research papers
Design and characterization of an undulator beamline optimized for smallangle coherent Xray scattering at the Advanced Photon Source
^{a}Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 021394307, USA, ^{b}Materials Science Division, Argonne National Laboratory, Argonne, IL 604394832, USA, and ^{c}Physics Department, McGill University, Montréal, Québec H3A 2T8, Canada
^{*}Correspondence email: larry@slate.imm.aps.anl.gov
An undulator beamline and smallanglescattering spectrometer have been implemented at the Advanced Photon Source. The beamline is optimized for performing smallangle widebandpass coherent Xray scattering measurements, and has been characterized by measuring static Xray speckle patterns from isotropically disordered samples. Statistical analyses of the speckle patterns have been performed from which the speckle widths and contrast are extracted versus wavevector transfer and sample thickness. The measured speckle widths and contrast are compared with an approximation to the intensity correlation function and found to be in good agreement with its predictions.
1. Introduction
Recent experiments have demonstrated the potential of Xray intensityfluctuation spectroscopy (XIFS), also known as Xray photoncorrelation spectroscopy (XPCS), to become a powerful probe of sample dynamics at low frequencies (<10^{4} Hz) and small length scales (<200 nm). For example, XIFS measurements have been made of the equilibrium dynamics of a binary alloy near its (Brauer et al., 1995), of the of gold (Dierker et al., 1995), palladium (ThurnAlbrecht et al., 1996) and antimony oxide (Tsui & Mochrie, 1998), colloids diffusing in glycerol, and of the equilibrium dynamics of blockcopolymer micelles in a matrix (Mochrie et al., 1997). New insights into nonequilibrium dynamics, such as in a binary glass (Malik et al., 1998), may also be obtained using this technique. All of these measurements were performed in a regime of wavevector and frequency space which extends beyond what is accessible to various other light, neutron or Xray scattering techniques (Dierker, 1995).
IFS with laser light has long been employed to investigate the dynamics of condensed matter on micrometre length scales in transparent media. Its principles are well known: a sample is illuminated by coherent laser light, resulting in a random speckle pattern which varies with time as a result of temporal fluctuations within the sample. The time autocorrelation function of the speckle pattern yields the characteristic times of the sample. Key to performing XIFS, therefore, is a sufficiently coherent and intense Xray beam illuminating the sample under study.
The beam coherence can conveniently be thought of as having two components: (i) transverse or lateral coherence, and (ii) longitudinal or temporal coherence. For a source with an approximately Gaussian intensity distribution, the `onesigma' transverse coherence lengths of the beam are given by in the horizontal direction and in the vertical direction, where λ is the Xray wavelength, R is the sourcetoobserver distance and σ is the onesigma source size. The usual transverse coherence lengths, defined by Goodman (1985) in terms of integrals of the source intensity distribution, are related to the onesigma coherence lengths by = = and = = . For Xrays produced by undulator A at the Advanced Photon Source (APS), the source sizes are σ_{z} ≃ 50 µm in the vertical direction and σ_{x} ≃ 350 µm in the horizontal direction so that, for λ ≃ 0.16 nm and for a sample at R ≃ 55 m, the transverse coherence lengths are ≃ 50 µm in the vertical direction and ≃ 7 µm in the horizontal direction.
The longitudinal or temporal coherence length of the beam is given by Λ ≃ λ(E/ΔE), where E is the energy of the Xray beam (7.66 keV), ΔE is the full width at halfmaximum (FWHM) of the energy spectrum, and the precise numerical factor relating the longitudinal coherence length to the FWHM depends on the energy spectrum (Goodman, 1985). Typically, ΔE/E takes values of 0.0002 for crystal monochromatization or 0.026 for pink beam. To observe coherentscattering effects, the optical path difference of the scattered Xrays must not be too much larger than the longitudinal coherence length of the beam. In particular, for a smallangle scattering experiment (wavevector transfers less than approximately 0.2 nm^{−1}), the energy bandwidth for which coherence effects can be observed is relatively large, i.e. ∼1%. Recent work in this field has exploited this fact (Dierker et al., 1995; ThurnAlbrecht et al., 1996; Tsui & Mochrie, 1998; Mochrie et al., 1997; Tsui et al., 1998; Abernathy et al., 1998; Malik et al., 1998).
Smallangle coherent Xray scattering experiments are facilitated in two ways at a thirdgeneration undulatorbased synchrotron Xray source. First, the ^{4} more transversely coherent Xrays through a unit aperture per unit relative bandwidth than has been available previously at secondgeneration bendingmagnet synchrotron Xray sources. Second, the energy bandwidth of an undulator harmonic approximately matches the allowed energy bandwidth of the Xray beam as derived from optical pathlengthdifference considerations. Provided that the other undulator harmonics are `filtered' away, it is then possible to use the entire undulator harmonic for a smallangle coherentscattering experiment with a concomitant increase in flux.
of transversely coherent Xrays is directly proportional to the source An undulator at the APS therefore routinely provides a factor in excess of 10Motivated by these considerations, we have implemented an Xray beamline and a spectrometer optimized for performing smallangle widebandpass coherent Xray scattering experiments. We have incorporated several novel features into the design of our apparatus which we describe in detail in §2. §3 outlines the alignment of these various beamline components and our characterization of the radiation source.
A crucial diagnostic for optimizing our beamline for XIFS measurements is a detailed characterization of its performance under static conditions. In particular, we produce static speckle patterns using isotropically disordered aerogels and analyse the scattered intensity using a statistical treatment which is described in detail by Abernathy et al. (1998). Our results are compared with a theory developed here. §4 details the results of the statistical analysis of the static speckle patterns and §5 contains a summary and our conclusions.
2. Experiment
The scattering measurements described in this paper were carried out at the MIT–McGill–IBM insertiondevice beamline at the APS (8ID). Table 1 lists the distances from the centre of the undulator to various key components, and Fig. 1 is a schematic plan view of the beamline. From righttoleft in Fig. 1, the principal components are as follows.

APS undulator A (Lai et al., 1993; Dejus et al., 1994; Ilinski et al., 1996) is the radiation source. It is a 72pole 2.4 mlong insertion device located in the downstream half of the sector8 5 m straight section. The radiation produced by the undulator is peaked at a series of harmonic energies that can be adjusted by changing the gap between the magnet poles. A deflection parameter, K, characterizes the behaviour at a given gap, and is related to the effective magnetic field seen by the particle beam via K = 0.934 λ_{u} [cm] B_{eff} [T]. Here, λ_{u} is the undulator period and B_{eff} is the effective magnetic field seen by the particle beam. The energy range of the undulator first harmonic spans 3–13 keV (deflection parameter K = 2.57–0.37); we typically operate with the undulator gap set to 18.0 mm, which places the first harmonic at a measured energy of E = 7.66 keV (K = 1.288). The measured FWHM bandwidth for the first undulator harmonic is ΔE/E = 0.026 FWHM; the source size is σ_{x} = 350 µm in the horizontal and σ_{y} = 50 µm in the vertical (Cai et al., 1996). This yields FWHM source sizes of 870 µm × 110 µm in the horizontal and vertical directions, respectively, assuming the source profile is Gaussian. The Xray beam divergences at 4% coupling are expected to be σ_{x′} = 25 µrad in the horizontal and σ_{y′} = 5 µrad in the vertical (Cai et al., 1996).
Radiation propagates from the straight section to our beamline via the APS front end (FE) (Kuzay, 1993). The FE connects to our beamline via a windowless differential pump. Immediately downstream of the differential pump are whitebeam slits. Currently, the maximum aperture of these slits is 275 µm in both the vertical and horizontal directions. This restriction will be waived in the near future. However, the use of such a small aperture limits the heat load on the downstream optics.
The next component is a horizontally deflecting mirror. The mirror is made from a polished singlecrystal silicon flat and its glancing incidence angle is 2.6 mrad. The top half of the mirror's optical face is uncoated (Si) and the bottom half is coated with Pt yielding critical energies of 10 keV and 30 keV for reflection from the top and bottom, respectively. Undulator radiation up to the .
is reflected from the mirror while higherenergy radiation is absorbed or transmitted through the mirror. Using a mirror as the first optic has several advantages: (i) the heat load on downstream optics is reduced, (ii) the entire undulator spectrum (<30 keV) can be delivered to downstream experiment stations for experiments requiring very high photon fluxes, and (iii) radiation shielding in downstream stations is facilitated. Characterization of the undulator beam reflected from the mirror and delivered downstream to the experiment station (enclosure 8IDE) is described in detail in §3.2The remaining items in our setup constitute the smallangle coherent Xray scattering spectrometer. A pair of crossed slits is used to select a partially transversely coherent portion of the Xray beam. The slit assembly is a custom design which allows the slit blades to be placed within a few centimetres of the sample and to be entirely contained within a vacuum. The slit blades are made of 2 mmthick tantalum and their beamdefining edges are rounded and polished to reduce parasitic scattering. The slits have independent adjustable openings in the horizontal and vertical directions from ∼1 µm to 2.5 mm. Independent horizontal and vertical apertures allow us to choose slit openings that best match the different transverse coherence lengths produced by the very different horizontal and vertical source sizes. The resolution and repeatability of the slit blades are ∼0.2 µm and ∼1 µm, respectively.
The samples we studied were contained in a small chamber that is integrated with the incident and exit flight path (no windows). Silica aerogels were used as canonical static speckleproducing samples for the following reasons: (i) they have well characterized scattering over a range of small angles (Sandy et al., 1997); (ii) they scatter strongly; and (iii) they span a variety of thicknesses, which is useful for testing models of speckle.
Immediately upstream of the sample, and inside the sample chamber, is a guard slit made of 3 mmthick tantalum. Like the slits, its beamtrimming edge was polished. It was used to reduce the parasitic scattering produced by the slits. We used only a single guard slit and mounted it on the inboard (storagering) side of the beam with its edge vertical. Likewise, the CCD camera (see below) was mounted so that it only collected scattering in the inboard direction. By working only in the inboard halfplane, we greatly simplified the alignment of the guard slit.
A Princeton Instruments (Princeton Instruments, Trenton, NJ) model EEV37 thermoelectrically cooled deepdepletion CCD chip with 1152 × 1242 22.5 µmsquare pixels directly detected the intensity scattered from the aerogel samples. Its et al. (1995).
at the Xray energy we used was about 20%. Data presented in this paper are the average of between 100 and 200 CCD exposures, each of duration 0.01–1.0 s. Readout time between each of the exposures was 1.5 s. Obtaining data as a time series prevents saturation of the CCD. In addition, it allows the CCD performance to be characterized as described by DufresneThe CCD was protected from the direct Xray beam by a beamstop made of tungsten. Both the beamstop and the guard slit are mounted on motorized stages so, by observing in real time the scattered intensity recorded by the CCD, we were able to optimize their positions in order to minimize the parasitic contribution to the measured scattering.
3. Beamline alignment and characterization
3.1. Alignment
The alignment or `steering' of the central cone of the undulator beam with respect to the fixed 3.0 mmexitdiameter mask that is part of the sector 8 differential pump can be determined by measuring angular profiles of the Xray beam at fixed energy (Ilinski, 1998). In particular, we note that for detuned energies, E, less than but in the vicinity of an undulator harmonic E_{n}, the maxima in the angular distribution of the synchrotron radiation appear symmetrically displaced from the centre axis at an angle given approximately by (Ilinski et al., 1996)
where γ is the particle beam energy, , divided by its rest mass, λ_{u} is the period of the undulator, and K is the deflection parameter. At the APS, = 7 GeV, λ_{u} = 3.3 cm and K varies from ∼2.2 to 0.4.
As an example, Fig. 2(a) is a profile obtained by scanning a 0.27 mmdiameter pinhole in the vertical direction through the undulator beam and recording the intensity transmitted through it with an Si(111) analyser crystal. The analyser crystal was set so that the accepted energy was E = 7.26 keV; the undulator gap was set so that the first harmonic was peaked at E = 7.66 keV. The measured profile has a local minimum at a vertical pinhole position of ∼0.4 mm and relatively hard cutoffs at ±1.5 mm. The hard cutoffs are the shadow of the 3 mm aperture differential pump, which limits the size of the Xray beam accepted by the sector 8 undulator beamline. The local minimum at the detuned energy corresponds to the onaxis position of the undulator at the harmonic energy. These observations allow any necessary beamsteering correction to be easily and reliably determined. For example, in Fig. 2(a), the deviation of the central cone from the centre of the fixed 3 mmdiameter aperture is ∼0.4 mm and the 0.27 mmdiameter analysing pinhole is ∼27 m from the source, yielding a steering error of 14 µrad in the vertical direction. A steering correction of 15 µrad down was requested from the APS control room. The result, shown in Fig. 2(b), demonstrates that, in the vertical direction at least, the undulator beam is steered correctly into the 3 mmdiameter fixed aperture.
A similar procedure was used to check the Xray beam steering in the horizontal direction. Now, however, because of the increased emittance of the beam in the horizontal direction, the second harmonic must be used. Fig. 3 is a profile in the horizontal direction of undulator A measured through a 0.27 mmdiameter pinhole and at an energy detuned ∼15% from the peak of the second harmonic. The figure shows that the Xray beam is correctly steered through the fixed 3 mmdiameter aperture; the `onaxis' position is the central minimum in the scan. Hard cutoffs at ±1.5 mm are the shadows of the 3 mmdiameter upstream aperture.
3.2. Undulator characterization
Fig. 4 shows undulator spectra, for an 18.00 mm undulator gap, plotted on an absolute scale. Solid and dashed lines are measured spectra after reflection from the Si and Pt stripes, respectively, of the mirror – socalled `pink' beams. (A closeup of the measured first harmonic profile is plotted in Fig. 10.) The dotdashed line is a calculated spectrum. The measured spectra were obtained by scanning the acceptance energy of an Si(111) analysercrystal assembly in enclosure 8IDE and recording the reflected intensity in a nitrogenfilled ion chamber. The whitebeam slits (27.5 m from the source) restricted the accepted beam size to a diameter of 0.27 mm. Glitches in the measured spectra at ∼10, 11.5 and 16 keV are Pt edges and higherorder Si(111) analysercrystal reflections. The increasing signal at higher energies for the Sireflected beam is directbeam scatter entering the ion chamber at small Si(111) analysercrystal angles. The calculated spectrum was obtained using the program us (Dejus & Luccio, 1994) with parameters set to match our beamline configuration. A comparison of the measured and calculated spectrum is given in Table 2.

Evidently, our beamline performs very close to its ideal. For the remaining measurements described in this paper, the mirror was set to reflect from its Si stripe and the undulator gap was set to 18.00 mm (E_{1} = 7.66 keV). As demonstrated by the solid line in Fig. 4, the mirror then acts as a lowpass filter with a rolloff energy just above the first harmonic; the entire undulator first harmonic is delivered to the experiment with minimal higherorder harmonic contamination.
3.3. Sourcesize characterization
We have found that a straightforward means to measure the effective undulator beam source size is as follows. [More precise measurements are described by Cai et al. (1996)]. The whitebeam slits are approximately equidistant from the centre of the 8ID undulator (the source) and our experiment apparatus (∼27.5 m). If the whitebeam slits are made very small they effectively function as a pinhole camera for imaging the source. Fits to the derivatives of the profiles obtained by scanning a `knifeedge' across the beam in the horizontal and vertical directions at the experiment position then yield the effective source sizes without any further deconvolution.
Fig. 5 illustrates this idea. Open circles are the numerical derivative of the profile obtained by scanning a single blade of a slit vertically across the Xray beam at the sample position in the experiment enclosure at sector 8. The inset to Fig. 5 is the original profile. The vertical gap of the whitebeam slits was set to 30 µm for these measurements. Fits by Gaussian profiles to the peak and the trough in the data (solid line) yield a onesigma source size in the vertical of 60 µm. A similar procedure for a scan in the horizontal (with the whitebeam slits defining an aperture of 120 µm in the horizontal) yields a onesigma source size in the horizontal of 330 µm. These results are in good agreement with the more careful measurements of Cai et al. (1996) which yielded onesigma source sizes of 50 µm and 350 µm in the vertical and horizontal directions, respectively.
4. Results and discussion
Fig. 6 shows the smallangle scattering produced by (a) 7 µm × 21 µm (H × V) and (b) 15 µm × 31 µm partially coherent Xray beams incident upon a 1.2 mmthick aerogel and (c) a 6 µm × 21 µm partially coherent Xray beam incident upon a 2.7 mmthick aerogel. Typical through the slits was of the order of 10^{12} photons s^{−1} and scaled with the aperture area. The whitebeam slits, which can produce an effectively smaller source size if partially closed (see later), were open for all three sets of these measurements so that the transverse coherence lengths of the Xray beam are determined by the sourcetosample distance, the source sizes and the collimatingslit apertures. For all the images presented in Fig. 6, dark regions indicate more recorded scattering and light regions indicate less. The bottom panels are the entire range of scattering that was collected while the top panels are closeups in the range of Q indicated. Each image is the average of 100 CCD exposures (frames). The exposure time per frame for (a) was 0.1 s, for (b) 0.03 s, and for (c) 0.05 s. All the data presented in Fig. 6 have been converted to wavevector transfer () using the relation Q = , where k = 2π/λ = 38.8 nm^{−1} is the wave number, is a vector from a point on the CCD image plane to the beamzero position on the CCD image plane, and R_{d} is the separation between the sample and the detector (see Table 1). The direct beam (stopped by a beam stop) is at = Q = 0, Q_{x} is in the horizontal direction and Q_{z} is in the vertical direction.
There are several notable qualitative features apparent in Fig. 6. First, for all three patterns the measured scattering is grainy – this is speckle! Second, for all three patterns the speckles appear streaked in the radial direction. This is on account of the nonzero energy bandwidth of the Xray beam. Third, we observe that for increased sample thickness the longitudinal widths of the speckles decreases. This is especially apparent when we compare the specklepattern closeup for the 1.2 mmthick aerogel [top panel of Fig. 6(a)] with the specklepattern closeup for the 2.7 mmthick aerogel [top panel of Fig. 6(c)]. Finally, the 15 µm × 31 µm beam incident upon a given sample produces many more speckles compared with the 7 µm × 21 µm beam [compare the bottom panels of Figs. 6(a) and 6(b)]. This observation is quite consistent with the smaller speckle size expected for larger slit settings.
In order to quantify these observations, we now turn to a statistical analysis of the static Xray speckle patterns. We omit detailed explanations of this formalism as it has recently been presented by Abernathy et al. (1998). A similar analysis was also recently given by Tsui et al. (1998).
First, the detector response is characterized by calculating the spatial autocorrelation of each CCD frame in a region where the measured intensity is very weak (isolated photon `hits') (Dufresne et al., 1995), in order to determine the spatial resolution of the detector independent of complicating intensity variations due to the speckle pattern itself. From this analysis we find that the FWHM detector resolution is 2.0 pixels or 3.8 × 10^{−4} nm^{−1} in both the vertical and horizontal directions.
Key to the remainder of the analysis is the twopoint intensity correlation function,
where r_{i} are points in the CCD image plane, and the angle brackets indicate a spatial average over a suitable region of the time average of a series of CCD frames. The spatial autocorrelation function is maximized for r_{1} = r_{2} [1 < C(r,r) ≤ 2], while for large separations it decays to 1. The contrast of the speckle pattern – an important figureofmerit for a scattering experiment with partially coherent Xrays – is the backgroundsubtracted value of the maximum of the spatial autocorrelation function, β(r) = C(r,r) − 1, and consequently takes on values [0 ≤ β(r) ≤ 1]. The FWHMs of the backgroundsubtracted intensitycorrelation function, C(r_{1},r_{2}) − 1, yield the speckle widths. As an example, Fig. 7 shows the intensity autocorrelation at 0.060 nm^{−1} calculated for the speckle pattern produced by 1.2 mmthick aerogel illuminated by a 7 µm × 21 µm (H × V) partially coherent Xray beam [top panel of Fig. 6(a)]. Open circles are data in the longitudinal direction and solid circles are data in the transverse direction. Evidently the speckles are considerably broader in the longitudinal direction than in the transverse direction. This feature is also obvious from inspection of the top panel of Fig. 6(a).
The results from analysing those portions of the three speckle patterns in Fig. 6 that lie within a region bounded by lines extending from the origin at ±15° from the vertical are plotted in Figs. 8 and 9. Plotted symbols in Fig. 8 are the experimentally determined contrasts versus wavevector transfer for the three different sets of sample and Xray beam parameters described above. Specifically, open circles are the measured contrast for the 7 µm × 21 µm (H × V) Xray beam incident upon the 1.2 mmthick aerogel, open squares are the measured contrast for the 15 µm × 31 µm Xray beam incident upon the 1.2 mmthick aerogel, and open triangles are the measured contrast for the 6 µm × 21 µm Xray beam incident upon the 2.7 mmthick aerogel. For all three sets of measurements, the measured contrast decreases continuously with increasing wavevector transfer. Moreover, for the same collimating slit setting, we see that the measured contrast decreases with increasing sample thickness. Lastly, we see that, especially for increasing wavevector transfer, the contrast from a sample of given thickness decreases with increasing collimating slit aperture.
Open and solid plotted symbols in Fig. 9 are the experimentally determined speckle widths versus wavevector transfer in the radial and transverse directions, respectively, for the three different sets of sample and Xray beam parameters described above. Specifically, open circles are the measured radial speckle widths for the 7 µm × 21 µm (H × V) Xray beam incident upon the 1.2 mmthick aerogel, open squares are for the 15 µm × 31 µm Xray beam incident upon the 1.2 mmthick aerogel, and open triangles are for the 6 µm × 21 µm Xray beam incident upon the 2.7 mmthick aerogel. The same sequence of symbols but now solid apply for the transverse speckle widths. As anticipated previously, the transverse speckle widths are considerably less than the radial speckle widths. Moreover, the transverse speckle widths appear to be independent of wavevector transfer while the radial speckle widths do not.
There are several approaches available for predicting the contrast and speckle widths of speckle patterns as a function of wavevector transfer, sample thickness and so forth (Abernathy et al., 1998; Tsui et al., 1998; Sinha et al., 1998). An approximation that we have found effective and will further elaborate here has been given by Pusey (1976). He derives that in the Fraunhofer limit the contrast of a speckle pattern is given by
where E(r, t) is the incident electric field at position r and time t, Q is the scattering vector, k is the wave number and c is the speed of light. [ = (x, 0, z) is the component of r perpendicular to the incident Xray wavevector.] The integration volume, V, is the illuminated sample volume, which we take to be a solid rectangle of dimensions L along x (x is horizontal and orthogonal to the incident Xray wavevector), M along z (z is vertical) and W along y. The incident wavevector is k_{i} = (0, k, 0) and the scattered wavevector is k_{f} = . The wavevector transfer is therefore Q = .
Equation (3) assumes that the correlation length of density (dielectric constant) fluctuations in the sample is much less than the sample dimensions and the transverse and longitudinal coherence lengths of the incident radiation. It also assumes that the coherence time of the incident electric field is much less than the correlation time of the sample. Both of these assumptions are valid for our measurements.
Further assuming, first, that the coherence properties of the radiation can indeed be factored into independent spatial and temporal parts [i.e. that the radiation exhibits `crossspectral purity' (Born & Wolf, 1980)], second, that the source is incoherent and shows a Gaussian intensity distribution, and, third, that the power spectrum of the radiation is Lorentzian, giving rise to an exponential time correlation function, then the electric field correlation function in (3) can be written as
where Λ is the longitudinal coherence length, and
is the optical path difference between two points within the sample at r_{1} = (x_{1}, y_{1}, z_{1}) and r_{2} = (x_{2}, y_{2}, z_{2}).
With regard to the longitudinal coherence, the power spectrum of the electric field fluctuations corresponding to an exponential correlation function with correlation time τ is
The FWHM of the power spectrum is 1/(πτ). It follows for a Lorentzian spectrum that the longitudinal coherence length is Λ = cτ = (kΔλ/2λ)^{−1}, where c is the speed of light, i.e. Λ = (λ/π)(E/ΔE), where ΔE is the FWHM of the energy spectrum of the undulator. In fact, the undulator spectrum in the vicinity of the first harmonic is not a Lorentzian; we discuss below why this approximation seems to work well.
To make comparisons with our experimental measurements, we must determine how the coherence lengths are related to the source sizes (σ_{x}, σ_{z}), the Xray wavelength (λ) and the sourcetosample distance (R). For an incoherent source with a Gaussian intensity distribution, the van Cittert–Zernicke theorem (Born & Wolf, 1980) tells us that the spatial part of the electric field correlation function appearing in equation (4) is given by
Comparing equations (4) and (7) shows that the onesigma coherence lengths are related to the onesigma undulator A source sizes by ξ = λR/2πσ_{x} ≃ 4 µm and ζ = λR/2πσ_{z} ≃ 28 µm.
To derive the contrast from equation (3), we use the sample and beam dimensions described above. We also assume that the sample is immediately adjacent to the collimating slits (in the `nearfield') or, more quantitatively, that λD/d < d, where D is the slittosample distance (∼7 cm) and d is the slit aperture. This condition is satisfied for both our horizontal and vertical slits. At zero azimuth (φ = 0), the sixdimensional integral in equation (3) factors into the product of a fourdimensional integral involving x_{1}, x_{2}, y_{1} and y_{2} (β_{r}) and a twodimensional integral involving z_{1} and z_{2} (β_{z}). Strictly, our statistical analysis calculates the speckle contrast and widths averaged within a range of ±15° about zero azimuth. However, we believe that the speckle widths and contrast calculated for φ = 0 should be a good approximation to this average and, for calculational convenience, we make this approximation.
The integral over z_{1} and z_{2} is
where erf() is the error function.
The integral over x_{1}, x_{2}, y_{1} and y_{2} is
where A = = = , and B = = . Making use of the transformations given by Welling & Griffin (1982), the fourdimensional integral for β_{r} may be transformed into a twodimensional integral,
which may be evaluated numerically. The final result, therefore, for the contrast is
By generalizing Pusey's discussion, it is possible to calculate how the intensity scattered at a wavevector Q is correlated with the intensity at a different wavevector Q′ = Q + κ. Equation (2.37) in Pusey (1976) should be rewritten with the exponential factor replaced by . An additional factor then appears in the second factorial moment of the intensity. We may therefore introduce the second factorial moments as a function of ,
where
and
Equations (13) and (14) characterize the average speckle shape in threedimensional The corresponding speckle widths (Δ_{x}, Δ_{y} and Δ_{z}) are derived by numerical evaluation of (13) and (14). A CCD camera makes a particular cut through such that the wavevector difference between pixels corresponding to different scattering angles at fixed azimuth (κ_{r}) and the wavevector difference corresponding to the same scattering angle at different azimuths (κ_{t}) are related to κ_{x}, κ_{y} and κ_{z} via
It should be noted that and . Equation (15) at φ = 0 in conjunction with (13) and (14) may be used to calculate the perfectly resolved widths of the autocorrelations in the radial and tangential directions (Δ_{r} and Δ_{t}, respectively).
Finally, in order to compare the predictions with our experimentally measured speckle widths and contrast, we must include in the model the smearing produced by the detector resolution. Following the approach described by Abernathy et al. (1998), we obtain
and
The superscripts `0' denote the perfectly resolved widths and contrasts derived above [equations (11), (13) and (14)], Δ_{d} is the detector resolution = 2 pixels = 3.80 × 10^{−4} nm^{−1}.
Before making detailed comparisons with our measurements, we note a few general features and simplifications of the model. First, speckle widths in the z (transverse) direction are predicted to be independent of Xray beam bandwidth, wavevector transfer and sample thickness. Our specklewidth data (Fig. 9) are consistent with the latter two observations (the bandwidth prediction was not tested). Second, the equation [equation (9)] for the perfectly resolved radial contribution to the contrast simplifies considerably in the limit of small wave vectors and/or vanishing relative bandwidth. Specifically, we find
which is the same as equation (8). In these limits and when , then . Similarly, for , . Therefore, in these limits,
i.e. the coherence area divided by the illuminated sample area.
Solid, dotdashed and dashed lines in Fig. 8 are the model predictions for the contrast for the experiment conditions corresponding to the plotted open circles, the open squares and the open triangles, respectively. The model predicts the correct wavevector dependence for the contrast and evidently also predicts the correct trends in the contrast as the sample thickness is varied and the slit sizes are changed. It does an adequate job of predicting the overall value of the contrast but the measured values of the contrast are systematically less than the predicted values. Possibly the discrepancy arises because the recorded scattering is contaminated by a small incoherent background arising from the slits and the beamline windows (Abernathy et al., 1998). In fact, Abernathy et al. (1998) propose that the beam incident upon the sample can be thought of as being composed of a coherent fraction (α) and an incoherent fraction (1 − α). They then show that the expected contrast needs to be scaled by α^{2}. In the context of this idea, we therefore see that the beam delivered to the sample has a coherent fraction of at least 0.8 over the entire range of measured wavevector transfers. Alternatively, the discrepancy may originate in the azimuthal averaging over ±15° that is not accounted for in our calculation.
Now we consider the speckle widths. Lines plotted in Fig. 9 are the model's predictions for the speckle widths for the various experiment conditions corresponding to the plotted symbols. Near the top of the figure, the solid, dotdashed and dashed lines are the predicted radial speckle widths versus wavevector transfers for a 1.2 mmthick sample illuminated by a 7 µm × 21 µm (H × V) partially coherent Xray beam, a 1.2 mmthick sample illuminated by a 15 µm × 31 µm beam, and a 2.7 mmthick sample illuminated by a 6 µm × 21 µm beam, respectively. The model evidently predicts the correct trends as the sample thickness, slit aperture and wavevector transfer are varied. Moreover, the predicted values of the radial speckle widths are in reasonable agreement with the measured values.
The variation of the longitudinal speckle width versus wavevector transfer for a particular sample thickness can be intuitively understood by considering the two contributions to the effective slit size versus Q as seen at the detector. The first effect is the foreshortening of the slit aperture with increasing scattering angle which leads to a quadratic increase in the longitudinal speckle width with increasing Q. The second effect results from the increase in the apparent sample thickness with increasing Q. The effective sample thickness is proportional to an effective slit width which leads to a 1/Qdependence for the speckle width versus increasing Q. The first effect dominates at small Q and the second at larger Q leading to the maximum observed in all three radial specklewidth curves plotted in Fig. 9. We also note that the first effect is independent of the sample thickness while the second is not and predicts that thicker samples yield narrower speckles. In this way we can understand the overall decrease in the longitudinal speckle widths with increasing sample thickness which is also observed in Fig. 9.
Near the bottom of Fig. 9 the solid line is the predicted transverse speckle width for a sample (of arbitrary thickness) illuminated by a 6 or 7 µm × 21 µm (H × V) partially coherent Xray beam and the dotdashed line is the prediction for a sample illuminated by a 15 µm × 31 µm beam. The predicted transverse speckle widths are independent of Q and sample thickness and, for the range of vertical slit openings plotted in Fig. 9, dominated by the resolution of the CCD detector (0.38 × 10^{−3} nm^{−1}). The predicted transverse speckle widths are in good agreement with the measured speckle widths.
Before concluding we wish to further comment on our use of a Lorentzian to approximate the power spectrum of the undulator harmonic [see the discussion of equation (6) above] and why it seems to produce results in agreement with our measurements. Goodman (1985) has given a general expression for the longitudinal coherence length,
It then follows from Parseval's theorem that the longitudinal coherence length is related to the power spectrum via
The measured energy spectrum of undulator A (plotted on an absolute scale) in the vicinity of the first harmonic is shown in Fig. 10. Whitebeam slits (27.5 m from the source) restricted the accepted beam size to a diameter of 0.27 mm. We may determine Λ for the `pinkbeam' spectrum produced at 8ID by numerically evaluating (21). Specifically, we have calculated
where S(E) is now the unnormalized spectrum shown in Fig. 10 (truncated at E = 8500 eV) and h is Planck's constant. The result is that Λ = 1/(0.013k), which is indistinguishable from Λ = (λ/π)(E/ΔE) obtained for the Lorentzian distribution. It is this coincidence that renders a Lorentzian spectrum a good approximation.
5. Conclusions
We have implemented a beamline and a spectrometer optimized for performing smallangle coherent Xray scattering experiments. A horizontally deflecting mirror is used to isolate the first harmonic of the undulator from the `white' incident spectrum and produce a `pink' beam incident upon the smallangle coherentscattering spectrometer. The spectrometer illuminates the sample under study with a partially coherent Xray beam and records the scattered intensity via an area detector.
We have used this setup to measure static speckle patterns from isotropically disordered samples under a variety of conditions. The static speckle patterns have been analysed using a statistical approach from which we obtain the speckle widths and contrast versus wavevector transfer. The results have been compared with an approximation, described in detail, to the intensity correlation function. We find that the predicted speckle contrast and widths agree very well with the observed values. We conclude that our setup is performing at close to its optimum level. Nevertheless, it should be noted that the speckle contrast is quite low (∼0.05). In principle, it would be possible to improve the speckle constrast by a factor of ten by reducing the bandwidth by a factor of ten, which would be valuable for XIFS experiments on radiationsensitive samples, reducing by tenfold the total on the sample, while leaving the coherent unchanged. In practice, however, monochromators yielding a relative bandwidth of ∼0.003 FWHM are not readily available at present.
Acknowledgements
We thank Douglas Abernathy and Gerhard Grübel of the European Synchrotron Radiation Facility, Peter Ilinski of the APS, and Matt Borthwick, Peter, Falus and Dirk Lumma of MIT for illuminating conversations. We appreciate Norbert Mulders of the University of Delaware for supplying the aerogel samples used in this work. We are especially grateful to Douglas Abernathy for providing many of the `Yorick' routines that were used as the basis for the speckle analysis presented here. We also acknowledge the expert technical assistance of Harold Gibson. This work was supported by NSF grant DMR9312543 and DOE BES grant DEFG0296ER45593 to MIT, the ANL LDRD program and NSERC. Use of the APS was supported by the US DOE (BES, OER) under Contract No. W31109Eng38.
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