliminary study was performed in order to choose the best way to consider multiple scattering. The first question is how important is the multiple scattering process itself and the second is how important is each effect (elastic with interference, elastic without interference and inelastic scattering) in the composition of multiple scattered. To answer these questions we performed calculations with the Monte Carlo code for glandular tissue, carrying out the complete experimental
Fig. 3. Number of scattered photons obtained for glandular tissue corrected for background and polarization (open circle symbol) compared with the expected number of scattered photons that undergo multiple scattering, calculated considering: (1) inelastic plus elastic scattering and considering interference effects (solid line); (2) inelastic plus elastic scattering without considering interference effects (IAM) (dotted line) and (3) just multiple inelastic scattering (dashed line).
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(i), the difference being larger at low scattering angles, where the elastic–inelastic and elastic– elastic processes are more probable than for larger
angles. The error bars correspond to the statistical uncertainty. Results obtained on other energies and geometries suggest that the number of photons that undergo multiple scattering (MS ) could be invariant with the scattering angle [11,16]. Our results (Fig. 3) show that a significant variation is obtained when elastic scattering is considered.
5. Geometric and attenuation correction
Fig. 4. Ratio between the number of multiple scattered photons calculated in different ways: MSI : inelastic plus elastic scattering and considering interference effects; MSX ð1Þ: inelastic plus elastic scattering without considering interference effects (IAM) (empty square symbols); MSX ð2Þ: just multiple inelastic scattering (dashed line).
The weight functions g and g0 , together with the angular spread, are shown in Fig. 5 obtained by fixing the detector at h ¼ 25° (a) and h ¼ 60° (b). At h ¼ 25° all curves are symmetric. At h ¼ 60° the curves for g and g0 are no long symmetric, but the angular spread is still symmetric around an angle slightly lower than the nominal one (0.13°), a minor shift that occurred at 25° (0.11°). All final results of cross-sections are given as a function of a corrected x, obtained with the mean scattering angle.
Fig. 5. Angular dispersion and weight function for elastic (gðhÞ – square symbols) and inelastic (g0 ðhÞ – circle symbols) single scattering at h ¼ 25° and h ¼ 60°.
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Fig. 6. Attenuation factors for elastic (GðhÞ) and inelastic (G0 ðhÞ) single scattering plotted as a function of scattering angle, obtained by: (1) Monte Carlo simulation (solid and dash line) and (2) analytical calculation ignoring beam profile and geometric effect (dot and dash-dot line).
We have compared our results for G and G0 with a widely used approximation that considers a monochromatic and parallel incoming beam together with a parallel outgoing beam. This approach allows the double-integration in Eq. (6) to be expressed in an analytical form [13], the method being usual for thin target geometry. In the present case, due to the complexity of the cylindrical geometry, a numerical method was used in order to obtain the analytical result for G and G0 (shown in the Fig. 6). These results agree (over the considered angular range) with the correspondent Monte Carlo ones within 3%.
6. Evaluation of the error bars The resulting experimental uncertainties in the measured differential elastic cross-section (vertical error bars) have been estimated by error propagation in Eq. (9) considering statistical uncertainty of counting rate, uncertainty from the Monte Carlo calculations and also from normalization procedures, assuming that all variables are noncorrelated. The statistical errors in N (Eq. (1)) range between 4% and 5%. The statistical uncertainties from simulation (MS and G, the latter including uncertainty in the sample thickness) are
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Fig. 7. Experimental elastic differential cross-section (multiplied by the number of molecules per volume), nv dr=dXelas , as function of momentum transfer, x, together the resulting vertical and horizontal error bar.
between 2% and 3%. The error from the normalization procedure (K) is around 3%. The resulting vertical error bars range between 5% and 7% and are displayed in Fig. 7, together with the obtained elastic differential cross-section times the number of atoms per volume. The horizontal error bar associated with the experimental momentum transfer is given by 2
2 1=2
Dx ¼ Af½ðE=2Þ cosðh=2ÞDh þ ½sinðh=2ÞDE g
;
where E is the photon energy, h is the scattering 1 keV1 ). The angular angle, and A ¼ 1=12:398 (A uncertainty was between 0.4° (at small-angle) and 0.8° (at high-angle) and includes inaccuracy of the mechanical sample positioning (0.02°). The resulting momentum-transfer uncertainties are of Dx ¼ 0:006 (at small-angle) and Dx ¼ 0:01 (at high-angle) which is sufficiently narrow to avoid overlapping of peaks in the experimental amorphous form factor. The results are also displayed in Fig. 7.
7. Conclusions We have presented an accurate method to obtain elastic scattering cross-section for amorphous samples from X-ray scattering experiments
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(scattering profiles) combining a series of complementary measurements and Monte Carlo simulations. Correcting geometric self-attenuation with Monte Carlo methods represents an improvement when compared with the standard analytical methods, which ignores beam profile and geometric effect, resulting in a difference around 3%. The results show that multiple scattering must be considered even in thin samples (for glandular tissue, in the considered geometry and thickness around 1=l, MS contribution could be around 10%). Another conclusion is that the multiple scattering in this energy depends on the scattering angle, but on the other hand, interference due to atoms or molecule positions is not important when considering multiple scattering. The process was applied for other geometries and materials, including thin foil and solid samples. The results should be published soon [9]. The method here described lead to a precise evaluation of elastic cross-section with trustable uncertainties for complex compounds, especially for material of medical interest, e.g. tissues and tissue-equivalent materials, creating new possibilities for material characterization and medical applications.
Acknowledgements This work was supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq), Coordenacß~ ao de Aperfeicßoamento de Pessoal de Nıvel Superior (CAPES) and Fundacß~ ao de Amparo a Pesquisa do Estado do Rio de Janeiro (FAPERJ). We would
like to thank Dr. Michael Farquharson for the careful revision and useful suggestions.
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