This article is the second part of a series dealing with the description and visualization of mathematical functions used to describe a powder diffraction pattern for teaching and education purposes. ...The first part dealt with the instrumental and sample contributions to the profile of a Bragg peak Dinnebier & Scardi (2021). J. Appl. Cryst.54, 1811–1831. The second part, here, deals with the mathematics and physics of the intensity in X‐ray powder diffraction. Scholarly scripts are again provided using the Wolfram language in Mathematica.
The most commonly used functions for calculating or correcting step‐scan and integrated intensities of a powder diffraction pattern are presented in an educational manner with the support of Mathematica scripts. The scripts can be easily used by interested readers to explore the effects of different instrumental and sample parameters.
A method of ab initio crystal structure determination from powder diffraction data for organic and metal–organic compounds, which does not require prior indexing of the powder pattern, has been ...developed. Only a reasonable molecular geometry is required, needing knowledge of neither unit‐cell parameters nor space group. The structures are solved from scratch by a global fit to the powder data using the new program FIDEL‐GO (`FIt with DEviating Lattice parameters ‐ Global Optimization'). FIDEL‐GO uses a similarity measure based on cross‐correlation functions, which allows the comparison of simulated and experimental powder data even if the unit‐cell parameters deviate strongly. The optimization starts from large sets of random structures in various space groups. The unit‐cell parameters, molecular position and orientation, and selected internal degrees of freedom are fitted simultaneously to the powder pattern. The optimization proceeds in an elaborate multi‐step procedure with built‐in clustering of duplicate structures and iterative adaptation of parameter ranges. The best structures are selected for an automatic Rietveld refinement. Finally, a user‐controlled Rietveld refinement is performed. The procedure aims for the analysis of a wide range of `problematic' powder patterns, in particular powders of low crystallinity. The method can also be used for the clustering and screening of a large number of possible structure candidates and other application scenarios. Examples are presented for structure determination from unindexed powder data of the previously unknown structures of the nanocrystalline phases of 4,11‐difluoro‐, 2,9‐dichloro‐ and 2,9‐dichloro‐6,13‐dihydro‐quinacridone, which were solved from powder patterns with 14–20 peaks only, and of the coordination polymer dichloro‐bis(pyridine‐N)copper(II).
A new method for the structure determination of molecular crystals from unindexed powder data has been developed and successfully applied. The method performs a global optimization using pattern comparison based on cross‐correlation functions.
Cu-paddle-wheel-based Cu3(BTC)2 (nicknamed Cu-BTC, where BTC ≡ benzene 1,3,5-tricarboxylate) is a metal organic framework (MOF) compound that adopts a zeolite-like topology. We have determined the ...pore-size distribution using the Gelb and Gubbins technique, the microstructure using small-angle neutron scattering and (ultra) small-angle X-ray scattering (USAXS\SAXS) techniques, and X-ray powder diffraction reference patterns for both dehydrated d-Cu-BTC Cu3(C9H3O6)2 and hydrated h-Cu-BTC Cu3(C9H3O6)2(H2O)6.96 using the Rietveld refinement technique. Both samples were confirmed to be cubic Fm
$\bar 3$
m (no. 225), with lattice parameters of a = 26.279 19(3) Å, V = 18 148.31(6) Å3 for d-Cu-BTC, and a = 26.3103(11) Å, and V = 18 213(2) Å3 for h-Cu-BTC. The structure of d-Cu-BTC contains three main pores of which the diameters are approximately, in decreasing order, 12.6, 10.6, and 5.0 Å. The free volume for d-Cu-BTC is approximately (71.85 ± 0.05)% of the total volume and is reduced to approximately (61.33 ± 0.03)% for the h-Cu-BTC structure. The d-Cu-BTC phase undergoes microstructural changes when exposed to moisture in air. The reference X-ray powder patterns for these two materials have been determined for inclusion in the Powder Diffraction File.
The X-ray diffraction powder patterns were prepared and the crystal structures were refined for the double-perovskite series of compounds, Sr2RSbO6 (R = Pr, Nd, Sm, Eu, Gd, Dy, Ho, Y, Er, Tm, Yb, and ...Lu). We found the structures of the entire Sr2RSbO6 series to be monoclinic with space group P21/n (no. 14), and Z = 2. From R = Lu to Pr, the lattice parameters “a” range from 5.7779(2) to 5.879 05(8) Å, “b” range from 5.7888(2) to 5.969 52(9) Å, “c” range from 8.1767(3) to 8.369 20(12) Å, “β” range from 90.112(2)° to 90.313(1)°, and “V” range from 273.483(4) to 293.714(7) Å3. These lattice parameters follow the well-established trend of “lanthanide contraction”. The R
3+ and Sb5+ ions are found to be fully ordered in the double-perovskite arrangement of alternating corner-sharing octahedra in a zigzag fashion. The SrO12, RO6, and SbO6 cages are all found to have distorted coordination environments. Powder diffraction patterns of these compounds have been prepared, submitted, and published in the Powder Diffraction File.
A new program for basic powder pattern manipulations and visualization is described. It provides a user‐friendly interface for comparison of spectra with each other and with simulated patterns based ...on single‐crystal data. The program contains all necessary tools for the preparation of routine images for qualitative phase analysis and can be downloaded free of charge from http://diffractwd.com.
Even for large quadrupolar interactions, the powder spectrum of the central transition for a half-integral spin is relatively narrow, because it is unperturbed to first order. However, the ...second-order perturbation is still orientation dependent, so it generates a characteristic lineshape. This lineshape has both finite step discontinuities and singularities where the spectrum is infinite, in theory. The relative positions of these features are well-known and they play an important role in fitting experimental data. However, there has been relatively little discussion of how high the steps are, so we present explicit formulae for these heights. This gives a full characterization of the features in this lineshape which can lead to an analysis of the spectrum without the usual laborious powder average.
The transition frequency, as a function of the orientation angles, shows critical points: maxima, minima and saddle points. The maxima and minima correspond to the step discontinuities and the saddle points generate the singularities. Near a maximum, the contours are ellipses, whose dimensions are determined by the second derivatives of the frequency with respect to the polar and azimuthal angles. The density of points is smooth as the contour levels move up and down, but then drops to zero when a maximum is passed, giving a step. The height of the step is determined by the Hessian matrix—the matrix of all partial second derivatives. The points near the poles and the saddle points require a more detailed analysis, but this can still be done analytically. The resulting formulae are then compared to numerical simulations of the lineshape.
We expand this calculation to include a relatively simple case where there is chemical shielding anisotropy and use this to fit experimental 139La spectra of La2O3.
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•Clearly reviews the positions of the features in a second-order perturbed lineshapes.•Presents equations for the heights of the steps for the first time.•Deals with a simple case of combined quadrupolar and anisotropic shielding.•Analyzes the La spectrum of La2O3 without the need for a powder average.•Fully general approach.
Methods of magnetic powder patterns (Bitter technique), optical microscopy, and transmission electron microscopy have been used to study the formation of martensite in a spherical sample of ...12Kh18N10T steel upon loading by quasi-spherical converging shock waves. Prior to loading, the steel exhibited carbide banding (TiC and Cr
23
C
6
carbides were located in bands along the axis of the initial sample). It has been shown that, upon loading, disperse crystals of martensite are formed in these carbide-containing bands. No martensite was formed in the central part of the solid sphere, in the surface layers, and in the Altshuler-pattern rays. It has been concluded that, in these regions, the temperature exceeded the
M
d
temperature, above which the deformation does not cause a martensitic transformation.
The International Centre for Diffraction Data (ICDD) produces the Powder Diffraction File (PDF). This paper discusses some of the seminal events in the history of producing this primary reference for ...powder diffraction. Recent key events that center on collaborative initiatives have led to an enormous jump in entry population for the PDF. Collective efforts to editorialize the PDF are ongoing and provide enormous added value to the file. Recently, the ICDD has created a new series of the PDF, designated PDF‐4. These relational database structures are being used to house the PDF of the future. The design and benefits of the PDF‐4 are described.
A systematic study of the chemical interaction of Ba
2YCu
3O
6+
y
and Gd
3NbO
7 was conducted under two processing conditions: purified air (21%
p
o
2
), and 100
Pa
p
o
2
(0.1%
p
o
2
). Phases ...present along the pseudo-binary join Ba
2YCu
3O
6
z
and Gd
3NbO
7 were found to be in two five-phase volumes within the
BaO
−
1
2
Y
2
O
3
−
1
2
Gd
2
O
3
−
Nb
2
O
5
−
CuO
y
system. Three common phases that are present in all samples are (Y,Gd)
2Cu
2O
5, Ba(Y,Gd)
2CuO
5 and Cu
2O or CuO (depending on the processing conditions). The assemblies of phases can be categorized in three regions, with Ba
2YCu
3O
6+
y
: Gd
3NbO
7 ratios of (I)<5.5:4.5; (II)=5.5:4.5; and (III)>5.5:4.5. The lowest melting temperature of the system was determined to be ≈938
°C in air, and 850
°C at 100
Pa
p
o
2
. Structure determinations of two selected phases, Ba
2(Gd
x
Y
1−
x
)NbO
6 (
Fm3¯
m, No. 225), and (Gd
x
Y
3−
x
)NbO
7 (
C222
1, No. 20 and
Ccmm, No. 63), were completed using the X-ray Rietveld refinement technique. Reference X-ray powder diffraction patterns for selected phases of Ba
2(Gd
x
Y
1−
x
)NbO
6 (
x=0.2, 0.4, 0.6, and 0.8) and (Gd
x
Y
3−
x
)NbO
7 (
x=0.6, 1.2, 1.8, 2.4 and 3) have been prepared for inclusion in the Powder Diffraction File (PDF).
Crystal structure for (Gd
x
Y
3−
x
)NbO
7 showing the partial layered feature. The alternate stacking of distorted NbO
6 octahedra and (Gd,Y)O
7 polyhedra are illustrated. The (Gd,Y)O
8 polyhedra are omitted for clarity.
PDF
