Cation order in the crystal structure of ‘minasgeraisite-(Y)’

The crystal structure of ‘minasgeraisite-(Y)’, triclinic P1, a = 9.994(4), b = 7.705(3), c = 4.764(2) Å, α = 90.042(9), β = 90.218(14), γ = 90.034(9) (°), V = 366.8(5) Å3 and Z = 1, has been refined to an R1 index of 2.86% for 4170 observed (|Fo| > 4σF) reflections. Significant observed (|Fo| > 40–60 σF) reflections violate the presence of a 21-screw axis and an a-glide plane, negating the space group P21/a previously found for minerals of the gadolinite–datolite group. Averaging of the X-ray data in Laue groups 2/m and $\bar 1$ gives the following agreement indices: 2/m (9.68%) and $\bar 1$ (5.68%). The internal agreement index from averaging of identical reflections collected at multiple positions along the diffraction vector is significantly lower than that for the Laue group $\bar 1$: Rpsi = 2.40%, where 13,109 reflections were collected, 4288 are unique for P1 symmetry, and Rpsi is based on a mean data redundancy factor of > 3. Both the data merging and an |E2–1| value of 0.773 indicate that P1 is the correct space group. The general formula for the gadolinite–datolite group is W2XZ2T2O8V2 (Z = 2) which we have expanded to 20 anions (Z = 1) to show the W-site cation ordering present in ‘minasgeraisite-(Y)’. Bismuth, Ca and REE are ordered over four W sites, with Bi dominant at W1, Ca dominant at W2, and Y dominant at W3 and W4. The dominant constituent at the X sites is a vacancy, and Ca does not occur at the X sites. Significant B and Si are assigned to the Be-dominant Z sites, and the T sites are occupied by Si. The simplified ‘minasgeraisite-(Y)’ formula (Z = 1) is BiCa(Y,Ln)2(□,Mn)2(Be,B,Si)4Si4O16 [(OH),O]4. ‘Minasgeraisite-(Y)’ should be assigned to a triclinic subgroup of the gadolinite–datolite group, and its lower symmetry suggests that Ca-substituted gadolinites and hingganites should be examined for evidence of triclinic symmetry associated with cation order at the W sites.

A comparison of X-ray stress measurement methods based on the fundamental equation

Stress measurement methods using X-ray diffraction (XRD) methods are based on so-called fundamental equations. The fundamental equation is described in the coordinate system that best suits the measurement situation, and so making a comparison between different XRD methods is not straightforward. However, by using the diffraction vector representation, the fundamental equations of different methods become identical. Furthermore, the differences between the various XRD methods reside in the choice of diffraction vectors and the way of calculating the stress from the measured data. The stress calculation methods can also be unified using the general least-squares method, which is a common least-squares method of multivariate analysis. Thus, the only difference between these methods turns out to be in the choice of the set of diffraction vectors. In the light of these ideas, three commonly used XRD methods are compared: the sin2ψ method, the XRD2 method and the cosα method, using the estimation of the measurement errors. The XRD2 method with 33 frames (data acquisitions) shows the best accuracy. On the other hand, the accuracy of the cosα method with three frames is comparable to that of the XRD2 method.

Algorithms of Two-Dimensional X-Ray Diffraction

ABSTRACTX-ray diffraction pattern collected with two-dimensional detector contains the scattering intensity distribution as a function of two orthogonal angles. One is the Bragg angle 2θ and the other is the azimuthal angle about the incident x-ray beam, denoted by γ. A 2D diffraction pattern can be integrated to a conventional diffraction pattern and evaluated by most exiting software and algorithms for conventional applications, such as, phase identification, structure refinement and 2θ-profile analysis. However, the materials structure information associated to the intensity distribution along γ direction is lost through the integration. The diffraction vector approach has been approved to be the genuine theory in 2D data analysis. The unit diffraction vector used for 2D analysis is a function of both 2θ and γ. The unit diffraction vector for all the pixels in the 2D pattern can be expressed either in the laboratory coordinates or in the sample coordinates. The vector components can then be used to derive fundamental equations for many applications, including stress, texture, crystal orientation and crystal size evaluation.

Interferometric Diffraction from Amorphous Double Films

We explore the interference fringes that arise in diffraction patterns from double-layer amorphous samples where there is a substantial separation, up to about a micron, between two overlapping thin films. This interferometric diffraction geometry, where both waves have interacted with the specimen, reveals phase gradients within microdiffraction patterns. The rapid fading of the observed fringes as the magnitude of the diffraction vector increases confirms that displacement decoherence is strong in high-energy electron scattering from amorphous samples. The fading of fringes with increasing layer separation indicates an effective illumination coherence length of about 225 nm, which is consistent with the value of 270 nm expected for the heated Schottky field emitter source. A small reduction in measured coherence length is expected because of the additional energy spread induced in the beam after it passes through the first layer.

X-Ray Absorption Coefficient Behavior Depending on Disposition of Diffraction Vector and Temperature Gradient Vector

The behavior of the interference absorption coefficient of X-rays in Laue geometry depending upon the disposition of diffraction vector and temperature gradient vector in the perpendicular direction to the reflecting atomic planes family was experimentally studied. The study was carried out for the different thicknesses of quartz single crystal for atomic planes. It was shown that in the case of anti-parallel disposition of the diffraction vector and temperature gradient vector the absorption coefficient of X-rays sharply decreases with the increase of temperature gradient and in the case of the parallel disposition of the diffraction vector and the temperature gradient vector the absorption coefficient firstly increases and then decreases. The theoretical calculation corresponding to the experiment conditions have been done. The physical explanation of the obtained experimental results has been made. The obtained results are in good correspondence with the experiment.

Focusing of hard X-ray beams in quartz crystal under the temperature gradient

The research of the focusing and defocusing of the diffracted X-rays with different wave fronts in crystals under the influences of the temperature gradient and the ultrasonic vibrations is given in the works [1,2]. This work is dedicated experimentally investigation of Laue diffraction of the hard X-rays in quartz single crystals under the influence of the temperature gradient. It is shown that the reflected beams under the influence of the temperature gradient are focused. When the vector of the temperature gradient is anti parallel to the diffraction vector, in accordance to the increasing value of the temperature gradient the focus point continuously approaches the crystal. When the vector of the temperature gradient is parallel to the diffraction vector, in accordance with the increasing value of the temperature gradient the focus point continuously distances itself from the crystal, and at a certain value of the temperature gradient it becomes a plane wave. Parallel to the further increasing of the temperature gradient we see an imaginary focus, which continuously approaches the entry surface of the crystal. It is shown that by using a thicker single crystal it is possible to focus and pump a larger angular and spectral width of X-rays in the direction of diffraction.

Algorithms for Two-dimensional XRD Data Evaluation

The diffracted x-rays from a polycrystalline (powder) sample form a series diffraction cones in space since large numbers of crystals oriented randomly in the space are covered by the incident x-ray beam. Each diffraction cone corresponds to the diffraction from the same family of crystalline planes in all the participating grains. When a two-dimensional (2D) detector is used for x-ray powder diffraction, the diffraction cones are intercepted by the 2D detector and the x-ray intensity distribution on the sensing area is converted to an image-like diffraction pattern. The 2D pattern contains the scattering intensity distribution as a function of two orthogonal angles. One is the Bragg angle 2θ and the other is the azimuthal angle about the incident x-ray beam, denoted by γ. A 2D diffraction pattern can be analyzed directly or by data reduction to the intensity distribution along γ or 2θ. The γ-integration can reduce the 2D pattern into a diffraction profile analogs to the conventional diffraction pattern which is the diffraction intensity distribution as a function of 2θ angles. This kind of diffraction pattern can be evaluated by most exiting software and algorithms for conventional applications, such as, phase identification, structure refinement and 2θ-profile analysis. However, the materials structure information associated to the intensity distribution along γ direction is lost through γ-integration. The intensity distribution and 2θ variations along γ contain more information, such as the orientation distribution, strain states, crystallite size and shape distribution. In order to understand and analyze 2D diffraction data, new approaches and algorithms are necessary. The diffraction vector approach has been approved to be the genuine theory in 2D data analysis. The unit diffraction vector used for 2D analysis is a function of both 2θ and γ. The unit diffraction vector for all the pixels in the 2D pattern measured in the laboratory coordinates can be transformed to the sample coordinates. The vector components can then be used to derive fundamental equations for many applications, including stress, texture, crystal orientation and crystal size evaluation by γ-profile analysis. The unit diffraction vector is also used in polarization and absorption correction.

Influence of Chemical Heterogeneities on Line Profiles

The chemical composition fluctuation in a material may cause line broadening due to the variation of the lattice parameter, which yields a distribution of the profile centers scattered from different volumes of the material. The nature of line broadening induced by chemical heterogeneities is similar to a microstrain-like broadening in the sense that the peak width increases with the magnitude of the diffraction vector. However, the dependence of compositional broadening on the orientation of diffraction vector (i.e. the anisotropic nature of this effect) differs very much from other types of strain broadening (e.g. from that caused by dislocations). The anisotropic line broadening caused by composition fluctuation is parameterized for different crystal systems and incorporated into the evaluation procedures of peak profiles. This chapter shows that the composition probability distribution function can be determined from the moments of the experimental line profiles using the Edgeworth series. The concentration fluctuations in decomposed solid solutions can also be determined from the intensity distribution in the splitted diffraction peaks.

Dislocation Generated by Electron Irradiation

Dislocation loop was generated by electron irradiation in nickel aluminum alloy. It is important to know dislocation characteristics obtained from a high energetic electron irradiation. If b is the Burger vector of a dislocation loop and g is the diffraction vector, dislocation loop will appear larger, smaller or disappear for g.b>0, g.b<0 or g.b=0, respectively. Dislocation loop was determined as follows – first, the appearance of dislocation loops is arranged in observation table. Second, based on type of dislocation loop, Burger vector and diffraction vector, appearance of dislocation loop is arranged in calculation table. Third, based on observation and calculation table, Burger vector and type of dislocation loop is determined. The results show that dislocation loops consist of perfect dislocation loops and Frank dislocation loops. The perfect dislocation loops have Burger vectors of ½[0 ] and ½[ 0] while Frank dislocation loops have Burger vectors of ⅓[1 1], ⅓[11 ], ⅓[ 11], ⅓[111], ⅓[1 1], ⅓[11 ] and ⅓[ 11]. All dislocation loops are interstitial types.