X-ray and Neutron total scattering
Rather than collecting all the Bragg peaks at discrete angles as is done in single-crystal structure determination, in a total scattering measurement the scattered intensity is measured as continuous function of the scattering angle. This is most conveniently expressed in terms of the wavevector Q that is defined, for elastic scattering, as: (2.1).
Where θ is the scattering angle and λ is the wavelength of radiation used. For elastic scattering, is the difference between the incident wavevector and the scattered wavevector as ( ) (Figure 2.1).
Coherent Scattering Intensity
X-ray and neutron total scattering measurements have been used for a long time to study the isotropic scattering for a variety of materials from powder, liquid or glass. As usual, in diffraction experiments we measure only the scattered intensity and not the sample scattering amplitude directly. It is important, however, that the measured sample is truly isotropic to avoid texture and preferential powder orientation problems. In this case, the scattered intensity depends only on the amplitude of Q and not on its direction and is described by the standard scattering equation . (2.2).
Where f denotes the Q-dependent atomic form factor for X-ray scattering, and the constant scattering length (b) for neutron scattering. Defining rij = ri-rj as the length of interatomic vector between the atoms i and j, the well-known Debye equation  is obtained by performing the powder average: (2.3).
Note that the sum in this equation is divided into two parts, namely terms for which i=j, the so-called “self-scattering” terms and terms for which i≠j, the “distinct” or “interference” terms. The self-scattering terms will carry no information on the relative distribution of atoms because it incorporates only the zero distances of an atom whose Q-dependence is due to its form factor f. Figure 2.2 shows Icoh(Q) of amorphous SiO2 gel measured by X-ray diffraction and the corresponding “self-scattering” contribution. One can clearly observe that the intensity of the diffracted beam falls off with increasing Q. This is a typical feature for X-ray measurements where the atomic scattering amplitude depends fairly strongly on Q and is usually denoted as f(Q).
Relation with Pair Distribution Function (PDF)
A conceptually simple way to examine the underlying atomic arrangement from the diffraction data is to take the sine Fourier transform of the reduced total structure factor F(Q) which yields the atomic pair distribution function (PDF) . The mathematical definition of the PDF, G(r), is: (2.11) Where ρ(r) and ρ0 are the local number density and average number density, respectively, and r is the radial distance. From Equation 2.11 we see that, as r→0, G(r)→-4πρ0. In general, G(r) has a baseline of slope -4πρ0 at low r. (Figure 2.5).
G(r) can be considered as a histogram of the atom-atom distances in a material. It gives the probability of finding two atoms separated by the distance r inside the sample. The peaks in the PDF diagram occur thus at values of r that correspond to the average interatomic distances in a material and since(r) oscillates aroundo , G(r) should oscillate around zero. The relative intensities of the peaks are related to the number of such interatomic distances in the sample. Thus in principal the PDF contains all the information about the structure of the material. It is noteworthy to mention that the PDF is radially averaged and is a one dimensional function. The PDF is similar to a Patterson map, the difference being that the former takes into account the total scattering (both the Bragg peaks and the diffuse scattering intensities), while the latter only considers the Bragg scattering and so a Patterson map only yields interatomic vectors within a unit cell. While the Bragg scattering yields only the average structure, the PDF method is thus capable of probing both the local and average structure. This combination of local and average structural analysis from the same scattering data represents a clear advantage for a wide range of applications.
According to Eq. 2.11, getting high Qmax with good statistics is crucial for obtaining accurate PDF with minimized termination ripples and a high r-resolution, which may be approximately given by (δr~π/Qmax) . For the highest resolution and accurate measurements the use of synchrotron radiation is preferred, however, perfectly acceptable PDFs can be obtained from laboratory measurements as we discuss in chapter 4. For example, using X-ray laboratory sources such as Molybdenum or Silver, scattering data can be collected up to Qmax~17 Å -1 and ~21 Å -1, respectively corresponding to maximum real space resolution δr of ~0.18 Å and ~0.15 Å respectively. However, the real space resolution is significantly reduced to ~0.39 Å when the Cu K radiation is used as a consequence of a finite Qmax of ~8 Å -1. From these examples it can be seen that the Qmax termination effects can be significantly reduced by using available in-house sources with shorter wavelength giving access to a higher Qmax. However, care must be taken since on going from Mo to Ag the diffraction power significantly drops in intensity. A good compromise between the resolution and higher diffracted intensities has therefore to be found for PDF measurements depending on the sample characteristics.
Data statistics and Collection time
The intensity of the diffracted beam falls off with increasing Q. As a result the signal becomes progressively dominated by noise at higher Q-values, which limits the accessible range of data that can be used for the sample characterization. The signal-to-noise ratio can be improved by using appropriate data collection strategy as we will discuss in the following.
Ideally the experimental measurements would involve a diffractometer that can resolve all the details of the scattered intensity Icoh(Q). However, real equipments have a limited resolution which, depending on the samples, may have a major effect on the PDF . The effect of the instrumental resolution is equivalent to multiplying the PDF by a function such as a Gaussian, which causes the PDF to decay at high-r values. For amorphous materials the effect of this damping is negligible, because these materials are dominated by short range order such that their PDF at high r is negligible before the Q-resolution starts to take effect. For crystalline materials, where theoretically the PDF extends to infinity, this issue becomes important, especially when investigating the high-r region. The same applies for the materials that are composed of domain sizes of limited structural coherence. In these cases it becomes crucial to determine if the decay in the PDF is due to the structure (domain size) or due to the instrument resolution. This requires a high resolution PDF experiment from which we can specify the coherent domain sizes for the material under study.
The total scattering measurement can be carried out from different powder diffractometer geometries such as a flat-plate symmetric reflection geometry or transmission geometry. The details on these experimental set ups are well documented in the literature . The choice between reflection and transmission configuration depends on the studied sample. In the case of absorbing samples, reflection geometry has the benefit that the absorption and active volume corrections can be derived fairly straightforward for the approximation of a parallel incident beam. However, care must be taken when making the flat plate samples, they should have very uniform thickness and density and the surface should also be as flat and smooth as possible. Another important consideration is that the beam footprint does not extend over the edge of the sample surface area. This will result in an angle dependent drop off in the intensity of the sample especially at low angles which is very difficult to correct. If the studied samples are significantly transparent, then the transmission geometry provides stronger scattering particularly at high angles. This configuration is also more robust than reflection geometry against alignment errors, especially at low angles. For these considerations, the transmission geometry is often used at high energy scattering beamlines. In PDF measurements it is crucial to collect enough points that the step size is always smaller than the instrument resolution, so all of the Q-space is probed. Adequate collimation before and after the sample is also important to minimize background from air scattering and sample environments. Furthermore, care must be taken to ensure that the focused beam corresponds to the center of the diffractometer and the sample is well aligned.
Table of contents :
Chapter 1 Introduction
Chapter 2 Experimental methods for molecular and nanoscale structural determination
2.1 X-ray and neutron total scattering
2.1.1 Coherent Scattering Intensity
2.1.2 The Total Structure Function
2.1.3 Relation with Pair Distribution Function (PDF)
2.1.4 Data Collection
188.8.131.52 Reduce the Experimental Errors
184.108.40.206.1 Termination Errors
220.127.116.11.2 Data statistics and Collection time
18.104.22.168.3 Instrumental resolution
22.214.171.124 Diffraction configurations :
2.1.5 Data Reduction
126.96.36.199 Multiplicative corrections
188.8.131.52.1 Polarization correction
184.108.40.206.1 Considering to Laboratory Source
220.127.116.11 Additive corrections
18.104.22.168.1 Incoherent scattering
22.214.171.124.2 Multiple scattering
126.96.36.199.3 Background Subtraction
2.2 Solid state NMR spectroscopy
2.3 IR and UV spectroscopy
Chapter 3 Structural Modeling
3.1 The Debye Function analysis (DFA)
3.2 Pair distribution function (PDF)
3.2.1 Calculating PDF from a model
3.2.2 Consider finite Q-range
3.2.3 Thermal motion
3.2.4 Instrumental resolution
3.2.5 PDF of nanoparticles
3.2.6 Related functions
3.3 The reverse Monte Carlo Method
Chapter 4 Synthesis and multiscale structural analysis of photoswitchable molecular hybrid materials
4.1 Sample preparation
4.2 Average crystal structure of the bulk material Na2[Fe(CN)5NO]·2H2O
4.3 Evidence of photoswitchable properties
4.4 The PDF and d-PDF experiments for the bulk and SNP@1nm SiO2 monolith
4.4.1 PDFs of bulk SNP
4.4.2 PDFs of amorphous SiO2 matrix
4.4.3 Fingerprint of SNP@SiO2 hybrids
4.5 The Solid State NMR….
4.5.1 Structure of amorphous SiO2 host
4.5.2 Structure and dynamics of the SNP guest
4.5.3 Hydration of the nanocomposite SNP@SiO2 xerogel
4.6 Multiscale structural analysis of SNP@SiO2 (1nm, 5nm, 11nm): the structural aspect-pore size and organisation relationship
4.6.1 Problems of SNP crystallized on monolith surface
4.6.2 SNP@SiO2 (5nm) .
4.6.3 Pre-doping SNP@SiO2 nanocomposites
Chapter 5 Structural investigation and photoluminescent properties of embedded neodymium complex
5.1 Sample preparation
5.2 Average crystal structure of the bulk material [NdCl2(H2O)6]Cl-
5.3 Optical properties
5.3.1 UV-visible spectroscopy
5.3.2 IR spectroscopy
5.4 Total scattering methods
5.4.1 Differential pattern
5.4.2 Structure of embedded Nd3+ complexes
5.4.3 Debye function analysis
General conclusion and perspectives