Bragg-peaks

The Bragg peak intensity reduction due to atomic displacements is described by the well-known temperature factors. Assuming that the position can be decomposed into an average position, ,) and an infinitesimal displacement, M = 8R = Ri — (R,) then the X-ray structure factors can be expressed as follows ... 

Displacements correlated within unit cells but not between them lead to very diffuse scattering that is not associated with the Bragg peaks. This can be conveniently explored... 

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Figure 1 Bragg diffraction. A reflected neutron wavefront (D, Dj) making an angle 6 wKh planes of atoms will show constructive interference (a Bragg peak maxima) whan the difference in path length between Df and (2CT) equals an integral number of wavelengths X. From the construction, XB = d sin 6.

Figure 5 Raw diffraction data at the start (bottom) and completion (top) of the in-sHu decomposition of slag experiments. Most of the peaks in the pattern are due to the parent slag phase. Bragg peaks due to titanium oxide (T) and iron metai (Fe) are marked.

Powder X-ray diffraction and SAXS were employed here to explore the microstructure of hard carbon samples with high capacities. Powder X-ray diffraction measurements were made on all the samples listed in Table 4. We concentrate here on sample BrlOOO, shown in Fig. 27. A weak and broad (002) Bragg peak (near 22°) is observed. Well formed (100) (at about 43.3°) and (110) (near 80°) peaks are also seen. The sample is predominantly made up of graphene sheets with a lateral extension of about 20-30A (referring to Table 2, applying the Scherrer equation to the (100) peaks). These layers are not stacked in a parallel fashion, and therefore, there must be small pores or voids between them. We used SAXS to probe these pores. 

Fig. 30. Calculated (002) Bragg peaks for various single layer fraetions of the sample from referenee 12. The ealeulations assumed that a fraetion, f, of the earbon was in single layers and that fractions 2/3(l-f) and l/3(l-f) were included in bilayers and trilayers respectively.

The Bragg peaks indicated an ordered local structure within the sample film, and the interlayer spacings were reproduced compared with the bulk samples, with only... 

Surprinslngly, we observe an drastic effect of the concentration on the SRO contribution (figure 2) indeed, in PtaV, the maxima are no longer located at a special point of the fee lattice but the (100) intensity is splltted perpendicularly in the (010) direction and presents a saddle point at (100) position. Notice that these two maxima are not located just above Bragg peaks of the ordered state the A B ground state presents Bragg peaks at ( 00) and equivalent positions whereas the SRO maxima peak between ( 00) and (100). 

Such off-zone-centre, soft-mode systems offer the most favourable conditions for a test of the hypothesis that the central peak is a precursor to a Bragg reflection in the transformation phase. Zone-centre softening, such as occurs in NbaSn, results in the central mode scattering emerging from an existing Bragg peak, which ultimately splits in the lower symmetry transformation structure, which presents a problem with resolution. 

Fig. 1-18. The L spectra of platinum. Reference to the L levels labeled at the lower right-hand corner of this figure shows that none of the lines occurs on the short-wravelength side of the corresponding absorption edge. -Note also that some of the Bragg peaks contain lines from more than one L level. os...

In this section we will discuss in some detail the application of X-ray diffraction and IR dichroism for the structure determination and identification of diverse LC phases. The general feature, revealed by X-ray diffraction (XRD), of all smectic phases is the set of sharp (OOn) Bragg peaks due to the periodicity of the layers [43]. The in-plane order is determined from the half-width of the inplane (hkO) peaks and varies from 2 to 3 intermolecular distances in smectics A and C to 6-30 intermolecular distances in the hexatic phase, which is characterized by six-fold symmetry in location of the in-plane diffuse maxima. The lamellar crystalline phases (smectics B, E, G, I) possess sharp in-plane diffraction peaks, indicating long-range periodicity within the layers. 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

Figure 27.4 shows a charge curve for the cell when it was charged over a period of 12 hours from 3.0 to 5.2 V. During the charge, 16 XRD scans were collected continuously. The time at which each spectrum was acquired is shown in Fig. 27.4. The time for each scan was about 45min. The corresponding spectra are shown in Fig. 27.5. Four hexagonal phases, HI, H2, Ola, and Ol, were observed and their Bragg peaks were indexed in Fig. 27.5. The phase transition from HI to H2 started in scan 4 and...

The observed profile P(20) of a Bragg peak at do = dhu is a convolution product between the instrumental profile r(20) and the physical profile p(20) ... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

T. Egami, S. Billinge (eds.) Underneath the Bragg Peaks Structural Analysis of Complex Materials, Elsevier Science, Amsterdam, 2003. 

Vol. 7 Underneath the Bragg Peaks Structural Analysis of Complex Materials... 

With the new VME/UNIX control system on the polarised hot-neutron normal-beam diffractometer D3 at ILL, each measurement cycle for both peak and background intensities lasts 2 s, and the (+)/(-) counting-time fractions are defined with a 1 MHz clock. There are two detector scalers and two monitor scalers ((+) and (-) states). In Table 1, we compare the flipping ratio measured for the strong 200 and the weak 600 Bragg peak reflections of a CoFe sample. As expected, the standard deviation cr (if) is improved in the case of the strong reflection (16%). 

For the case of surface truncation rods, the technique is based on the detection of diffraction peaks between Bragg peaks. Although this requires careful alignment and some a priori knowledge of the structure, monolayer sensitivity can be achieved. In fact, Samant et a/.138 have recently performed an in situ surface diffraction study of lead underpotentially deposited on silver employing this technique along with grazing incidence diffraction. It is clear that this technique will also find widespread use in the near future. 

Fig. 68 Comparison of temperature-dependent intensity of first-order Bragg peak for bare matrix copolymer (A) containing 0.5 wt% nanocomposites with plate-like (V), spherical (o) and rod-like ( ) geometry. Data are vertically shifted for clarity. Inset dependence of ODT temperature on dimensionality of fillers (spherical 0, rod-like 1, plate-like 2). Vertical bars width of phase transition region. Pure block copolymer is denoted matrix . From [215]. Copyright 2003 American Chemical Society...

Then, when the precipitation of the hybrid material is observed after 23 minutes (figure 1-c), the (100) Bragg peak of the 2D hexagonal packing is viewed in the SANS patterns. The position of the peak corresponds to a distance between the cylinders of 14.2 nm. 

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