Scattering vs. microscopy
Scattering data is frequently difficult to understand and analyze. Thus, a fair question is: Why not just use microscopy instead? The answer is that scattering provides a number of unique advantages, compared to realspace miscroscopy techniques (TEM, STEM, SEM, AFM, STM, etc.). In reality, of course, both scattering and microscopy methods have their own unique advantages and disadvantages. Thus the techniques are complementary: one should ideally use both techniques to get a complete understanding of a material. (In fact, the two techniques can be combined: electron microscopes frequently have the option of performing an electron diffraction measurement; conversely scattering experiments are now being performed in realspace mapping modes using microbeams or nanobeams.)
Below we compare and contrast scattering and microscopy methods.
Data: Scattering experiments yield reciprocal-space data, which is non-intuitive and abstract. By comparison, microscopy generates a realspace image of the sample, making spatial arrangement obvious and unambiguous.
Data analysis: Understanding scattering data is typically difficult. The data is a Fourier transform of the realspace electron-density distribution, but the data cannot simply be inverted. Instead, the data must be analyzed and fit using physical models (although there are many qualitative rules for interpreting typical features). By comparison, microscopy images can be directly understood (if image quality is good).
Statistical validity: A key advantage of scattering is the statistical averaging inherent in the technique. The size-scale probed by scattering is dependent on the wavelength of the radiation (and the scattering angle), and can range from angstroms to microns. However, the scattering pattern represents an average over the entire sample area probed by the beam (which may be microns to mm). By comparison, microscopy is inherently local. It is difficult to know whether the pattern one observes in a microscopy image is truly representative of the entire sample, or simply an outlier. (Worse, both instrument and experimenter bias may lead to preferentially selecting outlier images during microscopy studies.) Thus, scattering is an invaluable tool for quantifying the predominant order in a sample, in a statistically-robust way.
Sample thickness: Microscopy is typically two-dimensional and cannot visualize a thick sample. Many microscopies are limited to the surface (SEM, AFM, STM), or require a very thin section of material (TEM). This limits the kinds of questions that can be answered. By comparison, scattering techniques can typically penetrate deeply through materials (microns to cm, depending on material and type/energy of radiation), allowing thick samples to be probed. There are microscopy techniques that overcome this limitation (confocal imaging, 3D reconstruction via tomography), which may introduce other tradeoffs.
Sample preparation: Microscopy investigation frequently requires special sample preparation; e.g. casting on special substrates or laboriously generating a thin cross-section sample. Microscopy also frequently benefits from staining the sample to improve contrast. These manipulations are time-consuming, and may perturb the sample. Scattering experiments can frequently be performed on unmodified samples. (Staining can be used in scattering also, but is less frequently necessary.)
In-situ: Because x-ray and neutron beams can penetrate through relatively thick layers of materials (microns to cm), it is possible to perform in-situ experiments. This is advantageous because materials can be probed in their native states. For instance, proteins can be studied in aqueous environments at room or physiological temperatures (whereas electron microscopy typically requires sectioning and cryo-cooling of biological samples). In fact, experiments can be performed in-operando, where the material is being actively modulated or converted. It is certainly possible to develop in-situ and in-operando sample-cells for microscopy, and there are exciting developments in this field. Nevertheless, the relative speed and simplicity of in-situ scattering experiments makes them attractive.
Disorder: There are many samples which will look indistinct in microscopy. In particular, highly disordered materials are difficult to interpret and quantify in microscopy, whereas there are scattering techniques that can quantitatively interpret disorder (e.g. diffuse scattering).
Complex order: Highly-complex order can be difficult to understand in microscopy. A complex, three-dimensional unit cell, viewed in projection, can be difficult to understand. Scattering, however, can probe such a structure and give rise to a well-defined reciprocal-lattice whose symmetry and spacing are immediately identifiable. With appropriate data, the realspace unit cell can be reconstructed.
Contrast variation: Scattering experiments can benefit from simple contrast-variation mechanisms, such as varying the incident wavelength of the radiation, or substituting one element for another. In neutron scattering, different isotopes have wildly different scattering power, allowing for contrast variation, to highly a specific sub-structure, without perturbing the overall material chemistry. A popular trick is to vary the ratio of normal water to heavy water, allowing one to contrast-match different components. Contrast variation is also used in microscopy, but it is a powerful method in scattering, to resolve structural ambiguities