Application of ODF to the Rietveld Profile Refinement of Polycrystalline Solid

1993 ◽  
Vol 37 ◽  
pp. 49-57
Author(s):  
C. S. Choi ◽  
E. F. Baker ◽  
J. Orosz
Keyword(s):  
Three Dimensional ◽  
Inverse Pole Figure ◽  
Orientation Distribution ◽  
Pole Density ◽  
Diffraction Profile ◽  
Refinement Method ◽  
Pole Figures ◽  
Diffraction Patterns ◽  
Characterization Of Materials ◽  
Profile Refinement

The Rietveld profile refinement method is probably the most popular technique used for the crystallographic characterization of materials including crystal structures and phase analysis, but it has been used mostly with ideal powder sample, not with textured polycrystals, because effects of strong and complex textures. Most technological materials are fabricated by using thermo-mechanical forming processes, which inevitably produce strong and complex preferential orientations of the crystallites. Consequently, the diffraction patterns of a given technological material are not unique but vary considerably with the measuring direction, with intensity variations as large as factors of hundreds, depending on the degree of texture. The texture effect on the diffraction pattern of a certain sample direction is directly proportional to the pole density of the corresponding inverse pole figure, which can be obtained from the three-dimensional orientation distribution function (ODF) of the material. The ODFs of materials with high crystal symmetry, such as cubic, hexagonal, tetragonal, and orthorhombic, can be determined quite precisely, using modern texture analysis techniques (for example, Bungel, Wenk, and Kallend et al.). The pole density distributions of the inverse pole figures can be used in the diffraction profile calculation of a highly textured sample.

scholarly journals A New Approach to Describing Three-Dimensional Orientation Distribution Functions in Textured Materials–Part I: Formation of Pole Density Distribution on Model Pole Figures

1993 ◽  
Vol 22 (2) ◽  
pp. 73-85 ◽  
Author(s):  
V. N. Dnieprenko ◽  
S. V. Divinskii
Keyword(s):  
Coordinate System ◽  
Texture Component ◽  
Three Dimensional ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Pole Density ◽  
Texture Axis ◽  
Pole Figures ◽  
Orientation Distribution Functions ◽  
Textured Materials

New method for simulation of orientation distribution functions of textured materials has been proposed. The approach is based on the concept to describe any texture class by a superposition of anisotropic partial fibre components. The texture maximum spread is described in a “local” coordinate system connected with the texture component axis. A set of Eulerian angles γ1,γ2,γ3 are introduced with this aim. To specify crystallite orientations with respect to the sample coordinate system two additional sets of Eulerian angles are introduced besides γ1,γ2,γ3. One of them, (Ψ0,θ0,ϕ0), defines the direction of the texture axis of a component with respect to the directions of the cub. The other set, (Ψ1,θ1,ϕ1), is determined by the orientation of the texture component and its texture axis in the sample coordinate system. Analytical expressions approximating real spreads of crystallites in three-dimensional orientation space have been found and their corresponding model pole figures have been derived. The proposed approach to the texture spread description permits to simulate a broad spectrum of real textures from single crystals to isotropic polycrystals with a high enough degree of correspondence.

Defining the hematite topotaxial crystal growth in magnetite–hematite phase transformation

2020 ◽  
Vol 53 (4) ◽  
pp. 896-903
Author(s):  
Flávia Braga de Oliveira ◽  
Gilberto Álvares da Silva ◽  
Leonardo Martins Graça
Keyword(s):  
Electron Backscatter Diffraction ◽  
Inverse Pole Figure ◽  
Orientation Relationships ◽  
Transformation Matrices ◽  
Hematite Phase ◽  
Pole Figures ◽  
Magnetite Crystal ◽  
Diffraction Patterns ◽  
Backscatter Diffraction ◽  
Specific Orientation

Magnetite and hematite iron oxides are minerals of great economic and scientific importance. The oxidation of magnetite to hematite is characterized as a topotaxial reaction in which the crystallographic orientations of the hematite crystals are determined by the orientation of the magnetite crystals. Thus, the transformation between these minerals is described by specific orientation relationships, called topotaxial relationships. This study presents electron-backscatter diffraction analyses conducted on natural octahedral crystals of magnetite partially transformed into hematite. Inverse pole figure maps and pole figures were used to establish the topotaxial relationships between these phases. Transformation matrices were also applied to Euler angles to assess the diffraction patterns obtained and confirm the identified relationships. A new orientation condition resulting from the magnetite–hematite transformation was characterized, defined by the parallelism between the octahedral planes {111} of magnetite and rhombohedral planes \{10\bar {1}1\} of hematite. Moreover, there was a coincidence between one of the octahedral planes of magnetite and the basal {0001} plane of hematite, and between dodecahedral planes {110} of magnetite and prismatic planes \{11\bar {2}0\} of hematite. All these three orientation conditions are necessary and define a growth model for hematite crystals from a magnetite crystal. A new topotaxial relationship is also proposed: (111)Mag || (0001)Hem and (\bar {1}\bar {1}1)_{\rm Mag} || (10\bar {1}1)_{\rm Hem}.

scholarly journals Calculation of the Normalization Factor of Incomplete Pole Figures by Cubic Extrapolation

1986 ◽  
Vol 6 (3) ◽  
pp. 167-179 ◽  
Author(s):  
M. Dahms ◽  
H.-J. Bunge
Keyword(s):  
Spline Interpolation ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Pole Density ◽  
Normalization Factor ◽  
Texture Coefficients ◽  
Cubic Materials ◽  
Pole Figures ◽  
Experimental Errors ◽  
Normalization Factors

The calculation of orientation distribution functions from incomplete pole figures can be carried out by a least squares approximation of the texture coefficients Clμν and the normalization factors Nhkl to the available experimental data. This procedure is less susceptable to instabilities due to experimental errors if the normalization factors can be calculated independently of the coefficients Clμν. In the case of cubic materials, the relationship F20 = 0 to be fulfilled by pole figure values provides an independent condition for the calculation of the normalization factor. This condition can still be improved by taking the slopes of the pole density curves at α = αmax⁡ and α = 90° into account. An economic way to consider the slope in the pole figures is to use a cubic spline interpolation.

Texture of extruded uranium alloy by neutron diffraction

1985 ◽  
Vol 18 (6) ◽  
pp. 413-418 ◽  
Author(s):  
C. S. Choi ◽  
H. J. Prask
Keyword(s):  
Neutron Diffraction ◽  
Orientation Distribution ◽  
Fiber Texture ◽  
Pole Density ◽  
Ti Alloy ◽  
Composite Alloy ◽  
Density Distributions ◽  
Uranium Alloy ◽  
Pole Figures ◽  
Combination Function

The pole-density distributions of two hydrostatically extruded samples, a U–0.75 wt.% Ti alloy and a U–0.75 wt % Ti/W composite alloy, were studied by neutron diffraction methods. Analysis of U 112, U 131 and U 111 pole figures revealed that the α-U phases of both samples possess a [010]/[340] duplex fiber texture with a probability ratio of approximately 2.8:1 in favor of the [010] direction. The W phase of the composite sample had a [110] fiber texture. The orientation distribution profiles of the fiber axes obtained from the rocking curves (as a function of the tilt angle) were represented best by a Gaussian–Lorentzian combination function. The full widths at half maximum of the distributions were approximately 21, 11, and 5° for the U [010], U [340] and W [110] fiber axes, respectively.

scholarly journals Cyclic Textures in Aluminium Wires

Texture ◽  
1972 ◽  
Vol 1 (1) ◽  
pp. 31-49 ◽  
Author(s):  
U. Schläfer ◽  
H. J. Bunge
Keyword(s):  
Neutron Diffraction ◽  
Three Dimensional ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Axially Symmetric ◽  
Fibre Texture ◽  
Pole Figures ◽  
Processing Variables ◽  
Orientation Distribution Functions ◽  
The Wire

Three-dimensional orientation distribution functions were calculated from neutron diffraction pole figures of unwound cylinders taken at different distances from the centre of cold drawn Al-wires. Their features change from the axially symmetric type at the very centre of the wire towards a texture near to the rolling type at the surface. Relations between the three-dimensional function and ordinary fibre texture pole figures were used to study the dependence of the textures on certain processing variables for cold drawn as well as recrystallized wires.

The iterative series-expansion method for quantitative texture analysis. II. Applications

1992 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Distribution Function ◽  
Series Expansion ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Pole Density ◽  
General Outline ◽  
Positivity Condition ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts, \tilde {f}(g) and \tildes {f}(q), expressed by even- and odd-order terms of the series expansions. For the calculation of the even part \tilde {f}(g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time. In the previous paper in this series [Dahms & Bunge (1989) J. Appl. Cryst. 22, 439–447] a general outline of the method was given. This, the second part, gives details of the system of programs used as well as typical examples showing the versatility of the method.

Improvements in the Quantitative Evaluation of Three-Dimensional Texture. II. The Pole-Figure-Multiplying Method

1995 ◽  
Vol 28 (5) ◽  
pp. 532-533 ◽  
Author(s):  
L.-G. Yu ◽  
H. Guo ◽  
B. C. Hendrix ◽  
K.-W. Xu ◽  
J.-W. He
Keyword(s):  
Distribution Function ◽  
Texture Analysis ◽  
Pole Figure ◽  
Quantitative Evaluation ◽  
Orientation Distribution Function ◽  
Three Dimensional ◽  
Orientation Distribution ◽  
Simple Method ◽  
Pole Figures

A new simple method is proposed for determining the orientation distribution function (ODF) for three-dimensional texture analysis in a polycrystal based on the reality that the accuracy of an ODF is dependent on both the accuracy of each measured pole figure and the number of pole figures.

Improvements in the Quantitative Evaluation of Three-Dimensional Texture. I. The Nature of the Information Obtained from Pole Figures

1995 ◽  
Vol 28 (5) ◽  
pp. 527-531 ◽  
Author(s):  
L.-G. Yu ◽  
H. Guo ◽  
B. C. Hendrix ◽  
K.-W. Xu ◽  
J.-W. He
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Orientation Distribution Function ◽  
Angular Resolution ◽  
Three Dimensional ◽  
Orientation Distribution ◽  
High Angular Resolution ◽  
Area Theorem ◽  
Pole Figures ◽  
Experimental Resolution

The sources of indefiniteness in the orientation-distribution-function (ODF) description of crystalline texture are shown to result from the integral nature of the pole-figure measurement. An equipartition-area theorem is proved and it is shown that current methods use too few pole figures, which are measured to an unnecessarily high angular resolution. The experimental resolution is considered and the number of pole figures needed for ODF analysis is calculated as a function of the required ODF resolution.

Quantitative neutron diffraction texture measurement applied to α-phase alumina and Ti3AlC2

2011 ◽  
Vol 44 (5) ◽  
pp. 1062-1070 ◽  
Author(s):  
JianFeng Zhang ◽  
Erich H. Kisi ◽  
Oliver Kirstein
Keyword(s):  
Single Crystal ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Pole Density ◽  
Texture Measurement ◽  
Area Detector ◽  
Neutron Diffractometer ◽  
Α Phase ◽  
Pole Figures ◽  
Crystal Properties

Orientation distribution functions, essential for making a quantitative connection between single-crystal and polycrystal properties, have been determined for extruded α-phase alumina, hot-pressed Ti3AlC2and cold isostatically pressed Ti3AlC2using experimental pole figures recorded on the fixed-wavelength neutron diffractometer KOWARI. Some practical improvements to the calculation of the pole-figure density from the raw area-detector data, and for constructing pole figures on ann×n° hemispherical grid, are presented. The textures give some insight into particle flow during manufacture. Directly measured material textures were compared with one-dimensional pole density functions, such as the March and Rietveld functions commonly used for the correction of preferred orientation in Rietveld refinements, as a means of assessing the utility of the latter for the computation of diffraction elastic constants and other polycrystal properties from a given set of single-crystal properties.

scholarly journals Texture and Lamellae Distribution Functions in Lamellar Eutectics

1986 ◽  
Vol 6 (4) ◽  
pp. 265-287 ◽  
Author(s):  
H. J. Bunge
Keyword(s):  
Three Dimensional ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
X Rays ◽  
Two Dimensional ◽  
Absorption Factor ◽  
X Ray Diffraction ◽  
Diffraction Vector ◽  
X Ray ◽  
Pole Figures

The crystallographic orientation distribution and the geometrical lamellae orientation distribution in lamellar eutectics are, in general, not independent of each other. The combined orientation-lamellae distribution function depends on five angular parameters. X-ray diffraction in such eutectics may exhibit an anisotropic macroscopic absorption factor if the penetration depth of the X-rays is large compared with their planar size. As a consequence, the reflected X-ray intensity may depend on a third angle γ, i.e. a rotation of the sample about the diffraction vector s additionally to the usual pole figure angles α, β which describe the orientation of the diffraction vector s with respect to the sample coordinate system. It is thus necessary to measure three-dimensional generalized pole figures instead of conventional two-dimensional ones.

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