Pole Figure Inversion Research Articles

An iterative reconstruction algorithm for pole figure inversion using total variation regularization

This paper describes a new discrete method for inverting X-ray pole figures to estimate the orientation distribution function (ODF). The method employs the equal-volume `cubochoric' representation for uniform discretization of orientation space, SO(3). The forward-projection model is combined with an anisotropic total variation term to iteratively determine the ODF. The efficacy of the new method is evaluated with both model and experimental data and compared with existing discrete and series expansion methods.

Probing orientation information using 3-dimensional reciprocal space volume analysis.

The crystallographic texture of polycrystalline materials is the result of how these materials are processed and what external forces materials have experienced. Neutron and X-ray diffraction are standard methods to characterize global crystallographic textures. However, conventional neutron and X-ray texture analyses rely on pole figure inversion routines derived from intensity analysis of individual reflections or powder Rietveld analysis to reconstruct and model the orientation distribution from slices through reciprocal space. In this work, we describe an original approach to directly probe the crystallographic texture information of rolled aluminum from the intensity distribution in 3-dimensional reciprocal space volumes measured simultaneously. Using the TOPAZ time-of-flight Laue neutron diffractometer, reciprocal space analysis allowed determination of "pole spheres" with <1° angular resolution. These pole spheres are compared with reconstructed pole figures from classic texture analysis.

Special Section on Texture and Microstructure

The International Materials Research Congress (IMRC) is organized yearly at Cancun, Mexico, by the Sociedad Mexicana de Materiales (SMM) and the Materials Research Society (MRS). The conference is structured in plenary sessions, poster sessions and specialized symposia. In 2016, the XXVth IMRC conference included a topical symposium devoted to “Texture and Microstructure”. It is worth underlining that this was the very first conference ever organized in Latin America on a topic related to crystallographic texture. This symposium ran over 2 days. The first day was comprised of presentations on Recrystallization, Deformation, and Substructure. The second day continued on Deformation & Substructure, Unique Applications, and Data Acquisition & Processing. Thirteen invited and seventeen regular lectures were presented as well as 3 posters. The symposium co-organizers are pleased that many of the symposium participants have accepted the invitation to submit their work for a Special Section of Advanced Engineering Materials entitled “Texture and Microstructure”. Fifteen papers were submitted for review, eleven of which were accepted for publication. In addition to contributions from the symposium, several articles were invited to complement the research presented in Cancun and we also are pleased that the authors of six additional relevant papers submitted directly to Advanced Engineering Materials accepted our invitation for being included in this Special Section. The study of crystallographic textures was developed to a semi-quantitative theory by the pioneering work of Guenter Wasserman and Johanna Grewen, “Texturen Metallischer Werkstoffe” (Springer-Verlag, 1962), who studied mainly industrial relevant problems by means of the analysis of two-dimensional pole figures. Later, the reference book of Hans Bunge, “Texture Analysis in Materials Science − Mathematical Methods”, offered a rigorous and complete methodology to determine crystallographic textures by means of a statistical Orientation Distribution Function ODF, calculated by pole-figure inversion methods in the frame of spherical harmonic series. Bunge also developed a systematic method to calculate physical properties of textured polycrystals, as a weighted average of the single crystal properties, the weighting function being the ODF. The next milestone was the determination of individual orientations with Electron Backscattered Diffraction EBSD, where David Dingley made much of the pioneering work. After Brent Adams’ team (including his PhD student Stuart Wright) developed the first fully automated EBSD system they coined the term “Orientation Imaging Microscopy” to describe the technique. Dingley and Adams founded the start-up TSL to provide the first commercial OIM systems. Several companies currently offer automated EBSD systems capable of measuring the local orientation of the grains, the distribution and characteristics of the grain boundaries, the various phases, the local deformation, etc. However, the scientists interested in texture analysis should keep in mind that the statistical population sampled by EBSD is not the same as that by X-rays, since the penetration depth of X-rays is at least hundred times larger than that of electrons. But conversely, the electrons offer much better local resolution enabling the texture to be linked directly to the microstructure. The topic “textures” enjoys special attention from AEM. After the Special Issue “Texture and Microscopy” in 2003, the Special Issue “Texture and Anisotropy” in honor of Prof. Richard Penelle, University Orsay Paris Sud in 2010, with this Special Section “Texture and Microstructure”, all scientists interested in this topic will have in their hand a collection of papers offering a broad and actual covering of the topic. The Special Section opens with two review papers, one on fine-grained magnesium alloys by SPD, the other on electromagnetic coupling, followed by a communication on fatigue cracking resistance of bimodal Ti-6Al-4V alloys. Then, the reader will find 19 full papers covering all measurement techniques, neutron diffraction and Time of Flight neutron diffraction, X-ray diffraction and EBSD, including EBSD coupled with XEDS. The papers cover a broad variety of processing techniques, rolling, wire drawing, hot temperature extrusion, various techniques of Severe Plastic Deformation, e.g. Accumulated Roll Bonding, hot rectangular extrusion, ECASD. The reader will also find information on heat treatment and on epitaxial deposition of thin films. The materials studied are steel, aluminum composites, titanium alloys, magnesium alloys, and the currently widely studied Multi-Component Alloys (NiCoAlFeCu(Cr,Ti) in this Special Section). We are confident that with the papers included in this Special Section and the many references offered in the literature quoted, the reader will find useful information and inspiration for his own research work on “Texture and Microstructure”. Finally, this Special Section could not have been achieved without the constant, efficient and generous help of Sandra Kalveram, Editor-in-Chief at AEM. We express to her our deepest gratitude for her gigantesque help.

Three-Dimensional Crystal Plasticity Finite Element Simulation of Hot Compressive Deformation Behaviors of 7075 Al Alloy

Three-dimensional crystal plasticity finite element (CPFE) method is used to investigate the hot compressive deformation behaviors of 7075 aluminum alloy. Based on the grain morphology and crystallographic texture of 7075 aluminum alloy, the microstructure-based representative volume element (RVE) model was established by the pole figure inversion approach. In order to study the macroscopic stress-strain response and microstructural evolution, the CPFE simulations are performed on the established microstructure-based RVE model. It is found that the simulated stress-strain curves and deformation texture well agree with the measured results of 7075 aluminum alloy. With the increasing deformation degree, the remained initial weak Goss texture component tends to be strong and stable, which may result in the steady flow stress. The grain orientation and grain misorientation have significant effects on the deformation heterogeneity during hot compressive deformation. In the rolling-normal plane, the continuity of strain and misorientation can maintain across the low-angle grain boundaries, while the discontinuity of strain and misorientation is observed at the high-angle grain boundaries. The simulated results demonstrate that the developed CPFE model can well describe the hot compressive deformation behaviors of 7075 aluminum alloy under elevated temperatures.

A novel pole figure inversion method: specification of theMTEXalgorithm

A novel algorithm for ODF (orientation density function) estimation from diffraction pole figures is presented which is especially well suited for sharp textures and high-resolution pole figures measured with respect to arbitrarily scattered specimen directions,e.g.by area detectors. The estimated ODF is computed as the solution of a minimization problem which is based on a model of the diffraction counts as a Poisson process. The algorithm applies discretization by radially symmetric functions and fast Fourier techniques to guarantee smooth approximation and high performance. An implementation of the algorithm is freely available as part of the texture analysis softwareMTEX.

Domain Switching During Electromechanical Poling in Lead Zirconate Titanate Ceramics

In many polycrystalline piezoelectric ceramics, domain switching during the poling process leads to the development of a macroscopic polarization and piezoelectric behavior. Traditionally, poling involves the application of electric fields across two parallel electrodes. In the present work, a radial mechanical compressive stress is applied transverse to the electric field, increasing the potential for domain alignment during poling by taking advantage of ferroelasticity. Experiments demonstrate that poling of lead zirconate titanate using a combination of an electric field and a transverse mechanical compressive stress increases the d33 coefficient from 435 to 489 pC/N. Using neutron diffraction and pole figure inversion methods, the degree of non-180° domain switching is described using pole density distributions of the tetragonal c-axis (002). The degree of 002 domain alignment parallel to the electric field after the electromechanical poling process increases from 1.30 multiples of a random distribution (mrd) to >1.40 mrd at stresses exceeding 40 MPa.

A novel optimization-based pole-figure inversion method: comparison with WIMV and maximum entropy methods

An optimization-based method for pole-figure (PF) inversion that utilizes the orientation distribution function (ODF) gradient for conditional control of the solution is presented. The novel PF inversion method, coined the hybrid {\cal H}^1-seminorm minimization (HHSM), is empirically shown to be versatile, general and robust in the presence of simulated experimental errors. Finite elements (FE) and Rodrigues space are used for the representation and parameterization of the orientation space throughout. The versatility of the FE representation is significantly enhanced from previous implementations by introducing a method for obtaining discrete approximations to spherical harmonic modes from the local FE basis functions. A comparative study with similar implementations of the basic WIMV algorithm and the maximum entropy method is undertaken using several model ODFs of varying sharpness and symmetry. Randomly distributed noise is added to the synthetic PFs to simulate experimental errors and assess the stability of each method. Solution consistency is assessed by inverting two sets of measured PFs, one complete, one incomplete, using several meshes on the orientation space with an increasing number of degrees of freedom. The HHSM method is shown to compare favorably in tests with both the WIMV method and the maximum entropy method.

Using texture components in crystal plasticity finite element simulations

We discuss methods to map crystallographic textures in crystal plasticity finite element simulations. Fourier-type series expansion methods which use spherical harmonic library functions as well as the direct pole figure inversion methods are not well suited to reproduce texture information in a sufficiently localized spherical form onto finite element grids. Mathematically compact Gauss-shaped spherical texture components represent a better approach for including textures in finite element models since they represent an excellent compromise between discreteness (spherical localization), compactness (simple functions), mathematical precision (very good approximation also of complex orientation distribution functions already with small sets of texture components), scalability (the number of used texture components can be systematically varied according to the desired exactness of the texture fit), conceptual simplicity (simple mathematical handling), and physical significance (texture components can be directly linked to characteristic metallurgical mechanisms). The use of texture component functions has also advantages over the use of large sets of discrete single orientations with equal scatter and height since they are more compact, practical, and provide better physical insight into microstructural mechanisms and composition sensitive effects. The article presents a new approach for the mathematical reproduction of such crystallographic texture components in crystal plasticity finite element simulations. It explains in some detail why they are particularly suited for this purpose and how they can be used to map and recover textures in/from plasticity simulations.

A six-dimensional approach to microtexture analysis

A mathematical framework is presented for microtexture derivation and analysis. A six-dimensional space is introduced, defined as the product of direct space (x, y, z) and orientation space, parametrized by the Rodrigues vector (r 1, r 2, r 3). The framework is adapted to diffraction experiments, where orientation information from many grains or deformation microstructures is recorded in parallel, with the prime aim of allowing fast measurements and thus the ability to study microstructure dynamics. Assuming the use of a monochromatic beam the geometry of the projection surfaces is deduced. The derivation of the microtexture is formulated as an inverse problem similar to reconstruction problems in absorption or emission tomography. The applicability of the two major types of reconstruction algorithm, namely transform methods and algebraic methods, is discussed. An explicit solution in terms of the iterative algebraic reconstruction technique algorithm is given. The choice of basis function is discussed. Furthermore, the framework is applied to a number of important simplifying cases. These include grain-by-grain reconstruction, classical pole figure inversion and the generation of three-dimensional maps of non-deformed grains.

Volume Texture of a Deformed Quartzite Observed With U-Stage Microscopy and Neutron Diffractometry

The volume texture of a naturally deformed quartzite from the Kaoko belt, North-West Namibia, has been analysed by both universal stage microscopy and neutron diffraction. Universal stage microscopy is restricted to the determination of the base pinacoid preferred orientation in quartzite. For a more complete description of the texture, the orientations of additional crystal planes, such as first and second order prisms as well as positive and negative rhombs, must be known. Neutron methods allow the evaluation of pole figures of all Bragg reflecting planes, of which those of the first order prisms being considered to be the most active slip planes, are of particular interest. Drawbacks of neutron diffraction, i.e. the faking of an eventually absent inversion centre and lack of resolution, can be overcome by pole figure inversion and subsequent calculation of desired pole figures. Both, universal stage microscopy and neutron diffraction yield well comparable results, of course only with respect to the pole figure of the c-axis.

ODF Computer Program for High-Resolution Texture Analysis of Low-Symmetry Materials

A program for 3-dimensional texture analysis using the harmonic method has been developed. Pole figure inversion as well as ODF representation of individual orientation data are provided up to a series expansion degree of 150. The number of orientations or pole figure points is not limited. All point groups for crystal and sample symmetry, including the icosahedral group for investigation of quasicrystals are supported. Pole figure inversion is carried out using the positivity method with separation of partial or total coincidences. Several autocorrelation functions e.g. the OCF (crystal and sample symmetry) and the reduced MODF as well as correlations between different data sets can be calculated and, as an additional tool, averaging of tensor properties with arbitrary rank and symmetry is enabled. The graphical representation contains Euler cuts of the orientation distribution functions, stereographic or equal area projections of pole figures and inverse pole figures as well as diagrams showing mean values and variances of the C coefficients.

Partial texture analysis by the modified maximum entropy method

The orientation distribution function (ODF), being a continuous function of three variables, requires a large number of values or series coefficients for its complete description. A partial texture analysis, using a fixed angle texture analyzer, i.e., less pole figure data, gives a limited number of values of the texture function, i.e., an approximate ODF. This partial texture analysis is very important both for theory and for its application to on-line texture analysis, used to refine a manufacturing process and optimize materials properties. Quite recently, the modified maximum entropy method (MMEM), which improved the primary maximum entropy method with series coefficients as constraints, introduced the maximum entropy principle directly into the least square equation of pole figure inversion proposed by Bunge. Because the positivity condition is automatically satisfied, less data are needed to obtain an accurate solution with the MMEM. In the present paper, the MMEM is used of find the partial ODF from only a few data measured on the pole figure. The validity of the new algorithm will be proved by the partial ODF analysis of cold-rolled copper sheets with moderate-to-strong textures.

Physical Versus Mathematical Aspects in Texture Analysis

Texture deals with the orientational aspects of the crystal lattice in polycrystalline aggregates. This comprises the classical orientation distribution function ODF as well as higher‐order textural quantities. The quantitative treatment of these quantities requires a good deal of mathematical methods. This concerns particularly the representation of orientation data including all kinds of symmetries, the transformation of experimentally measured raw data into the required distribution functions, as well as mathematical models of texture formation by physical processes and of the texture‐property relationship.When physical facts are idealized in terms of a mathematical description or by mathematical models then the physical limits, within which these are valid, must be known. Such physical limits are, for instance, definition of crystal orientation by the crystal lattice which leads to an unsharpness relationship between location and orientation resolution as well as a relationship between statistical relevance and angular resolving power. Other physical limits are given by the “fuzzyness” of sample symmetry. A central problem is pole figure inversion i.e. the inversion of the projection equation. This problem may have a “physical” solution even if it does not have a mathematical one. Finally, in the problem of rationalizing orientation distribution functions in terms of a low number of “components”, mathematical aspects may be quite different from physical ones.In all these problems it is thus necessary to keep the mathematical aspects apart from physical aspects.

The Modified Maximum Entropy Method (MMEM) in QTA for Lower Symmetry Polycrystalline Aggregates

A new algorithm of quantitative texture analysis (QTA), which is called the modified maximum entropy method (MMEM), has been proposed and applied to determination of textures in polycrystalline samples of lower crystal symmetry with overlapping diffraction peaks (Wang and Xu, 1995a). By introducing directly the maximum entropy principle into the least square procedure of pole figure inversion (Bunge, 1969), then both minimizing the differences between experimental and postulated pole figure data, and maximizing entropy may be satisfied simultaneously. Thus, the maximum entropy principle is applied to the entire process of QTA in frame of the harmonic method (HM). The detailed comparisons among the three pole figure inversion methods, i.e. the traditional HM, the primary maximum entropy method (MEM) and the MMEM, are given through a model example of simulated fiber texture. It is shown that the precise and stable solution of inverse pole figure for the polycrystalline samples with smooth or sharp textures will be obtained by the MMEM even using a less number of pole figures. The minimum range of polar angle and the least number of pole figures, which are needed in the QTA for pretended tetragonal materials by the MMEM, are discussed in detail.

Determination of fiber texture vector from overlapping pole figures by the maximum entropy method

In recent years, there has been an increasing interest in low symmetry and multiphase materials such as intermetallics, ceramics, polymers and minerals. To overcome the difficulties of texture analysis for these materials arising from the overlapping Bragg-reflections, the peak profile analysis and the pole figure inversion adapted to overlapping pole figures are mostly used. The concept of maximum entropy has received wide application in the information theory and the image reconstruction. Quite recently, it was introduced to texture analysis for cubic and hexagonal materials. While the positive condition is automatically satisfied, the assumption of maximum entropy may give an access, with less number of pole figures, to the optimization of the pole figure inversion within the variation width of solutions. In this paper, the maximum entropy method (MEM) is applied to the determination of fiber texture vector from overlapping pole figures. An algorithm is developed for tetragonal materials and demonstrated through a simulation example.

Applying texture analysis to materials engineering problems

Textures in materials have been studied extensively since the 1930s following the pioneering work of Wassermann.1,2 The modern era of texture measurement started in 1949 with the development of the x-ray pole figure technique for texture measurement by Schultz.3 Finally, modern texture analysis was initiated with the publication by Bunge4 and Roe5 of a mathematical method of pole figure inversion, which is now used to calculate the orientation distribution function (ODF). This article cannot summarize such an extensive body of work, but it does endeavor to provide the background necessary to understand texture analysis; it also illustrates several applications of texture.

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

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 &amp; 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.

The iterative series-expansion method for quantitative texture analysis. I. General outline

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density distribution 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. Rather it contains a range of possible solutions (kernel) which 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 \tilde{\tilde f}(g) 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.

Some basic concepts of texture analysis and comparison of three methods to calculate orientation distributions from pole figures

Several methods have been developed which derive the orientation distribution (ODF) in a polycrystalline sample from pole figures measured by X-ray or neutron diffraction techniques. The theoretical backgrounds of the conventional harmonic method, of the vector method and of the method of Williams [J. Appl. Phys. (1968). 39, 4329–4335], Imhof [Textures Microstruct. (1982). 5, 73–86] and Matthies & Vinel [Phys. Status Solidi B. (1982). 112, K111–K120] (WIMV) are reviewed. A quantitative comparison is then made using the same input data and the same computer to evaluate resolution, errors and efficiency. The input data consist of standard functions, Taylor predictions and measured pole figures covering a realistic range of possibilities for both cubic and trigonal crystal symmetries. Comprehensive error criteria are introduced, and it is proposed to use both integral errors (RP) and difference pole figures to assess the quality of the pole-figure inversion. The harmonic method and WIMV are able to reproduce the original pole figures from the ODF within computer roundoff errors. Resolution of the vector method, particularly for low crystal symmetry, is considerably worse owing to the large-volume cells in orientation space. Computing time is optimal for the conventional harmonic method (for medium termination L), slightly worse for WIMV and about an order of magnitude higher for the vector method. Whereas the conventional harmonic method only reproduces the ghost-afflicted part \tilde{f}(g) of the ODF, the vector method satisfies automatically the non-negativity criterion; however, only WIMV provides a general (conditional) ghost correction. In the examples chosen the ghost-corrected ODF f(g) closely coincides with the starting model (model with standard functions or Taylor prediction) supporting its physical relevance. An attractive feature of WIMV is that it leads to results of satisfactory quality using fewer pole figures than the harmonic method. This is particularly important for low crystal symmetries. Furthermore, the treatment of incomplete pole figures is straightforward.

Determination of the Complete Orientation Distribution Function by the Zero‐Range Method

Because of the superposition of pole figures corresponding to symmetrically equivalent crystal directions, only the reduced orientation distribution function can be obtained directly by pole figure inversion. The additional information contained in the positivity condition of the ODF allows, however, the determination of an approximation to the “indeterminable” part and hence of the complete ODF f(g), if the texture has sufficiently large zero‐ranges. The application of the method and the accuracy of the results was tested using two theoretical and one experimental textures. The accuracy of the complete ODF depends on the size of the zero‐range, the errors in its determination, and on the errors, experimental and truncational, of the reduced ODF. The “physical zero” used in order to determine the zero‐range is defined according to the statistical error of the pole figure measurement.