International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 6.1, pp. 164-166   | 1 | 2 |

Section 6.1.4. Beam conditioning

U. W. Arndta

aLaboratory of Molecular Biology, Medical Research Council, Hills Road, Cambridge CB2 2QH, England

6.1.4. Beam conditioning

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The primary X-ray beam from the source is conditioned by a variety of devices, such as filters, mirrors and monochromators, to produce the appropriate properties for the beam incident on the sample.

6.1.4.1. X-ray mirrors

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It is usually necessary to focus the X-ray beam in two orthogonal directions. This can be achieved either by means of one mirror with curvatures in two orthogonal planes or by two successive reflections from two mirrors which are curved in one plane and planar in the other; the two planes of curvature must be at right angles to one another. In the arrangement adopted by Kirkpatrick & Baez (1948[link]) and by Franks (1955[link]), the two mirrors lie one behind the other (Fig. 6.1.4.1[link]) and thus produce a different degree of collimation in the two planes. Instead of this tandem arrangement, the mirrors can lie side-by-side, as proposed by Montel (1957[link]), to form what the author calls a `catamegonic roof' (Fig. 6.1.4.2[link]). The mirrors are then best made from thicker material, and the reflecting surfaces are ground to the appropriate curvature. The same arrangement has been used by Osmic Inc. (1998[link]) for their Confocal Max-Flux Optics, in which the curved surfaces are coated with graded-spacing multilayers.

[Figure 6.1.4.1]

Figure 6.1.4.1 | top | pdf |

Production of a point focus by successive reflections at two orthogonal curved mirrors. Arrangement due to Kirkpatrick & Baez (1948[link]) and to Franks (1955[link]).

[Figure 6.1.4.2]

Figure 6.1.4.2 | top | pdf |

The `catamegonic' arrangement of Montel (1957[link]), in which two confocal mirrors with orthogonal curvatures lie side-by-side.

Flat mirror plates can be bent elastically to a desired curvature by applying appropriate couples. Fig. 6.1.4.3[link] shows the bending method adopted by Franks (1955[link]). A cylindrical curvature results from a symmetrical arrangement that produces equal couples at both ends. With appropriate unequal couples applied at the two ends of the plate, the curvature can be made parabolic or elliptical. Precision elliptical mirrors have been produced by Padmore et al. (1997[link]); unequal couples are applied in this way. Cylindrically curved mirrors can be produced by applying a force at the tip of a triangular plate whose base is firmly anchored (Fig. 6.1.4.4[link]). Lemonnier et al. (1978[link]) first used this method for making curved-crystal monochromators. Milch (1983[link]) described X-ray mirrors made in this way; the effect of the linear increase of the bending moment along the plate is compensated by the linear increase of the plate section so that the curvature is constant. An elliptical or a parabolic curvature results if either the width or the thickness of the plate is made to vary in an appropriate way along the length of the plate. Arndt, Long & Duncumb (1998[link]) described a monolithic mirror-bending block in which the mirror plates are inserted into slots cut to an elliptical curvature by ion-beam machining. The solid angle of collection is made four times larger than for a two-mirror arrangement by providing a pair of horizontal mirrors and a pair of vertical mirrors in tandem in one block (Fig. 6.1.4.5[link]).

[Figure 6.1.4.3]

Figure 6.1.4.3 | top | pdf |

Mirror bender (after Franks, 1955[link]). The force exerted by the screw produces two equal couples which bend the mirror into a circular arc. The slotted rods act as pivots and also as beam-defining slits.

[Figure 6.1.4.4]

Figure 6.1.4.4 | top | pdf |

Triangular mirror bender as described by Lemonnier et al. (1978[link]) for crystal plates and by Milch (1983[link]) for glass mirrors. The base of the triangular plate is clamped and the bending force is applied at the apex along the arrow.

[Figure 6.1.4.5]

Figure 6.1.4.5 | top | pdf |

Mirror holder with machined slots for two orthogonal pairs of curved mirrors (after Arndt, Duncumb et al., 1998[link]).

Mirror plates for these benders are usually made from highly polished glass, quartz, or silicon plates which are coated with nickel, gold, or iridium.

Mirrors for synchrotron beam lines that focus the radiation in the vertical plane are most often ground and polished to the correct shape, rather than bent elastically. Much longer mirrors can be made in this way.

The collecting efficiency of specularly reflecting mirrors depends on the reflectivity of the surface and on the solid angle of collection; this, in turn, is a function of the maximum glancing angle of incidence, which is the critical angle for total external reflection, θc. For X-rays of wavelength λ, measured in Å,[\theta_{c} \simeq 2.32 \times 10^{-3} (Z\rho/A)^{1/2} \lambda, ]where Z is the atomic number, A is the atomic mass and ρ is the specific gravity of the reflecting surface.

Thus, for Cu Kα radiation and a gold surface, θc ≃ 10 mrad. The reflectivity of the mirror surface is strongly dependent on the surface roughness; for the reflectivity to be more than 50%, the r.m.s. roughness must not exceed 10 Å.

It is not possible to design a reflecting collimator with a planar angle of collection greater than about 3θc. For the shorter wavelengths, in particular, variable-spacing multilayer mirrors (Schuster & Göbel, 1997[link]) hold considerable promise. If the spacing at the upstream end of the mirror is 30 Å, the largest angles of incidence will be 26 and 17 mrad for 1.54 and 1.0 Å X-rays, respectively. By comparison, the critical angles at a gold surface for these radiations are 10 and 6.5 mrad, respectively.

6.1.4.2. Focusing collimators for microfocus sources

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In most arrangements that include conventional X-ray tubes, the planar angle of collection is very small. A more efficient use is always made of the radiation from the target by a focusing collimator, which forms an image of the source on the sample (Fig. 6.1.4.6[link]). The angle of collection should be as large as possible, while the cross fire, i.e. the angle of convergence, is kept small, say, at about 10−3 rad. It is possible to design focusing collimators based on gold-surfaced toroids of revolution (Elliott, 1965[link]), which afford a planar angle of collection of about three times the critical angle for total external reflection, that is, about 30 × 10−3 rad.

[Figure 6.1.4.6]

Figure 6.1.4.6 | top | pdf |

Ellipsoidal mirror for use with a microfocus X-ray tube, where x1 is ∼15 mm. The major axis, 2a, may be up to 600 mm, whereas the exit aperture, 2y2, lies in the region 0.8–1.4 mm. The angle γ determines the cross fire on the sample and is less than 1 rad.

Consequently, the mirror should magnify about 30 times, and if the image diameter, determined by a typical sample size, is to be 300 µm, the size of the focus should be about 10 µm. The solid angle of collection of such an imaging toroid is about 8 × 10−4 steradians, that is, more than 1000 times greater than the solid angle of a simple non-imaging collimator. The averaged mirror reflectivity achieved at present is about 0.3, so the microfocus tube and toroidal mirror combination produces a similar intensity at the sample as the conventional tube with a non-focusing collimator at about 300 times the power. Future increases of the reflectivity are likely as the surface roughness of the mirrors is improved.

A suitable microfocus tube has been described by Arndt, Long & Duncumb (1998[link]); mirrors used with this tube were discussed by Arndt, Duncumb et al. (1998[link]). The tube design allows the distance between the source and the mirror to be as little as 10 mm in order to achieve the necessary magnification without making the distance between the tube and the sample inconveniently long.

6.1.4.3. Other focusing collimators

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There has been very active development in recent years of tapering capillaries for focusing X-rays, either as individual capillaries (see the review by Bilderback et al., 1994[link]), or in the form of multicapillary bundles. The latter were first described by Kumakhov & Komarov (1990[link]); since then, they have undergone great improvements in the form of fused bundles (Bly & Gibson, 1996[link]) (Fig. 6.1.4.7[link]). Single capillaries have found the greatest use as X-ray concentrators, where a larger-diameter beam of X-rays enters the large end of a tapered capillary and is concentrated to a diameter of a few µm. Fused polycapillary bundles have been employed as focusing collimators for protein crystallography (MacDonald et al., 1999[link]). Both types of capillary optics are usually designed as multi-bounce devices, in which the X-rays undergo several, or many, reflections at the walls of the capillary; consequently the cross-fire half-angle at the output end has a value about equal to the critical angle for reflection at a glass surface or, perhaps, 4 mrad. This is sometimes too great for producing diffraction patterns with an optimum signal-to-background ratio.

[Figure 6.1.4.7]

Figure 6.1.4.7 | top | pdf |

A polycapillary collimator (after Bly & Gibson, 1996[link]).

Other methods of focusing X-rays, such as zone plates (Kirz, 1974[link]) and refractive optics, are being investigated, but at present none of them can compare with toroidal reflectors for data collection from single crystals of macromolecules.

6.1.4.4. Crystal monochromators

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When the X-rays from the tube target are specularly reflected by a mirror, the spectrum is cut off for X-rays below the shortest wavelength for which the critical angle is equal to the smallest angle of incidence on the mirror. For a typical mirror designed for Cu Kα radiation, this cutoff wavelength might be about 0.75 Å, and the harder X-rays can be further attenuated by a β-filter. Of course, the more nearly monochromatic the radiation falling on the sample, the lower the radiation damage and the higher the spot-to-background ratio in the recorded patterns.

White radiation is almost completely eliminated by reflecting the primary X-ray beam using a natural or artificial (multilayer) crystal. The most commonly used type of plane monochromator for macromolecular crystallography is a single crystal of graphite. This material (HOPG, or highly ordered pyrolytic graphite) has a relatively large mosaic spread, typically about 0.4°, and it cannot separate the Kα doublet. This separation is essential in most small-molecule investigations, but is unnecessary for macromolecular crystals, which rarely diffract beyond 1.5 Å, and disadvantageous where a high intensity of the beam reflected by the monochromator is the main consideration.

The intensity of the diffraction pattern obtained with a graphite monochromator is only about two or three times lower than that resulting from a β-filtered pinhole-collimated beam. The situation is different at synchrotron beam lines, which must incorporate a monochromator in order to select the desired X-ray energy band. Curved focusing crystals collect X-rays over a relatively large horizontal angular range and thus produce a beam with a horizontal convergence angle of up to several milliradians. Much more nearly parallel beams are produced by reflection at several crystals in tandem, often in the form of monolithic channel-cut monochromators. In present-day storage rings, the power density at the first optical element is of the order of 10 W mm−2 at wiggler and undulator beam lines. This amount of power can be dissipated by careful design of water-cooling channels (Quintana & Hart, 1995[link]; van Silfhout, 1998[link]). In addition, the monochromator crystal, usually of silicon or germanium, may be profiled to minimize distortions as a result of thermal stresses.

The next generation of insertion devices will subject the optical elements to loads of several hundred W mm−2. Possible engineering solutions to the very severe heat-loading problem include the use of diamond crystals as reflecting elements. This material has a very high thermal conductivity, especially at low temperatures.

References

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