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Instrumentation for SAXS and SANS
C H A P T E R
7 Instrumentation for SAXS and SANS An extended review on instrumentation for small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) was compiled by Pedersen.177,178 Here we report only on the central points about sources and cameras, followed by a mathematical description of instrumental broadening effects. Their correction will be discussed in Chapter 8, Numerical Methods.
X-RAYS For most of the applications in research laboratories, X-ray tubes with copper anodes are used. They emit X-rays at a wavelength of 0.154 nm (or 8.05 keV, Kα line), and the weaker Kβ line with a wavelength of 0.139 nm (or 8.91 keV). The intensity contribution of the Kβ line in the sample is between 20% and 35% of the Kα line, and depends on the operating time of the tube.179 Each tube also emits a continuous spectrum, the so-called Bremsstrahlung. This contribution can be suppressed electronically with a discriminator. The remaining wavelength distribution is shown in Fig. 7.1. Other anode materials generate other wavelengths. Molybdenum radiation (λ Kα 5 0.071 nm) is widely used in materials research, due to its much lower absorption coefficient than that of copper. X-ray tubes are operated with a high voltage of typically 40 kV. Higher levels of voltage will essentially only increase the continuous high-energy background radiation, but not the intensity of the Kα line.180 The maximum power is limited by the heating of the anode to about 2 kW (50 mA) current. This power limit for sealed tubes is given by the restricted heat transfer from the hot anode spot to the cooling fluid (water), and depends on the individual tube type. Rotating anodes are a special solution to this problem. This cylindrical, water-cooled anode must rotate in the high vacuum of the tube which is a nontrivial technical problem. Such rotating anodes have a much higher power ( . 10 kW), but are much more expensive, need more maintenance, and are less reliable. X-ray tubes, able to generate very small focal spot sizes, typically below 50 μm in diameter, are called microfocus X-ray tubes. Solid-anode microfocus X-ray tubes are in principle
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FIGURE 7.1 Wavelength distribution of copper radiation, measured with a xenon-filled proportional counter. Units of the abscissa are relative to the Cu Kα line.
very similar to standard tubes, but the electron beam is focused into a very small spot on the anode in the range 520 μm. This leads to a much higher source brightness. The brightness is the amount of photons emitted per unit time, per unit area, and per solid angle. In order to avoid anode melting, the electron-beam power density must be below a maximum value in the range 0.40.8 Wμm21, depending on the anode material.181 The gain in flux compared to a standard sealed tube can be up to one order of magnitude, depending on the beam shape. The cost of the source, including focusing mirrors, is double that of a sealed-tube system. In metal-jet anode microfocus X-ray tubes, the solid metal anode is replaced with a jet of liquid metal, which acts as the electron-beam target. The advantage of the metal-jet anode is that the maximum electron-beam power density is significantly increased, to about 3 2 6 Wμm21 for gallium and tin anodes.182,183 Gallium has an emission at Kα 5 9.2 keV (close to that of copper), while an indium rich alloy also generates a highenergy Kα line at 24.2 keV.184 Metal-jet anodes are quite expensive, however, they may increase the flux in the sample by up to one order of magnitude. Synchrotrons and storage rings are other sources of X-rays with increasing importance for scientific research. In these installations, charged particles (positrons or electrons) are forced into a (partial) circular path by bending magnets or special devices like wigglers and undulators.185 The necessary radial acceleration causes the emission of radiation in the X-ray regime, because the particles move with high energy (close to the speed of light). The radiation is emitted in tangential directions (like sparks from a grinding wheel). The radiation has a very high intensity and is reasonably well collimated in one direction. The radiation is “white” (broad wavelength distribution), and has a pulsed time structure. The desired wavelength is selected by a monochromator. The main disadvantage of these radiation sources is the high radiation level in the experimental position. All experiments
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FIGURE 7.2 Dependence of the scattering intensity I on sample thickness t. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
must therefore be performed under remote control in a shielded hutch, which makes the alignment of the system a somewhat difficult task.
Optimum Sample Thickness In order to get as much scattered intensity as possible, it is necessary to optimize the sample thickness. The intensity I scattered at small angles is proportional to te2μt (t being the thickness of the sample, and μ the absorption coefficient). This expression has its maximum at topt. 5 1/μ (see Fig. 7.2). For Cu-Kα radiation this thickness is a little bit less than 1 mm for H2O, and about 76 μm for Al (corresponding values are 8.3 mm and 718 μm for Mo-Kα!). Aqueous solutions are filled in very thin quartz capillaries with a diameter of about 1 mm. The absorption of quartz is about 10 times higher than that of water. The typical wall thickness is therefore about 50 μm or less.
SMALL-ANGLE X-RAY SCATTERING CAMERAS Instruments for the measurement of SAXS curves are usually called cameras. The difference between these devices and regular cameras for taking pictures is the fact that there are no lenses available for X-rays. The refractive index n0 is nearly equal to one, far above the resonance frequency. Slits or pinholes are frequently used to create a geometrically
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well-defined narrow primary beam. Collimating or focusing systems are difficult to realize. Bent mirrors or crystal monochromators with a special cut can be used for this purpose. Such systems are very expensive, and are used for small-angle scattering stations at synchrotrons, or sometimes in combination with rotating anodes in laboratory systems. Many modern laboratory systems use so-called Go¨bel mirrors to collimate or focus the primary beam (see below).
Pinhole or Slit Cameras The simplest system is a pair of two parallel slits. The narrower the slits and the greater the distance between the pair of slits, the higher is the resolution of the camera. As the resolution of the camera we define the minimum q value at which we can measure the scattering from the sample with negligible background scattering from the camera (the remaining contribution is from the primary beam, which has in any case to be blocked by a primary beam stop to protect the detector). The simple two-slit system has an essential drawback: each edge of the slit is itself a source of a considerable amount of secondary scattering (parasitic scattering) into the small-angle region, and thus reduces the possible resolution. This design can be improved by a third slit (“guard slit”) behind the second slit, adjusted in such a way that it is just not hit by the direct beam, but absorbs as much as possible of the parasitic scattering that originates from the second slit (Fig. 7.3). In the plane of registration, PR, the zone P is the direct beam, the zones, s, correspond to regions of strong, and the zones w to regions of weak, parasitic scattering. This weak parasitic scattering or background scattering can be highly reduced by so-called scatterless hybrid metalsingle-crystal slits.187 These systems can also help to increase the flux of X-rays in the sample by a factor of three. The same idea can also be used for pin-hole systems. FIGURE 7.3 Slit camera with three slits, each consisting of a pair of edges running perpendicular to the plane of the paper. The dimensions in the vertical direction are greatly enlarged. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
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The resolution can be increased by increasing the distance between the slits. Stasiecki and Stuhrmann188 installed such a system, with a length of 50 meters below the roof of the institute building at the University of Mainz! With this camera, they were able to measure the size of red blood cells in the regime of several micrometers. This installation was obviously to show the principle possibility to measure such large systems with SAXS.
The Block Camera The block camera, designed by O. Kratky186 and often referred to as the Kratky Camera (Fig. 7.4), uses blocks to define the size of the primary beam. Unlike the slit system, it does not allow measurements above and below the direct beam. The system is formed by a U-shaped middle part, M, a bridge, B, and an entrance slit (or block), E. The main idea is to allow full parasitic scattering below the primary beam, but to have negligible parasitic scattering above the beam; this is the half-plane used for the measurement (Fig. 7.5).
FIGURE 7.4 Threedimensional schematic drawing of the block system with the plane of registration, PR, and the primary beam stop, PS. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
FIGURE 7.5 Schematic drawing of a section through the block system with the focus, F, and with an intensity distribution in the plane of registration.
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We can see from the above figures that there is, under ideal conditions, no parasitic scattering above the main section, H. The direct beam is blocked by the beam stop, PS, to protect the detector. The resulting primary beam passing through the sample, P, is a thin long line (typical thickness or width 100 μm, typical length 1 cm). This system has a rather high intensity of X-ray light shining on the sample that is necessary for the measurement of colloidal systems with low contrast. The disadvantage is the strong instrumental broadening effect coming from the length of the primary beam (smearing effect). This effect can, however, easily be eliminated numerically for isotropic, nonoriented samples. Air would be another source of parasitic scattering. In nearly all modern laboratory systems, the whole space between the entrance window (between the X-ray tube window and the collimation system) and the detector window can be evacuated. Special refillable capillary holders allow the positioning of liquid samples in the vacuum.
Comparison of Pinhole and Block Cameras In a very simple picture, a pinhole camera is characterized by a point-like primary beam, while a block or long slit camera has a linear primary beam like many pinholes parallel to each other. When considering a pinhole system with a beam diameter in the order of 100 μm, a long slit of 1 cm, and a thickness of 100 μm, we see immediately that the beam intensity is about 100 times higher in the long slit system. The disadvantage of the long slit or block system is the geometrical smearing effect: we have the superposition of about 100 scattering experiments with increasing linear offset. These smearing effects have to be corrected or desmeared. For a long time this was a big challenge for the experimenter. Modern computer routines such as indirect Fourier transformation have made this an easy task, once the intensity distribution across the primary beam has been determined experimentally. For further details, see Pedersen (2002)178 and Chapter 8, Numerical Methods.
Go¨bel Mirrors X-rays are emitted from the anode of an X-ray tube in all directions. Only a small portion of the photons will pass the pinholes, slits, or blocks. This situation could be improved by using focusing elements that convert the diverging beam into a collimated parallel beam, when using parabolic mirrors, or into a focused beam when using elliptical mirrors. Contrary to total-reflection mirrors (Franks mirrors),189191 where the X-ray beam is reflected at incident angles below 0.1 degree, Bragg diffraction from crystals can change the direction and divergence of the incident X-ray beam at higher angular values. The principle that a particular wavelength is diffracted at a particular Bragg angle led to the development of conventional monochromator crystals. Because of the divergent-beam characteristic of laboratory X-ray sources, these crystals do not work efficiently (most of the radiation emitted from the source is cut away prior to diffraction). In contrast, Go¨bel mirrors are graded multilayer crystals produced such that the spacing d between the layers varies as shown in Fig. 7.6a. The d-spacing gradient depends on factors that include wavelength and the location of the mirror with respect to the focus of
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FIGURE 7.6 (A) The mirror is made of graded multilayers consisting of A silicon and B tungsten. (B) The parabolically bent mirror collimates divergent X-rays into a parallel beam.
the source. The combination of layered materials with a large contrast in electron density (e.g., W and Si) results in high integral reflectivity. Go¨bel mirrors are parabolically bent to yield a diffracted monochromated parallel beam.192,193 The radiation is monochromated to a high-intensity Cu Kα primary beam (in the case of Cu anodes), while Cu Kβ and Bremsstrahlung are suppressed or do not pass the collimation system. If a single parabolically bent Go¨bel mirror is placed in the beam path, so that the line focus of the X-ray source is at the focus of the parabola, then approximately 1 degree of the divergent primary beam will be collected and transformed into a monochromatic parallel beam (Fig. 7.6b). Such Go¨bel mirrors have been combined with the Kratky block collimator to create a very high-flux camera with low background.194 In the meantime, Go¨bel mirrors are available as a one-dimensional mirror with parabolic (collimating) and elliptical (focusing) bending. Twodimensional mirrors collimate or focus the X-rays into a pinhole system. In so-called Montel optics, the two mirrors are mounted side-by-side at 90 degrees to each other.195
The BonseHart Camera The BonseHart Camera196 is based on multiple reflections of the primary beam and the scattered light from opposite sides of a channel that is cut into a single crystal of germanium or silicon. Only light rays of a certain direction (and wavelength) are reflected. The suppression of other rays increases with the number of reflections. The first crystal is fixed and
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FIGURE 7.7 Schematic drawing of the BonseHart camera: (1) source; (2) monochromator crystal; (3) sample; (4) analyzer crystal; (5) detector. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
defines the dimension of the primary beam. The second one rotates with the detector around the sample, and defines the scattering angle. This system creates a line-shaped primary beam like the Kratky system, but the light intensity in the sample is much lower. This is the reason why BonseHart systems are used with synchrotrons, and sometimes with rotating anodes. The BonseHart system has a much better resolution than the Kratky system; the minimum q values are comparable with that available in light scattering. Such experiments are, therefore, often called ultrasmall-angle X-ray scattering (USAXS) (Fig. 7.7). It should be mentioned that it is also possible have a point-collimation USAXS system by using two crystals on each side of the sample.197 Such a four-crystal system can only be used for strong scatterers, or for a two-dimensional detection of oriented samples.
MATHEMATICAL DESCRIPTION OF THE INSTRUMENTAL EFFECTS The ideal experimental design, i.e., a point-like strictly monochromatic primary beam with high energy, point-like sample and point-like detector can hardly be reached in a real experiment. The various effects caused by the most common experimental designs are discussed in the following section. Instrumental broadening is caused by the finite dimensions of the primary beam and of the detector. In the Kratky block system, the primary beam has the shape of a thin long slit (see Fig. 7.8). The detector may also have the shape of a slit. All possible rays in the dashed channel in this figure are registered by the detector at the position m. Obviously, the actual scattering angles in the sample may be quite different for all these rays. Let I0(x,t) be the two-dimensional intensity distribution of the primary beam entering the sample, where t is the coordinate from the center of the primary beam in the direction of the slit length, and x is the coordinate perpendicular to t (slit width). Simple geometrical considerations show that the intensity distribution I0(x,t) can be written as a product of two independent distributions: I0 ðx; tÞ 5 PðtÞ:QðxÞ
(7.1)
This independence is correct with sufficient accuracy for most practical applications. The total deflection of a ray through an arbitrary point (t, x) into the center of the detector can easily be found with the help of Fig. 7.9, and is given by the expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm2xÞ2 1 t2 (7.2)
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FIGURE 7.8 Ray channel to the detector in a slit camera.
FIGURE 7.9 Ray geometry in a slit camera.
The influence of the finite detector size will be discussed later. The intensity I(m) at the position m is the sum (integral) over all contributions from all possible t and x positions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN ðN IðmÞ 5 2 dx dt:PðtÞ:QðxÞ:I ðm2xÞ2 1 t2 (7.3) 2N
o
where 2sin θ/2 D m/a and a are the distance between the sample and the plane of registration. The deflection m in the plane of registration is related to the length of the scattering vector q by: q5
2π m λa
(7.4)
which follows directly from above and Eq. (1.5). If we assume, for the moment, that one of the two dimensions of the primary beam is negligible, we get one-dimensional effects: 1. slit-length effect: I~ðmÞ 5 2
ðN
PðtÞ:I
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 t2 dt
0
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(7.5)
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2. slit-width effect: I^ðmÞ 5
ðN 2N
QðxÞ:I ðm 2 xÞdx
(7.6)
It can be shown that the two-dimensional effect in Eq. (7.3) can be calculated by a sequential calculation of the slit-length effect Eq. (7.5), followed by the slit-width effect Eq. (7.6).198
Influence of the Size of the Detector Every detector has a finite size. Modern position-sensitive detectors are characterized by pixel size. In the case of a two-dimensional detector combined with a long slitcollimation system, the signal may be integrated along a virtual slit to reduce the noise of the data. A one-dimensional detector may have a very narrow pixel size in the measuring direction (m), but a larger window width.
Detector Length The detector (slit) length has the same effect as an extension of the primary beam P(t). However, it is not necessary to evaluate a new smearing integral. One can show that it is possible to combine the two effects (detector slit length and P(t)) by a convolution of the profile P(t) with the detector-length sensitivity function, which can in most cases be approximated by a step function. This convolution is already performed experimentally if the profile P(t) is recorded with the same detector window.
Detector Slit Width Like the detector slit length, the detector slit width causes an additional smearing effect. Since the primary beam profile Q(x) is usually measured with the same detector slit width as the scattering curve, it is not necessary to make additional corrections.
Measurement of the Beam Profiles P(t) and Q(x) As the beam is scattered in the sample, it is obvious that one would like to measure the beam profiles in the sample position to know the intensity at any point x and t. In the above equations, however, we have assumed that the primary beam entering the sample has no divergence, i.e., it is assumed to be a bundle of rays parallel to the central axis. This condition is not perfectly fulfilled. It is practically impossible to measure this divergence, but we can take it into account by measuring the profile P(t) in the plane of registration and using this profile in our equations, i.e., using it as if it corresponds to a nondivergent beam at the sample position. In any case, we have to attenuate the primary beam for these measurements to protect the detector from these high intensities. This situation is illustrated in Fig. 7.10 for an arbitrary beam. The beam deflection is given by the length of the sections (b 1 c). The section b corresponds to the coordinate t in the sample,
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FIGURE 7.10 Divergence of the primary beam.
and c is caused by the divergence. A beam parallel to the axis at t 5 b 1 c would have exactly the same deflection.
The Wavelength Effect The smearing effect caused by a wavelength distribution W(λ0 ) in the primary beam can be described by the wavelength integral13: ðN I ðmÞ 5 W ðλ0 Þ:I m=λ0 dλ0 (7.7) 0 0
where λ 5 λ/λ0 and λ0 is the mean or nominal wavelength used in Eq. (1.5), etc. This integration must be performed before the geometrical integrations of Eqs. (7.5) and (7.6). It should be mentioned that the influence of the wavelength smearing is fully equivalent to the effect of polydispersity in the sample. From Eq. (7.7), we see that the wavelength effect is zero for q (or m) equal to zero, and that the effective smearing interval increases linearly with the scattering angle. This wavelength integral is most important for neutron scattering where W(λ0 ) may have a half width of 10% and more. For X-rays without monochromator, this distribution can be approximated by a sum of two delta functions to represent the Kα and the Kβ line, the weight of the Kβ line can be estimated according to Zipper (1969).179 The angular dependence of the relative ranges of integration of the three smearing integrals is shown in Fig. 7.11. As already mentioned above, the wavelength integration width increases linearly with m (or q), the slit width integral is independent of the scattering angle, while the integration range decreases with increasing m-value for the slit length effect.
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FIGURE 7.11 Angular dependence of the ranges of integration of the smearing integrals: &&& wavelength integral, OOO slit-length integral (slit length L), Δ Δ Δ slit-width integral.
For gas-filled position-sensitive counters one would also find a smearing effect, due to the depth of the detector, that arises from rays not entering perpendicular to the wire axis,24 but the angles are so small for SAXS experiments that this effect can be neglected. In summary, we can describe the instrumental smearing effects by Eqs. (7.5)(7.7), or by one single combined formula in which t and x are given in units of q according to Eq. (7.4): 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðN ðN ðN ðq-xÞ2 1 t2 A dλ0 dt dx QðxÞ:PðtÞ:Wðλ0 Þ:I @ (7.8) Iexp ðqÞ 5 2 λ0 -N 0 0 A numerical calculation method for the exact determination of the resolution functions for radial symmetric collimation and scattering was described by Barker and Pedersen (1995).199
NEUTRONS The two main sources of neutrons are steady-state reactors and spallation sources. During the fission process in a neutron reactor, there is a continuous production of neutrons. At the very beginning these neutrons have a high kinetic energy that is reduced by the moderator fluid that is required to enhance fission, but the energy still corresponds to a wavelength of less than 0.1 nm. For SANS experiments, a further reduction of the kinetic energy is needed to have a high flux of neutrons in the necessary wavelength range. These cold moderators (cold source) are typically filled with liquid hydrogen or deuterium, and operate at around 20 K. They have to be placed in the highest neutron flux possible (i.e., near the reactor core). Outgoing neutrons are coupled to bent neutron guides that only transport the “cold” neutrons with a wavelength in the regime of 0.22 nm to the SANS experiment. Hard γ radiation does not follow the bent neutron guides, and is absorbed by the reactor shielding. High-flux research reactors are currently operational in many places around the world like in France, Grenoble, ILL; Germany, Garching, FRM-II; USA,
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Gaithersburg, NIST; and Lucas Heights, Australia, ANSTO with a neutron flux of about 1.5 3 10154 3 1014 neutrons per cm2 and s. In a nuclear spallation source, a particle accelerator is used to produce a beam of neutrons. Accelerated protons are smashed onto a heavy metal target such as tantalum. Each collision produces some neutrons and other particles. The secondary particles hit surrounding nuclei and create even more neutrons; in total about 2030 fast neutrons per proton. These neutrons are then slowed in moderators filled with liquid hydrogen or liquid methane to the energies that are needed for the scattering instruments. Spallation sources provide a pulsed neutron beam, and therefore a time-of-flight method is used on such sources. Spallation neutron sources are currently operational in places like Chicago, USA, IPNS; Oxford, UK, ISIS, PSI-Villigen, Switzerland, SINQ; Los Alamos, USA, LANSCE; and Oak Ridge, USA, SNS.
SMALL-ANGLE NEUTRON SCATTERING CAMERAS It is already clear from the section on neutron sources that SANS experiments can only be done in rather big installations which exist in increasing numbers in a variety of places around the world. Fortunately, there is a world directory of SANS instruments: (http://www.ill.eu/instruments-support/instruments-groups/groups/lss/more/ world-directory-of-sans-instruments/). Fig. 7.12 displays a schematic presentation of the SANS instrument D11 in Grenoble, one of the first large-scale facilities, which was commissioned in 1972200 and updated in 2012. The D11 is the archetype of a long, pinhole geometry instrument for SANS with high resolution and low background. It was designed for the study of large-scale structures in soft-matter systems, chemistry, biology, solid-state physics, and materials science. The mechanical velocity selector selects the wavelength of the neutrons with a polydispersity of about 10%, neutron guides (collimators) and diaphragms define the primary beam, the sample position is situated in a short air gap and allows for a variety of different sample cells and holders. The position sensitive two-dimensional detector can be positioned in an evacuated vacuum tube of 40 m length to allow for different resolution of the experiments. A most modern compact time-of-flight SANS instrument optimized for measurements of small sample volumes at the European Spallation Source has been published recently.201 It should be mentioned that there are also Bonse 2 Hart type USANS instruments.202
FIGURE 7.12 Schematic view of the pinhole SANS instrument D11 in Grenoble. Source: Figure taken with permission from homepage of ILL Grenoble.
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CONCLUSION In this chapter we discussed possible sources and cameras for SAXS and SANS. Basically, one has to differentiate between laboratory sources (only available for SAXS) and large-scale installations like synchrotrons and storage rings for SAXS and reactors, or spallation sources for SANS. Details of the instruments (often called “cameras”) are important to understand the limitations in flux, resolution, and sample thickness. Instrumental broadening or smearing functions are affecting the data evaluation process. Pinhole cameras are absolutely necessary for oriented samples, while slit cameras offer a higher primary beam power. Low background is important, and is available for pinhole systems when using scatter-less slits, while the block collimation system has a low background under slit conditions. Go¨bel mirrors help to focus the X-ray beam into the sample position. Detectors must have a high efficiency, a wide dynamic range, and a small channel or pixel size. Pinhole cameras need a two-dimensional detector, while onedimensional detectors can be used in slit cameras without real disadvantage.
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7 Instrumentation for SAXS and SANS An extended review on instrumentation for small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) was compiled by Pedersen.177,178 Here we report only on the central points about sources and cameras, followed by a mathematical description of instrumental broadening effects. Their correction will be discussed in Chapter 8, Numerical Methods.
X-RAYS For most of the applications in research laboratories, X-ray tubes with copper anodes are used. They emit X-rays at a wavelength of 0.154 nm (or 8.05 keV, Kα line), and the weaker Kβ line with a wavelength of 0.139 nm (or 8.91 keV). The intensity contribution of the Kβ line in the sample is between 20% and 35% of the Kα line, and depends on the operating time of the tube.179 Each tube also emits a continuous spectrum, the so-called Bremsstrahlung. This contribution can be suppressed electronically with a discriminator. The remaining wavelength distribution is shown in Fig. 7.1. Other anode materials generate other wavelengths. Molybdenum radiation (λ Kα 5 0.071 nm) is widely used in materials research, due to its much lower absorption coefficient than that of copper. X-ray tubes are operated with a high voltage of typically 40 kV. Higher levels of voltage will essentially only increase the continuous high-energy background radiation, but not the intensity of the Kα line.180 The maximum power is limited by the heating of the anode to about 2 kW (50 mA) current. This power limit for sealed tubes is given by the restricted heat transfer from the hot anode spot to the cooling fluid (water), and depends on the individual tube type. Rotating anodes are a special solution to this problem. This cylindrical, water-cooled anode must rotate in the high vacuum of the tube which is a nontrivial technical problem. Such rotating anodes have a much higher power ( . 10 kW), but are much more expensive, need more maintenance, and are less reliable. X-ray tubes, able to generate very small focal spot sizes, typically below 50 μm in diameter, are called microfocus X-ray tubes. Solid-anode microfocus X-ray tubes are in principle
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FIGURE 7.1 Wavelength distribution of copper radiation, measured with a xenon-filled proportional counter. Units of the abscissa are relative to the Cu Kα line.
very similar to standard tubes, but the electron beam is focused into a very small spot on the anode in the range 520 μm. This leads to a much higher source brightness. The brightness is the amount of photons emitted per unit time, per unit area, and per solid angle. In order to avoid anode melting, the electron-beam power density must be below a maximum value in the range 0.40.8 Wμm21, depending on the anode material.181 The gain in flux compared to a standard sealed tube can be up to one order of magnitude, depending on the beam shape. The cost of the source, including focusing mirrors, is double that of a sealed-tube system. In metal-jet anode microfocus X-ray tubes, the solid metal anode is replaced with a jet of liquid metal, which acts as the electron-beam target. The advantage of the metal-jet anode is that the maximum electron-beam power density is significantly increased, to about 3 2 6 Wμm21 for gallium and tin anodes.182,183 Gallium has an emission at Kα 5 9.2 keV (close to that of copper), while an indium rich alloy also generates a highenergy Kα line at 24.2 keV.184 Metal-jet anodes are quite expensive, however, they may increase the flux in the sample by up to one order of magnitude. Synchrotrons and storage rings are other sources of X-rays with increasing importance for scientific research. In these installations, charged particles (positrons or electrons) are forced into a (partial) circular path by bending magnets or special devices like wigglers and undulators.185 The necessary radial acceleration causes the emission of radiation in the X-ray regime, because the particles move with high energy (close to the speed of light). The radiation is emitted in tangential directions (like sparks from a grinding wheel). The radiation has a very high intensity and is reasonably well collimated in one direction. The radiation is “white” (broad wavelength distribution), and has a pulsed time structure. The desired wavelength is selected by a monochromator. The main disadvantage of these radiation sources is the high radiation level in the experimental position. All experiments
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FIGURE 7.2 Dependence of the scattering intensity I on sample thickness t. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
must therefore be performed under remote control in a shielded hutch, which makes the alignment of the system a somewhat difficult task.
Optimum Sample Thickness In order to get as much scattered intensity as possible, it is necessary to optimize the sample thickness. The intensity I scattered at small angles is proportional to te2μt (t being the thickness of the sample, and μ the absorption coefficient). This expression has its maximum at topt. 5 1/μ (see Fig. 7.2). For Cu-Kα radiation this thickness is a little bit less than 1 mm for H2O, and about 76 μm for Al (corresponding values are 8.3 mm and 718 μm for Mo-Kα!). Aqueous solutions are filled in very thin quartz capillaries with a diameter of about 1 mm. The absorption of quartz is about 10 times higher than that of water. The typical wall thickness is therefore about 50 μm or less.
SMALL-ANGLE X-RAY SCATTERING CAMERAS Instruments for the measurement of SAXS curves are usually called cameras. The difference between these devices and regular cameras for taking pictures is the fact that there are no lenses available for X-rays. The refractive index n0 is nearly equal to one, far above the resonance frequency. Slits or pinholes are frequently used to create a geometrically
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well-defined narrow primary beam. Collimating or focusing systems are difficult to realize. Bent mirrors or crystal monochromators with a special cut can be used for this purpose. Such systems are very expensive, and are used for small-angle scattering stations at synchrotrons, or sometimes in combination with rotating anodes in laboratory systems. Many modern laboratory systems use so-called Go¨bel mirrors to collimate or focus the primary beam (see below).
Pinhole or Slit Cameras The simplest system is a pair of two parallel slits. The narrower the slits and the greater the distance between the pair of slits, the higher is the resolution of the camera. As the resolution of the camera we define the minimum q value at which we can measure the scattering from the sample with negligible background scattering from the camera (the remaining contribution is from the primary beam, which has in any case to be blocked by a primary beam stop to protect the detector). The simple two-slit system has an essential drawback: each edge of the slit is itself a source of a considerable amount of secondary scattering (parasitic scattering) into the small-angle region, and thus reduces the possible resolution. This design can be improved by a third slit (“guard slit”) behind the second slit, adjusted in such a way that it is just not hit by the direct beam, but absorbs as much as possible of the parasitic scattering that originates from the second slit (Fig. 7.3). In the plane of registration, PR, the zone P is the direct beam, the zones, s, correspond to regions of strong, and the zones w to regions of weak, parasitic scattering. This weak parasitic scattering or background scattering can be highly reduced by so-called scatterless hybrid metalsingle-crystal slits.187 These systems can also help to increase the flux of X-rays in the sample by a factor of three. The same idea can also be used for pin-hole systems. FIGURE 7.3 Slit camera with three slits, each consisting of a pair of edges running perpendicular to the plane of the paper. The dimensions in the vertical direction are greatly enlarged. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
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The resolution can be increased by increasing the distance between the slits. Stasiecki and Stuhrmann188 installed such a system, with a length of 50 meters below the roof of the institute building at the University of Mainz! With this camera, they were able to measure the size of red blood cells in the regime of several micrometers. This installation was obviously to show the principle possibility to measure such large systems with SAXS.
The Block Camera The block camera, designed by O. Kratky186 and often referred to as the Kratky Camera (Fig. 7.4), uses blocks to define the size of the primary beam. Unlike the slit system, it does not allow measurements above and below the direct beam. The system is formed by a U-shaped middle part, M, a bridge, B, and an entrance slit (or block), E. The main idea is to allow full parasitic scattering below the primary beam, but to have negligible parasitic scattering above the beam; this is the half-plane used for the measurement (Fig. 7.5).
FIGURE 7.4 Threedimensional schematic drawing of the block system with the plane of registration, PR, and the primary beam stop, PS. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
FIGURE 7.5 Schematic drawing of a section through the block system with the focus, F, and with an intensity distribution in the plane of registration.
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We can see from the above figures that there is, under ideal conditions, no parasitic scattering above the main section, H. The direct beam is blocked by the beam stop, PS, to protect the detector. The resulting primary beam passing through the sample, P, is a thin long line (typical thickness or width 100 μm, typical length 1 cm). This system has a rather high intensity of X-ray light shining on the sample that is necessary for the measurement of colloidal systems with low contrast. The disadvantage is the strong instrumental broadening effect coming from the length of the primary beam (smearing effect). This effect can, however, easily be eliminated numerically for isotropic, nonoriented samples. Air would be another source of parasitic scattering. In nearly all modern laboratory systems, the whole space between the entrance window (between the X-ray tube window and the collimation system) and the detector window can be evacuated. Special refillable capillary holders allow the positioning of liquid samples in the vacuum.
Comparison of Pinhole and Block Cameras In a very simple picture, a pinhole camera is characterized by a point-like primary beam, while a block or long slit camera has a linear primary beam like many pinholes parallel to each other. When considering a pinhole system with a beam diameter in the order of 100 μm, a long slit of 1 cm, and a thickness of 100 μm, we see immediately that the beam intensity is about 100 times higher in the long slit system. The disadvantage of the long slit or block system is the geometrical smearing effect: we have the superposition of about 100 scattering experiments with increasing linear offset. These smearing effects have to be corrected or desmeared. For a long time this was a big challenge for the experimenter. Modern computer routines such as indirect Fourier transformation have made this an easy task, once the intensity distribution across the primary beam has been determined experimentally. For further details, see Pedersen (2002)178 and Chapter 8, Numerical Methods.
Go¨bel Mirrors X-rays are emitted from the anode of an X-ray tube in all directions. Only a small portion of the photons will pass the pinholes, slits, or blocks. This situation could be improved by using focusing elements that convert the diverging beam into a collimated parallel beam, when using parabolic mirrors, or into a focused beam when using elliptical mirrors. Contrary to total-reflection mirrors (Franks mirrors),189191 where the X-ray beam is reflected at incident angles below 0.1 degree, Bragg diffraction from crystals can change the direction and divergence of the incident X-ray beam at higher angular values. The principle that a particular wavelength is diffracted at a particular Bragg angle led to the development of conventional monochromator crystals. Because of the divergent-beam characteristic of laboratory X-ray sources, these crystals do not work efficiently (most of the radiation emitted from the source is cut away prior to diffraction). In contrast, Go¨bel mirrors are graded multilayer crystals produced such that the spacing d between the layers varies as shown in Fig. 7.6a. The d-spacing gradient depends on factors that include wavelength and the location of the mirror with respect to the focus of
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FIGURE 7.6 (A) The mirror is made of graded multilayers consisting of A silicon and B tungsten. (B) The parabolically bent mirror collimates divergent X-rays into a parallel beam.
the source. The combination of layered materials with a large contrast in electron density (e.g., W and Si) results in high integral reflectivity. Go¨bel mirrors are parabolically bent to yield a diffracted monochromated parallel beam.192,193 The radiation is monochromated to a high-intensity Cu Kα primary beam (in the case of Cu anodes), while Cu Kβ and Bremsstrahlung are suppressed or do not pass the collimation system. If a single parabolically bent Go¨bel mirror is placed in the beam path, so that the line focus of the X-ray source is at the focus of the parabola, then approximately 1 degree of the divergent primary beam will be collected and transformed into a monochromatic parallel beam (Fig. 7.6b). Such Go¨bel mirrors have been combined with the Kratky block collimator to create a very high-flux camera with low background.194 In the meantime, Go¨bel mirrors are available as a one-dimensional mirror with parabolic (collimating) and elliptical (focusing) bending. Twodimensional mirrors collimate or focus the X-rays into a pinhole system. In so-called Montel optics, the two mirrors are mounted side-by-side at 90 degrees to each other.195
The BonseHart Camera The BonseHart Camera196 is based on multiple reflections of the primary beam and the scattered light from opposite sides of a channel that is cut into a single crystal of germanium or silicon. Only light rays of a certain direction (and wavelength) are reflected. The suppression of other rays increases with the number of reflections. The first crystal is fixed and
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FIGURE 7.7 Schematic drawing of the BonseHart camera: (1) source; (2) monochromator crystal; (3) sample; (4) analyzer crystal; (5) detector. Source: Figure taken with permission from Kratky, O., Experimental Technique. In Small Angle X-ray Scattering, Glatter, O.; Kratky, O., Eds. Academic Press: London, 1982; pp 53-83.186
defines the dimension of the primary beam. The second one rotates with the detector around the sample, and defines the scattering angle. This system creates a line-shaped primary beam like the Kratky system, but the light intensity in the sample is much lower. This is the reason why BonseHart systems are used with synchrotrons, and sometimes with rotating anodes. The BonseHart system has a much better resolution than the Kratky system; the minimum q values are comparable with that available in light scattering. Such experiments are, therefore, often called ultrasmall-angle X-ray scattering (USAXS) (Fig. 7.7). It should be mentioned that it is also possible have a point-collimation USAXS system by using two crystals on each side of the sample.197 Such a four-crystal system can only be used for strong scatterers, or for a two-dimensional detection of oriented samples.
MATHEMATICAL DESCRIPTION OF THE INSTRUMENTAL EFFECTS The ideal experimental design, i.e., a point-like strictly monochromatic primary beam with high energy, point-like sample and point-like detector can hardly be reached in a real experiment. The various effects caused by the most common experimental designs are discussed in the following section. Instrumental broadening is caused by the finite dimensions of the primary beam and of the detector. In the Kratky block system, the primary beam has the shape of a thin long slit (see Fig. 7.8). The detector may also have the shape of a slit. All possible rays in the dashed channel in this figure are registered by the detector at the position m. Obviously, the actual scattering angles in the sample may be quite different for all these rays. Let I0(x,t) be the two-dimensional intensity distribution of the primary beam entering the sample, where t is the coordinate from the center of the primary beam in the direction of the slit length, and x is the coordinate perpendicular to t (slit width). Simple geometrical considerations show that the intensity distribution I0(x,t) can be written as a product of two independent distributions: I0 ðx; tÞ 5 PðtÞ:QðxÞ
(7.1)
This independence is correct with sufficient accuracy for most practical applications. The total deflection of a ray through an arbitrary point (t, x) into the center of the detector can easily be found with the help of Fig. 7.9, and is given by the expression: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm2xÞ2 1 t2 (7.2)
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FIGURE 7.8 Ray channel to the detector in a slit camera.
FIGURE 7.9 Ray geometry in a slit camera.
The influence of the finite detector size will be discussed later. The intensity I(m) at the position m is the sum (integral) over all contributions from all possible t and x positions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN ðN IðmÞ 5 2 dx dt:PðtÞ:QðxÞ:I ðm2xÞ2 1 t2 (7.3) 2N
o
where 2sin θ/2 D m/a and a are the distance between the sample and the plane of registration. The deflection m in the plane of registration is related to the length of the scattering vector q by: q5
2π m λa
(7.4)
which follows directly from above and Eq. (1.5). If we assume, for the moment, that one of the two dimensions of the primary beam is negligible, we get one-dimensional effects: 1. slit-length effect: I~ðmÞ 5 2
ðN
PðtÞ:I
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 t2 dt
0
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2. slit-width effect: I^ðmÞ 5
ðN 2N
QðxÞ:I ðm 2 xÞdx
(7.6)
It can be shown that the two-dimensional effect in Eq. (7.3) can be calculated by a sequential calculation of the slit-length effect Eq. (7.5), followed by the slit-width effect Eq. (7.6).198
Influence of the Size of the Detector Every detector has a finite size. Modern position-sensitive detectors are characterized by pixel size. In the case of a two-dimensional detector combined with a long slitcollimation system, the signal may be integrated along a virtual slit to reduce the noise of the data. A one-dimensional detector may have a very narrow pixel size in the measuring direction (m), but a larger window width.
Detector Length The detector (slit) length has the same effect as an extension of the primary beam P(t). However, it is not necessary to evaluate a new smearing integral. One can show that it is possible to combine the two effects (detector slit length and P(t)) by a convolution of the profile P(t) with the detector-length sensitivity function, which can in most cases be approximated by a step function. This convolution is already performed experimentally if the profile P(t) is recorded with the same detector window.
Detector Slit Width Like the detector slit length, the detector slit width causes an additional smearing effect. Since the primary beam profile Q(x) is usually measured with the same detector slit width as the scattering curve, it is not necessary to make additional corrections.
Measurement of the Beam Profiles P(t) and Q(x) As the beam is scattered in the sample, it is obvious that one would like to measure the beam profiles in the sample position to know the intensity at any point x and t. In the above equations, however, we have assumed that the primary beam entering the sample has no divergence, i.e., it is assumed to be a bundle of rays parallel to the central axis. This condition is not perfectly fulfilled. It is practically impossible to measure this divergence, but we can take it into account by measuring the profile P(t) in the plane of registration and using this profile in our equations, i.e., using it as if it corresponds to a nondivergent beam at the sample position. In any case, we have to attenuate the primary beam for these measurements to protect the detector from these high intensities. This situation is illustrated in Fig. 7.10 for an arbitrary beam. The beam deflection is given by the length of the sections (b 1 c). The section b corresponds to the coordinate t in the sample,
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FIGURE 7.10 Divergence of the primary beam.
and c is caused by the divergence. A beam parallel to the axis at t 5 b 1 c would have exactly the same deflection.
The Wavelength Effect The smearing effect caused by a wavelength distribution W(λ0 ) in the primary beam can be described by the wavelength integral13: ðN I ðmÞ 5 W ðλ0 Þ:I m=λ0 dλ0 (7.7) 0 0
where λ 5 λ/λ0 and λ0 is the mean or nominal wavelength used in Eq. (1.5), etc. This integration must be performed before the geometrical integrations of Eqs. (7.5) and (7.6). It should be mentioned that the influence of the wavelength smearing is fully equivalent to the effect of polydispersity in the sample. From Eq. (7.7), we see that the wavelength effect is zero for q (or m) equal to zero, and that the effective smearing interval increases linearly with the scattering angle. This wavelength integral is most important for neutron scattering where W(λ0 ) may have a half width of 10% and more. For X-rays without monochromator, this distribution can be approximated by a sum of two delta functions to represent the Kα and the Kβ line, the weight of the Kβ line can be estimated according to Zipper (1969).179 The angular dependence of the relative ranges of integration of the three smearing integrals is shown in Fig. 7.11. As already mentioned above, the wavelength integration width increases linearly with m (or q), the slit width integral is independent of the scattering angle, while the integration range decreases with increasing m-value for the slit length effect.
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FIGURE 7.11 Angular dependence of the ranges of integration of the smearing integrals: &&& wavelength integral, OOO slit-length integral (slit length L), Δ Δ Δ slit-width integral.
For gas-filled position-sensitive counters one would also find a smearing effect, due to the depth of the detector, that arises from rays not entering perpendicular to the wire axis,24 but the angles are so small for SAXS experiments that this effect can be neglected. In summary, we can describe the instrumental smearing effects by Eqs. (7.5)(7.7), or by one single combined formula in which t and x are given in units of q according to Eq. (7.4): 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðN ðN ðN ðq-xÞ2 1 t2 A dλ0 dt dx QðxÞ:PðtÞ:Wðλ0 Þ:I @ (7.8) Iexp ðqÞ 5 2 λ0 -N 0 0 A numerical calculation method for the exact determination of the resolution functions for radial symmetric collimation and scattering was described by Barker and Pedersen (1995).199
NEUTRONS The two main sources of neutrons are steady-state reactors and spallation sources. During the fission process in a neutron reactor, there is a continuous production of neutrons. At the very beginning these neutrons have a high kinetic energy that is reduced by the moderator fluid that is required to enhance fission, but the energy still corresponds to a wavelength of less than 0.1 nm. For SANS experiments, a further reduction of the kinetic energy is needed to have a high flux of neutrons in the necessary wavelength range. These cold moderators (cold source) are typically filled with liquid hydrogen or deuterium, and operate at around 20 K. They have to be placed in the highest neutron flux possible (i.e., near the reactor core). Outgoing neutrons are coupled to bent neutron guides that only transport the “cold” neutrons with a wavelength in the regime of 0.22 nm to the SANS experiment. Hard γ radiation does not follow the bent neutron guides, and is absorbed by the reactor shielding. High-flux research reactors are currently operational in many places around the world like in France, Grenoble, ILL; Germany, Garching, FRM-II; USA,
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Gaithersburg, NIST; and Lucas Heights, Australia, ANSTO with a neutron flux of about 1.5 3 10154 3 1014 neutrons per cm2 and s. In a nuclear spallation source, a particle accelerator is used to produce a beam of neutrons. Accelerated protons are smashed onto a heavy metal target such as tantalum. Each collision produces some neutrons and other particles. The secondary particles hit surrounding nuclei and create even more neutrons; in total about 2030 fast neutrons per proton. These neutrons are then slowed in moderators filled with liquid hydrogen or liquid methane to the energies that are needed for the scattering instruments. Spallation sources provide a pulsed neutron beam, and therefore a time-of-flight method is used on such sources. Spallation neutron sources are currently operational in places like Chicago, USA, IPNS; Oxford, UK, ISIS, PSI-Villigen, Switzerland, SINQ; Los Alamos, USA, LANSCE; and Oak Ridge, USA, SNS.
SMALL-ANGLE NEUTRON SCATTERING CAMERAS It is already clear from the section on neutron sources that SANS experiments can only be done in rather big installations which exist in increasing numbers in a variety of places around the world. Fortunately, there is a world directory of SANS instruments: (http://www.ill.eu/instruments-support/instruments-groups/groups/lss/more/ world-directory-of-sans-instruments/). Fig. 7.12 displays a schematic presentation of the SANS instrument D11 in Grenoble, one of the first large-scale facilities, which was commissioned in 1972200 and updated in 2012. The D11 is the archetype of a long, pinhole geometry instrument for SANS with high resolution and low background. It was designed for the study of large-scale structures in soft-matter systems, chemistry, biology, solid-state physics, and materials science. The mechanical velocity selector selects the wavelength of the neutrons with a polydispersity of about 10%, neutron guides (collimators) and diaphragms define the primary beam, the sample position is situated in a short air gap and allows for a variety of different sample cells and holders. The position sensitive two-dimensional detector can be positioned in an evacuated vacuum tube of 40 m length to allow for different resolution of the experiments. A most modern compact time-of-flight SANS instrument optimized for measurements of small sample volumes at the European Spallation Source has been published recently.201 It should be mentioned that there are also Bonse 2 Hart type USANS instruments.202
FIGURE 7.12 Schematic view of the pinhole SANS instrument D11 in Grenoble. Source: Figure taken with permission from homepage of ILL Grenoble.
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CONCLUSION In this chapter we discussed possible sources and cameras for SAXS and SANS. Basically, one has to differentiate between laboratory sources (only available for SAXS) and large-scale installations like synchrotrons and storage rings for SAXS and reactors, or spallation sources for SANS. Details of the instruments (often called “cameras”) are important to understand the limitations in flux, resolution, and sample thickness. Instrumental broadening or smearing functions are affecting the data evaluation process. Pinhole cameras are absolutely necessary for oriented samples, while slit cameras offer a higher primary beam power. Low background is important, and is available for pinhole systems when using scatter-less slits, while the block collimation system has a low background under slit conditions. Go¨bel mirrors help to focus the X-ray beam into the sample position. Detectors must have a high efficiency, a wide dynamic range, and a small channel or pixel size. Pinhole cameras need a two-dimensional detector, while onedimensional detectors can be used in slit cameras without real disadvantage.
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