The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals

2018 ◽  
Vol 74 (5) ◽  
pp. 586-594 ◽  
Author(s):  
Takashi Saka
Keyword(s):  
Riemann Surface ◽  
Analytical Method ◽  
Bragg Condition ◽  
Approximation Characteristic ◽  
Dynamical Diffraction ◽  
X Ray ◽  
The Real ◽  
Dispersion Surface ◽  
Beam Approximation ◽  
Characteristic Features

The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals is analysed using a Riemann surface. In the conventional two-beam approximation, each branch or wing of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities that specify the degree of departure from the exact Bragg condition and the reflection strength. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters with no approximation. Characteristic features of the dispersion surface are also revealed by geometrical considerations with respect to the Riemann surface.

Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface

2016 ◽  
Vol 72 (3) ◽  
pp. 338-348 ◽  
Author(s):  
Takashi Saka
Keyword(s):  
Riemann Surface ◽  
Complex Analysis ◽  
Dynamical Theory ◽  
Bragg Condition ◽  
X Ray Diffraction ◽  
X Ray ◽  
The Real ◽  
Dispersion Surface ◽  
Complex Planes ◽  
Characteristic Features

The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.

Properties of X-ray resonant scattering in the Bragg case revealed on the Riemann surface

2016 ◽  
Vol 72 (4) ◽  
pp. 472-479 ◽  
Author(s):  
Takashi Saka
Keyword(s):  
Riemann Surface ◽  
Numerical Analysis ◽  
Strength Characteristic ◽  
Resonant Scattering ◽  
Dynamical Theory ◽  
Acta Cryst ◽  
Rocking Curve ◽  
X Ray ◽  
Dispersion Surface ◽  
Characteristic Features

Continuing the work described in the previous paper [Saka (2016).Acta Cryst.A72, 338–348], the dynamical theory for perfect crystals in the Bragg case is reformulated using the Riemann surface. In particular, diffraction under resonant scattering conditions is investigated. The characteristic features of the dispersion surface and the rocking curve are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg conditions and the reflection strength. Characteristic properties that have been deduced through numerical analysis are derived analytically using these four parameters. Visualization of the geometric relationships between the four parameters on the Riemann surface is useful for understanding many properties such as symmetry and sharpness of the rocking curve under special conditions. Therefore, employing the Riemann surface is instructive for numerical analysis and useful for understanding dynamical diffraction in the Bragg case.

Phase of Pendellösung oscillations in X-ray dynamical diffraction for perfect crystals

2020 ◽  
Vol 76 (2) ◽  
pp. 132-136
Author(s):  
Takashi Saka
Keyword(s):  
Phase Shift ◽  
Perfect Crystal ◽  
Dynamical Diffraction ◽  
X Ray ◽  
The Real ◽  
Diffracted Waves

Formulations are given for the intensities of transmitted and diffracted waves in the Laue case of a perfect crystal. This is applicable irrespective of the magnitudes of both the real and imaginary parts of the Fourier components of the crystal polarizability. The phase shift of the Pendellösung oscillations of the transmitted and diffracted waves is analyzed in detail for a symmetrical Laue case. The phase is determined to shift continuously from out of phase to in phase for absorbing crystals.

Bulk standards for low-temperature biological x-ray microanalysis

1984 ◽  
Vol 42 ◽  
pp. 282-283
Author(s):  
P. Echlin ◽  
M. McKoon ◽  
E.S. Taylor ◽  
C.E. Thomas ◽  
K.L. Maloney ◽  
...  
Keyword(s):  
Phase Separation ◽  
Low Temperature ◽  
Analytical Method ◽  
Salt Solutions ◽  
Absolute Concentration ◽  
Cooling Process ◽  
X Ray ◽  
The Absolute ◽  
Analytical Procedures ◽  
Electrical Conductors

Although sections of frozen salt solutions have been used as standards for x-ray microanalysis, such solutions are less useful when analysed in the bulk form. They are poor thermal and electrical conductors and severe phase separation occurs during the cooling process. Following a suggestion by Whitecross et al we have made up a series of salt solutions containing a small amount of graphite to improve the sample conductivity. In addition, we have incorporated a polymer to ensure the formation of microcrystalline ice and a consequent homogenity of salt dispersion within the frozen matrix. The mixtures have been used to standardize the analytical procedures applied to frozen hydrated bulk specimens based on the peak/background analytical method and to measure the absolute concentration of elements in developing roots.

A Calculation real abundance of gaseous elements of the comet PanSTARRS C/2011 S4

2020 ◽  
Vol 18 (45) ◽  
pp. 21-31
Author(s):  
Salman Zaidan Khalaf ◽  
Khaleel Abrahim ◽  
Imad Kassar Akeab
Keyword(s):  
Solar Wind ◽  
Physical Properties ◽  
Atomic Number ◽  
Mathematic Model ◽  
Emission Energy ◽  
X Ray ◽  
The Real

    X-ray emission contains some of the gaseous properties is produced when the particles of the solar wind strike the atmosphere of comet ISON and PanSTARRS Comets. The data collected with NASA Chandra X-ray Observatory of the two comets, C/2012 S1 (also known as Comet ISON) and C/2011 S4 (Comet PanSTARRS) are used in this study.    The real abundance of the observed X-ray spectrum elements has been extracted by a new simple mathematic model. The study found some physical properties of these elements in the comet’s gas such as a relationship between the abundance with emitted energy. The elements that have emission energy (2500-6800) eV, have abundance (0.1-0.15) %, while the elements that have emission energy (850-2500) eV and (6800-9250) eV have abundance (0.2-0.3) %.    The relation between interacted energy and atomic number is form two sets.  The interacted energy of each element is increased as the atomic number increased. This case has been seen in both comets

How to Use the Features of Total Reflection of X-Rays for Energy Dispersive XRF

1988 ◽  
Vol 32 ◽  
pp. 105-114 ◽  
Author(s):  
H. Schwenke ◽  
W. Berneike ◽  
J. Knoth ◽  
U. Weisbrod
Keyword(s):  
Thin Films ◽  
Penetration Depth ◽  
Fluorescence Analysis ◽  
Total Reflection ◽  
X Rays ◽  
X Ray ◽  
Characteristic Features ◽  
Respective Characteristic ◽  
Nanometer Regime ◽  
Sensitive Method

AbstractThe total reflection of X-rays is mainly determined by three parameters , that is the orltical angle, the reflectivity and the penetration depth. For X-ray fluorescence analysis the respective characteristic features can be exploited in two rather different fields of application. In the analysis of trace elements in samples placed as thin films on optical flats, detection limits as low as 2 pg or 0.05 ppb, respectively, have been obtained. In addition, a penetration depth in the nanometer regime renders Total Reflection XRF an inherently sensitive method for the elemental analysis of surfaces. This paper outlines the main physical and constructional parameters for instrumental design and quantitation in both branches of TXRF.

Calcium Carbodiimide Compounds Revisited – Syntheses, Single Crystal Structure Determination and Vibrational Spectra of Ca11N6[CN2]2, Ca4N2[CN2] and Ca[CN2]

2008 ◽  
Vol 63 (5) ◽  
pp. 530-536 ◽  
Author(s):  
Olaf Reckeweg ◽  
Francis J. DiSalvo
Keyword(s):  
Crystal Structure ◽  
Raman Spectrum ◽  
Single Crystal ◽  
Single Crystals ◽  
Vibrational Spectra ◽  
Structure Determination ◽  
X Ray ◽  
Characteristic Features ◽  
Reported Data ◽  
Correct Data

Single crystals of Ca11N6[CN2]2 (dark red needles, tetragonal, P42/mnm (no. 136), a = 1456.22(5), and c = 361.86(2) pm, Z = 2), Ca4N2[CN2] (transparent yellow needles, orthorhombic, Pnma (no. 62), a = 1146.51(11), b = 358.33(4), and c = 1385.77(13) pm, Z = 4) and Ca[CN2] (transparent, colorless, triangular plates, rhombohedral, R3̅m (no. 166), a = 369.00(3), and c = 1477.5(3) pm, Z = 3) were obtained by the reaction of Na2[CN2], CaCl2 and Ca3N2 (if demanded by stoichiometry) in arc-welded Ta ampoules at temperatures between 1200 - 1400 K. Their crystal structures were re-determined by means of single crystal X-ray structure analyses. Additionally, the Raman spectra were recorded on these same single crystals, whereas the IR spectra were obtained with the KBr pellet technique. The title compounds exhibit characteristic features for carbodiimide units with D∞h symmetry (d(C-N) = 121.7 - 123.8 pm and ∡ (N-C-N) = 180°). The vibrational frequencies of these units are in the expected range (Ca11N6[CN2]2: νs = 1230, νs = 2008; δ = 673/645/624 cm−1; Ca4N2[CN2]: νs = 1230, νs = 1986; δ = 672/647 cm−1; Ca[CN2]: νs = 1274, νs = 2031, δ = 668 cm−1). The structural results are more precise than the previously reported data, and with the newly attained Raman spectrum of Ca11N6[CN2]2 we correct data reported earlier.

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