International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 4.2, pp. 229-241

Section 4.2.5. Filters and monochromators

D. C. Creaghb

4.2.5. Filters and monochromators

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4.2.5.1. Introduction

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All sources of X-rays, whether they be produced by conventional sealed tubes, rotating-anode systems, or synchrotron-radiation sources, emit over a broad spectral range. In many cases, this spectral diversity is of concern, and techniques have been developed to minimize the problem. These techniques involve the use of filters, mirrors, and Laue and Bragg crystal monochromators, chosen so as to provide the best compromise between flux and spectral purity in a particular experiment. In other chapters, authors have discussed the use of techniques to improve the spectral purity of X-ray sources. This section does not purport to be a comprehensive exposition on the topic of filters and monochromators. Rather, it seeks to point the reader towards the information given elsewhere in this volume, and to add complementary information where necessary. A search of the Subject Index will find references to filters and monochromators that are not explicitly mentioned in the text of this section.

The ability to select photon energies, or bands of energies, depends on the scattering power of the atoms from which the monochromator is made and the arrangement of the atoms within the monochromator. The scattering powers of the atoms and their dependence on the energy of the incident photons were discussed in Sections 4.2.3[link] and 4.2.4[link] and are discussed more fully in Section 4.2.6[link]. In brief, the scattering power of the atom, or atomic scattering factor, is defined, for a given incident photon energy, as the ratio of the scattering power of the atom to that of a free Thomson electron. The scattering power is denoted by the symbol [f(\omega, \Delta)] and is a complex quantity, the real part of which, [f^{\prime }(\omega, \Delta)], is related to the elastic scattering cross section, and the imaginary part of which, [f^{\prime \prime }(\omega, \Delta)], is related directly to the photoelectric scattering cross section and therefore the linear attenuation coefficient [\mu _l].

At an interface between, say, air and the material from which the monochromator is made, reflection and refraction of the incident photons can occur, as dictated by Maxwell's equations. There is an associated refractive index n given by [ n=(1+\chi)^{1/2}, \eqno (4.2.5.1)]where [ \chi =-(r_e\lambda ^2/\pi)\textstyle \sum \limits _jN_j\;f_j(\omega, \Delta), \eqno (4.2.5.2)][r_e] is the classical radius of the electron, and [N_j] is the number density of atoms of type j.

An angle of total external reflection [\alpha _{c}] exists for the material, which is a function of the incident photon energy, since [f_{j}(\omega, \Delta)] is a function of photon energy. Thus, a polychromatic beam incident at the critical angle of one of the photon energies (E) will reflect totally those components having energies less than E, and transmit those components with energies greater than E. Fig. 4.2.5.1[link] shows calculations by Fukumachi, Nakano & Kawamura (1986[link]) for the reflectivity of single layers of aluminium, copper and platinum as a function of incident energy for a fixed angle of incidence (0.2°). For the aluminium specimen, the reflectivity curve shows the rapid decrease in reflectivity as the critical angle is exceeded. The reflectivity in this region varies as [E^{-2}]. The effect of increasing atomic number can be seen: the higher the atomic factor [f(\omega, \Delta)], the greater the energy that can be reflected from the surface. Also visible are the effects of the dispersion corrections [f^{\prime }(\omega, \Delta)] and [f^{\prime \prime }(\omega, \Delta)] on reflectivity. For copper, the K shell is excited, and for platinum the [L_{\rm I}], [L_{\rm II}] and [L_{\rm III}] shells are excited by the polychromatic beam.

[Figure 4.2.5.1]

Figure 4.2.5.1 | top | pdf |

The variation of specular reflectivity with incident photon energy is shown for materials of different atomic number and a constant angle of incidence of 0.2°. (a) Aluminium: note the rapid decrease of reflectivity with energy. (b) Copper: the sudden decrease of reflectivity is due to the modification of the scattering-length density owing to absorption at the K-absorption edge. (c) Platinum: the three discontinuities in the reflectivity curve are due to absorption at the LI-, LII-, and LIII-absorption edges.

Interfaces can therefore be used to act as low-pass energy filters. The surface roughness and the existence of impurities and contaminants on the interface will, however, influence the characteristics of the reflecting surface, sometimes significantly.

4.2.5.2. Mirrors and capillaries

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Whilst neither of these classes of X-ray optical device is strictly speaking a monochromator, they nevertheless form component parts of monochromator systems in the laboratory and at synchrotron-radiation sources.

4.2.5.2.1. Mirrors

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In the laboratory, mirrors are used in conjunction with conventional sealed tubes and rotating-anode sources, the emission from which consists of Bremsstrahlung upon which is superimposed the characteristic spectrum of the anode material (Subsection 2.3.5.2[link] ). The shape of the Bremsstrahlung spectrum can be significantly modified by mirrors, and the intensity emitted at harmonics of the characteristic wavelength can be significantly reduced. More importantly, the mirrors can be fashioned into shapes that enable the emitted radiation to be brought to a focus. Ellipsoidal, logarithmic spiral, and toroidal mirrors have been manufactured commercially for use with laboratory X-ray sources. Since the X-rays are emitted isotropically from the anode surface, it is important to devise a mirror system that has a maximum angle of acceptance and a relatively long focal length.

At synchrotron-radiation sources, the high intensities that are generated over a very broad spectral range give rise to significant heat loading of subsequent monochromators and therefore degrade the performance of these elements. In many systems, mirrors are used as the first optical element in the monochromator, to reduce the heat load on the primary monochromator and to make it easier for the subsequent monochromators to reject harmonics of the chosen radiation. Shaped mirror geometries are often used to focus the beam in the horizontal plane (Subsection 2.2.7.3[link] ). A schematic diagram of the optical elements of a typical synchrotron-radiation beamline is shown in Fig. 4.2.5.2[link] . In this, the primary mirror acts as a thermal shunt for the subsequent monochromator, minimizes the high-energy component that may give rise to possible harmonic content in the final beam, and acts as a vertical collimator. The radii of curvature of mirrors can be changed using a mechanical four-point bending system (Oshima, Harada & Sakabe, 1986[link]). More recent advances in mirror technology enable the shape of the mirror to be changed through use of the piezoelectric effect (Sussini & Labergerie, 1995[link]).

[Figure 4.2.5.2]

Figure 4.2.5.2 | top | pdf |

The use of mirrors in a typical synchrotron-radiation beamline. The X-rays are emitted tangentially to the orbit of the stored positron beam. They pass through a beam-defining slit onto a mirror that serves three purposes, viz energy discrimination, heat absorption, and focusing, by means of a mechanical four-point bending system. The beam then passes into a double-crystal monochromator, which selects the desired photon energy. The second element of this monochromator is capable of being bent sagittally using a mechanical four-point bending system to focus the beam in the horizontal plane. The beam is then refocused and redirected by a second mirror.

4.2.5.2.2. Capillaries

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Capillaries, and bundles of capillaries, are finding increasing use in situations where a focused beam is required. The radiation is guided along the capillary by total external reflection, and the shape of the capillary determines the overall flux gain and the uniformity of the focused spot. Gains in flux of 100 and better have been reported. There is, however, a degradation in the angular divergence of the outgoing beam. For single capillaries, applications are laboratory-based protein crystallography, microtomography, X-ray microscopy, and micro-X-ray fluorescence spectroscopy. The design and construction of capillaries for use in the laboratory and at synchrotron-radiation sources has been discussed by Bilderback, Thiel, Pahl & Brister (1994[link]), Balaic & Nugent (1995[link]), Balaic, Nugent, Barnea, Garrett & Wilkins (1995[link]), Balaic et al. (1996[link]), and Engström, Rindby & Vincze (1996[link]). They are usually used after other monochromators in these applications and their role as a low-pass energy filter is not of much significance.

Bundles of capillaries are currently being produced commercially to produce focused beams (ellipsoidally shaped bundles) and half-ellipsoidal bundles are used to form beams of large cross section from conventional laboratory sources (Peele et al., 1996[link]; Kumakhov & Komarov, 1990[link]).

4.2.5.2.3. Quasi-Bragg reflectors

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For one interface, the reflectivity (R) and the transmissivity (T) of the surface are determined by the Fresnel equations, viz: [ R=\left | (\theta _{1}-\theta _{2})/(\theta _{1}+\theta _{2})\right | ^{2}, \eqno (4.2.5.3)]and [ T=\left | 2\theta _{1}/(\theta _{1}+\theta_{2})\right | ^{2}, \eqno (4.2.5.4)]where [\theta _{1}] and [\theta _{2}] are the angles between the incident ray and the surface plane and the reflected ray and the surface plane, respectively.

If a succession of interfaces exists, the possibility of interference between successively reflected rays exists. Param­eters that define the position of the interference maxima, the line breadths of those maxima, and the line intensity depend inter alia on the regularity in layer thickness, the interface surface roughness, and the existence of surface tilts between successive interfaces. Algorithms for solving this type of problem are incorporated in software currently available from a number of commercial sources (Bede Scientific, Siemens, and Philips). The reflectivity profile of a system having a periodic layer structure is shown in Fig. 4.2.5.3[link] . This is the reflectivity profile for a multiple-quantum-well structure of alternating aluminium gallium arsenide and indium gallium arsenide layers (Holt, Brown, Creagh & Leon, 1997[link]). Note the interference maxima that are superimposed on the Fresnel reflectivity curve. From the full width at half-maximum of these interference lines, it can be inferred that the energy discrimination of the system, ΔE/E, is 2%. The energy range that can be reflected by such a multilayer system depends on the interlayer thickness: the higher the photon energy, the thinner the layer thickness.

[Figure 4.2.5.3]

Figure 4.2.5.3 | top | pdf |

The reflectivity of a multiple-quantum-well device is shown. This consists of 40 alternating layers of AlGaAs and InGaAs. Shown also, but shifted downwards on the vertical scale for the purpose of clarity, is the theoretical prediction based on standard electromagnetic theory.

Commercially available multilayer mirrors exist, and hitherto they have been used as monochromators in the soft X-ray region in X-ray fluorescence spectrometers. These monochromators are typically made of alternating layers of tungsten and carbon, to maximize the difference in scattering-length density at the interfaces. Whilst the energy resolution of such systems is not especially good, these monochromators have a good angle of acceptance for the incident beam, and reasonably high photon fluxes can be achieved using conventional laboratory sources.

A recent development of this, the Goebel mirror system, is supplied as an accessory to a commercially available diffractometer (Siemens, 1996a[link],b[link],c[link]; OSMIC, 1996[link]). This system combines the focusing capacity of a curved mirror with the energy selectivity of the multilayer system. The spacing between layers in this class of mirror multilayers can be laterally graded to enhance the incident acceptance angle. These multilayers can be fixed to mirrors of any figure to a precision of 0.3′ and can therefore can be used to form parallel beams (parabolic optical elements) as well as focused beams (elliptical optical elements) of high quality.

4.2.5.3. Filters

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It is usual to consider only the cases where a quasimonochromatic beam is to be extracted from a polychromatic beam. Before discussing this class of usage, mention must be made of two simple forms of filtering of radiation.

In the first, screening, a thin layer of absorbing foil is used to reduce the effect of specimen fluorescence on photon counting, film and imaging-plate detectors. A typical example is the use of aluminium foil in front of a Polaroid camera used in a Laue camera to reduce the K-shell fluorescence radiation from a transition-metal crystal when using a conventional sealed molybdenum X-ray source. A 0.1 mm thick foil will reduce the fluorescent radiation from the crystal by a factor of about five, and this radiation is emitted isotropically from the specimen. In contrast, the wanted Laue-reflected beams are emitted as a nearly parallel beam, and the signal-to-noise ratio in the resulting photograph is much increased.

The second case is the ultimate limiting case of filtering, shielding. If it is necessary to shield an object completely from a polychromatic incident beam, a sufficient thickness of absorbing material, calculated using the data in Section 4.2.4[link], to reduce the beam intensity to the level of the ambient background is inserted in the beam. [The details of how shielding systems are designed are given in reference works such as the Handbook of Radiation Measurement and Protection (Brodsky, 1982[link]).] In general, the use of an absorber of one atomic species will provide insufficient shielding. The use of composite absorbers is necessary to achieve a maximum of shielding for a minimum of weight. This is of utmost importance if one is designing, say, the shielding of an X-ray telescope to be carried in a rocket or a balloon (Grey, 1996[link]). To produce shielding that satisfies the requirements of minimum weight, good mechanical rigidity, and ability to be constructed to good levels of mechanical tolerance, shielding must be constructed using a number of layers of different absorbers, chosen such that the highest-energy radiation is just stopped in the first layer, the L-shell fluorescent radiation created in the absorption process is stopped in the second, and the lower-energy L and M fluorescent radiation is stopped by the next layer, and so on until the desired radiation level is reached.

In the usual case involving filters, the problem is one of removing as much as possible of the Bremsstrahlung radiation and unwanted characteristic radiation from the spectrum of a laboratory sealed tube or rotating-anode source whilst retaining as much of the wanted radiation as is possible. To give an example, a thin characteristic radiation filter of nickel of appropriate thickness almost completely eliminates the Bremsstrahlung and Kβ radiation from an X-ray source with a copper target, but reduces the intensity in the Cu Kα doublet by only about a factor of two. For many applications, this is all that is necessary to provide the required degree of monochromatization. If there is a problem with the residual Bremsstrahlung, this problem may be averted by making a second set of measurements with a different filter, one having an absorption edge at an energy a little shorter than that of the desired emission line. The difference between the two sets of measurements corresponds to a comparatively small energy range spanning the emission line. This balanced-filter method is more cumbersome than the single-filter method, but no special equipment or difficult adjustments are required. In general, if the required emission is from an element of atomic number Z, the first foil is made from material having atomic number Z − 1 and the second from atomic number Z + 1. A better balance can be achieved using three foils (Young, 1963[link]). The use of filters is discussed in more detail in §2.3.5.4.2[link] . Data for filters for the radiations in common use are given in Tables 2.3.5.2[link] and 2.3.5.3[link] . The information necessary for choosing filter materials and estimating their optimum thicknesses for other radiations is given in Sections 4.2.2[link], 4.2.3[link], and 4.2.4[link].

It should be remembered that filtration changes the wavelength of the emission line slightly, but this is only of significance for measurements of lattice parameters to high precision (Delf, 1961[link]).

4.2.5.4. Monochromators

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4.2.5.4.1. Crystal monochromators

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Even multifoil balanced filters transmit a wide range of photon energies. Strictly monochromatic radiation is impossible, since all atomic energy levels have a finite width, and emission from these levels therefore is spread over a finite energy range. The corresponding radiative line width is important for the correct evaluation of the dispersion corrections in the neighbourhood of absorption edges (§4.2.6.3.3.2[link]). Even Mössbauer lines, originating as they do from nuclear energy levels that are much narrower than atomic energy levels, have a finite line width. To achieve line widths comparable to these requires the use of monochromators using carefully selected single-crystal reflections.

Crystal monochromators make use of the periodicity of `perfect' crystals to select the desired photon energy from a range of photon energies. This is described by Bragg's law, [ 2d_{hkl}\sin \theta =n\lambda, \eqno (4.2.5.5)]where [d_{hkl}] is the spacing between the planes having Miller indices hkl, [\theta] is the angle of incidence, n is the order of a particular reflection (n = 1, 2, 3[\ldots]), and λ is the wavelength.

If there are wavelength components with values near λ/2, λ/3, [\ldots], these will be reflected as well as the wanted radiation, and harmonic contamination can result. This can be a difficulty in spectroscopic experiments, particularly XAFS, XANES and DAFS (Section 4.2.3[link]).

Equation (4.2.5.5)[link] neglects the effect of the refractive index of the material. This is usually omitted from Bragg's law, since it is of the order of 10−5 in magnitude. Because the refractive index is a strong function of wavelength, the angles at which the successive harmonics are reflected are slightly different from the Bragg angle of the fundamental. This fact can be used in multiple-reflection monochromators to minimize harmonic contamination.

As can be seen in Fig. 4.2.5.4[link] , each Bragg reflection has a finite line width, the Darwin width, arising from the interaction of the radiation with the periodic electron charge distribution. [See, for example, Warren (1968[link]) and Subsection 7.2.2.1[link] .] Each Bragg reflection therefore contains a spread of photon energies. The higher the Miller indices, the narrower the Darwin width becomes. Thus, for experiments involving the Mössbauer effect, extreme back-reflection geometry is used at the expense of photon flux.

[Figure 4.2.5.4]

Figure 4.2.5.4 | top | pdf |

In (a), the schematic rocking curve for a silicon crystal in the neighbourhood of the 111 Bragg peak is shown. The full curve is due to the crystal set to the true Bragg angle, and the dotted curve corresponds to a surface tilted at an angle of 2′′ with respect to the beam prior to the acquisition of the rocking curve. Only the 111 and 333 reflections are shown for clarity. The 222 reflection is very weak because the geometrical structure factor is small. The separation of the 111 and 333 peaks occurs because the refractive index is different for these reflections. In a double-crystal monochromator, white radiation from the source will produce the scattered intensity given by the full curve. If that intensity distribution now falls on the second crystal, which is tilted with respect to the first, an angle of tilt can be found for which the Bragg condition is not fulfilled in the second crystal, and the 333 radiation cannot be reflected. The resultant reflected intensity is shown in (b). Note that this is an idealized case, and in practice the existence of tails in the reflectivity curve can allow the transmission of some harmonic radiation through the double-crystal monochromator.

If the beam propagates through the specimen, the geometry is referred to as transmission, or Laue, geometry. If the beam is reflected from the surface, the geometry is referred to as reflection, or Bragg, geometry. Bragg geometry is the most commonly used in the construction of crystal monochromators. Laue geometry has been used in only a relatively few applications until recently. The need to handle high photon fluxes with their associated high power load has led to the use of diamond crystals in Laue configurations as one of the first components of X-ray optical systems (Freund, 1993[link]). Phase plates can be created using the Laue geometry (Giles et al., 1994[link]). A schematic diagram of a system used at the European Synchrotron Radiation Facility is shown in Fig. 4.2.5.5[link] . Radiation from an insertion device falls on a Laue-geometry pre-monochromator and then passes through a channel-cut (multiple-reflection) monochromator. The strong linear polarization from the source and the monochromators can be changed into circular polarization by the asymmetric Laue-geometry polarizer and analysed by a similar Laue-geometry analysing crystal.

[Figure 4.2.5.5]

Figure 4.2.5.5 | top | pdf |

A schematic diagram of a beamline designed to produce circularly polarized light from initially linearly polarized light using Laue-case reflections. Radiation from an insertion device is initially monochromated by a cooled diamond crystal, operating in Laue geometry. The outgoing radiation has linear polarization in the horizontal plane. It then passes through a silicon channel-cut monochromator and into a silicon crystal of a thickness chosen so as to produce equal amounts of radiation from the σ and π branches of the dispersion surface. These recombine to produce circularly polarized radiation at the exit surface of the crystal.

More will be said about polarization in §4.2.5.4.4[link] and Section 6.2.2[link] .

4.2.5.4.2. Laboratory monochromator systems

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Many laboratories use powder diffractometers using the Bragg–Brentano configuration. For these, a sufficient degree of monochromatization is achieved through the use of a diffracted-beam monochromator consisting of a curved-graphite monochromator and a detector, both mounted on the 2θ arm of the diffractometer. Such a device rejects the unwanted Kβ radiation and fluorescence from the sample with little change in the magnitude of the Kα lines. Incident-beam monochromators are also used to produce closely monochromatic beams of the desired energy. Single-reflection monochromators used for the reduction of spectral energy spread are described in Subsections 2.2.7.2[link] and 2.3.5.4[link] .

For most applications, this simple means of monochromatization is adequate. Increasingly, however, more versatility and accuracy are being demanded of laboratory diffractometer systems. Increased angular accuracy in both the θ and 2θ axes, excellent monochromatization, and parallel-beam geometry are all demands of a user community using improved techniques of data collection and data analysis. The necessity to study thin films has generated a need for accurately collimated beams of small cross section, and there is a need to have well collimated and monochromatic beams for the study of rough surfaces. This, coupled with the need to analyse data using the Rietveld method (Young, 1993[link]), has caused a revolution in the design of commercial diffractometers, with the use of principles long since used in synchrotron-radiation research for the design of laboratory instruments. Monochromators of this type are briefly discussed in §4.2.5.4.3[link].

4.2.5.4.3. Multiple-reflection monochromators for use with laboratory and synchrotron-radiation sources

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Single-reflection devices produce reflected beams with quite wide, quasi-Lorentzian, tails (Subsection 2.3.3.8[link] ), a situation that is not acceptable, for example, for the study of small-angle scattering (SAXS, Chapter 2.6[link] ). The effect of the tails can be reduced significantly through the use of multiple Bragg reflections.

The use of multiple Bragg reflections from a channel cut in a monolithic silicon crystal such that the channel lay parallel to the (111) planes of the crystal was shown by Bonse & Hart (1965b[link]) to remove the tails of reflections almost completely. This class of device, referred to as a (symmetrical) channel-cut crystal, is the most frequently used form of monochromator produced for modern X-ray laboratory diffractometers and beamlines at synchrotron-radiation sources (Figs. 4.2.5.2[link], 4.2.5.5[link]).

The use of symmetrical and asymmetrical Bragg reflections for the production of highly collimated monochromatic beams has been discussed by Beaumont & Hart (1973[link]). This paper contains descriptions of the configurations of channel-cut monochromators and combinations of channel-cut monochromators used in modern laboratory diffractometers produced by Philips, Siemens, and Bede Scientific. In another paper, Hart (1971[link]) discussed the whole gamut of Bragg reflecting X-ray optical devices. Hart & Rodriguez (1978[link]) extended this to include a class of device in which the second wafer of the channel-cut monochromator could be tilted with respect to the first (Fig. 4.2.5.6[link] ), thereby providing an offset of the crystal rocking curves with the consequent removal of most of the contaminant harmonic radiation (Fig. 4.2.5.4[link]). The version of monochromator shown here is designed to provide thermal stability for high incident-photon fluxes. Berman & Hart (1991[link]) have also devised a class of adaptive X-ray monochromators to be used at high thermal loads where thermal expansion can cause a significant degradation of the rocking curve, and therefore a significant loss of flux and spectral purity. The cooling of Bragg-geometry monochromators at high photon fluxes presents a difficult problem in design.

[Figure 4.2.5.6]

Figure 4.2.5.6 | top | pdf |

A schematic diagram of a Hart-type tuneable channel-cut monochromator is shown. The monochromator is cut from a single piece of silicon. The reflecting surfaces lie parallel to the (111) planes. Cuts are made in the crystal block so as to form a lazy hinge, and the second wafer of the monochromator is able to be deflected by a force generated by a current in an electromagnet acting on an iron disc glued to the upper surface of the wafer. Cooling of the primary crystal of the monochromator is by a jet of water falling on the underside of the wafer. This type of system can tolerate incident-beam powers of 500 W mm−2 without significant change to the width of the reflectivity curve.

Kikuta & Kohra (1970[link]), Matsushita, Kikuta & Kohra (1971[link]) and Kikuta (1971[link]) have discussed in some detail the performance of asymmetrical channel-cut monochromators. These find application under circumstances in which beam widths need to be condensed or expanded in X-ray tomography or for micro X-ray fluorescence spectroscopy. Hashizume (1983[link]) has described the design of asymmetrical monolithic crystal monochromators for the elimination of harmonics from synchrotron-radiation beams.

Many installations use a system designed by the Kohzu Company as their primary monochromator. This is a separated element design in which the reference crystal is set on the axis of the monochromator and the first crystal is set so as to satisfy the Bragg condition in both elements. One element can be tilted slightly to reduce harmonic contamination. When the wavelength is changed (i.e. θ is changed), the position of the first wafer is changed either by mechanical linkages or by electronic positioning devices so as to maintain the position of the outgoing beam in the same place as it was initially. This design of a fixed-height, separated-element monochromator was due initially to Matsushita, Ishikawa & Oyanagi (1986[link]). More recent designs incorporate liquid-nitrogen cooling of the first crystal for use with high-power insertion devices at synchrotron-radiation sources. In many installations, the second crystal can be bent into a cylindrical shape to focus the beam in the horizontal plane. The design of such a sagittally focusing monochromator is discussed by Stephens, Eng & Tse (1992[link]). Creagh & Garrett (1995[link]) have described the properties of a monochromator based on a primary monochromator (Berman & Hart, 1991[link]) and a sagittally focusing second monochromator at the Australian National Beamline at the Photon Factory.

A recent innovation in X-ray optics has been made at the European Synchrotron Radiation Facility by the group led by Snigirev (1994[link]). This combines Bragg reflection of X-rays from a silicon crystal with Fresnel reflection from a linear zone-plate structure lithographically etched on its surface. Hanfland et al. (1994[link]) have reported the use of this class of reflecting optics for the focusing of 25 to 30 keV photon beams for high-pressure crystallography experiments (Fig. 4.2.5.7[link] ).

[Figure 4.2.5.7]

Figure 4.2.5.7 | top | pdf |

A schematic diagram of the use of a Bragg–Fresnel lens to focus hard X-rays onto a high-pressure cell. The diameter of the sample in such a cell is typically 10 μm. The insert shows a scanning electron micrograph of the surface of the Bragg–Fresnel lens.

Further discussion on these monochromators is to be found in this volume in Subsection 2.2.7.2[link] , §2.3.5.4.1[link] , Chapter 2.7[link] , and Section 7.4.2[link] .

4.2.5.4.4. Polarization

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All scattering of X-rays by atoms causes a probable change of polarization in the beam. Jennings (1981[link]) has discussed the effects of monochromators on the polarization state for conventional diffractometers of that era. For accurate Rietveld modelling or accurate charge-density studies, the theoretical scattered intensity must be known. This is not a problem at synchrotron-radiation sources, where the incident beam is initially almost completely linearly polarized in the plane of the orbit, and is subsequently made more linearly polarized through Bragg reflection in the monochromator systems. Rather, it is a problem in the laboratory-based systems where the source is in general a source of elliptical polarization. It is essential to determine the polarization for the particular monochromator and the source combined to determine the correct form of the polarization factor to use in the formulae used to calculate scattered intensity (Chapter 6.2[link] ).

References

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