2.1. Cylindrical edges

In Fig. 1 the irregular scattering is due to two reasons. The first one relates to surface roughness and inhomogeneity of sintered tungsten blades which hardly surrender the polishing procedures. The second reason is the X-ray scattering from the sharp blade edges. The geometry of cylindrical edges is sketched in Fig. 2. Let be the critical angle for total reflection. If an X-ray strikes the second cylinder with an angle of incidence , then (i) either , and the X-ray is reflected at an angle , or (ii) , and it is absorbed, provided that .

Figure 2 Sketch of two cylinders (dimensions not to scale). Some intensity is reflected by the cylinder surface for . If , X-rays are strongly absorbed.

For molybdenum at 8 keV, = 0.44°, and = 6.44 µm. For R = 1 mm, = 16 µm. Transmission of X-rays can be neglected here but, for higher X-ray energies, tantalum slits are necessary.

Cylinders of a hard metal like Mo can be easily polished with diamond powder to a state of high surface quality. The absence of sharp edges effectively removes parasitic scattering. In this situation the surface acts as a mirror for case (i). The part of the incident beam involved in this reflection corresponds to a width of about

at each cylinder edge. This is a small fraction of a few µm beam size. The average reflecting angle is large (about half a degree) and guard slits can conveniently remove this contribution to the background.

Cylinders were mounted on high-precision slits, with a reproducibility of about 0.2 µm. The mechanical design of these slits was developed at the ESRF.

The diffraction pattern of a single set of slits consisting of four molybdenum cylinders with an aperture of 2 µm × 2 µm is displayed in Fig. 3. The incoming beam was limited by primary slits to 200 µm × 200 µm at 10 m, just after the optics of the ID01 beamline of the ESRF. Under these conditions, in the absence of perturbation, the beam can be considered as a fully coherent plane wave. This measurement was performed at E = 8 keV with the detector located 1.75 m from the slits. A direct-illumination CCD camera was used on account of its high resolution (22 µm), and the `droplet algorithm' (Livet et al., 2000) ensured low noise and large dynamic range for X-ray registration. The first pattern (Fig. 3a) was obtained in air, with Al attenuators to protect the CCD camera from the direct beam. To reduce the background, the same measurement was performed completely under vacuum with no attenuator or Be window (Fig. 3b). A 0.5 mm-diameter beamstop was used to protect the CCD camera.

Figure 3 (a) Diffraction pattern from the cylindrical slit system. The cylinders were polished with 1/4 µm diamond paste. (b) Same diffraction pattern in completely evacuated surroundings, without attenuators or Be window. A 0.5 mm-diameter beamstop was used to protect the CCD from the direct beam. Strong asymmetry is visible. These patterns were recorded at 1.75 m from the slit (E = 8 keV).

Both figures display asymmetric diffraction patterns with a large number of interference fringes. The contrast (visibility) between fringes is strong and diagonal fringes are visible. On the right-hand side of Fig. 3, the period of the horizontal fringes decreases with increasing distance from the central peak. This asymmetry is due to the asymmetry of the slit (Vlieg et al., 1997; Libbert et al., 1997). The analytical expression of this asymmetric profile is given in the following section.

2.2. Coherent diffraction of a single slope slit

Diffraction of a plane wave by a single rectangular aperture is well known (Born & Wolf, 1980) and can be solved analytically in the far-field regime (Fraunhofer approximation). The near-field/far-field boundary is here defined by

where is again the beam size.

A sketch of the slits configuration is shown in Fig. 4. A 3 mm distance between opposite cylinders means that some modifications to the coplanar slits calculation of Born & Wolf (1980) are required. Using the same notations as Born & Wolf (1980), the scattered amplitude at the point P(x,y) on the detector coming from a rectangular aperture of sides 2a and 2b can be written as

where r = is the distance between the source O and a point of the aperture, and s = , where P is a point on the detector.

Figure 4 Sketch of asymmetrical slits (not to scale). For , the aperture appears smaller than for , giving rise to more widely spaced fringes.

This integral can be calculated by expanding the optical path (r+s) as a function of and .

It can be assumed that the line between the slit edges is almost horizontal, with an inclination . The optical path r+s decomposes into

where s_o and r_o are the source–slit and slit–detector distances, respectively. Despite the fact that, in the previous equation, = 1.5 mm is much larger than in the symmetric case, the distance above which the second-order terms in and can be neglected is the same as in the symmetric case: the near-field/far-field distance is ≃ 3 cm for ≃ 2 µm, much smaller than s_o (≃ 1.75 m) and r_o (≃ 10 m).

It can be shown that the first-order terms can be written

and

with = x/s_o. The asymmetric intensity is deduced,

with

To first order,

Now, the effective aperture = a-(xh/4s_o) depends on the position x on the detector. For then and the fringes appear more widely spaced. For () the fringes appear closer (see Fig. 4).

2.3. Computer calculation of slit scattering

Analytic calculations compare well with the experiment (see Fig. 5), but they are restricted by simple situations. In order to obtain results in more complex experimental set-ups, we performed computer simulations of the diffracted intensity. To fit experimental conditions, a point source was assumed to lie at a distance of 10 m away from our experimental set-up. As this distance is large, the incoming beam can be considered as a plane wave. In the vicinity of the sample, this wave is limited by slits or pinholes. The wave propagating across the asymmetrical set of slits is calculated in a plane perpendicular to the beam for each slit edge. The wave amplitude A₀([(r₀)\vec]) in a plane can be deduced from the wave amplitude A( [(r)\vec] ) in a previous plane at a distance D,

This equation corresponds to the Huygens–Fresnel principle (Born & Wolf, 1980; §8.2). Amplitudes are integrated in a `nearly semi-infinite' plane, limited on one side by the edge of the slit. Limits in other directions are chosen large enough, and the integration step is chosen small enough so that oscillations due to cut-off and to finite integration steps become negligible. All these choices correspond to Nyquist sampling theorem considerations.

Figure 5 Comparison between (a) experimental diffraction pattern (corresponding to Fig. 3) and (b) fit using equation (6). The fit indicates an aperture of 2.49 µm (H) × 2.28 µm (V). (c) Fit along the horizontal profile (log units).

This means that four successive surface integrals are calculated for the estimate of wave amplitudes across asymmetrical slits. Corresponding intensities in the detector yield satisfactory agreement with experiment, as shown in Fig. 6.

Figure 6 Measured (a) and simulated (b) scattering from 2 µm asymmetric slits.

In this figure, simulated results are scaled to equalize the maximum intensities (33000 counts in the central pixel) and to set the lower level equal to unity. Vertical scales are logarithmic and the diffraction is plotted only for the central square of the CCD detector of side 50 pixels. The simulations assumed a 2 µm × 2 µm square aperture, and, apart from experimental Poisson noise, the main discrepancy between simulation and experiment can be attributed to an underestimate of the slit aperture: a value of 2.2 µm yields a better fit, as observed in Fig. 5.

3.1. Wide-angle parasitic scattering from slits

In the situation where the beam is coherent and the resolution of the detector is sufficiently good, beautiful oscillating intensities can be observed, as in Fig. 6. These were obtained with a slit aperture d₁ = 200 µm after the ID01 optics hutch at a distance D₁ = 10 m before the experimental set-up. A square slit aperture = 2 µm was placed close to the goniometer centre and the slit-to-detector distance was D₂ = 1.75 m. It can be shown that the coherence properties of the experiment depend on a dimensionless ratio z (Livet et al., 1998), the value of which is z₁ = ≃ 0.4 for the beam and z₂ = × 0.3 for the detector. Low values of z provide good conditions for the observation of interference effects. As these two ratios are significantly smaller than unity, the quality of the resulting Fraunhofer fringes is excellent (see Fig. 6).

The oscillations discussed earlier in this paper correspond to strong `wide'-angle scattering. This parasitic scattering is the consequence of slit diffraction, and is not limited to the case of coherent beams. This can be seen from Fig. 7, where the scattering is measured for various apertures of the slits. In all cases, streaks are observed although, for 5 µm, coherence is poor and no significant oscillations are observed by the detector. The circularly averaged wide-angle scattering of the slits can be estimated. If the intensity is normalized [I(q = 0) = 1], the q dependence of I can be expressed as

where J₀ is a Bessel function and S is the area of the aperture. The isotropically averaged Patterson function can be written to first order in terms of the total length l (= ) of the slit edges,

The isotropically averaged asymptotic intensity is then

This latter can be compared with the total intensity crossing the slits,

This asymptotic behaviour can be observed from a plot of I(q)q³, as shown in Fig. 8.

Figure 7 Scattering from square asymmetrical slits. Apertures are (a) 10 µm, (b) 7 µm, (c) 5 µm and (d) 3 µm, detector at 1.75 m, 200 × 200 pixels of size 22 µm.

Figure 8 Asymptotic behaviour observed from slits of various sizes (isotropic average). From bottom to top: 10 µm, 7 µm, 5 µm, 3 µm and 2 µm.

Table 1 lists the values of N₀ (maximum number of counts), and , normalized by the estimate of N₀, as in equations (13) and (12).

Table 1 Estimates of the total peak intensity, the normalized integral (equation 13), the asymptotic behaviour (equation 12) and the ratio (equation 14) for various slit apertures Slit aperture (µm) N0 q3I(q)/I(q = 0) estimated (µm) 10 33500 1.275×10-8 1.8×10-14 14 7 29600 1.365×10-8 3.0×10-14 9 5 31200 2.07×10-8 8.0×10-14 5. 3 35800 3.88×10-8 3.0×10-13 2.7 2 32700 7.15×10-8 7.0×10-13 2.1

For the measurements of Fig. 7, to normalize I(q), a precise estimate of N₀ is required. This is difficult to obtain here for 5 µm, because the detector resolution is low.

For this reason, the ratio

was estimated (last column of Table 1). This should be equal to , and does not depend on the estimate of N₀.

In the q range discussed here ( Å^-1), the majority of the overall experimental background signal can be attributed to the length l of the slit edges. The ratio in equation (14) only compares the total beam intensity with the asymptotic behaviour. It does not depend on detector resolution, and the beam needs not necessarily to be fully coherent. It is an intrinsic experimental feature. In the measurement of Fig. 7, scattering from air remains negligible, and the observed background is explained by the cut-off introduced by the slits.

In the case of unpolished slits, like in Fig. 1, edge imperfections can increase the background. However, the main effect is probably `erratic scattering', and the directional intensity, perpendicular to the straight edges, measured in Fig. 8 is spread among various directions. Increase of the angular average background can occur mainly if l in equations (11)–(14) increases due to surface roughness. As these results are also valid if the beam is not coherent, equation (14) places a lower bound on the scattering from slits when used for beam definition. Here, it is assumed that the incoming beam is homogeneous, which is obviously not the case if the slits are used as guard slits.

3.2. Background and guard slits

The results discussed in the first part of this paper correspond to experiments where the beam is fully coherent, and the calculation of slit diffraction assumes plane waves crossing the aperture. In this case the source is considered as a point source.

In the case of a coherent beam, small-angle X-ray scattering (SAXS) experiments require reduced parasitic scattering in the measured range of the q vector. This is achieved by inserting guard slits between pinhole and sample (Livet et al., 2001; Geissler et al., 2000), which causes a distortion of the shape of the wave after the pinhole. It seems obvious that the guard slit configuration must be intermediate between the two simple limiting cases:

(i) if the guard slits are closed to the same size as the pinhole aperture, they generate strong diffraction, equivalent to that of the pinhole alone;

(ii) if the guard slits are opened wide they do not reduce significantly the small-angle diffraction from the pinhole.

Moreover, the relative positions of the pinhole (or defining slits), guard slits and sample are important parameters in the experimental set-up.

These parameters can be used to estimate the feasibility of SAXS measurements in the very low q region, i.e. having a background sufficiently low to use area detectors and an efficient beamstop. For this aspect, the relevant distance between pinhole and guard slits is in equation (3).

In a practical synchrotron experiment, if the pinhole-to-sample distance is significantly larger than , the effective beam size is increased at the sample. This causes a contraction of the speckle structure, making it difficult to observe. On the other hand, to achieve optimum background reduction, the guard slits must be sufficiently far from the pinhole. Fulfilling these conditions leads to a sample position as close as possible to the guard slits. For this reason, the discussion is limited to two parameters in the vicinity of the sample: the aperture d of the guard slits and the distance D between aperture and guard slits. The dependence of the background intensity on angle was estimated in a standard configuration used for coherent SAXS measurements, with a pinhole–detector distance of 2.9 m and where, as previously, the pinhole consists of slits closed to 10 µm.

Fig. 9 shows typical results (D = 0.4 m) of the calculations. These were obtained in the horizontal direction, where the background intensity is maximum (see Fig. 7). The top curve corresponds to the normal Fraunhofer slit pattern, and the effect of guard slits can be appreciated from the two other curves. For a guard slit aperture of 24 µm, a rapid decrease in intensity is found, with the background decreasing by two orders of magnitude for  Å^-1, after which it runs parallel to the diffraction from the single 10 µm slit. For the 48 µm guard slit, the background decreases by a further factor of three, but only from  Å^-1.

Figure 9 Intensities from a 10 µm aperture pinhole and guard slit apertures of (triangles), 24 µm (open circles) and 48 µm (filled circles) with D = 0.4 m, calculated in the detection plane 2.5 m after the guard slits.

The normalized two-dimensional background intensity can, to a very good approximation, be written as

In a `diagonal' direction of the area detector, the decrease in intensity is much faster. For Q = (q_x,q_x), i.e. = q_x2^1/2, the decrease must be squared and parasitic intensity can be reduced by four to five orders of magnitude in these examples, for either  Å^-1 or  Å^-1.

The asymptotic directional behaviour of the background scattering of this set-up can be estimated, assuming that the guard slits are in a far-field position. For q_y = 0 (as in Fig. 9), we find

This equation compares well with the results of Fig. 9: the fast oscillatory factor has a q_x^-2 behaviour at large q, similar to that calculated without guard slits, and the oscillations correspond to diffraction from the guard slits: their period is significantly shorter because the apertures (24 µm and 48 µm) are much larger. Obviously, such oscillations cannot be observed because the resolution of our CCD is too low, and a finite source dimension will further damp them. For the same reason, in practice, can be replaced by 0.5. The main result of equation (16) is that the scattering from the 10 µm slit is reduced by a factor , i.e. 1.4×10^-2 and 3.6×10^-3, respectively, in the cases illustrated in Fig. 9. These values are in good agreement with the numerical results.

Although equation (16) initially assumes a relatively large distance, = , it provides a good estimate for the background at `large' angles ( rad) provided that (here, D = 0.4 m and = 0.6 m). As equation (15) still holds, it is obvious that the introduction of a second set of slits after those limiting the beam increases the relative contrast between the horizontal or vertical directions and the `diagonal' directions: a factor of 10 reduction in the horizontal direction can give a factor of 100 in the diagonal direction.

Experiments with a coherent beam can be considered as a limiting case in a high-resolution experiment. Conversely, the diffraction calculation can easily be extended to the case of beams of low coherence. This can be found from the above calculations with an extended source.

Results of Fig. 10 were obtained with an extended incoherent secondary source of size 300 µm × 300 µm at 10 m, and can be compared with Fig. 9, where a point source was assumed. Neglecting oscillations, the two figures are similar. This shows that to estimate background in high-resolution experiments, diffraction effects must be taken into account. This is indeed observed on the ESRF BM2 beamline: after the slits are carefully polished, the background has the shape of a cross, even in incoherent experiments. Provided other background sources (windows etc.) are low enough, the background becomes strongly anisotropic.

Figure 10 Intensities from a 10 µm aperture and guard slits with apertures of (triangles), 24 µm (open circles) and 52 µm (filled circles) with D = 0.4 m, calculated in the detection plane at 2.5 m after the guard slits. The source is assumed to be 300 µm × 300 µm at 10 m.

3.3. Beamstop

In SAXS experiments, measurements at very low q values (USAXS) are difficult with area detectors. The main reason is the strong increase of intensity observed, which saturates the electronics, and which can even destroy the detector. Usually, background intensity is considered in terms of experimental imperfections. Our results, as seen in Figs. 9 and 10, show that background is intrinsic and corresponds to diffraction from the optical elements.

In the case where the slits are carefully polished and aligned, the scattering can be calculated from first principles. The more general case of an incoherent beam can be modelled by the superposition of intensities from all points of an incoherent extended source.

The shape of the scattering pattern from slits is strongly anisotropic and this anisotropy increases when guard slits are added, even if total parasitic scattering is reduced. This suggests a new shape for beamstops. In Fig. 11 the parasitic SAXS scattering is shown as a function of guard slit aperture for configurations similar to those of Fig. 9. All background intensities are normalized by the total intensity crossing the first set of slits. A 1 mm square detector is assumed to be located at a distance of 2.5 m from the guard slits, as in Figs. 9 and 10. The intensity, shown as a function of guard slit aperture d, is the parasitic scattering integrated over this small detection area, after elimination of pixels corresponding to two different beamstops: a circular beamstop of diameter 0.5 mm and a cross-shaped beamstop, which can be idealized as two wires, one horizontal and one vertical, crossing at the centre of the beam, of diameter 400 µm.

Figure 11 Calculated background intensities received on a 1 mm × 1 mm detector from a 10 µm aperture with guard slits of various apertures d at D = 0.4 m. The detection plane is 2.5 m beyond the guard slits. Circular beamstop: filled circles; cross-shaped beamstop: open circles. Results for an extended source (300 µm × 300 µm at 10 m) are also shown (large filled and open circles).

In both cases, the q range of measurement is 0.5–1.0 ×10^-3 Å^-1.

Two calculations were carried out. (i) In the first case the beam is assumed to be fully coherent, and results similar to those of Fig. 9 (point source at 10 m, square slits of size 10 µm, guard slits of variable sizes at a 0.4 m distance) were summed over the area of the 1 mm × 1 mm detector. (ii) To simulate an incoherent experiment, the same source size as in Fig. 10 (300 µm × 300 µm at 10 m) was assumed.

In case (i), the results are a strongly oscillatory function of d, in which the oscillations arise from the full coherence of the beam. They come from the term in equation (16). The oscillations are damped in the case of beams of low coherence. If the oscillations are neglected, the same behaviour is observed in the two cases:

(i) the background is three orders of magnitude lower with the cross-shaped beamstop;

(ii) for large values of d the background increases (the limits for d = are 0.022 for the circular beamstop and 3.0×10^-4 for the cross-shaped beamstop with a point source);

(iii) for small values of d, a rapid increase of the background is observed.

Experimentally, this behaviour is routinely observed. In SAXS experiments, careful adjustment of the guard slits is an important parameter. An initial decrease in background on closing the slits is followed by a rapid increase when the beam `hits' the slits. The conventional interpretation of this effect is reflection from the surface of the slits. If this were the case, it would be strongly dependent on slit shape and quality. From our results, the increase in background intensity when the beam hits the guard slits is of the order of 0.5% of the beam intensity in the narrow region of detection of Fig. 11, and this is essentially a streak in the direction perpendicular to the slit edges. Such diffraction from slit edges is different from surface scattering, whose contribution for perfectly polished slits of diameter 2 mm corresponds to a negligible fraction of the incoming beam. After the guard slits, the reflected intensity corresponds to an angle  rad ≃ . This angle must replace in equation (2), and, correspondingly, only a few angströms of the incident beam width will contribute to the background.

According to our calculations, on average, the increase of the background intensity weakly depends on the detailed shape of the slits, and is essentially due to diffraction phenomena. This background is irregularly scattered in the case of unpolished blades (like in Fig. 1); the average scattering remains of the same order of magnitude (see equation 12).