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Bragg angle

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Schematic illustration of electron channelling effects over a Bragg angle on the domain A with a small positive value of deviation parameter (s), s > 0, “channelling condition” and the domain B a small negative value, s < 0, “Backscattering condition” at a subgrain boundary (e.g., Figs. 9 and 14) and the corresponding contrast variations of BSE through the angle variation of domains. Note: The inversion of contrast with the tilt is at the Bragg angle.

Schematic illustration of electron channelling effects over a Bragg angle o... Available to Purchase

Published: 01 January 2018
Fig. 10 Schematic illustration of electron channelling effects over a Bragg angle on the domain A with a small positive value of deviation parameter ( s ), s  > 0, “channelling condition” and the domain B a small negative value, s  < 0, “Backscattering condition” at a subgrain boundary
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The relationship between Bragg angle, 𝛉, and scattering vector, Q, for diffraction from a lattice. Q may be defined as the difference between the wave vectors of the incoming and scattered rays, ki and ks.

The relationship between Bragg angle, 𝛉, and scattering vector, Q , for d... Available to Purchase

Published: 01 January 2006
Figure 4. The relationship between Bragg angle, 𝛉, and scattering vector, Q , for diffraction from a lattice. Q may be defined as the difference between the wave vectors of the incoming and scattered rays, k i and k s .
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(a) Evolution of the FeLα peak maximum (expressed as the Bragg angle sinθ measured with a TAP monochromator) as a function of the beam energy, for an almandine (pure Fe2+-bearing garnet) and an andradite (pure Fe3+-bearing garnet). The beam energy was varied from 2.5 to 15 keV. The self absorption increases with increasing beam voltage, which shifts the FeLα peak toward longer sinθ (i.e., lower energies). This phenomenon is particularly visible for Fe2+. The values extrapolated (by second-order polynomial fits) to the position of the Fe LIII edge give an estimation of the actual (i.e., the self-absorption free) peak positions. The actual Fe2+Lα peak position is found to be shifted by 0.21 eV toward lower energies compared to Fe3+Lα. (b) 15 keV wavelength scan profiles showing the shift in maximum position (≈1.2eV) between the almandine Fe2+Lα peak and the andradite Fe3+Lα peak.

( a ) Evolution of the Fe L α peak maximum (expressed as the Bragg angle si... Available to Purchase

Published: 01 April 2004
F igure 1. ( a ) Evolution of the Fe L α peak maximum (expressed as the Bragg angle sinθ measured with a TAP monochromator) as a function of the beam energy, for an almandine (pure Fe 2+ -bearing garnet) and an andradite (pure Fe 3+ -bearing garnet). The beam energy was varied from 2.5 to 15
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Background-subtracted energy dispersive X-ray spectra for two non-Bragg diffraction orientations around exact (111) reflection of the heated orthoclase. The two spectra in different shades are slightly displaced along the horizontal axis with respect to each other for clarity. Gray: The reflection angle is about 5 mrad smaller than the Bragg angle (Fig. 2a). Black: The reflection angle is about 5 mrad larger than the Bragg angle (Fig. 2b).

Background-subtracted energy dispersive X-ray spectra for two non-Bragg dif... Available to Purchase

Published: 01 January 2010
: The reflection angle is about 5 mrad smaller than the Bragg angle (Fig. 2a ). Black: The reflection angle is about 5 mrad larger than the Bragg angle (Fig. 2b ).
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The effect of mis-centering a crystal on a diffractometer. Diffraction from the same plane in a crystal always generates a diffracted beam at twice the Bragg angle (i.e., 2θ) to the incident beam irrespective of the position of the sample crystal. If the crystal is centered the correct Bragg angle is determined as θ=12arctan⁡(hcdd), where hc is the distance of the spot on the detector from the direct beam, and dd is the detector distance from the center of the diffractometer. The diffracted beam from a crystal displaced towards the X-ray source hits the detector at a greater distance ho from the direct beam. This position is then interpreted as diffraction coming from the center of the diffractometer at the correct distance (dd) from the detector (dashed line) and therefore as a larger incorrect Bragg angle θoff=12arctan⁡(hodd).

The effect of mis-centering a crystal on a diffractometer. Diffraction from... Open Access

Published: 01 July 2022
Figure 2. The effect of mis-centering a crystal on a diffractometer. Diffraction from the same plane in a crystal always generates a diffracted beam at twice the Bragg angle (i.e., 2θ) to the incident beam irrespective of the position of the sample crystal. If the crystal is centered the correct
Journal Article

Quantitative determination of the shock stage of L6 ordinary chondrites using X-ray diffraction Available to Purchase

Naoya Imae, Makoto Kimura
Published: 01 September 2021
American Mineralogist (2021) 106 (9): 1470–1479.
DOI: https://doi.org/10.2138/am-2021-7554
... from the analyses based on the Williamson–Hall plots, which depict the tangent Bragg angle and integral breadth β. The lattice strain in olivine, e Ol , is distributed from ~0.05% to ~0.25%, while that in orthopyroxene, e Opx , is distributed from ~0.1 to ~0.4%, where we selected the isolated peaks...
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Absolute quantification by powder X-ray diffraction of complex mixtures of crystalline and amorphous phases for applications in the Earth sciences Available to Purchase

Denis G. Rancourt, Mei-Zhen Dang
Published: 01 October 2005
American Mineralogist (2005) 90 (10): 1571–1586.
DOI: https://doi.org/10.2138/am.2005.1794
... ( A ⊥ ) of the incident (and outgoing) beam and the effective incident beam intensity ( I o ), including counter efficiency, beam path losses, etc. The problem of incomplete collection sphere integration (including the q = 0 region) is resolved by showing that all the results hold for a given Bragg angle range...
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Journal Article

A method for taking back reflection oscillation photographs of single crystals Available to Purchase

L. G. Berry, S. V. L. N. Rao
Published: 01 August 1965
American Mineralogist (1965) 50 (7-8): 1118–1121.
...L. G. Berry; S. V. L. N. Rao Abstract In x-ray diffraction studies “back reflection” geometry is utilized in powder cameras (Cullity, 1956; Azaroff and Buerger, 1958) and in Weis-senberg cameras (Buerger, 1937) for the accurate measurement of high Bragg angles, permitting the determination...
View PDF for article title, A method for taking back reflection oscillation photographs of single crystals
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Journal Article

A modified graphical method of absorption correction for the displacement of lines in a powder photograph Available to Purchase

K. Viswanathan
Published: 01 October 1964
American Mineralogist (1964) 49 (9-10): 1474–1480.
... silicate minerals, such as garnets, pyroxenes and amphiboles, have fairly large linear absorption coefficients, μ, and hence can be called, following Peiser et al. (1955), “strongly absorbing specimens.” Because of strong absorption, these substances give rise to errors in the measured Bragg angles owing...
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Journal Article

The absorption and refraction corrections and the lattice constant of chromium Available to Purchase

M. E. Straumanis, C. C. Weng
Published: 01 June 1956
American Mineralogist (1956) 41 (5-6): 437–448.
...M. E. Straumanis; C. C. Weng Abstract It has been shown by using x -radiations from a Cu and a Cr target that the absorption correction can be neglected at Bragg angles larger than 76°, if the powder mounts are thin enough (0.12 to 0.2 mm. in diameter), using a Lindemann glass hair as a core...
View PDF for article title, The absorption and refraction corrections and the lattice constant of chromium
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Precision lattice measurements of galena Available to Purchase

B. Wasserstein
Published: 01 February 1951
American Mineralogist (1951) 36 (1-2): 102–115.
... distortion which, however, is removed by gentle heating. The resultant sharp lines with very large Bragg angles enabled precision data to be obtained for the first time from x-ray powder photographs of galena. Such data made it clear that bismuth can substitute for lead causing a shrinkage of the unit cell...
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Strong variation of the channelling contrasts in a subgrain boundary of the Finero olivine tilting the specimen over the Bragg angle of the (004) Kikuchi band. (a) The stage tilt is at −3.0°. (b) The schematic drawing illustrates that BSE signal intensity changes due to a small angle gap (less than 1°, confirmed by using EBSD measurements, Supplementary material, Fig. S7a) over the Bragg angle of g = 004, which is addressed at the subgrain boundary having an orientation change by dislocation array. (c) The stage tilt is at −2.5°.

Strong variation of the channelling contrasts in a subgrain boundary of the... Available to Purchase

Published: 01 January 2018
Fig. 14 Strong variation of the channelling contrasts in a subgrain boundary of the Finero olivine tilting the specimen over the Bragg angle of the (004) Kikuchi band. (a) The stage tilt is at −3.0°. (b) The schematic drawing illustrates that BSE signal intensity changes due to a small angle gap
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Evolution of the powder XRD pattern during heating of butlerite from 20 to 400 °C in vacuum, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). Note that the sample amorphizes at temperatures between 200 and 260 °C. At higher temperature, the phase Fe(SO4)(OH) crystallizes. Two strong peaks between 43° and 45° 2θ and several weaker peaks at higher Bragg angles originate from diffraction from the holder material.

Evolution of the powder XRD pattern during heating of butlerite from 20 to ... Available to Purchase

Published: 01 March 2018
Fig. 1 Evolution of the powder XRD pattern during heating of butlerite from 20 to 400 °C in vacuum, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). Note that the sample amorphizes at temperatures between 200 and 260 °C. At higher
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Evolution of the powder XRD pattern during heating of parabutlerite from 20 to 400 °C in vacuum, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). Note that the sample amorphizes at temperatures around 260 °C. At higher temperature, the phase Fe(SO4)(OH) crystallizes. Two strong peaks between 43° and 45° 2θ and several weaker peaks at higher Bragg angles originate from diffraction from the holder material.

Evolution of the powder XRD pattern during heating of parabutlerite from 20... Available to Purchase

Published: 01 March 2018
Fig. 2 Evolution of the powder XRD pattern during heating of parabutlerite from 20 to 400 °C in vacuum, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). Note that the sample amorphizes at temperatures around 260 °C. At higher temperature
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Evolution of the powder XRD pattern during heating of amarantite from 20 to 300 °C in air, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). The sample persists to 80 °C and then transforms to metahohmannite that amorphizes and slowly transforms to poorly crystalline Fe(SO4)(OH). Two strong peaks between 43° and 45° 2θ and several weaker peaks at higher Bragg angles originate from diffraction from the holder material.

Evolution of the powder XRD pattern during heating of amarantite from 20 to... Available to Purchase

Published: 01 March 2018
Fig. 3 Evolution of the powder XRD pattern during heating of amarantite from 20 to 300 °C in air, with the intensities at each Bragg angle encoded by colors (blue = low intensities, red = high intensities). The sample persists to 80 °C and then transforms to metahohmannite that amorphizes
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Origin of diffraction peak broadening in the kinematical approximation. (a) Large crystal with orientation just off the Bragg angle. For each different plane there is a slight phase shift in the scattered radiation. For some plane well below the top of the crystal surface there is a plane scattering out of phase with it. Hence for large crystals the reflection peak is quite sharp. (b) For small crystals scattering from the top and bottom-most planes do not create such a large phase shift when the crystal is set off the Bragg angle. Hence the range of angles where the scattered radiation is cancelled is reduced and the reflection broadens. From Schultz (1982) with permission of Prentice-Hall.

Origin of diffraction peak broadening in the kinematical approximation. (a)... Available to Purchase

Published: 01 January 2001
Figure 19. Origin of diffraction peak broadening in the kinematical approximation. (a) Large crystal with orientation just off the Bragg angle. For each different plane there is a slight phase shift in the scattered radiation. For some plane well below the top of the crystal surface
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The experimental high-resolution spectroscopic setup at FAME-UHD beamline used in this study. The autoclave is tilted by ~10 ° from the vertical position to match the required Bragg angle of the crystal analyzers to selectively probe the AuLα1 fluorescence line.

The experimental high-resolution spectroscopic setup at FAME-UHD beamline u... Open Access

Published: 01 March 2022
Figure 2. The experimental high-resolution spectroscopic setup at FAME-UHD beamline used in this study. The autoclave is tilted by ~10 ° from the vertical position to match the required Bragg angle of the crystal analyzers to selectively probe the Au L α 1 fluorescence line.
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Result of Rietveld analysis for kozoite-(Nd). Observed and calculated diffraction profiles (top) are given as plus signs and as a line, respectively. Short bars indicate Bragg angles, and the pattern at the bottom indicates the difference between the observed and calculated intensities.

Result of Rietveld analysis for kozoite-(Nd). Observed and calculated diffr... Available to Purchase

Published: 01 July 2000
F igure 3. Result of Rietveld analysis for kozoite-(Nd). Observed and calculated diffraction profiles (top) are given as plus signs and as a line, respectively. Short bars indicate Bragg angles, and the pattern at the bottom indicates the difference between the observed and calculated
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The FWHM values from pseudo-Voigt fitting: garnet solid solutions with intermediate compositions show much broader X-ray peaks than the two end-members pyrope and grossular. XRD peak broadening changes linearly with Bragg angle over this limited range of 2θ.

The FWHM values from pseudo-Voigt fitting: garnet solid solutions with inte... Available to Purchase

Published: 01 January 2016
Figure 3 The FWHM values from pseudo-Voigt fitting: garnet solid solutions with intermediate compositions show much broader X-ray peaks than the two end-members pyrope and grossular. XRD peak broadening changes linearly with Bragg angle over this limited range of 2θ.
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a) X-ray fluorescence spectra and two-dimensional μXRD patterns (λ= 1.968 Å) from four selected points-of-interest for the nodule presented in Figure 30. b) μSXRD (negative contrast) maps of lithiophorite and goethite. c) One-dimensional μXRD patterns obtained by integrating intensities of 2D patterns at constant Bragg angle.

a) X-ray fluorescence spectra and two-dimensional μXRD patterns (λ= 1.968 A... Available to Purchase

Published: 01 January 2002
intensities of 2D patterns at constant Bragg angle.