A common theory of line broadening and rocking curves

2015 ◽  
Vol 48 (2) ◽  
pp. 418-430 ◽  
Author(s):  
Gyula Zilahi ◽  
Tamás Ungár ◽  
Géza Tichy
Keyword(s):  
Rigid Body ◽  
Electron Backscatter Diffraction ◽  
Strain Tensor ◽  
Theoretical Description ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
Rocking Curve ◽  
X Ray ◽  
Peak Broadening ◽  
Individual Grains

X-ray diffraction peak broadening is discussed in terms of line broadening and rocking-curve broadening in a novel theoretical description. The nonlocal strain tensor is factorized by using the method of polar decomposition instead of the more conventional separation into symmetrical and antisymmetrical components. A number of X-ray line-broadening and rocking-curve experiments on the same single crystals or individual grains in bulk polycrystals prove that plastic deformation produces strained subgrains mutually rotated by rigid-body rotations. The novel theoretical description appropriately accounts for the rigid-body rotation and strain at the same time and provides straightforward separation of the two effects of line and rocking-curve broadening in the radial and normal directions of the diffraction vector. The mathematical results are discussed in terms of experiments of X-ray diffraction, Laue asterism and electron backscatter diffraction. From the experimental results it is shown that the simultaneous evaluation of line and rocking-curve broadening provides qualitative information about the redundant and geometrically necessary character of dislocations, not available if only one or the other is accessible.

Submicron Resolution X-Ray Strain Measurements On Patterned Films: Some Hows And Whys

1996 ◽  
Vol 428 ◽  
Author(s):  
M. A. Marcus ◽  
A. A. Macdowell ◽  
E. D. Isaacs ◽  
K. Evans-Lutterodt ◽  
G. E. Ice
Keyword(s):  
Mechanical Behavior ◽  
Strain Tensor ◽  
Energy Measurement ◽  
X Ray Diffraction ◽  
Strain Measurements ◽  
X Ray ◽  
Crystal Perfection ◽  
Patterned Films ◽  
Individual Grains ◽  
Interconnect Technology

AbstractAs interconnect technology moves deeper into the submicron realm, the physics governing the mechanical behavior, hence the reliability of metal lines changes. Events at the level of individual grains become more important. One would like to be able to measure the strain and orientation of single grains, and the distribution of strain across a grain. Such data may help us understand such effects as sunken grains, the erratic occurrence of stress voiding and Blechlength effects in electromigration. To get this information, techniques involving X-ray diffraction with submicron beams are being developed. These experiments involve detecting Laue spots from selected areas and energy-analyzing them to find the strain tensor. Motivations for such measurements, the specifications for a useful instrument, some designs now being developed and the intrinsic limitations of these approaches will be discussed. Design issues in the choice of focussing optics and energy measurement will be highlighted with calculations of how the Laue spots would look for different conditions of crystal perfection.

scholarly journals Grain size dependent high-pressure elastic properties of ultrafine micro/nanocrystalline grossular

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jin S. Zhang ◽  
T. Irifune ◽  
M. Hao ◽  
D. Zhang ◽  
Y. Hu ◽  
...  
Keyword(s):  
Grain Size ◽  
Elastic Properties ◽  
Elastic Behavior ◽  
Ambient Conditions ◽  
X Ray Diffraction ◽  
X Ray ◽  
Inhomogeneous Strain ◽  
Peak Broadening ◽  
Volume Measurements ◽  
Individual Grains

AbstractWe have performed sound velocity and unit cell volume measurements of three synthetic, ultrafine micro/nanocrystalline grossular samples up to 50 GPa using Brillouin spectroscopy and synchrotron X-ray diffraction. The samples are characterized by average grain sizes of 90 nm, 93 nm and 179 nm (hereinafter referred to as samples Gr90, Gr93, and Gr179, respectively). The experimentally determined sound velocities and elastic properties of Gr179 sample are comparable with previous measurements, but slightly higher than those of Gr90 and Gr93 under ambient conditions. However, the differences diminish with increasing pressure, and the velocity crossover eventually takes place at approximately 20–30 GPa. The X-ray diffraction peaks of the ultrafine micro/nanocrystalline grossular samples significantly broaden between 15–40 GPa, especially for Gr179. The velocity or elasticity crossover observed at pressures over 30 GPa might be explained by different grain size reduction and/or inhomogeneous strain within the individual grains for the three grossular samples, which is supported by both the pressure-induced peak broadening observed in the X-ray diffraction experiments and transmission electron microscopy observations. The elastic behavior of ultrafine micro/nanocrystalline silicates, in this case, grossular, is both grain size and pressure dependent.

Strain Mapping in InGaAsP Epitaxial Films by An X-Ray Diffraction Technique

1985 ◽  
Vol 54 ◽  
Author(s):  
Jharna Chaudhuri ◽  
William E. Mayo ◽  
Sigmund Weissmann
Keyword(s):  
Electronic Materials ◽  
Strain Tensor ◽  
Diffraction Method ◽  
Epitaxial Films ◽  
X Ray Diffraction ◽  
Strain Mapping ◽  
Rocking Curve ◽  
X Ray ◽  
Inp Substrate ◽  
Elastic Strain Tensor

ABSTRACTA new x-ray diffraction method is developed to determine the full elastic strain tensor and its distribution about a strain center in single crystal materials. It is based on the recently developed Computer Aided Rocking Curve Analyzer and is particularly well suited for analysis of thin film structures common to electronic materials. This technique will be described in detail, and its application in measuring the non-uniform strains in InGaAsP epitaxial film on InP substrate will be presented. Also, possibility of using this method to measure the uniformity of film thickness will be discussed.

A Survey of Diverse Approximations for Microstructural Characterization using Powder Diffraction Data: β-Ni(OH)2, a Case Study

2006 ◽  
Vol 972 ◽  
Author(s):  
Monste Casas-Cabanas ◽  
Juan Rodriguez-Carvajal ◽  
Judith Oro-Sole ◽  
M. Rosa Palacin
Keyword(s):  
Rietveld Refinement ◽  
Microstructural Characterization ◽  
Diffraction Peak ◽  
Powder Diffraction Data ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
X Ray ◽  
Peak Broadening ◽  
Transmission Electron

AbstractThe microstructural features of diverse samples are studied by powder X-ray diffraction using different methods for the analysis of the diffraction peak broadening. The results obtained are thoroughly analyzed taking into account the assumptions and simplifications done in each of the chosen methods (Scherrer, Stokes and Wilson, Williamson-Hall and Rietveld refinement) and direct observation of the studied specimens by transmission electron microscopy is used in order to contrast the results obtained. Classic simple methods that consider only size effects (Scherrer, Stokes and Wilson) or the combined effects of size and strains (Williamson-Hall) provide a rapid overview of the origins of line-broadening but the most reliable results are obtained when the whole diffraction pattern is taken into account (Rietveld refinement).

X-ray diffraction properties of curved crystal diffractors

1992 ◽  
Vol 50 (2) ◽  
pp. 1734-1735
Author(s):  
W. Z. Chang ◽  
D. B. Wittry
Keyword(s):  
Diffraction Theory ◽  
X Rays ◽  
Diffraction Image ◽  
X Ray Diffraction ◽  
Rocking Curve ◽  
X Ray ◽  
Bent Crystal ◽  
Diffraction Geometry ◽  
Crystal Parameter ◽  
Quantitative Procedure

Since Du Mond and Kirkpatrick first discussed the principle of a bent crystal spectrograph in 1930, curved single crystals have been widely utilized as spectrometric monochromators as well as diffractors for focusing x rays diverging from a point. Curved crystal diffraction theory predicts that the diffraction parameters - the rocking curve width w, and the peak reflection coefficient r of curved crystals will certainly deviate from those of their flat form. Due to a lack of curved crystal parameter data in current literature and the need for optimizing the choice of diffraction geometry and crystal materials for various applications, we have continued the investigation of our technique presented at the last conference. In the present abstract, we describe a more rigorous and quantitative procedure for measuring the parameters of curved crystals.The diffraction image of a singly bent crystal under study can be obtained by using the Johann geometry with an x-ray point source.

scholarly journals Stable and Metastable Phase Equilibria Involving the Cu6Sn5 Intermetallic

2021 ◽  
Author(s):  
A. Leineweber ◽  
M. Löffler ◽  
S. Martin
Keyword(s):  
Phase Equilibria ◽  
Electron Backscatter Diffraction ◽  
Metastable Phase ◽  
Stable Phase ◽  
X Ray Diffraction ◽  
X Ray ◽  
Heat Treated ◽  
Ordered Phases ◽  
Backscatter Diffraction ◽  
Kinetics Of

Abstract Cu6Sn5 intermetallic occurs in the form of differently ordered phases η, η′ and η′′. In solder joints, this intermetallic can undergo changes in composition and the state of order without or while interacting with excess Cu and excess Sn in the system, potentially giving rise to detrimental changes in the mechanical properties of the solder. In order to study such processes in fundamental detail and to get more detailed information about the metastable and stable phase equilibria, model alloys consisting of Cu3Sn + Cu6Sn5 as well as Cu6Sn5 + Sn-rich melt were heat treated. Powder x-ray diffraction and scanning electron microscopy supplemented by electron backscatter diffraction were used to investigate the structural and microstructural changes. It was shown that Sn-poor η can increase its Sn content by Cu3Sn precipitation at grain boundaries or by uptake of Sn from the Sn-rich melt. From the kinetics of the former process at 513 K and the grain size of the η phase, we obtained an interdiffusion coefficient in η of (3 ± 1) × 10−16 m2 s−1. Comparison of this value with literature data implies that this value reflects pure volume (inter)diffusion, while Cu6Sn5 growth at low temperature is typically strongly influenced by grain-boundary diffusion. These investigations also confirm that η′′ forming below a composition-dependent transus temperature gradually enriches in Sn content, confirming that Sn-poor η′′ is metastable against decomposition into Cu3Sn and more Sn-rich η or (at lower temperatures) η′. Graphic Abstract

Temperature-induced structural phase transitions in RERhSn (RE = Y, Gd-Tm, Lu)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Simon Engelbert ◽  
Rolf-Dieter Hoffmann ◽  
Jutta Kösters ◽  
Steffen Klenner ◽  
Rainer Pöttgen
Keyword(s):  
Rare Earth ◽  
Phase Transitions ◽  
Driving Force ◽  
Room Temperature ◽  
Structural Phase ◽  
Structural Phase Transitions ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
Temperature Dependent ◽  
X Ray

Abstract The structures of the equiatomic stannides RERhSn with the smaller rare earth elements Y, Gd-Tm and Lu were reinvestigated on the basis of temperature-dependent single crystal X-ray diffraction data. GdRhSn crystallizes with the aristotype ZrNiAl at 293 and 90 K. For RE = Y, Tb, Ho and Er the HP-CeRuSn type (approximant with space group R3m) is already formed at room temperature, while DyRhSn adopts the HP-CeRuSn type below 280 K. TmRhSn and LuRhSn show incommensurate modulated variants with superspace groups P31m(1/3; 1/3; γ) 000 (No. 157.1.23.1) (γ = 3/8 for TmRhSn and γ = 2/5 for LuRhSn). The driving force for superstructure formation (modulation) is a strengthening of Rh–Sn bonding. The modulation is expressed in a 119Sn Mössbauer spectrum of DyRhSn at 78 K through line broadening.

X-Ray Diffraction Study of Hep Metals. I. Line Broadening in Polycrystalline Cd Powder

1974 ◽  
Vol 29 (12) ◽  
pp. 1771-1777 ◽  
Author(s):  
N. C. Haider ◽  
S. H. Hunter
Keyword(s):  
Room Temperature ◽  
Small Deformation ◽  
Correction Factors ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
Fault Probability ◽  
Large Particle Size ◽  
X Ray ◽  
Melting Temperatures ◽  
Diffraction Patterns

Powder Cd of 99.999% purity was prepared at room temperature (25 °C) and x-ray diffraction patterns were obtained using CuKaα radiation with Ni-filter. The line broadening was analyzed after incorporating the appropriate correction factors. At room temperature Cd was found to have large particle size (653 A), small root mean square strain (.001), small deformation fault probability a (.003). and negligible growth fault probability β(0). Compared to other hep metals which have been studied earlier and which have higher melting temperatures, metal Cd is much less affected by mechanical deformation at room temperature.

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