Diffraction topography

Diffraction topography (short: "topography") is an imaging technique based on Bragg diffraction. Diffraction topographic images ("topographies") record the intensity profile of a beam of X-rays (or, sometimes, neutrons) diffracted by a crystal. A topography thus represents a two-dimensional spatial intensity mapping (image) of the X-rays diffracted in a specific direction, so regions which diffract substantially will appear brighter than those which do not. This is equivalent to the spatial fine structure of an Laue reflection. Topographs often reveal the irregularities in a non-ideal crystal lattice. X-ray diffraction topography is one variant of X-ray imaging, making use of diffraction contrast rather than absorption contrast which is usually used in radiography and computed tomography (CT). Topography is exploited to a lesser extent with neutrons, and is the same concept as dark field imaging in an electron microscope. Topography is used for monitoring crystal quality and visualizing defects in many different crystalline materials. It has proved helpful e.g. when developing new crystal growth methods, for monitoring growth and the crystal quality achieved, and for iteratively optimizing growth conditions. In many cases, topography can be applied without preparing or otherwise damaging the sample; it is therefore one variant of non-destructive testing.

Diffraction topography (short: "topography") is an imaging technique based on Bragg diffraction. Diffraction topographic images ("topographies") record the intensity profile of a beam of X-rays (or, sometimes, neutrons) diffracted by a crystal. A topography thus represents a two-dimensional spatial intensity mapping (image) of the X-rays diffracted in a specific direction, so regions which diffract substantially will appear brighter than those which do not. This is equivalent to the spatial fine structure of an Laue reflection. Topographs often reveal the irregularities in a non-ideal crystal lattice. X-ray diffraction topography is one variant of X-ray imaging, making use of diffraction contrast rather than absorption contrast which is usually used in radiography and computed tomography (CT). Topography is exploited to a lesser extent with neutrons, and is the same concept as dark field imaging in an electron microscope. Topography is used for monitoring crystal quality and visualizing defects in many different crystalline materials. It has proved helpful e.g. when developing new crystal growth methods, for monitoring growth and the crystal quality achieved, and for iteratively optimizing growth conditions. In many cases, topography can be applied without preparing or otherwise damaging the sample; it is therefore one variant of non-destructive testing. \Delta \theta(\vec r) = \frac{1}{\vec h \cdot \cos \theta_B} \frac{\partial}{\partial s_{\vec h}} \left[ \vec h \cdot \vec u(\vec r)\right] where \vec u(\vec r) is the displacement vector field, and \vec s_0 and \vec s_{h} are the directions of the incident and diffracted beam, respectively. In this way, different kinds of disturbances are "translated" into equivalent misorientation values, and contrast formation can be understood analogously to orientation contrast. For instance, a compressively strained material requires larger Bragg angles for diffraction at unchanged wavelength. To compensate for this and to reach diffraction conditions, the sample needs to be rotated, similarly as in the case of lattice tilts. A simplified and more "transparent" formula taking into account the combined effect of tilts and strains onto contrast is the following: \Delta \theta(\vec r) = -\tan \theta_B \frac{\Delta d}{d}(\vec r) \pm \Delta \varphi(\vec r) \Delta x = S \cdot \frac{d}{D} = \frac{S}{D} \cdot d where I is the spread of the image of one sample point in the image plane, D is the source-to-sample distance, and d is the sample-to-image distance. The ratio S/D corresponds to the angle (in radians) under which the source appears from the position of the sample (the angular source size, equivalent to the incident divergence at one sample point). The achievable resolution is thus best for small sources, large sample distances, and small detector distances. This is why the detector (film) needed to be placed very close to the sample in the early days of topography; only at synchrotrons, with their small S and (very) large D, could larger values of d finally be afforded, introducing much more flexibility into topography experiments. Thirdly, even with perfect detectors and ideal geometric conditions, the visibility of special contrast features, such as the images of single dislocations, can be additionally limited by diffraction effects. A dislocation in a perfect crystal matrix gives rise to contrast only in those regions where the local orientation of the crystal lattice differs from average orientation by more than about the Darwin width of the Bragg reflection used. A quantitative description is provided by the dynamical theory of X-ray diffraction. As a result, and somehow counter-intuitively, the widths of dislocation images become narrower when the associated rocking curves are large. Thus, strong reflections of low diffraction order are particularly appropriate for topographic imaging. They permit topographists to obtain narrow, well-resolved images of dislocations, and to separate single dislocations even when the dislocation density in a material is rather high. In more unfavourable cases (weak, high-order reflections, higher photon energies), dislocation images become broad, diffuse, and overlap for high and medium dislocation densities. Highly ordered, strongly diffracting materials – like minerals or semiconductors – are generally unproblematic, whereas e.g. protein crystals are particularly challenging for topographic imaging. Apart from the Darwin width of the reflection, the width of single dislocation images may additionally depend on the Burgers vector of the dislocation, i.e. both its length and its orientation (relative to the scattering vector), and, in plane wave topography, on the angular departure from the exact Bragg angle. The latter dependence follows a reciprocity law, meaning that dislocations images become narrower inversely as the angular distance grows. So-called weak beam conditions are thus favourable in order to obtain narrow dislocation images. ===Digital topography=== The use of electronic detectors such as X-ray CCD cameras, replacing traditional X-ray film, facilitates topography in many ways. CCDs achieve online readout in (almost) real-time, dispensing experimentalists of the need to develop films in a dark room. Drawbacks with respect to films are the limited dynamic range and, above all, the moderate spatial resolution of commercial CCD cameras, making the development of dedicated CCD cameras necessary for high-resolution imaging. A further, decisive advantage of digital topography is the possibility to record series of images without changing detector position, thanks to online readout. This makes it possible, without complicated image registration procedures, to observe time-dependent phenomena, to perform kinetic studies, to investigate processes of device degradation and radiation damage, and to realize sequential topography (see below). TOPO Beamline at ANKA, Karlsruhe (Germany) Beamline F1 of HASYLAB at DESY, Hamburg (Germany) National Synchrotron Light Source (NSLS), Kansas (USA) SPring-8, near Himeji (Japan)

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