Higher Bragg Angles Research Articles

Stress measurement in polymeric materials by X‐ray diffraction

AbstractExperiments are described which show that stresses in amorphous polymers can be measured by diffracting X‐rays at high Bragg angles from a filler consisting of crystalline particle or powders. The method is applicable, when calibrated, to both applied and residual stresses, to stress relaxation studies, to both tensile and compressive stresses, to both interior and surface positions in an object, and to composites of various types as well as to polymers that are substantially homogeneous. In the lower (Hookian) range, strains and stresses in metallic embedded particles increase linearly with applied stresses and strains in the matrix. When applied stresses exceed an apparent yield point, which correlates with the yield strength of the metallic filler, the elastic strains in the particles increase only slightly or even decrease as the matrix strains are increased, and with constant applied strain, the particles reveal changes due to relaxation. Strains in the particles are found to be smaller, and stresses higher, than in the matrix. Tests of graphite‐fiber epoxy composites with embedded filings of silver or aluminum alloy show that compressive residual stresses from curing at 350°C and cooling to room temperature are registered in the particles as well as stresses externally applied to the cured composite. Precision of the order of 1000 psi (0.7 kg/mm2 or 6.9 M Pa) was obtained with measurements made and evaluated rapidly and simply; with more refined techniques, much higher precision would be possible.

Intensitätsmessung bei ruhender Probe am Neutronen-Einkristalldiffraktometer

In neutron single-crystal diffractometry integrated intensities are usually measured by rotating the specimen through the reflecting position. It is possible, however, to produce a monochromatic beam with properties that allow intensity measurement with a fixed specimen. For this bent monochromators are especially suitable, but plane mosaic-crystal monochromators in antiparallel (+) position can also be used. The production, suitability and use of a plastically bent germanium monochromator is discussed. By means of a structure determination of the room-temperature phase of NH4Br, intensity measurements with a fixed specimen are compared with measurements of integrated intensities. With regard to accuracy, the methods are equivalent. With a bent germanium monochromator, the intensity measurement with a fixed specimen leads, especially in the range of high Bragg angles, to a reduction of the measuring time by a factor of at least two.

Two new experimental diffraction methods for a precise measurement of crystal perfection

Two instruments have been developed for the independent measurement of lattice strains, tilts and changes in the lattice constant using diffraction of X-rays at very high (89.5°) and of γ-rays at very low (0.5°) Bragg angels.The X-ray instrument consists of a double crystal diffractometer on which two crystals are arranged in the n: - n-position thus eliminating the high wavelength dispersion occuring at high Bragg angles. Wavelength from the continuous spectrum are used. By measuring both broadening and position of the reflection curve, lattice strains and changes in the lattice constant can be determined at the same time with an accuracy of 10−6 and 10−7 respectively which is limited only by the dynamical diffraction properties of the crystal.The γ-ray instrument represents a single crystal diffractometer. The 412 keV γ-radiation of a gold source (λ = 3.02 × 10−2 Å ΔΛ/Λ = 10−6) is collimated into a beam of a cross-section of 0.2x10 mm2 and an angular divergence of 10 seconds of arc. Because of the low Bragg angle only lattice tilts contribute to the broadening of the rocking curve. Theoretical linewidths calculated for a perfect crystal are of the order of magnitude of only 0.1 seconds. Consequently the sensitivity of detecting lattice defects is very high. However, to determine lattice tilts of less than 10′ the integrated reflection power has to be used. A further advantage of the short wavelength employed is the weak absorption. Therefore, a larger volume will be seen by the γ-rays and thick (up to 20 mm) crystal samples can be studied without cutting and surface preparation thus allowing a rapid control of crystal perfection, perhaps even during the process of crystal growth. Some typical reflection curves are given for both instruments.

Measurements of centroid and peak shifts due to dispersion and the Lorentz factor at very high Bragg angles

Differential response errors due to dispersion and the Lorentz factor have tan3 θ dependence for the powder diffractometer and should thus become critical for Bragg angles greater than ~ 155°, 2θ. The magnitude of these errors is demonstrated by measuring centroids and peaks at Bragg angles up to 170°, 2θ using Seeman–Bohlin geometry. Lattice parameters calculated from a single high angle centroid fall up to 0.0008 A below the values obtained by extrapolation using the data of many profiles. No significant deviation using peak positions was observed within the limits of experimental error. While these errors are considered to be significant for residual stress measurements they can be ignored for routine lattice parameter determinations.

A Small Bragg Diffraction X-Ray Spectrometer

A small, easily constructed, and high efficiency Bragg angle x-ray spectrometer has been developed. Employing either a lead stearate or a KAP crystal, it has been used to measure x rays in the 150–1000 eV region from ion-atom collisions. At the carbon-K energy (277 eV), it has an FWHM of 14 eV and an efficiency of 10−2 counts per incident x ray. Electronics and controls for fully automatic scans are described.

A contribution to the theory of x-ray monochromators with approximate focusing of Johann's type

Monochromating properties of x-ray monochromators can be described completely by the characteristic filter function ψ, for the determination of which it is necessary to know the important physical characteristics of the monochromating unit (the distribution of the emissivity in the focus and the reflection curve of the crystal) and the geometry of the experimental arrangement. The analytical form of this function is given for Johann's monochromators with approximate focusing, and the properties of these monochromators are demonstrated on a series of computations of the transmitted wavelength distribution for lower and higher Bragg angles θ. The general results of computations are summarized in several rules for the choice of parameters of monochromating units.

A Versatile Bragg-Brentano/Seeman-Bohlin Powder Diffractometer

An interchangeable Bragg-Brentano:Seeman-Bohlin diffractometer, with variable specimen setting, has been constructed. A study of the influence of the geometry on the resolution, angular displacement and intensity of diffraction profiles clearly demonstrates that the standard Bragg-Brentano method is to be preferred when recording Bragg reflections below 40°,2θ. At higher Bragg angles, which permit the specimen to be set further away from the X-ray source, it is demonstrated that the diffracted intensity of the Seeman-Bohlin method can be increased above that of the Bragg- Brentano because the use of an increased divergent angle does not reduce the resolution. A study of the effect of specimen setting on the systematic errors associated with the Seeman- Bohlin geometry indicates that a setting of 60° (half-angle from the X-ray source) gives optimum resolution with minimum displacement error.

ИЗМЕРЕНИЕ ИНТЕНСИВНОСТЕЙ ДИФРАКЦИЙ ПОЛИКРИСТАЛЛИЧЕСКИ Х ВЕЩЕСТВ, ОСОБЕННО У БОЛЬШИХ УГЛОВ БРЭГГА

The paper describes a simple experimental method for interplacing the background at diffractions having high Bragg angles, based on the use of single crystals of the same material.

A Note on Peak Displacements in X-ray Diffractometry

It is shown that at moderately high Bragg angles the shifts Δ(2θ) of the peaks of x-ray diffraction maxima resulting from geometrical aberrations of the apparatus are to a first approximation the same as those of the centroids. A second approximation to Δ(2θ) is obtained by adding Vh'/2h to the centroid shift, where V is the variance of the aberration and h and h' are the second and third derivatives of the observed line profile at its maximum. Forms of the correction suitable for practical application are discussed. The peak shifts resulting from dispersion and the Lorentz factor cancel to a first approximation. Physical aberrations, such as absorption, that multiply the line profile I(λ)dλ by a factor f(λ) depending on wavelength, shift the peak by approximately Δ(2θ)=-2f'Itanθ/λfI, where f, f', I, I are the values of the functions and their derivatives at the peak.

Absorption Correction in Precision Determination of Lattice Parameters

X-ray powder and rotation patterns of six cubic substances (W, Si, Cr, TiO, Al, and V) using specimens of different thickness were made; the patterns were measured, the lattice constants calculated and plotted against the θ angle or against the Nelson and Riley function. Such curves for thin mounts, semitransparent to x-rays (below 0.2 mm in diam), consist of two parts: a region of high slope (due to absorption) and one approaching a zero slope at high Bragg angles. The latter, varying in extent with the size and nature of the mount, indicates that due to the rapid decrease in absorption, the lines in this region are shifted, little if any, from their correct positions. Hence, lattice constants calculated from the reflections of the high θ angles lying on the horizontal portion of the curve need not be corrected for absorption. The constant, with negligible error, can be calculated from a single last strong α1 or β reflection. If an extrapolation is applied, the curve must be traced through points in the high back reflection region. Otherwise any extrapolation, disregarding the horizontal part, would lead to lattice constants too high in value. The amount of line displacement due to absorption in thin samples (below 0.2 mm in diam) agrees with the geometrical concept of Hadding and Buerger.

The effect of variations in ambient temperature upon the optical alinement of an X-ray fluorescence spectrometer

During proving tests made on an X-ray fluorescence spectrometer, it was found that variations in ambient temperature produced a significant effect upon the dispersion of the instrument. Experimental evidence indicated that this was due to changes in the interplanar spacing of the diffracting crystal. The magnitude of the effect is shown to be in good agreement with theoretical considerations. Since the angular dispersion is an inverse function of d cos θ, the effect is most marked at high Bragg angles. The significance of the observed changes in angular dispersion, in relation to the accuracy of analysis, is pointed out.

Accurate determination of the cell dimensions of magnetite

SummaryAn analysed magnetite from Bisperg, Säter, Dalecarlia, Sweden, gave the formula, which corresponds closely to the ideal composition. A precise determination of the cell dimensions gavea8·3963±0·0005 Å. at 18° C. Gentle heating of the powdered magnetite in evacuated silica glass tubes, before X-ray examination, resulted in sharply defined lines with high Bragg angles; an improvement which enabled precision data to be obtained. Four other new analyses of magnetites are also given and their lattice parameters are determined in the same way and are found to range from 8·3960 Å. to 8·3970 Å. The Bisperg magnetite being the purest sample examined, the value of its cell edge is taken as representative of pure magnetite.The effect of the different ionic substitutions on the cell dimensions of natural magnetites is discussed. An attempt is also made to explain the great variations among the published values of cell dimensions of artificial preparations; one main reason being the presence of defect structures with varying oxygen contents in excess of the formula requirements.

X-ray line broadening from filed aluminium and wolfram

Methods of analysis previously used in the interpretation of line broadening are discussed and are shown to be inadequate; more reliable methods being outlined. An analysis of published results using one of these methods suggests that the observed effects can be attributed to simultaneous small particle size and strain broadening. Measurements of the changes in intensity distribution have been made, using a Geiger counter spectrometer, in the spectra of cold worked aluminium and wolfram. The line breadths may be attributed to simultaneous small particle size and strain broadening, the latter predominating, particularly at the higher Bragg angles, and it is shown that the observed effects are produced by dislocations or some similar structural fault. The observed rise in the breadths of the high angle lines from annealed materials suggests that some dislocations remain after annealing. Fourier analysis of the line shapes in general merely confirm the results of the analysis of the line breadths, but in the case of the recovered specimens it suggests that the dislocations form into walls (“polygonization”).

A Rapid X-ray Method for the Determination of the Orientation of Single Crystals or Large-grained Polycrystalline Aggregates

A type of goniometer is described which enables a crystal to be set in a given orientation. The method makes use of strong, high Bragg angle, reflexions which can be received on a fluorescent screen. The orientation of any given face of the crystal can readily be calculated from the readings of two angle scales on the instrument, and can be plotted on a stereographic projection. The calculations necessary for plotting the crystal face are given in an Appendix.