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

International Tables for Crystallography (2006). Vol. C, ch. 2.7, pp. 115-117

Section Transmission topographs

A. R. Langa

aH. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, England Transmission topographs

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The term `X-ray topograph' was introduced by Ramachandran (1944[link]) who took transmission topographs of cleavage plates of diamond using essentially the arrangement shown in Fig.[link]. (In this case, S was a 0.3 mm diameter pinhole placed in front of the window of a W-target X-ray tube so as to form a point source of diverging continuous radiation.) Ramachandran adopted a distance a = 0.3 m and ratio a/b of about 12, which produced images of about 25 µm geometrical resolution having the characteristics of Fig.[link], i.e. sensitive to diffraction contrast but not to orientation contrast. For each reflection under study, the film was inclined to the incident beam with that obliquity calculated to produce an undistorted image of the specimen plate. Guinier & Tennevin (1949[link]) studied both diffraction contrast and orientation contrast in continuous-radiation transmission topograph images. Their minimum b/a ratio was set by the need to avoid overlap of Laue images of the crystal produced by different Bragg planes.

Collimated characteristic radiation is used in the methods of `section topographs' (Lang, 1957[link]) and `projection topographs' (Lang, 1959a[link]), the latter being sometimes called `traverse topographs'. Fig.[link] explains both techniques. When taking a section topograph, the specimen CD, usually plate shaped, is stationary (disregard the double-headed arrow in the figure). The ribbon-shaped incident beam issuing from the slit P is Bragg reflected by planes normal, or not far from normal, to the major surfaces of the specimen. As drawn, the Bragg planes make an angle α with the normal to the X-ray entrance surface of the specimen, the positive sense of α being taken in the same sense as the deviation [2\theta_B] of the Bragg-reflected rays. If the crystal is sufficiently perfect for multiple scattering to occur within it (with or without loss of coherence), then the multiply scattered rays associated with the Bragg reflection excited will fill the volume of the triangular prism whose base is ORT, the `energy-flow triangle' or `Borrmann triangle', contained between OT and OR whose directions are parallel to the incident wavevector, [{\bf K}_0], and diffracted wavevector, [{\bf K}_h], respectively. Both the [{\bf K}_0] and [{\bf K}_h] beams issuing from the X-ray exit surface of the crystal carry information about the lattice defects within the crystal. However, it is usual to record only the [{\bf K}_h] beam. This falls on the film, F, in a strip extending normal to the plane of incidence, of height equal to the illuminated height of the specimen multiplied by the axial magnification factor (a + b)/a, and forms the section topograph image. The screen, Q, prevents the [{\bf K}_0] beam from blackening the film but has a slot allowing the diffracted beam to fall on F. A diffraction-contrast-producing lattice defect cut by OT at I will generate supplementary rays parallel to [{\bf K}_h] and will produce an identifiable image on F at [I'], the `direct image' or `kinematic image' of the defect. The depth of I within CD can be found via the measurement of [I'T'/R'T']. From a series of section topographs taken with a known translation of the specimen between each topograph, a three-dimensional construction of the trajectory of defect I (e.g. a dislocation line) within the crystal can be built up. To obtain good definition of the spatial width of the ribbon incident beam cutting the crystal, the distance between P and the crystal is kept small. The minimum practicable opening of P is about 10 µm. If diffraction is occurring from planes perpendicular to the X-ray entrance surface of the specimen, i.e. symmetrical Laue case diffraction, the width [R'T'] of the section topograph image is simply [2t\sin\theta_B], t being the specimen thickness, and neglecting the contribution from the width of the ribbon incident beam. With asymmetric transmission, as drawn in the figure, [R'T'=t\sec\,(\theta_B-\alpha)\sin2\theta_B]. The distance b is made as small as is permitted by the specimen shape and the need to separate the emerging [{\bf K}_0] and [{\bf K}_h] beams. Suppose b is 10 mm. Then, with a source S having axial extension 100 µm, the distance a = SP should be not less than 0.5 m in order to keep the geometrical resolution in the axial direction better than 2 µm, and should be correspondingly longer with larger source sizes.


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Arrangements for section topographs and projection topographs.

To take a projection topograph, the specimen CD and the cassette holding the film F are together mounted on an accurate linear traversing mechanism that oscillates back and forth during the exposure so that the whole area of interest in the specimen is scanned by the ribbon beam from P. The screen Q is stationary. If the specimen is plate shaped, the best traverse direction to choose is that parallel to the plate, as indicated by the double-headed arrow, for then the diffracted beam will have minimum side-to-side oscillation during the traverse oscillation, the opening of Q can be held to a minimum, and thereby unwanted scattering reaching F kept low. The projection topograph image is an orthographic projection parallel to [{\bf K}_h] of the crystal volume and its content of diffraction-contrast-producing lattice defects. If the specimen is plate-like, of length L in the plane of incidence, then, with F normal to [{\bf K}_h], the magnification of the topograph image in the direction parallel to the plane of incidence is [L\cos(\theta_B+\alpha)]. There will generally be a small change of axial magnification (a + b)/a along L. The loss of three-dimensional information occurring through projection can be recovered by taking stereopairs of projection topographs. The first method (Lang, 1959a[link], b[link]) used hkl, [\bar h\bar k\bar l] pairs of topographs as stereopairs. One disadvantage of this method is that the convergence angle is fixed at [2\theta_B], which may be unsuitably large for thick specimens. The method of Haruta (1965[link]) obtains two views of the specimen using the same [hkl] reflection, by making a small rotation of the specimen about the h vector between the two exposures, and has the advantage that this rotation, and hence the stereoscopic sensitivity, can be chosen at will. When taking projection topographs, the slit P can be wider than the narrow opening needed for high-resolution section topographs, but not so wide as to cause unwanted [K\alpha_2] reflection to occur. Best use of the X-ray source is made when the width of P is the same as or somewhat greater than S.

In certain investigations, the methods of [{\bf K}_h]-beam `limited projection topographs' (Lang, 1963[link]) and of [{\bf K}_0]-beam section topographs and projection topographs are useful; Fig.[link] shows the arrangement of screens and diffracted-beam slits then adopted. The limited projection topograph technique can be employed with a plate-shaped specimen [CDD'C'], as in the following examples. Suppose the surface of the plate contains abrasion damage that cannot be removed but that causes diffraction contrast obscuring the images of interior defects in the crystal. The diffracted-beam slit (equivalent to the opening in Q shown in Fig.[link]), which is opened to the setting [S_1] for a standard projection topograph, may now be closed down to setting [S_2] so as to cut into the [RR'] and [TT'] edges of the [{\bf K}_h] beam, and thereby prevent direct images of near-surface defects located between [CC'] and [XX'], and between [YY'] and [DD'], from reaching F. As another example, it may be desired to receive direct images from a specimen surface and a limited depth below it only (e.g. when correlating surface etch pits with dislocation outcrops). Then, setting [S_3] of the diffracted-beam slit is adopted and only the direct images from defects lying between depth [ZZ'] and the surface [DD'] reach F.


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Arrangements for limited projection topographs and direct-beam topographs.

To record the [{\bf K}_0] beam image, either in a section topograph or a projection topograph, some interception of the [{\bf K}_0] beam on the [OTT''] side is needed to avoid intense blackening of the film G by radiation coming from the source, which will generally contain much energy in wavelengths other than those undergoing Bragg diffraction by the crystal. Screen S4, critically adjusted, performs the required interception. Recording both [{\bf K}_0]-beam and [{\bf K}_h]-beam images is valuable in some studies of dynamical diffraction phenomena, such as the `first-fringe contrast' in stacking-fault fringe patterns (Jiang & Lang, 1983[link]). Such recording can be done simultaneously, on separate films, normal to [{\bf K}_0] and [{\bf K}_h], respectively, when [2\theta_B] is sufficiently large.

When collimated characteristic radiation is used, recording projection topographs of reasonably uniform density becomes difficult when the specimen is bent. To keep the ω axis oriented at the peak of the Bragg reflection while the specimen is scanned, several devices for `Bragg-angle control' have been designed, for example the servo system of Van Mellaert & Schwuttke (1972[link]). The signal is taken from a detector registering the Bragg reflection through the film F, but this precludes use of glass-backed emulsions if X-ray wavelengths such as that of Cu K and softer are used. An alternative approach with thin, large-area transmission specimens is to revert to the geometry of Fig.[link] and deliberately elastically bend the crystal to such radius as will enable its whole length to Bragg diffract a single wavelength diverging from S, similar to a Cauchois focusing transmission monochromator (Wallace & Ward, 1975[link]). No specimen traversing is then needed, but b cannot be made small if the wide [{\bf K}_0] and [{\bf K}_h] beams are to be spatially separated in the plane of F.

Quite simple experimental arrangements can be adopted for taking transmission topographs under high-absorption conditions, when only anomalously transmitted radiation can pass through the crystal. The technique has mainly been used in symmetrical transmission, as shown with the specimen [CC'D'D] in Fig.[link] . When the specimen perfection is sufficiently high for the Borrmann effect to be strongly manifested, and μt [\gt] 10, say, the energy flow transmitted within the Borrmann triangle is constricted to a narrow fan parallel to the Bragg planes, the fan opening angle being only a small fraction of [2\theta_B] [see IT B (2005[link], Part 5[link] )]. Radiation of a given wavelength coming from a small source at S and undergoing Bragg diffraction in [CC'D'D] will take the path shown by the heavy line in Fig.[link], simplifying the picture to the case of extreme confinement of energy flow to parallelism with the Bragg planes. At the X-ray exit surface DD′, splitting into [{\bf K}_0] and [{\bf K}_h] beams occurs. A slit-less arrangement, as shown in the figure, may suffice. Then, when S is a point-like source of [K\alpha] radiation, and distance a is sufficiently large, films F1 and F2 will each record a pair of narrow images formed by the [\alpha_1] and [\alpha_2] wavelengths, respectively. A wider area of specimen can be imaged if a line focus rather than a point focus is placed at S (Barth & Hosemann, 1958[link]), but then the [\alpha_1] and [\alpha_2] images will overlap. Under conditions of high anomalous transmission, defects in the crystal cause a reduction in transmitted intensity, which appears similarly in the [{\bf K}_0] and [{\bf K}_h] images. Thus, it is possible to gain intensity and improve resolution by recording both images superimposed on a film F3 placed in close proximity to the X-ray exit face DD′ (Gerold & Meier, 1959[link]).


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Topographic techniques using anomalous transmission.


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