Secondary Extinction Correction Research Articles

Secondary extinction correction used in a novel method for improved XRD characterizations of textured materials: The case of thin films

Secondary extinction correction used in a novel method for improved XRD characterizations of textured materials: The case of thin films

Secondary extinction correction used in a novel method for improved XRD characterizations of textured materials: The case of thin films

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Texture and secondary extinction measurements in Al\\Ti stratified films by X-ray diffraction

In the present work the effects of secondary extinction has been studied using standard X-ray diffractometer and texture diffractometer techniques. For this purpose the texture of Al\\Ti stratified films have been measured and evaluated. It is shown that these films have a very sharp single component texture. Therefore the evaluation and the interpretation of the measurement data is only possible when secondary extinction correction is carried out. The effect of secondary extinction is evaluated on the 111 pole figures of the stratified films with different layer pair numbers. The volume fraction of the main texture component is calculated from pole figures and it is shown that the grains of the films are practically completely oriented with 〈111〉 crystallographic direction parallel to the film surface normal. Moreover in the case of pole density measurements the secondary extinction values themselves are simultaneously calculated.

Secondary Extinction Used in Thickness and Pole Density Measurements of Textured Films by X-Ray Diffraction

A new X-ray diffraction method for a more accurate determination of film thickness and pole densities is developed by introducing a secondary extinction correction. For the purpose an equation based on the kinematic theory of X-ray diffraction and the mosaic crystal model has been derived. The discussion of results shows that the correct characterization of textured films requires in general the secondary extinction to be taken into account.

Secondary Extinction Correction Used in Pole Density Measurements by X-Ray Diffraction

A new X-ray diffraction method for a more accurate determination of film pole densities is developed by introducing a secondary extinction correction. To this end an equation based on the kinematic theory of X-ray diffraction and the mosaic crystal model is derived. Pole densities were computed either by using the two order reflections techniques or by the least-square method. The discussion of results shows that the correct study of film structure by X-ray diffraction requires in general the secondary extinction to be taken into account.

MXD: a general least-squares program for non-standard crystallographic refinements

MXD is a new least-squares program for non-standard crystallographic refinement. It uses an original procedure of formula management that is derived from up-to-date compiler programming techniques. While still offering all the least-squares possibilities of most traditional programs, e.g. SHELX, SDP, XRAY etc., MXD has been fully designed for complex crystallographic, magnetic and/or modulated structure refinement. The user enters all the specific formulae describing his own problem into the input data file. Thus, MXD is very convenient when dealing with very complex problems including fitting of modulated structures (magnetic and/or charge density wave), twin or magnetic domain distribution and severe secondary extinction correction. Moreover, if the experimental data are accurate enough (especially from using polarized neutrons), form-factor fitting is accessible. Independently collected data, using various wavelengths, powder or/and single-crystal data, polarized and unpolarized neutron information etc., can be treated simultaneously.

On the treatment of primary extinction in diffraction theories based on intensity coupling

Czochralski-grown silicon crystals of approximately 10 cm diameter and 1 cm thickness have been annealed at 1470 K in order to create a homogeneous defect structure, which is a basic condition for all statistical treatments of extinction. Absolute values of the integrated reflecting power of the 220, 440 and 660 reflections have been measured with 0.0392 A γ-radiation in symmetrical Laue geometry for sample thicknesses between 1 and 3 cm. The amount of extinction in the experimental data varies between y ≃ 0.95 and y ≃ 0.05. Darwin's extinction theory has been used to describe the thickness dependence of the data sets. Despite some shortcomings of the model, it is shown that the assumption of a physically unrealistic Lorentzian mosaic distribution models the effect of primary extinction in an extinction theory based on the energy-transfer model. The sharp central part of the Lorentzian distribution produces a reduction of the effective sample thickness due to primary extinction, whereas the wings of the distribution dominate the correction for secondary extinction in the remaining part of the sample. A more flexible mosaic distribution function is proposed, which should be useful in cases of severe extinction.

The application of an empirical correction for absorption and secondary extinction to neutron data for crystals of brushite, CaHPO4 · 2H2O

The application of an empirical correction for absorption and secondary extinction to neutron data for crystals of brushite, CaHPO₄ · 2H₂O

The application of an empirical correction for absorption and secondary extinction to neutron data for crystals of brushite, CaHPO4· 2H2O

The application of an empirical correction for absorption and secondary extinction to neutron data for crystals of brushite, CaHPO₄· 2H₂O

Di-μ-bromo-bis[(pyridine)(triphenylphosphine)copper(I)

Cu2Br2(CsHsN)2{P(C6H5) 3}), M r = 970.0, monoclinic, C2/c, a -- 26.188 (7), b = 14.222 (3), c =11.259(4)A, fl=94.51(2) ° , V=4180(2)A 3, z =4, Dx= 1.54Mgm -3, 2(MoKct)=0.71073./k, /z= 2.90 mm -1, F(000) = 1944, T = 296 K, R = 0.047 for 1271 observed reflections. The compound is in a dimeric form; the metal atom is tetrahedrally coordi- nated to pyridine (Cu-N 2.053 (9)A), two Br atoms (Cu-Br 2.538 (2), 2.507 (2)A; Br-Cu-Br 108.50 (8) ° ) and triphenylphosphine (Cu-P 2.209 (3)A) and is slightly distorted, with a dihedral angle between the (BrCuBrl and (PCuN) planes of 90.9 (2) °. The Cu-..Cu' distance is 2.948 (2) A. Introduction. Monodentate tertiary phosphines form many complexes with Cu ~ that show a variety of stoichiometries and structures (Gill, Mayerle, Welcker, Lewis, Ucko, Barton, Stowens & Lippard, 1976). Considerable interest exists in structural studies of copper(I) derivatives, not only because of their stereochemistry but also for their importance in oxidation-reduction reactions in enzymes containing copper (Peisach, Aisen & Blumberg, 1966) and in organic synthesis (Tsuda, Fujii, Kawasaki & Saegusa, 1980). Some years ago, Jardine, Rule & Vohra (1970) reported the preparation of halogenocopper(I) com- plexes using tertiary phosphines as ligands. We synthe- sized one of them, the title compound, and pro- ceeded with its crystal structure determination because the only characterizations in the Jardine et al. (1970) paper were analytical data and molecular weight and to elucidate their hypothesis that the compound, though monomeric in solution, may be dimeric in the solid state. Experimental. Irregular colourless crystals from butanol at 277 K, max. and min. linear dimensions 0.30, 0-15 mm; Nonius CAD-4 diffractometer, graphite-monochromated Mo Ka; cell parameters by least squares on setting angles for 22 reflections, 9 < 0< 30°; w--20 scans, scan width (0.80 + 0.35 tan0) °, scan speed 6.7 ° min-~; max. range of hkl: -27 <h < 27, k< 15, l< 11, 0max=22°; standards 060, 10,~,,2 varied +2.4% of mean intensities over data collection; 2086 reflections measured, 2553 unique, Rin t = 0.014, 1271 observed above 3tr(/), Lp correc- tions; structure solved by direct methods. In final cycles of full-matrix least-squares refinement all non-hydrogen atoms anisotropic. Py H atoms included at constrained positions (C-H = 1.00 (1) A) based on those found in difference synthesis, all with fixed isotropic U= 0.06 A2; phenyl rings of triphenylphosphine as rigid bodies (C-C = 1.395, C-H = 1.08 A, all angles 120 °, H atoms with common U=0.06A2). Function minimized ~w(IFol--IFcl) 2 with w=(e2(Fo) + 0.00075Fo2)-1; max. A/e=O.O05; 135 parameters refined; excluding unobserved reflections R-----0-047, wR = 0.050; inspection of F c and Fo values indicated correction for secondary extinction required;* max. A/tr = 0.002, Ap excursions within --0.44 and 0.56 e A-3; scattering factors for non-H atom~ from Cromer & Mann (1968) with corrections for anomalous dispersion from Cromer & Liberman (1970), for H from Stewart, Davidson & Simpson (1965); programs used: MULTAN80 (Main, Fiske, Hull, Lessinger, Germain, Declercq & Woolfson, 1980); SHELX76 (Sheldrick, 1976) and ORTEP (Johnson, 1965). Most of the calculations were per- formed on a VAX 11/780 computer of the Instituto de Fisica e Quimica de S~o Carlos. Discussion. A projection of the crystallographically independent moiety is shown in Fig. 1. Positional

Structure of the primitive tetragonal ErRh4B4

Structure of the primitive tetragonal ErRh4B4

Experimental test of Kawamura and Kato's secondary extinction correction formula

Experimental test of Kawamura and Kato's secondary extinction correction formula

CRYSTAL STRUCTURE AND OPTICAL OBSERVATIONS OF BiV04

A single-crystal X-ray diffraction analysis has been performed on fergusonite BiVO4 synthesized by pulling method. At room temperature the crystal belongs to monoclinic system, with space group B2/b, z = 4, and cell parameters a = 7.247?, b=5.096?, c = 11.702? and r =134.18o. The structure has been analysed by two and three-dimensional Patterson procedure, and calculated through three-dimensional electron density. A full-matrix, least square refinement gave R=0.067, Rw =0.058 for 335 independent reflections. An absorption and secondary extinction correction gave final R = 0.058. The Bi and V atoms occupy a respective set of 4e special positions and the O atoms two sets of general positions. The oxygen atoms take part in four-fold coordinations around vanadium in the sense that they form regular VO4 tetrahedra. The structure also shows eight-fold oxygen coordinations around bismuth and form distorted Bi-O dodecahedron. The space groups for temperatures above and below Tc are found to be I41/a and 12/b (or B2/b) respectivelv. The phase transition from tetragonal to monoclinic does skip over a possible intermediate orthorhombic modification. The phase transition is apparently driven by a lone-pair distortion of the Bi-O polyhedron. Bi3+ and V5+ displace alternatively up and down along the c axis, resulting in a layer-like Character for monoclinic BiVO4 The character has been proved by optical observations. The BiVO4 crystal cleaves easily to give smooth (cleavage) faces parallel to plane perpendicular to the c axis.Three kinds of samples, i.e crystals with cell-like structure and disc shape, crystal ribbon with natural cleavage and powder crystal grain, were used for optical observation, and the relationship between the transmittance of BiVO4 and optical wave-length was studied.

Experimental correction for primary and secondary extinction. I. Method

The energy transfer equations are used to solve secondary extinction in a parallel-sided crystal slab in Laue geometry. Practical formulas are given for determination of the refiectivity per unit length, σ(∊), from the measured absolute powers of the diffracted and direct beams at rocking angle ∊ = θ- θB. The calculation assumes non-divergent beams, and the experimental tolerances are considered in detail. The primary extinction factor is defined by yp(∊) = σ(∊)/σkin(∊), where σkin is the kinematical reflectivity. When yp > 0.5 it may be approximated by yp ≃ exp[-(αδ)2], where α is about 0.5 and δ is the average size of the coherent domains when measured in units of the extinction distance Λ, δ = D/Λ. The extinction distance can be varied by changing the X-ray wavelength or polarization, and yp(∊) can be determined from two measurements.

Experimental correction for primary and secondary extinction. II. Application to beryllium

The profiles of the first three reflections from a parallel-sided crystal slab of Be are measured on an absolute scale with polarized Cu Kα radiation and unpolarized Mo Kα radiation. The effects of secondary extinction are corrected by measuring the transmitted intensity and using the conservation of energy. Primary extinction is determined by comparing two measurements where the extinction distances are different; this is realized by changing the wavelength or polarization of the incident radiation. The calculated average secondary-extinction factor, {\bar Y}s, ranges from 0.98 to 0.74, and the average primary extinction factor, {\bar Y}p, from 0.96 to 0.42. The room-temperature values of the structure factors, F(10.1) = 1.90 ± 0.01, F(00.2) = 3.37 ± 0.02, F(10.1) = 2.78 ± 0.03, differ substantially from earlier experimental values and from the values calculated for a free atom, but agree very closely with the results of a LCAO calculation.

Toward a secondary-extinction correction

The treatment of secondary extinction currently in general use by crystallographers is due to Becker & Coppens [Acta Cryst. (1974), A30, 129-147]. It is derived from approximate solutions to the Darwin transfer equations developed by Zachariasen [Acta Cryst. (1967), 23, 558-564]. Among the approximations invoked by Zachariasen is that the effects of scattering and absorption are to be treated separately. A scheme is here proposed which makes that unnecessary. It is further suggested that progress in the correction of experimental data for secondary- extinction effects is more likely if one begins with the exact solution to the transfer equations due to Werner [J. Appl. Phys. (1974), 45, 3246-3254]. That solution has so far not penetrated the diffraction literature, probably because it is in the form of a difficult infinite series of Bessel functions. In one special case, which nevertheless has attracted the general attention of those interested in this subject, it is shown that the Werner series may be summed, yielding an integral equation involving only the zero-order Bessel function. Excellent closed-form approximations to this integral, accurate to about 1% or less, are displayed.

Absolute measurement of the x-ray scattering factor ofCo0.91Fe0.09

Absolute measurements have been performed to determine the x-ray scattering factor of ${\mathrm{Co}}_{0.91}$${\mathrm{Fe}}_{0.09}$. This study has been made using two imperfect single crystals in symmetrical transmission, and applying a measured secondary extinction correction to the integrated intensities obtained with Ag $\mathrm{K}\ensuremath{\alpha}$ radiation. A measurement of the incident-beam intensity placed the results on an absolute scale. The experimental values of the scattering factors have been found to depend systematically on the reflection half-width, thus indicating the presence of primary extinction. The scattering factors for the (110) and (200) reflections, determined from an extrapolation to infinite half-width, have been found to be about 2% lower than the corresponding theoretical free-atom values. The measured asphericity of the total $d$ charge ($%{t}_{2g}=61.5$) agrees with that calculated from a polarized-neutron measurement of the magnetic form factor of the same alloy.

Refinement of α-lead azide by neutron diffraction

Pb(N3) 2, orthorhombic, Pnma; a = 6.63, b = 16.25, c = 11.31 A, Z = 12 [Azaroff (1956), Z. Kristallogr. 107, 362-3691. The four independent azide groups in the structure are all asymmetric, with N N bond distances, corrected by the riding model, ranging from 1-164 to 1.196 A, and N N N bond angles as small as 177.9 ° . Introduction. It is well known that the stability of azide compounds is strongly correlated with the degree of covalency of the azide bonds in crystals. The structure of a-Pb(N3) z, a well known primary explosive, was previously determined from a partial, threedimensional, neutron diffraction data set by Choi & Boutin (1969). A recent advance (Garrett, 1972) in techniques for growing lead azide crystals has provided us with additional crystals of a size and shape suitable for neutron diffraction. We have therefore redetermined the structure by refining a full, three-dimensional data set, in order to improve the accuracy of the structure parameters. Intensities were measured out to a limiting value of sin 0/2 = 0.65 /~-I by the procedure described by Prince (1972), on a computer-controlled four-circle diffractometer at the NBS reactor. Of 1466 independent reflections within the limiting sphere, there were observable intensities for 1328. Intensities were reduced to F values, and reflections were assigned weights according to W = [a~ + (0.005F)21 -~, where a~ is the estimated standard deviation of F due to counting statistics. The structure was refined by full-matrix leastsquares methods (Finger & Prince, 1975), including an isotropic secondary extinction parameter (Zachariasen, 1968) to a weighted R of 0.043 and an unweighted R of 0.038. The extreme secondary extinction correction (Fcorr/Fcalc) was 0.34 for the 004 reflection. The final least-squares atomic positional parameters are given in Table I.* Discussion. The four crystaltographically distinct azide groups, designated in the following discussion as azide I, azide II, azide III, and azide IV, are all distorted significantly from the linear, symmetric conformation found in more stable ionic azide compounds (Choi & Prince, 1976). Bond distances and angles are given in Table 1. Final least-squares atomic positional parameters ( x 104)for a-Pb(N 3)2

Hydrogen bonding in the crystalline state. The crystal structure of NaH2PO4.2H2O by X-ray and neutron diffraction

Sodium dihydrogen phosphate dihydrate (NaH2PO4.2H20) is orthorhombic, space group P212121, with a=7.275 (9), b=11-384 (6) and c=6-606 (4) A; Z=4. The structure was solved from X-ray measurements by direct methods and refined anisotropically with X-ray (R = 0.032 over 898 reflexions) and neutron data (R=0.059 over 751 reflexions); both sets of intensities were collected on singlecrystal diffractometers (Mo Ks radiation and 2= 1-035 A respectively). A correction for secondary extinction was applied to the neutron data. While the two acidic hydrogens form strong hydrogen bonds (2.577 and 2.536 ,&), one of those of the water molecules is involved in a very weak O H . . . O contact of 3-106 ,&. The P tetrahedra are linked by hydrogen bonds in a distorted diamond-type framework. Six systems of pseudo-hexagonal channels are present; the largest of them, along [001], accommodates a chain of Na octahedra. The configuration of the hydrogen-bonded groups is described. Bond-strength/bond-length correlations are discussed and their weakness for the case of strongly hydrogen-bonded atoms is pointed out. A comparison of X-ray and neutron results by Z 2 tests and half-normal probability plots shows significant differences for the thermal parameters only.

Absolute X-Ray Measurement of the Atomic Scattering Factor of Chromium

Absolute measurements were made to determine the atomic scattering factor of chromium by x-ray diffraction. This study was made at room temperature, using an imperfect single crystal in transmission and applying a measured secondary-extinction correction to the integrated intensities obtained with monochromatic $\mathrm{Mo}K\ensuremath{\alpha}$ radiation. The measurements were placed on an absolute scale using a value of the Debye temperature of 525\ifmmode^\circ\else\textdegree\fi{}K and a measured mass-absorption coefficient of 30.15 ${\mathrm{cm}}^{2}$/g. As in previous measurements on nickel, the apparent scattering factors of the (110) and (200) reflections were found to depend linearly on the reciprocal of the reflection half-width, thus indicating the presence of primary extinction. The true atomic scattering factors for the (110) and (200) reflections, determined from an extrapolation to zero reciprocal half-width, were found to be considerably lower than the theoretical values obtained from free-atom Hartree-Fock calculations. From the paired reflections (330)-(411) and (600)-(442) the ${t}_{2g}$ population of $3d$ electrons was found to be about 72%, thus indicating a marked departure from spherical symmetry.