Optoelectronic properties of nanoparticles depend on the chemistry of the material and on the dimensionality, which includes the size and shape of nanomaterials ranging from 0D spherical objects, 1D nanorods and 2D nanosheets to 3D superstructures. In-situ small-angle (SAXS) and wide-angle X-ray scattering (WAXS) measurements are relatively non-destructive analytical methods and provide structural information about the nucleation and growth of nanomaterials.[21]
This chapter gives an overview of ray scattering and a basic account of the analysis method. X-rays are electromagnetic waves with wavelength from 0.1 nm to 10 nm and can be used for the study of objects with very small dimensions, including atoms, molecules and nanocrystals.[84][85] In a typical scattering experiment, a collimated X-ray beam collides with a sample, where three interactions can be described. One fraction of X-rays will be absorbed, another fraction will pass through and a fraction will be scattered on the scattering objects.[84] Scattering can be divided into
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Compton scattering (inelastic scattering) and Thomson scattering (elastic scattering). Whereas Compton scattering does not carry structural information, Thomson scattering is interesting for fundamental structural studies. In this scattering event, the incoming photons hit the sample and scatter without energy transfer. The X-rays collide with the bound electrons of the scatting object and induce the oscillation of these electrons with the same frequency. This causes the emission of radiation with the same frequency of the incident wave. In case of an aperiodic scatterer, the emitted secondary waves are scattered in all directions.
Figure 7: Schematic representation of two incident X-rays (blue) on three atomic layers of a crystal, e.g.
atoms are separated by the distance d. The X-rays are reflected (red) from the adjacent planes with the path difference between two X-rays of 2𝑑 𝑠𝑖𝑛𝜃.
For a periodic scatterer (figure 7), coherent waves of neighbouring atoms can interfere and scatter in precise directions. The scattered X-rays are detected and the information is summarized in an interference pattern caused by constructive (bright spots) and destructive interference.[86] Bragg formulated a fundamental equation for the interference of X-rays which is defined as
𝑛𝜆 = 2𝑑 𝑠𝑖𝑛𝜃 (2.12) The interplay of the scattering angle θ, the distance d of the periodic planes and the wavelength λ of the X-rays are described by Braggꞌs law. For constructive interference, the secondary X-rays are in phase and the phase difference 2d sinθ must be an integer number n of the wavelength λ. The conditions for Braggꞌs law are met. The destructive interference occurs when the equation is not
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satisfied and this leads to dark spots in the interference pattern. The scattered X-rays are detected and lead to a 2D scattering pattern with structural information about the scatterer.
In the Fourier space, the Braggꞌs law is rewritten as 𝑞 =2𝜋
𝑑 =4𝜋
𝜆 𝑠𝑖𝑛𝜃 (2.13) and shows the reciprocal relation between the size of the particle d and the scattering angle θ. In contrast to the scattering angle θ, the ꞌlength of the scattering vectorꞌ q is independent of the applied wavelength and the resulting scattering profile shows gives structural information as function of the q-value. The dimension of q-values is one over length [1/nm] and therefore small objects scatter at large angles (q-values) and large objects scatterer at small angles (q-values).[84]
As mentioned above, the interference of secondary waves emitted from a certain object produce 2D scattering patterns in the plane of the detector. The amount of data can be reduced by producing a 1D scattering pattern which contains all the necessary information about the scatterer. This SAXS pattern consists of three regimes[87], the Guinier, Fourier and Porod regions.
The Guinier regime is located at very low q-values, and gives information about larger objects in real space. Analysis of this regime allows the direct estimation of the radius of gyration Rg. The Guinier plot is the line fit to the natural logarithm of the intensity as a function of q2 (ln I(q) vs. q2).
The structure of the particles can be determined, by calculating the volume and molecular weight of the scattering object.[84]
The Fourier region gives significant information about the particle shape and the size distribution.
The decay at large scattering angles is directly related to the particle shape. The Porod regime describes the interface and fractal dimensions of scattering objects. The inter-particle interference has no effect at large angles and the surface-to-volume ratio can be determined.
The particle size, volume, and electron density can be described as a function of the form factor F(q).[87] The scattering of the intra-particle structure in an ideal monodisperse and diluted solution is collected and no position or orientational correlation between single particles contribute to the experimental scattering curves. The particles are well-separated and the total intensity of the scattering profile is then the sum of each particle intensity for monodisperse particles. The scattering is produced by the atoms of the particle and those of all identical particles is summed up
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and the scattering pattern is the result of the form factor multiplied by the number of the detected particles.
At higher concentrations, stronger interaction between particles cannot be neglected and a densely packed particle system leads to rearrangement of particles and a long-range order (Van-der-Waals interaction, Coulomb interaction, hard-sphere interaction).[84] The Structure factor S(q) becomes important and provides information about these interactions.[85] The corresponding scattering curve shows Bragg peaks for periodic components and the degree of order of the particles (peak intensity), and the spatial extent of the ordered domain (peak width), give more information. The multiplication of the Form factor F(q) and the Structure factor S(q) contribute to the SAXS pattern and together contain the information about the internal density of the particles and interactions.
𝐼(𝑞) = 𝐹(𝑞)𝑆(𝑞) (2.14) The sample-to-detector distance, and the beam collimation will greatly influence the scattering profile. For the generation of high intensity X-rays (keV) a synchrotron radiation source is necessary where the rays are produced accelerated, collected and collimated into an intense X-ray beam (beamline).[88] With a shorter sample-to-detector distance, diffraction patterns at larger scattering angles can be observed. The method is therefore called wide-angle X-ray scattering (WAXS) and the crystallinity (crystal structure) on the atomic scale can be detected. WAXS is based on the same theoretical background as SAXS and requires analysis of scattering patterns caused by Bragg peaks.