research papers
Xray nearfield speckle: implementation and critical analysis
^{a}Department of Physics, Yale University, New Haven, CT 06511, USA, ^{b}Department of Applied Physics, Yale University, New Haven, CT 06511, USA, and ^{c}Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA
^{*}Correspondence email: xlu@bnl.gov
The newly introduced coherencebased technique of Xray nearfield speckle (XNFS) has been implemented at 8IDI at the Advanced Photon Source. In the nearfield regime of highbrilliance synchrotron Xrays scattered from a sample of interest, it turns out that, when the scattered radiation and the main beam both impinge upon an Xray area detector, the measured intensity shows lowcontrast speckles, resulting from interference between the incident and scattered beams. A micrometerresolution XNFS detector with a high numerical aperture microscope objective has been built and its capability for studying static structures and dynamics at longer length scales than traditional farfield Xray scattering techniques is demonstrated. Specifically, the dynamics of dilute silica and polystyrene colloidal samples are characterized. This study reveals certain limitations of the XNFS technique, especially in the characterization of static structures, which is discussed.
Keywords: Xray; near field; speckle; spectroscopy; scattering.
1. Introduction
Although smallangle Xray scattering (SAXS) and Xray photon correlation spectroscopy (XPCS) have succeeded in exploring the structure and dynamics of many interesting systems, the length scale of the observable systems is generally limited to a range from several nanometers to 100 nm, corresponding to a wavevector range of 10^{3} Å^{−1} to 0.1 Å^{−1} (an angular range of 0.1° to 10°) (Dierker et al., 1995; Mochrie et al., 1997; Pontoni et al., 2003; Falus et al., 2004; Narayanan et al., 2006; Lu et al., 2008). Special difficulties are encountered when exploring the lower limit of the angular range, since to isolate the weak scattering from the strong direct beam it is necessary to block the direct beam and extraneous scattering from slits, etc., which sets a boundary for the smallest detected angle. In order to probe longer length scales with Xrays, a Bonse–Hart camera is typically used, which can access a wavevector (q) range of 10^{4} Å^{−1} to 0.1 Å^{−1} (Diat et al., 1995; Ilavsky et al., 2002; Narayanan et al., 2001). A conventional Bonse–Hart camera is onedimensional collimated which is not suitable for anisotropic samples. At a cost of reduced scattering intensity, there are several papers describing a twodimensionalcollimated Bonse–Hart camera (Bonse & Hart, 1966; Konishi et al., 1997; Ilavsky et al., 2009). They demand a deconvolution procedure and are inefficient in comparison with the areadetectorbased method available for larger wavevectors. The scanning procedure also makes it difficult for timeresolved measurements. In addition, previous works (EhrburgerDolle et al., 2001; Shinohara et al., 2007) show that ultrasmall angles could be achieved by using a very long sampletodetector distance or a very small beam stop. In this case, additional interpolations with SAXS data are required because of the limitation of the field of view. The recently introduced coherencebased Xray nearfield speckle (XNFS) technique potentially offers an improved means of studying characteristically large length scale structures with Xrays (Cerbino et al., 2008). In addition, XNFS is able to extend Xray measurements to wavevectors (length scales) at least an order of magnitude smaller (larger) than may be achieved using a Bonse–Hart camera.
The principle of XNFS is as follows. When coherent or partially coherent radiation impinges on a disordered material consisting of a number of scatterers at random locations, a random set of phase shifts will be induced on the scattering beam. As a consequence, a grainy pattern will be observed in the scattered beam a certain distance away from the material. This pattern is called a speckle pattern (Sutton et al., 1991). In the nearfield region, under conditions where the scattered radiation and the transmitted beam simultaneously impinge upon an Xray area detector, highquality speckles can also be observed resulting from coherent interference between the incident and scattered beams. This speckle is called Xray nearfield speckle (XNFS) in analogy to the nearfield speckle (NFS) that was initially exploited using laser sources (Giglio et al., 2000).
If, instead of a laser, one uses a highbrilliance Xray source, it then becomes possible to study dense, optically turbid and/or absorbing media, in a range of length scales where no other Xray or optical methods are applicable. To date there exists a single manuscript describing the extension of NFS into the Xray regime, by Cerbino et al. (2008). They showed that the spatial power spectrum of XNFS is in principle simply and directly related to the sample's [S(q)] in the range of wavevectors from 10^{5} Å^{−1} or less to 10^{3} Å^{−1} or larger. Equivalently, XNFS measures the density–density correlation function [g_{1}(r)] from length scales of 6 ×10^{4} Å or more to 1 ×10^{3} Å or less. In addition, the evolution of the heterodyne speckle pattern in time determines the sample's intermediate scattering function [S(q,t)] and its spatial Fourier transform, g_{1}(r, t). Here, we will demonstrate the implementation of XNFS measurements at beamline 8IDI at the Advanced Photon Source (APS) at Argonne National Laboratory, and will explore and discuss the utility and drawbacks of this method for studies of colloidal suspensions.
2. Basic theory
In this section we present a derivation of what we may expect to measure in XNFS experiments. We envision a sample with density and a detector located in the plane z = , so that and specify a given detector pixel. Then, we may write for the amplitude scattered by volume element dx dy dz at (x, y, z) to the detector pixel at at time t,
where r_{0} is the Thomson radius, a_{0}exp[ikz − − ] is the amplitude of the incident wave at z, and
and are the real and imaginary parts of the Xray
respectively.For a sufficiently uniform sample, for which the zintegrals of and are independent of x and y, and ignoring the phase part, we may write
where is the Xray absorption length. Equation (1) represents a quite different regime than that used to interpret Xray imaging experiments. Such imaging experiments instead rely on the x and y dependence of and to create an image of the sample. Henceforth, we will neglect explicit mention of absorption, but otherwise take equation (1) as our expression for the scattered amplitude.
In the nearfield regime, in which scattered Xrays interfere with the incident beam within the coherence area of the incident beam, we are necessarily concerned with values of and that are of the order of 100 µm or less, and values of that are of the order of several millimeters or more, so that and . It follows that
and, therefore,
To determine the total amplitude scattered to (), it is simply necessary to integrate over the volume (V) of the sample, i.e.
Heterodyne nearfield speckle involves interference between the scattered beam and the incident beam of amplitude . Thus, the intensity at time t recorded at () is
where at the detector = and we have taken .
Therefore, the measured intensity is
where and are the real and imaginary (absorptive) parts of the electron density, respectively.
Equation (6) implicitly assumes perfect transverse coherence. To incorporate the effect of a finite transverse coherence length, it is necessary to introduce a mutual coherence function,
where and are the transverse coherence lengths in the x and ydirections, respectively. Incorporating the effect of partial coherence, equation (6) becomes
The first term of (8) is constant and the second term of (8) is a convolution in terms of realspace variables. Therefore, in Fourier space, the first term becomes a function at the origin, while the second term becomes a product. Therefore, in terms of the Fourier transform variables q and p ( and ), in the realistic case that the zvariations in the sample density occur on length scales less than k/q^{2}, which is typically hundreds of micrometers or more, it may further be shown that the Fouriertransformed intensity is given by
where is the thickness of the sample, is mixed in real and
and the phase factor is equal toNote that the tan^{1} terms in the phase factor describe an onaxis phase jump of a focused beam (Gouy effect) (Gouy, 1890) and underline the well known fact that field correlations propagate as the radiation field does. So these terms account for the change in the phase in the interference between the scattered beam and transmitted beam.
Equation (9), which stands as a onedimensional convolution, may be further simplified by the following argument: the zvariations in occur on length scales set by the sample's structure, namely of the order of tens of micrometers or less. On the other hand, the zvariations in the remainder of the integrand occur on a length scale given by k/q^{2}, which is typically many hundreds of micrometers or more. Therefore, in the integrand it is permissible to replace each by its mean value, i.e. by its zero Fourier component divided by the sample thickness, i.e. ≃ . This factor, which is now independent of z, may be then taken outside of the integral, yielding
Changing variable , equation (10) becomes
The sample thickness is about the diameter of the capillary equal to 0.7 mm, which is much smaller than the sampletodetector distance s ranging from 53 mm to 203 mm. Thus, we assume that the integrand varies negligibly within the range of z of the sample. As a result, equation (11) can be further simplified as
Here the term + s^{2}) + + s^{2})]} describes the effect of the partial coherence of the source beam on the speckle intensity. It is a product of two Gaussians with variances = . The full width at halfmaximum (FWHM) of the Gaussian distribution is . Fig. 1 plots the FWHM of coherence of the source beam versus (a) sampletodetector distance s and (b) coherence length . From Fig. 1(a) we observe that w_{q} decreases as s increases until it reaches a constant value at a certain distance,
which is the usual nearfield condition, called the Fresnel condition. However, for XNFS that requires that the way scattered radiation falls onto the sensor duplicates the actual angular distribution of the scattered intensity, a much stronger condition should be satisfied (Giglio et al., 2000),
where a is the size of the scattering particle. With this condition the source beam fills the whole field of view and the speckle size is related to the actual size of the probing material. In Fig. 1(b), w_{q} is plotted versus coherence length for different values of s. For the estimated coherence lengths at 8ID indicated via the dashed lines in (b), in principle, one expects to observe an sdependent change in the twodimensional speckle intensity. However, in reality, owing to the limited spatial resolution, which in turn limits the q range and, more critically, the sensor response (Alaimo et al., 2009), we have not been able to observe this sdependent variation, as we discuss in more detail below. When and as we expect at 8ID, then equation (12) may be simplified even further,
where the term is introduced as the spatial coherence transfer function by Cerbino et al. (2008).
Introducing = , which is always small, except at Xray energies near an as
and = , we may rewrite (15)where p and q are the wavevectors obtained in the x and ydirections, respectively, by numerically Fourier transforming the CCD image, I_{0} = , is the sample thickness, is the electron density in Fourier space, and where the latter equality defines the transfer function T(q,p). It is worth emphasizing that the transfer function with = (q,p) is written as
Equation (16) immediately allows us to calculate the static in terms of measured quantities. Specifically,
Similarly, it is straightforward to show that, in the context of XNFS, the normalized IFS is
According to (19), g_{1} is independent of the transfer function , and in turn does not depend on the sampletodetector distance s.
In summary, the intensity measured in the nearfield speckle experiments is proportional to the density of the sample rather than the modulus squared of the density as in conventional farfield speckle experiments like XPCS and SAXS. Thus, the time autocorrelation of I(Q) gives directly the intermediate scattering function g_{1} [equation (19)]. When the delay time is chosen to be zero, we obtained a quantity proportional to the static S(Q) [equation (18)] times the NFS transfer function .
3. Detector design
High spatial resolution and high
are key goals for imaging the Xray speckles in XNFS experiments. In XNFS experiments, in order to resolve micrometersized particles, it is necessary to employ detectors capable of resolving on the micrometer scale. A typical Xray imaging detector consists of a crystal Xray scintillator, a microscope objective and a fast highresolution largedynamicrange CCDbased camera. The scintillator converts Xrays into visible light; the objective collects the visible light and magnifies the visiblelight image; and finally the CCD camera records the image. Our aim was to design a detector with the best combination of scintillator and objective to achieve the optimal combination of spatial resolution and for XNFS experiments.A key characteristic of an objective is its numerical aperture (NA). The NA of an objective defines the largest angle of light acceptance as well as the lightcollecting power. The ^{2}. Thus, a large NA objective is necessary for high It is common to use immersion oil of high (n = 1.515) between the front lens of the objective and the scintillator to achieve a high NA. We employed a Nikon Plan Fluor 40× oil immersion microscope objective with NA = 1.3, a working distance of 0.2 mm and a field of view of diameter 0.67 mm. This objective uses an infinityfocused optical system with a reference focal length of 200 mm. In our case, with the implementation of a tube lens with adjustable focal length from 25 mm to 150 mm, the 40× objective gives a real magnification of 5–30 times.
scales as (NA)Generally, taking into account both the effects of diffraction and depth of focus, the spatial resolution [R] as a function of NA is given by (Martin & Koch, 2006)
where p = 0.18 µm and q = 0.075 µm are constants obtained by numerical simulations (Martin & Koch, 2006) and is the Xray absorption length of the scintillator. Based on (20), one can plot R versus NA for different , as shown in Fig. 2. It is clear from Fig. 2 that, in order to achieve a spatial resolution of a micrometer or less with high (NA 1.0), we have to choose a scintillator with an Xray absorption length of 10 µm or less.
Besides the Xray absorption length, several additional characteristics of the crystal scintillator are critical for Xray imaging, including high Xray n = 1.515); and a small thickness to minimize spherical aberration. At a minimum, the scintillator thickness should be smaller than the working distance of the objective (0.2 mm), so that the objective can focus to the upstream side of the scintillator.
high light yield; the emission wavelength being compatible with CCD readout (400 nm–700 nm); a similar to the immersion oil (As shown in Table 1, one potential candidate is YAG:Ce (Y_{3}Al_{5}O_{12}:Ce), which has been widely used in Xray imaging detectors. However, its Xray absorption length at 7.44 keV is about 25 µm, which is not suitable for a submicrometer resolution detector with a NA = 1.3 objective (Fig. 2). Another candidate is CdWO_{4}, whose Xray absorption length at 7.44 keV is only 6.5 µm. However, it is very difficult to obtain thin crystals of CdWO_{4}. It easily breaks before being thinned to the desired thickness (<0.2 mm) owing to its (010) cleavage plane. In addition, the CdWO_{4} (n = 2.2) is much different from that of immersion oil (n = 1.515), which will induce a relatively large spherical abberation. In our setup, we use LYSO (Lu_{1.8}Y_{0.2}SiO_{5}) which appears to be the most appropriate candidate overall. Its Xray absorption length at 7.44 keV is 9.5 µm, slightly larger than the Xray absorption length of CdWO_{4} but still good enough to produce high spatial resolution. Its light yield is 32 photons per keV, much higher than both CdWO_{4} and YAG:Ce. Its (n = 1.81) is not too far from that of immersion oil inducing less spherical abberation than CdWO_{4}. In addition, we found a manufacturer providing twosided polished LYSOs with a thickness of 0.15 mm, just appropriate for our objective, although even thinner would lead to reduced spherical aberrations (Koch et al., 1998; Born & Wolf, 1999).

4. Experimental setup
XNFS measurements were carried out at beamline 8IDI of the APS at Argonne National Laboratory, using Xrays of energy 7.44 keV and half the source beam size available at the 8IDI hutch, about 0.5 mm × 0.5 mm. Fig. 3 shows a sketch of the optical setup located in the 8IDI hutch. From right to left we have the beam source, the sample stage, the scintillator, the microscope objective and the CCD camera. In XNFS experiments, no spatial and spectral filtering of the direct beam are required (Cerbino et al., 2008). We inherit the exiting setup for XPCS which gives an energy resolution of ≃ 3 ×10^{4} and remove all the slits letting the full beam impinge onto the sample, and record the interference pattern of the transmitted and scattered beams by means of our detector placed at distances from the sample ranging from z = 53 mm to 203 mm.
As described before, we use twosided polished LYSO (Lu_{1.8}Y_{0.2}SiO_{5}) with a thickness of 0.15 mm to convert Xrays into visible light and a Nikon Plan Fluor 40× oil immersion microscope objective with a numerical aperture of NA = 1.3 to magnify the image. Both of them are mounted on a piezoelectric stage which has a mechanical manual adjustable coarse travel range of 4 mm and a piezoelectric travel range of 20 µm with a resolution of 20 nm. As a result, by carefully tuning the distance between the objective and the scintillator, we are able to focus the image which is in turn magnified 5 to 30 times by a tube lens with adjustable focal length and then recorded by a `CoolSNAP' CCD camera made by Photometrics. The camera features 1392 × 1040 pixels of size 6.45 µm × 6.45 µm and a maximum frame rate of 56 Hz. All the images soobtained are subsequently cropped to 1024 × 1024 pixels for convenience of twodimensional Fourier transformations in the data analysis.
There are several contributions to the detection resolution.
(i) The resolution of the scintillator, for Xrays incident onto the scintillator at a single point. According to Koch et al. (1998) and Martin & Koch (2006), this is typically 0.1 µm for 7–8 keV Xrays, smaller than optical limits on the resolution.
(ii) The resolution determined by the diffraction limit and the defect of focus of the objective and the scintillator via equation (3). With an objective with NA = 1.3 and LYSO with an Xray absorption length of 9.5 µm, we obtain a spatial resolution of 0.98 µm.
(iii) The reduction in resolution caused by the spherical aberration due to the use of a scintillator of mismatched et al., 1998; Born & Wolf, 1999). Since our objective is corrected for spherical aberration and the additional spherical aberration induced by replacing the glass coverslip by the scintillator can be compensated for by adjusting the oil thickness, this factor is not critical to the spatial resolution.
which is proportional to the thickness of the scintillator and the cube of NA (Koch(iv) The (demagnified) size of the CCD pixel, if too large, could limit the resolution. For a magnification of 30, the CoolSNAP can resolve a length scale as small as 6.45/30 = 0.215 µm which does not limit the resolution. As a result, the spatial resolution of the detector should be largely determined by factor (ii), which is about 1 µm.
Hence, we are able to estimate the maximum q range. With r_{min} = 0.98 µm, in q_{max} = ≃ 4 ×10^{4} Å^{−1}. On the other hand, the lower limit of the wavevector q_{min} should be determined by the largest accessible length scale. In principle, the largest length scale is the size of the measured scattering image equal to 1024 ×0.215 µm = 0.22 mm, so q_{min} = mm ≃ 3 ×10^{6} Å^{−1}. However, practically our data show an identical low q profile that is independent of which sample is being studied and of the sampletodetector distance. This is likely to be due to the beam structure on long length scales. So, the realistic useful q_{min} is of the order of 10^{5} Å^{−1}. Nevertheless, the qrange achieved is at least a decade below the range accessible to the conventional XPCS experiments.
5. Silica 0.45 µm
The first sample we measured at 8IDI, as an initial test of our XNFS setup, was a 10^{8} s^{1}. The static peak of this sample is expected to be located around 10^{3} Å^{−1}, which means that the sample has a uniform scattering profile in the q range accessible in our XNFS setup [i.e. S(q) ≃ a constant]. In other words, the intensity profile I(q) we measured from this silica suspension should simply result from the transfer function T(q) [equation (16)]. Thus, it should be an ideal sample to examine the sampletodetector distance (s) dependence of T(q).
of silica particles of diameter 0.45 µm and a () around 0.05 in water. The sample is injected into a 0.7 mmdiameter boron glass capillary for Xray measurements with an energy of 7.44 keV and a ofIllustrated in Fig. 4(a) is a typical raw image of the silica suspension measured at z = 53 mm. The image contains 1024 × 1024 pixels with a pixel size of d_{pix} = 6.45 µm. The magnification of the detector is set to 30. Thus this image corresponds to a region of size 0.22 mm × 0.22 mm in the sample. The speckle pattern appears quite obscure and weak owing to the largescale static background. After averaging over 1000 frames, the speckle pattern is washed away leaving the static beam profile fluctuation unchanged, as shown in Fig. 4(b). In order to remove the largescale static fluctuations, we perform a normalization of each raw image by dividing each one with an average image, averaged over 1000 frames, as shown in Fig. 4(b). Fig. 4(c) presents one example of the resultant image, which reveals a clear uniform speckle pattern [I(x,y)]. Next, a twodimensional discrete Fourier transform is performed via
which produces the qspace image, as shown in Fig. 4(d). Here, N = 1024. As a result, the corresponding q coordinates are given by
where is the size of one pixel, equal to d_{pix}/30 = 0.215 µm.
We report in Fig. 5 several examples of the magnified Fouriertransformed image corresponding to the region inside the dashed lines in Fig. 4(d). Different panels are obtained at different sampletodetector distances: (a) 53 mm, (b) 103 mm, (c) 153 mm and (d) 203 mm. In each image, prominant fringes can be seen. It is clear that the fringes become finer when the detector is moved away from the sample. This agrees with the theoretical prediction that the transfer function is proportional to a sine term whose frequency depends on the sampletodetector distance [equation (17)]. Note that the rings of the Fourier transformation are not azimuthally uniform. This effect may be related to asymmetry in the coherence of the source beam. However, the envelope of the asymmetry has no obvious sdependence, in contrast to what may be expected on the basis of equation (12). Thus, we do not understand this asymmetry in detail. In fact, examination of these data (Fig. 5), in the light of equation (12), suggests that the predicted effect of a finite coherence length is not playing a role in determining these data, presumably because the width of the Gaussian in equation (12) is greater than our accessible qrange of q_{max} ≃ 2 ×10^{3} Å^{−1}, even for the largest values of s studied. Alaimo et al. (2009) showed that by far the largest contribution to the q decay of the speckle power spectrum is due to the sensor transfer function. It is highly possible that these asymmetry rings are due to the sensor response. This is supported by the slightly elongated speckles in the direction orthogonal to that of the power spectra, as shown Fig. 4(c). Ideally, an accurate sensor transfer function should be obtained when the sample is placed close enough to the sensor. However, owing to the limitation of our setup, 53 mm is almost the closest distance we can reach.
To quantify , we plot in Fig. 6(a) the azimuthally averaged intensity profile (symbols) versus wavevector (q) for different values of s varying from 203 mm to 53 mm with a decrement of 10 mm. The thick solid lines in Fig. 6(a) are the fits of I(q) to the relation
with Q = = (q^{2} + p^{2})^{1/2} and T(Q) as a simplification of (17),
which consists of a product of three terms. The first factor is additional to equation (17) in order to take into account the contributions from the sensor transfer function (Alaimo et al., 2009); the second term derives from the partial coherence of the incident beam. How the coherence length enters is somewhat counterintuitive: it is proportional to the width of the Gaussian term in However, this term is set to unity for fitting, since it plays no significant role in the accessible q range, as we observed in the context of Fig. 5. The third term is a sine function that describes the fringes produced by the interference of the scattered beam and incident beam. In the Xray domain, this term is called the phasecontrast transfer function, and is related to the socalled Talbot effect in imaging measurements (Cerbino et al., 2008). Note that we assume = = for the simplification of a phase factor in the sine function.
Besides the contribution of T(Q) and background noise, the second term in (23), in the form of a stretched describes another experimentally significant contribution to the intensity profile: namely multiple scattering. The existence of multiple scattering is evident based on three observations. Firstly, the sinesquared term goes to zero periodically, which should make the intensity profile exhibit minima with the same magnitude. However, the measured minima decrease with q. The second piece of evidence pointing to the importance of multiple scattering comes from the dynamic data (see later), which display a qdependent decay rate and exponent that mirrors and antimirrors the form of T(Q) [Figs. 7(c) and 7(d)]. This indicates that we are measuring faster dynamics owing to multiple scattering where single scattering vanishes. Thirdly, very strongly scattering samples, i.e. deliberately multiplyscattering samples, show no minima at all.
Hence, we fit the intensity data in two steps. In the first step we focus on multiple scattering. Note that the singlescattering term vanishes at values of q satisfying (1/2)[(Q^{2}s/k) + = , where n = 1, 2, 3,…. As a consequence, we extract the intensity at the Q values corresponding to those minima in I(Q) and fit them with only the multiplescattering term plus the background. The v is fixed to 500 µm. The amplitude A_{2} and the exponent were allowed to vary. The resultant multiplescattering term is plotted as the thin solid lines in Fig. 6(a). The bestfit exponents are plotted in Fig. 6(b) versus the sampletodetector distance s. fluctuates around 0.345, giving an empirical stretchedexponential form for the intensity of the multiple scattering. Next, the remaining intensities after subtracting the filled multiple scattering and background are fitted with the transfer function [equation (24)]. We use the measured sampletodetector distance s and try to find one set of w, and that works for all the values of s. The only varying parameter is the overall amplitude. The set w = 1.8 µm, = 163 µm and = 0.026 yields a good fit for all values of s studied, as shown by the thick solid curves in Fig. 6(a). , the scale of the sensor transfer function, is about 11 µm, 17 pixels on the detector (Cerbino et al., 2008; Alaimo et al., 2009). The small value of indicates that the Xray absorption for this silica sample is essentially small. In general, the fitting reproduces the data with few fitting parameters, confirming the theoretical relation between the sampletodetector distance s and the transfer function T(q), and consequently confirming the feasibility of our experimental setup.
With the same principle of XPCS, the fluctuations of nearfield speckles should reflect the dynamics of the sample. As shown in equation (19), the time autocorrelation of the intensity gives rise to g_{1} instead of g_{2} in XNFS experiments. Hence, we have presented in Fig. 7(a) the normalized intermediate scattering function [] versus delay time () for between 0.4 s and 319 s corresponding to an exposure time of 0.2 s for different s at q = 8.180 ×10^{5} Å^{−1}. The values of g_{1} collapse into one curve for different s, which agrees with the theoretical prediction that g_{1} has no sdependence owing to the cancelation of T(q). However, for a larger wavevector of q = 1.420 ×10^{4} Å^{−1}, the g_{1} values do not overlap for different s, as shown in Fig. 7(b). To elucidate the reason for this discrepancy and quantify our observations, we have fitted g_{1} measured at different s and q to a stretched exponential form,
The bestfit relaxation rate () [equation (25)] versus wavevector q is illustrated in Fig. 7(c). The values of at successive s are displaced by a factor of 1.1 from the previous s value for clarity. Generally, at different s show a q^{2} behavior, illustrated by the dashed line in Fig. 7(c). However, peaks are observed at the q positions coinciding with the q positions of the dips in the transfer function T(q) (Fig. 6a).
Away from these multiplescattering peaks, increases as q^{2} versus q, which is reasonable for a SiO_{2} suspension undergoing Quantitatively, for we expect = D_{m}q^{2}. As a result, we derive the value of the D_{m} = ≃ 1.167 ×10^{12} m^{2} s^{−1}. According to the firstorder hydrodynamic interactions, D_{m} = (Batchelor, 1976), where the Stokes–Einstein D_{0} = ≃ 1.048×10^{12} m^{2} s^{−1} with k_{B} the T the room temperate equal to 293 K, the equal to 1×10^{3} kg m^{−1} s^{−1}, we obtain ≃ 0.07, which is reasonable.
At the q positions of the peaks, where the singlescattering amplitude goes to a minimum because of the zeros of the sinesquared term in T(q), we hypothesize that we are measuring multiple scattering of the sample. This theory explains why we obtain faster dynamics at those q positions (Berne & Pecora, 2000). Fig. 7(d) shows the corresponding bestfit exponent , which exhibits similar fluctuation patterns as T(q) and supports our hypothesis. Underlying this hypothesis is the idea that the rapid variations of T(q) versus q may be associated with single scattering, whereas the intensity of multiple scattering likely shows a relatively smooth qdependence. Accordingly, if, for a particular set of data, the scattering minima owing to T(q) are indistinct (do not send the scattering intensity to zero), then it follows that the XNFS data set in question suffers from multiple scattering. Hence, to calculate g_{1}, we have to pick q smaller than the first dip of T(q) so that the measured coherent scattering is reliable.
To further test this idea, we carried out measurements on a sample that could be expected to show very strong scattering and therefore strong multiple scattering, namely a 3 mmthick sample of Gillette Foamy shaving foam, which is know to consist of a dense foam of micrometersized air bubbles in aqueous liquid. Fig. 8(a) shows the scattering intensity from such a sample as a function of q, obtained using the XNFS prescription. However, in contrast to the more weakly scattering silica spheres, discussed above, evidently in this case there are not the oscillations in intensity that are expected for XNFS, i.e. there is no evidence that the XNFS is displayed in these data. We infer that this is indeed the result of multiple scattering and that the Xray scattering from the 3 mmthick foam is completely in the multiplescattering regime. This implies that is a signature of single scattering. We can also calculate g_{1} for the foam according to the XNFS prescription. This is shown in Fig. 8(b). The dynamics are rather slow, of the order of 0.1 s^{−1}.
These results point to another difficulty with the XNFS method (which is common to ultrasmallangle Xray scattering methods in general), namely that multiple scattering must be carefully considered and if possible eliminated. In the case of the foam, a sufficiently thin sample (much thinner than 3 mm) would have eventually reached the singlescattering regime. Interestingly, in the case of XNFS, in contrast to more traditional USAXS methods, the existence or not of multiple scattering may be straightforwardly and immediately recognized from the intensity profile, i.e. , as we discussed previously.
6. Polystyrene 4 µm
In this section we present the XNFS data obtained from a qrange accessible by our XNFS setup. Hence, we expect to observe more complicated intensity profiles with the contributions from both structure of the suspension and the transfer function. Illustrated in Fig. 9 are the scattering intensities (symbols) plotted versus q obtained by azimuthally averaging the Fouriertransformed scattering images over 1000 frames for sampletodetector distances s = 113 mm, 143 mm, 173 mm and 203 mm (from top to bottom). The corresponding transfer functions [T(q)] obtained by fitting data of a silica sample measured at the same s are plotted as solid lines with the same colors for easy comparisons for the peak positions of T_{talb}. The intensity data deviate from the lines. Firstly, the peak positions of the data match those of T(q). There might be one extra peak located at q around 10^{4} Å^{−1} for all s, which comes from the static peak. In addition, the peaks of this sample are less sharp than those of the silica sample. This indicates that the power spectra are largely suffered from the sensor transfer function.
of polystyrene particles of diameter 4 µm. Similar to the preparation of the silica sample, this sample is injected into the same boron glass capillary, thus resulting in a sample thickness of 0.7 mm. This sample is not as stable as the last sample, since particles with 4 µm undergo sedimentation. The static structurefactor peak of the polystyrene suspension of this size lies within theIllustrated in Fig. 10(a) are the normalized intermediate scattering functions (g_{1}) at q = 8.18 ×10^{5} Å^{−1} for delay times from 0.02 D_{0}q^{2} to 20 D_{0}q^{2} s and sampletodetector distances s from 203 mm to 113 mm with an interval of 10 mm. The g_{1} for the polystyrene suspension do not totally overlap for different s at this q position, but decay slightly faster when the detector moves closer to the sample stage. We reason that this is the result of the sedimentation of the polystyrene particles, which leads to a denser sample with faster dynamics.
Following the same procedure as for the silica sample, the bestfit relaxation rates (), obtained by fitting one of the 100 frames g_{1} with a single exponential form [equation (25)], are plotted versus wavevector q for different s in Fig. 10(b). The values of at different s are displaced for clarity. Similarly, peaks that correspond to dips of transfer function are observed, confirming our conclusion about the measurement of multiple scattering at the minima of the transfer function. In this case the peaks are more visible than those observed in the silica sample, indicating stronger multiple scattering in this sample with larger polystyrene particles.
7. Future work and conclusion
In conclusion, we have presented the implementation of the new coherent Xray technique, Xray nearfield speckle, as well as its applications and limitations. Clearly, XNFS is capable of obtaining ultrasmallangle Xray scattering and Xray photon correlation spectroscopy with its simple setup and direct relationship to the density correlation function. It effectively extends to wavevectors an order of magnitude smaller than the wavevector range covered by conventional SAXS and XPCS, and enables us to explore the static and dynamic structures of micrometersized samples. We believe this technique will be valuable for optically dense and turbid samples which induce strong multiple scattering optically.
Technically, XNFS is not difficult to realise. A speckle pattern is produced simply by letting both scattered beam and transmitted beam impinge onto the detector. It does not require spatial filtering as it did in XPCS, which allows us to use the whole source beam and in turn greatly enhance the speckle contrast. As a consequence, it does not require laborious alignments. All the efforts were devoted to the design of the detector. A high numerical aperture objective was employed to produce high spatial resolution and efficient light collection. The measurements give convincing results, which proves the feasibility of this setup. Improvements could be made on several aspects. One is to utilize a thinner scintillator, which will give rise to reduced spherical aberration. A faster CCD camera will improve the probing range of the dynamics of this technique.
One key difficulty of this technique is due to the transfer function T(q). It entangles with S(q). It is straightforward to characterize the structurefactor peaks and dips located at positions smaller than the q position of the first dip of T(q). However, this would make the reliable q range very small. If XNFS is to realise its full potential, it will be necessary to figure out an effective way to deconvolve the static from the transfer function in the future. One possible way is to use sampletodetector distances as small as possible, although with a cost in scattering contrast. Strong absorption samples might not be affected by this factor owing to the phase factor induced in the sine term of T(q) [equation (24)]. Another possible improvement might be made by measuring a control sample with exactly the same material but uniform S(q) in the accessible q window, then dividing the intensity profile of the interested sample by that of the control sample. Another important issue is multiple scattering. Evidence of the existence of multiple scattering comes from the intensity profile and sampletodetector dependent decay rate. Making the sample as thin as possible should solve this problem.
Acknowledgements
We thank T. Chiba, A. Mack, E. R. Dufresne, R. L. Leheny, C. O'Hern, S. Sanis, M. Spannuth and J. Wettlaufer for discussions, and the NSF for support via DMR 0906697. The APS is supported by the DOE.
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