1. Introduction

With the advent of modern synchrotron radiation sources and the prevalence of nanotechnology, X-ray scattering technique capable of yielding structural details at the length scale of nanometer and beyond has been widely used in a variety of research fields, such as structural biology ^[1,2], colloidal science ^[3-5] and polymer physics ^[6,7]. It is proven to be one of the most powerful structural characterization techniques. In contrast to the complexity of data reduction for light scattering with lasers as a consequence of multiple scatterings ^[8], X-rays interact weakly with matters and thus provide a relatively simple path to the knowledge of particle sizes, shapes etc. However, the intensity of X-ray scattering decreases rapidly with wavevector transfer and falls quickly into the level of scattering background. A proper way to handle this background is crucial for accurate structural determination, as any miscalculation would affect greatly the net scattering intensity and the final results.

X-ray scattering background can be divided into two categories: inherent (incoherent scattering, sample compressibility) and external to the sample (environmental background, scattering from sample holders, etc.). The background is usually handled by doing two separate experiments: with and without samples under the same experimental conditions ^[9-11]. The first experiment records X-ray scattering intensity (I_b), which composes of the scattering signal from samples and the background. The second experiment is done for subtracting the background scattering (I_b) from I_sb, so as to obtain the net scattering intensity I_s (X-ray scattering intensity from the sample itself):

where r_bg is the scaling factor. The conventional method to obtain r_bg is to measure the ratio between the attenuated fluxes of the direct beam under the same conditions through the sample holder with and without samples inside it ^[12-14]. This requires two ionization chambers installed directly in front and back of the sample, as shown in Fig. 1. However, gases filled in the ionization chambers bring additional background scattering and produce parasitic scattering. Also, instability of the incident X-ray beam will influence the accuracy of r_bg measured in a transmission experiment.

Fig. 1Full-screen View

A typical X-ray scattering setup with the sample positioned between two ionization chambers, and an area detector to record the scattering.

2. Iterative method

Here we introduce an iterative approach to handle X-ray scattering background and derive the scaling factor. This method is based on the assumption that the theoretical model used to interpret the scattering data can correctly represent the underlying physical structure of a sample. Thus, a scattering curve, calculated using a theoretical model to fit the experimental data with small divergence, gives a good interpretation to the data. And the ratio between the scattering intensities of any two points on the calculated and measured curves with the same wavevector transfer (q) values will be essentially the same. Our method applies this principle in a more effective way. Due to background noise, experimental data always show fluctuations, so, instead of calculating the ratio between single data points, we take ratio of the averaged values of neighboring points. In practice, three points at low q usually with good signal-to-noise ratio, and three points at high q without fast changing features like oscillations, are taken into account. We find that the mean of three data points is sufficient for the present study, while more points are required for experimental data with low signal-to-noise ratios in order to attain reliable results.

First, we compute the ratio r_hl of the averaged values for the theoretical scattering curve,

where I_calc(h3) and I_calc(l3) are average values of three data points at high and low q values, respectively. Assuming the theoretical curve gives a good fit to the experimental data, one can write

where I_s(h3) and I_s(l3) are average values of three points at high and low q of the net scattering curve, respectively. Substituting Eq. (2) into Eq. (3), the scaling factor r_bg can be obtained,

The net scattering signal, I_s, can be obtained by substituting r_bg into Eq. (1). This method can automatically separate scattering signal from background in the data fitting process and requires no ionization chamber readings, hence no parasitic scatterings nor influence of incident beam fluctuations. The selection of a suitable theoretical model to fit experimental data is essential for this method. Any poor interpretation to the sample structure will lead to unacceptable results. Recently, this method was applied successfully to the structural reconstruction of Pt-coated Au dumbbells ^[15] and Au plasmonic nanostructures ^[16].

3. Experimental and theoretical considerations

To validate the feasibility of the iterative background subtraction method we proposed, small angle X-ray scattering (SAXS) experiments were carried out on Beamline BL16B1 ^[17] of Shanghai Synchrotron Radiation Facility (SSRF). Two types of Au nanoparticles in small (Sample-S) and large (Sample-L) diameters were dispersed in a solution and measured separately with the incident X-ray beams at 10 keV, using an MAR165 area detector.

X-ray scattering amplitude from a sphere can be written as ^[18,19]

where r_e is the Thomson scattering length, ρ is the electron density difference between gold nanospheres and the solvent, r is the radius of gold nanospheres, and the Φ(x) function stands for

The detectable X-ray scattering intensity, I(q), is a total summation of the modulus square of F(q) multiplied by their corresponding statistical probability due to the size distribution of nanospheres,

where G(r) = exp[−(r−μ)²/(2σ²)] /[σ(2π)^1/2] is the Gaussian distribution of the nanospheres radius r with two variables of the mean value μ and the standard deviation σ.

In the fitting process, a target function J to assess the similarity between the experimental net scattering intensity I_s(q) and the calculated intensity I_calc(q), is given as

where the second term is the absolute value of normalized cross-correlation ^[20] and the over bar denotes the average. The J function has a range from 0 to 1, where “1” referring to no linear correlation while “0” to a strong linear proportional correlation or a perfect fit for the calculated and experimental data. A global optimization algorithm called Covariance Matrix Adaptation Evolution Strategy (CMAES) ^[21] was applied to optimize the model parameters. There are two fitting parameters μ and σ for conventional and iterative methods.

4. Results and discussions

The fitting results for Sample-S and Sample-L are showed in Fig. 2 with their parameters listed in Table 1. For Sample-S, the J function values obtained by the iterative method about six times smaller than that from the conventional method (Table 1), implying structural parameters obtained from the iterative method provide much better interpretations to the data than the conventional method (Fig. 2a). For the conventional method, the theoretical scattering curve deviates for q>1.15 nm⁻¹ from the experimental data, though they fits well for q ≤1.15 nm^–1. So, the ascribed r_bg = 0.99 in Table 1, obtained conventionally, is inaccurate. It has a strong influence for data points at large q values, because at this region, I_b and I_sb have the same magnitude and any small change to r_bg will make a big difference on the final data curve. For sample-L, the values of J and r_bg obtained by both methods are close to each other, and the theoretical and experimental scattering curves are in good agreement (Fig. 2b). These results demonstrate that the iterative method is effective in handling SAXS background, particularly for data points at high q region. In contrast, the conventional method is not always valid, because r_bg obtained directly from ionization chamber readings may be unreliable.

Fig. 2Full-screen View

Comparison of two scattering background-subtraction methods used in SAXS data fittings.

In Fig. 3, the particle size distributions of the two samples obtained by SAXS measurements, using the two background subtraction methods, are compared with the transmission electron microscopy (TEM) results. Some parts of the TEM images of Sample-S and Sample-L are depicted in the inserts of Figs. 3. For Sample-S, the mean radius and standard deviation are 3.06 and 0.41 nm as obtained from the Gaussian fit to the TEM distribution histogram. As shown in Fig. 3(a), the particle size distributions by the iterative method agree better with the Gaussian fit from the TEM data than the conventionally obtained particle size distributions, which shift slightly towards larger radii. For Sample-L, the mean radius is (4.33±0.38) nm as obtained from the Gaussian fit to the TEM data. The particle size distributions derived from both methods agree very well with the Gaussian fit obtained from the TEM histogram as shown in Fig. 3(b). The details of the particle structural parameters are summarized in Table 1.

Fig. 3Full-screen View

Comparison of the Au NP size distributions obtained from TEM measurements and SAXS data fittings using different background subtraction methods for Sample-S (a) and Sample-L (b). The insets show parts of TEM images of two types of Au NPs.

Next, the influence of scaling factor deviation from a proper value on the final fitting results is demonstrated with Sample-S. The fitting results with the scaling factor increased by 5% (r_bgⁱ⁵) and 10% (r_bgⁱ¹⁰), and decreased by 5% (r_bg^d5) and 10% (r_bg^d10), with respect to the value obtained from the iterative method (r_bg^IM) are shown in Fig. 4. Using r_bgⁱ⁵ and r_bg^d5, the J function values are about fivefold than those obtained from r_bg^IM, and the fits are not satisfactory (Fig. 4a). For r_bgⁱ¹⁰ and r_bg^d10, the J function values become even larger, and the fittings are not acceptable. Especially for r_bgⁱ¹⁰, the net scattering intensity drops sharply at the large q values (Fig. 4b) due to a large scaling factor being used to subtract the scattering background. The particle size distributions obtained using the four scaling factors are shown in Fig. 4(c). It can be seen that the particle radii obtained with r_bg^d5 and r_bg^d10 shift slightly towards large values as compared to those from r_bg^IM. For r_bgⁱ⁵ and r_bgⁱ¹⁰, the particle radii shift towards smaller values, especially for r_bgⁱ¹⁰. As summarized in Table 2, the results clearly illustrate on a simple system the importance of accurate determination of the scaling factor for SAXS background subtraction. The net scattering intensities depend sensitively on the scaling factor especially at high q regions and any slight deviation from an adequate value would significantly affect the final results.

Fig. 4Full-screen View

The fitting results for Sample-S with the scaling factor altered by 5% (a) and 10% (b) with respect to the value obtained from our method. The experimental and theoretical data are plotted with blue dots and red lines, respectively. (c) Particle size distributions of Sample-S obtained using different scaling factors for background subtraction.

5. Conclusion

In summary, we have proposed an iterative X-ray scattering background subtraction method with its feasibility and reliability demonstrated by SAXS studies of Au NPs. It is better than conventional method in fittings the experimental data in good agreements with the TEM results. Also, the influence of the scaling factor deviation on the final results is examined. While we demonstrate in the present study the validation of the iterative method to the Au nanoparticle model system, it has been successfully applied to the structural reconstructions of Pt-coated Au dumbbells ^[15] and Au plasmonic nanostructures ^[16]. In contrast to the microscopic techniques such as TEM where an object or a small part of the sample is magnified and investigated, for X-ray scattering the whole illuminated sample volume is studied and consequently the averaged structural parameters are obtained. For an unknown sample, both methods are required in order to capture a complete picture as they are complementary to one another. Moreover, X-rays can penetrate through gas, liquid and solid, which makes them ideal sources to carry out in situ measurements with specimens in their natural environments. Whereas it is certainly advantageous to explore sample properties under their genuine conditions, the surrounding media would bring additional scattering background and any miscalculation can render the final results unreliable. In this regard, the iterative method provides an effective way to handle X-ray scattering background and can be generally applied to a variety of scattering experiments. It will not only be beneficial to studying samples in gas or liquid environments but also resolving weak scatterings from samples like surfactant and polymer micelles and biological proteins.