#
Diffraction Methods for Qualitative and Quantitative Texture Analysis of Ferroelectric Ceramics^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

_{4.5}Na

_{0.5}Ti

_{4}O

_{15}and 0.91Bi

_{1/2}Na

_{1/2}TiO

_{3}-0.07BaTiO

_{3}-0.02K

_{0.5}Na

_{0.5}NbO

_{3}[18,19].

## 2. Experimental Methods

#### 2.1. Sample Preparation

_{1/2}Na

_{1/2}TiO

_{3}and 0.91Bi

_{1/2}Na

_{1/2}TiO

_{3}-0.07BaTiO

_{3}-0.02K

_{0.5}Na

_{0.5}NbO

_{3}(BNT-7BT-2KNN) ceramics were fabricated using conventional solid-state processing to produce randomly oriented materials or using the tape-casting method using a 5wt% BNT platelets for 00l oriented ceramics. Bi

_{1/2}Na

_{1/2}TiO

_{3}(BNT) platelets with 00l perpendicular to their large face were prepared using a two-step molten salt method. See [29] for details about the seed processing and [30,31] for further information about ceramic processing.

#### 2.2. Bragg-Brentano X-ray Diffraction

#### 2.3. Neutron Rocking Curves

#### 2.4. Angular Dependent Diffraction Data

_{max}<= 22) on the determined crystallographic texture. Neutron data were analyzed without an imposed sample symmetry to highlight the advantages of modeling textures using an ODF.

#### 2.5. Field Dependent Diffraction

## 3. Mathematical Methods

#### 3.1. Lotgering Factor

_{hkl}is the integrated intensity of the hkl pole, p

_{o}and p represent the fraction of the 00l poles for randomly oriented and templated samples, respectively. A θ-2θ scan is required to determine the Lotgering factor for a given sample accurately. Measuring the fraction aligned along the off-axis would require an uncoupled θ-2θ scan or a coupled θ-2θ scan with a χ-tilt. As the sample normal is the primary reference axis, the Lotgering technique would not capture the complex orientation symmetries associated with many deformation textures [1,22,23,41,42] and naturally occurring textures in biominerals [43,44].

#### 3.2. March-Dollase

#### 3.3. Orientation Distribution Functions

_{1},Φ,φ

_{2}) are the Bunge Euler angles. Thus, a pole figure inversion of a set of known poles, P

_{hkl}(α, β), allows the determination of a pole figure for an unknown pole. In recent years, ODF models have been incorporated into Rietveld refinement programs.

#### 3.3.1. Harmonic ODF

#### 3.3.2. Discrete ODF

_{i}is the multiplicity of the i-th pole, and N

_{o}is a normalization factor. Using a Rietveld analysis, the number of measured pole figures equals the number of poles in the refined spectra for a pole figure inversion. The final ODF is determined by minimizing the error between the calculated and experimental pole figures.

_{hkl}is the number of discretization points used for the integration of orientation space around pole hkl, w

_{hkl}weights the reflection to account for peak overlap, and ${P}_{hkl}^{i}({\left({\varphi}_{1},\mathsf{\Phi},{\varphi}_{2}\right)}^{-1},{h}_{m})$ is the calculated pole figure for the i-th iteration [44].

#### 3.4. Dipole Distribution

## 4. Results and Discussion

#### 4.1. Lotgering Factor

_{o}in Equation (1)). A p and p

_{o}of 0.845 (3) and 0.193 (3) were calculated for the textured and randomly oriented BNT ceramics. The high p for the textured sample is indicative that 00l dominantly contributed to the measured diffraction data. The textured ceramic was found to have a high f of 0.809 (2). A Lotgering factor for ideal p

_{o}-value for a theoretical pattern for the crystallographic structure of BNT was also determined. Table 1 compares the Lotgering factors determined for an experimental and predicted reference. Even though the p

_{o}for the predicted pattern is slightly lower (0.188 compared with 0.193 (3)), the resulting Lotgering factors are similar and within the estimated uncertainty. It should be noted that this indicator only considers a single set of diffraction data and does not provide a metric to determine crystallographic distributions within a given sample.

#### 4.2. March–Dollase

_{B}angular range because the sample blocks the diffracted X-rays (θ

_{B}> α) or incident X-rays are blocked (θ

_{B}< α). In principle, diffraction collected using an area detector or four-circle goniometer would allow an angular range of ±~80° before the sample would block the diffraction. Materials with a low-angle texture pole benefit from the low-angle orientation permitted range with an area detector. An alternative is to measure the pole density using neutron diffraction, where the low material absorption does not restrict the allowed angular range or require absorption corrections [2]. See Figure 2 for a comparison of a pole density before and after correcting for background contributions.

#### 4.3. Rietveld Texture Analysis

_{max}) between 4 and 22, the maximum cutoff implemented in MAUD. Using a standard n = 4 harmonic cutoff, the resulting refined ODF has a negative texture and high refinement error (R > 20%). Increasing the harmonic cutoff above n = 10 removes the negative texture in the 00l ODF, and the resulting refinement error is significantly improved. Increasing harmonic cutoff above L

_{max}= 4 has minimal impact on the final Rietveld refinement error, but the final refined texture increases linearly with harmonic cutoff. From L

_{max}= 10–22, the weighted refinement error and the MRD increased. In contrast, the overall texture and Rietveld refinement errors are far less EWIVM resolution, see Figure 4. EWIMV refines a higher overall texture and achieves a similar refinement error as a high harmonic cutoff (L

_{max}> 12). The EWIMV consistently determined a sharper overall texture. The apparent discrepancy between the harmonic and discrete methods could arise from the underlying differences in their formulation. The harmonic texture has implicit smoothing that can impact strong textures.

_{max}= 10 are captured with a L

_{max}= 20. Even with a high harmonic cutoff, the effect of Fourier smoothing is apparent. The refinement of an unconstrained harmonic texture model with a L

_{max}= 20 required ~12 h to reach 5 convergence iterations. A harmonic texture analysis using a different Rietveld program, such as GSAS, might allow for a better-reconstructed pole figure fitting the experimental data because GSAS allows up L

_{max}= 36. Both spherical harmonic pole figure reconstructions fit the experimental reconstruction better than the pole figure reconstruction using the exponential harmonic method.

_{max}of 10 and 20 without an imposed sample symmetry introduces 60 and 361 additional coefficients, introducing complexities for reaching convergence. Analyzing materials lower symmetry (e.g., orthorhombic, trigonal, or tetragonal) would further increase the number of coefficients.

#### 4.4. Ferroelastic Domain Texture

_{111}indicates that the fraction of domains present is less than expected before electric poling. These data evidence that extensive domain reorientation has occurred with BNT achieving an η

_{111}that suggests >40% of the available domains switched into the field direction.

_{111}compared to what was calculated using an experiment reference. While η

_{111}increased from 0.30 (6) to 0.44 (6) when the equivalent reference was used, the reduction is still within the estimated error. Although the estimated 111/$1\overline{1}1$ intensity ratio of 0.23 is well below 1/3 and would overestimate η

_{111}. It is noted that the use of a coarse integration range (15°) can introduce numerical errors associated with the discretization of the integral (Equation (17)). The resulting error could account for the apparent discrepancy. Decreasing the integration range would help reduce the discretization error.

## 5. Comparison of Methods

## Supplementary Materials

_{o}randomly oriented metric for an experimental (Powder) and simulated data (Structure Factor), Figure S1:Peak fit of measured data before application of electric field with the 111 (green) and $1\overline{1}1$ (magenta) peaks shown for reference with the measured data (x), modeled peak fits (red) and residual (black), Figure S2: Peak fit of measured data after application of electric field with the 111(green) and $1\overline{1}1$ (magenta) peaks shown for reference with the measured data (x), modeled peak fits (red) and residual (black). Note the substantial increase in the 111 intensity compared with Figure S1.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Xie, Y.X.; Lutterotti, L.; Wenk, H.-R.; Kovacs, F. Texture analysis of ancient coins with TOF neutron diffraction. J. Mater. Sci.
**2004**, 39, 3329–3337. [Google Scholar] [CrossRef] [Green Version] - Kocks, U.F.; Tome, C.N.; Wenk, H.-R. Texture and Anisotropy; Cambridge University Press: Cambridge, UK, 2000; ISBN 978-0-521-79420-6. [Google Scholar]
- Dehoff, R.R.; Kirka, M.M.; Sames, W.J.; Bilheux, H.; Tremsin, A.S.; Lowe, L.E.; Babu, S.S. Site specific control of crystallographic grain orientation through electron beam additive manufacturing. Mater. Sci. Technol.
**2015**, 31, 931–938. [Google Scholar] [CrossRef] - Messing, G.L.; Trolier-McKinstry, S.; Sabolsky, E.M.; Duran, C.; Kwon, S.; Brahmaroutu, B.; Park, P.; Yilmaz, H.; Rehrig, P.W.; Eitel, K.B.; et al. Templated grain growth of textured piezoelectric ceramics. Crit. Rev. Solid State Mater. Sci.
**2004**, 29, 45–96. [Google Scholar] [CrossRef] - Daniel, L.; Hall, D.A.; Koruza, J.; Webber, K.G.; King, A.; Withers, P.J. Revisiting the blocking force test on ferroelectric ceramics using high energy x-ray diffraction. J. Appl. Phys.
**2015**, 117, 174104. [Google Scholar] [CrossRef] - Chen, J.-H.; Hwang, B.-H.; Hsu, T.-C.; Lu, H.-Y. Domain switching of barium titanate ceramics induced by surface grinding. Mater. Chem. Phys.
**2005**, 91, 67–72. [Google Scholar] [CrossRef] - Kockelmann, W.; Siano, S.; Bartoli, L.; Visser, D.; Hallebeek, P.; Traum, R.; Linke, R.; Schreiner, M.; Kirfel, A. Applications of TOF neutron diffraction in archaeometry. Appl. Phys. A Mater. Sci. Process.
**2006**, 83, 175–182. [Google Scholar] [CrossRef] - Ghosh, D.; Sakata, A.; Carter, J.; Thomas, P.A.; Han, H.; Nino, J.C.; Jones, J.L. Domain Wall Displacement is the Origin of Superior Permittivity and Piezoelectricity in BaTiO
_{3}at Intermediate Grain Sizes. Adv. Funct. Mater.**2014**, 24, 885–896. [Google Scholar] [CrossRef] [Green Version] - Fancher, C.M.; Brewer, S.; Chung, C.C.; Röhrig, S.; Rojac, T.; Esteves, G.; Deluca, M.; Bassiri-Gharb, N.; Jones, J.L. The contribution of 180° domain wall motion to dielectric properties quantified from in situ X-ray diffraction. Acta Mater.
**2017**, 126, 36–43. [Google Scholar] [CrossRef] - Schenk, T.; Fancher, C.M.; Hyuk Park, M.; Richter, C.; Kunneth, C.; Kersch, A.; Jones, J.L.; Mikolajick, T.; Schroeder, U. On the Origin of the Large Remanent Polarization in La:HfO2. Adv. Electron. Mater.
**2019**, 5, 1900303. [Google Scholar] [CrossRef] - Bowman, K.J.; Chen, I.-W.W. Transformation Textures in Zirconia. J. Am. Ceram. Soc.
**1993**, 76, 113–122. [Google Scholar] [CrossRef] - Esteves, G.; Fancher, C.M.; Röhrig, S.; Maier, G.A.G.A.; Jones, J.L.; Deluca, M. Electric-field-induced structural changes in multilayer piezoelectric actuators during electrical and mechanical loading. Acta Mater.
**2017**, 132, 96–105. [Google Scholar] [CrossRef] - Daniels, J.E.; Jo, W.; Rődel, J.; Honkimaki, V.; Jones, J.L. Electric-field-induced phase-change behavior in (Bi
_{0.5}Na_{0.5})TiO_{3}-BaTiO_{3}-(K_{0.5}Na_{0.5})NbO_{3}: A combinatorial investigation. Acta Mater.**2010**, 58, 2103–2111. [Google Scholar] [CrossRef] - Jones, J.L.; Aksel, E.; Tutuncu, G.; Usher, T.-M.; Chen, J.; Xing, X.; Studer, A.J. Domain wall and interphase boundary motion in a two-phase morphotropic phase boundary ferroelectric: Frequency dispersion and contribution to piezoelectric and dielectric properties. Phys. Rev. B
**2012**, 86, 024104. [Google Scholar] [CrossRef] [Green Version] - Jones, J.L.; Hoffman, M.; Bowman, K.J. Saturated domain switching textures and strains in ferroelastic ceramics. J. Appl. Phys.
**2005**, 98, 024115. [Google Scholar] [CrossRef] - Jones, J.L.; Hoffman, M.; Daniels, J.E.; Studer, A.J. Direct measurement of the domain switching contribution to the dynamic piezoelectric response in ferroelectric ceramics. Appl. Phys. Lett.
**2006**, 89, 092901. [Google Scholar] [CrossRef] - Saito, Y.; Takao, H.; Tani, T.; Nonoyama, T.; Takatori, K.; Homma, T.; Nagaya, T.; Nakamura, M. Lead-free piezoceramics. Nature
**2004**, 432, 84–87. [Google Scholar] [CrossRef] [PubMed] - Jones, J.L.; Slamovich, E.B.; Bowman, K.J. Product and component grain and domain textures in ferroelectric ceramics. Mater. Sci. Forum
**2005**, 495–497, 1401–1406. [Google Scholar] [CrossRef] - Fancher, C.M.; Blendell, J.E.; Bowman, K.J. Decoupling of superposed textures in an electrically biased piezoceramic with a 100 preferred orientation. Appl. Phys. Lett.
**2017**, 110, 062901. [Google Scholar] [CrossRef] - Wenk, H.-R.; Grigull, S. Synchrotron texture analysis with area detectors. J. Appl. Crystallogr.
**2003**, 36, 1040–1049. [Google Scholar] [CrossRef] - Ischia, G.; Wenk, H.-R.; Lutterotti, L.; Berberich, F. Quantitative Rietveld texture analysis of zirconium from single synchrotron diffraction images. J. Appl. Crystallogr.
**2005**, 38, 377–380. [Google Scholar] [CrossRef] - Matthies, S.; Pehl, J.; Wenk, H.-R.; Lutterotti, L.; Vogel, S.C. Quantitative texture analysis with the HIPPO neutron TOF diffractometer. J. Appl. Crystallogr.
**2005**, 38, 462–475. [Google Scholar] [CrossRef] - Vogel, S.C.; Hartig, C.; Lutterotti, L.; Von Dreele, R.B.; Wenk, H.-R.; Williams, D.J. Texture measurements using the new neutron diffractometer HIPPO and their analysis using the Rietveld method. Powder Diffr.
**2004**, 19, 65–68. [Google Scholar] [CrossRef] [Green Version] - Fancher, C.M.; Hoffmann, C.M.M.; Frontzek, M.D.D.; Bunn, J.R.R.; Payzant, E.A.A. Probing orientation information using 3-dimensional reciprocal space volume analysis. Rev. Sci. Instrum.
**2019**, 90, 013902. [Google Scholar] [CrossRef] - Xu, P.; Harjo, S.; Ojima, M.; Suzuki, H.; Ito, T.; Gong, W.; Vogel, S.C.; Inoue, J.; Tomota, Y.; Aizawa, K.; et al. High stereographic resolution texture and residual stress evaluation using time-of-flight neutron diffraction. J. Appl. Crystallogr.
**2018**, 51, 746–760. [Google Scholar] [CrossRef] - Peterson, N.E.; Einhorn, J.R.; Fancher, C.M.; Bunn, J.R.; Payzant, E.A.; Agnew, S.R. Quantitative texture analysis using the NOMAD time-of-flight neutron diffractometer. J. Appl. Crystallogr.
**2021**, 54, 867–877. [Google Scholar] [CrossRef] - Gorfman, S. Sub-microsecond X-ray crystallography: Techniques, challenges, and applications for materials science. Crystallogr. Rev.
**2014**, 20, 210–232. [Google Scholar] [CrossRef] - Daniels, J.E.; Finlayson, T.R.; Studer, A.J.; Hoffman, M.; Jones, J.L. Time-resolved diffraction measurements of electric-field-induced strain in tetragonal lead zirconate titanate. J. Appl. Phys.
**2007**, 101, 094104. [Google Scholar] [CrossRef] - Zeng, J.T.; Kwok, K.W.; Tam, W.K.; Tian, H.Y.; Jiang, X.P.; Chan, H.L.W. Plate-like Na0.5Bi0.5TiO3 template synthesized by a topochemical method. J. Am. Ceram. Soc.
**2006**, 89, 3850–3853. [Google Scholar] [CrossRef] - Fancher, C.M.; Blendell, J.E.; Bowman, K.J. Poling effect on d33 in textured Bi0.5Na0.5TiO3-based materials. Scr. Mater.
**2013**, 68, 443–446. [Google Scholar] [CrossRef] - Fancher, C.M.; Jo, W.; Rödel, J.; Blendell, J.E.; Bowman, K.J. Effect of Texture on Temperature-Dependent Properties of K
_{0.5}Na_{0.5}NbO_{3}Modified Bi_{1/2}Na_{1/2}TiO_{3}-x BaTiO_{3}. J. Am. Ceram. Soc.**2014**, 8, 2557–2563. [Google Scholar] [CrossRef] - Newville, M.; Otten, R.; Nelson, A.; Ingargiola, A.; Stensitzki, T.; Allan, D.; Fox, A.; Carter, F.; Michał; Pustakhod, D.; et al. lmfit/lmfit-py 1.0.2 (Version 1.0.2). 2021. Available online: https://zenodo.org/record/4516651 (accessed on 5 June 2021).
- Chakoumakos, B.C.; Cao, H.; Ye, F.; Stoica, A.D.; Popovici, M.; Sundaram, M.; Zhou, W.; Hicks, J.S.; Lynn, G.W.; Riedel, R.A. Four-circle single-crystal neutron diffractometer at the High Flux Isotope Reactor. J. Appl. Crystallogr.
**2011**, 44, 655–658. [Google Scholar] [CrossRef] - Lutterotti, L. Total pattern fitting for the combined size-strain-stress-texture determination in thin film diffraction. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At.
**2010**, 268, 334–340. [Google Scholar] [CrossRef] - Lutterotti, L.; Matthies, S.; Wenk, H.-R.; Schultz, A.S.; Richardson, J.W. Combined texture and structure analysis of deformed limestone from time-of-flight neutron diffraction spectra. J. Appl. Phys.
**1997**, 81, 594–600. [Google Scholar] [CrossRef] - Matthies, S.; Lutterotti, L.; Wenk, H.R. Advances in Texture Analysis from Diffraction Spectra. J. Appl. Crystallogr.
**1997**, 30, 31–42. [Google Scholar] [CrossRef] - Matthies, S. 20 Years WIMV, History, Experience and Contemporary Developments. Mater. Sci. Forum
**2002**, 408–412, 95–100. [Google Scholar] [CrossRef] - Bernier, J.V.; Miller, M.P.; Boyce, D.E. A novel optimization-based pole-figure inversion method: Comparison with WIMV and maximum entropy methods. J. Appl. Crystallogr.
**2006**, 39, 697–713. [Google Scholar] [CrossRef] - Kieffer, J.; Karkoulis, D. PyFAI, a versatile library for azimuthal regrouping. J. Phys. Conf. Ser.
**2013**, 425, 202012. [Google Scholar] [CrossRef] - Lotgering, F. Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures—I. J. Inorg. Nucl. Chem.
**1959**, 9, 249–254. [Google Scholar] [CrossRef] - Lonardelli, I.; Gey, N.; Wenk, H.-R.; Humbert, M.; Vogel, S.C.; Lutterotti, L. In situ observation of texture evolution during α→β and β→α phase transformations in titanium alloys investigated by neutron diffraction. Acta Mater.
**2007**, 55, 5718–5727. [Google Scholar] [CrossRef] - Wenk, H.-R.; Lonardelli, I.; Pehl, J.; Devine, J.; Prakapenka, V.; Shen, G.; Mao, H.; Lonardeli, I. In situ observation of texture development in olivine, ringwoodite, magnesiowustite and silicate perovskite at high pressure. Earth Planet. Sci. Lett.
**2004**, 226, 507–519. [Google Scholar] [CrossRef] - Wenk, H.-R.; Cont, L.; Xie, Y.; Lutterotti, L.; Ratschbacher, L.; Richardson, J. Rietveld texture analysis of Dabie Shan eclogite from TOF neutron diffraction spectra. J. Appl. Crystallogr.
**2001**, 34, 442–453. [Google Scholar] [CrossRef] [Green Version] - Lonardelli, I.; Wenk, H.-R.; Lutterotti, L.; Goodwin, M. Texture analysis from synchrotron diffraction images with the Rietveld method: Dinosaur tendon and salmon scale. J. Synchrotron Radiat.
**2005**, 12, 354–360. [Google Scholar] [CrossRef] [PubMed] - March, A. Mathematical theory of regularity according to grain-form for affine deformation. Zeitschrift Krist. Krist. Krist. Krist.
**1932**, 81, 285–297. [Google Scholar] - Dollase, W.A. Correction of intensities for preferred orientation in powder diffractometry: Application of the March model. J. Appl. Crystallogr.
**1986**, 19, 267–272. [Google Scholar] [CrossRef] - Yilmaz, H.; Trolier-McKinstry, S.; Messing, G.L. Reactive templated grain growth of textured sodium bismuth titanate (Na
_{1/2}Bi_{1/2}TiO_{3}-BaTiO_{3}) ceramics—II dielectric and piezoelectric properties. J. Electroceramics**2003**, 11, 217–226. [Google Scholar] [CrossRef] - Bunge, H.J. Texture Analysis in Materials Science: Mathematical Methods; Butterworths: Boston, UK, 1982. [Google Scholar]
- Roe, R.-J.J.; Krigbaum, W.R. Description of crystallite orientation in polycrystalline materials having fiber texture. J. Chem. Phys.
**1964**, 40, 2608–2615. [Google Scholar] [CrossRef] - Roe, R.J. Description of crystallite orientation in polycrystalline materials. 3. general solution to pole figure inversion. J. Appl. Phys.
**1965**, 36, 2024–2031. [Google Scholar] [CrossRef] - Matthies, S.; Vinel, G.W. On the reproduction of the orientation distribution function of texturized samples from reduced pole figures using the conception of a conditional ghost correction. Phys. Status Solidi B Basic Solid State Phys.
**1982**, 112, K111–K114. [Google Scholar] [CrossRef] - Vanhoutte, P. A method for the generation of various ghost correction algorithms—The example of the positivity method and the exponential method. Textures Microstruct.
**1991**, 13, 199–212. [Google Scholar] [CrossRef] [Green Version] - Williams, R.O. Analytical methods for representing complex textures by biaxial pole figures. J. Appl. Phys.
**1968**, 39, 4329–4335. [Google Scholar] [CrossRef] - Imhof, J. Determination of a approximation of orientation distribution function using one pole figure. Zeitschrift Met.
**1977**, 68, 38–43. [Google Scholar] - Pawlik, K.; Pospieeh, J.; Lficke, K.; Pospiech, J.; Lucke, K. The odf approximation from pole figures with the aid of the adc method. Textures Microstruct.
**1991**, 14, 25–30. [Google Scholar] [CrossRef] - Schaeben, H. Entropy optimization in quantitative texture analysis. 2. application to pole-to-orientation density inversion. J. Appl. Phys.
**1991**, 69, 1320–1329. [Google Scholar] [CrossRef] - Li, S.; Huang, C.-Y.; Bhalla, A.S.; Cross, L.E. 90°-Domain reversal in Pb(Zr
_{x}, Ti_{1−x})O_{3}ceramics. Ferroelectr. Lett. Sect.**1993**, 16, 7–19. [Google Scholar] [CrossRef] - Wan, S.; Bowman, K.J. Modeling of electric field induced texture in lead zirconate titanate ceramics. J. Mater. Res.
**2001**, 16, 2306–2313. [Google Scholar] [CrossRef] - Jones, J.L.; Vogel, S.C.; Slamovich, E.B.; Bowman, K.J. Quantifying texture in ferroelectric bismuth titanate ceramics. Scr. Mater.
**2004**, 51, 1123–1127. [Google Scholar] [CrossRef] - Jones, J.L.; Slamovich, E.B.; Bowman, K.J. Domain texture distributions in tetragonal lead zirconate titanate by x-ray and neutron diffraction. J. Appl. Phys.
**2005**, 97, 034113. [Google Scholar] [CrossRef] - Vaudin, M.D.; Rupich, M.W.; Jowett, M.; Riley, G.N.; Bingert, J.F. A method for crystallographic texture investigations using standard X-ray equipment. J. Mater. Res.
**1998**, 13, 2910–2919. [Google Scholar] [CrossRef] [Green Version] - Seabaugh, M.M.; Vaudin, M.D.; Cline, J.P.; Messing, G.L. Comparison of texture analysis techniques for highly oriented alpha-Al
_{2}O_{3}. J. Am. Ceram. Soc.**2000**, 83, 2049–2054. [Google Scholar] [CrossRef] - Jones, J.L.; Slamovich, E.B.; Bowman, K.J. Critical evaluation of the Lotgering degree of orientation texture indicator. J. Mater. Res.
**2011**, 19, 3414–3422. [Google Scholar] [CrossRef] - Poterala, S.F.; Trolier-McKinstry, S.; Meyer, R.J.; Messing, G.L. Processing, texture quality, and piezoelectric properties of <001>
_{C}textured (1−x)Pb(Mg_{1/3}Nb_{2/3})TiO_{3}-xPbTiO_{3}ceramics. J. Appl. Phys.**2011**, 110, 8. [Google Scholar] - Matthies, S.; Priesmeyer, H.G.; Daymond, M.R. On the diffractive determination of single-crystal elastic constants using polycrystalline samples. J. Appl. Crystallogr.
**2001**, 34, 585–601. [Google Scholar] [CrossRef] [Green Version] - Wenk, H.-R.; Pawlik, K.; Pospiech, J.; Kallend, J.S. Deconvolution of Superposed Pole Figures by Discrete ODF Methods: Comparison of ADC and WIMV for Quartz and Calcite with Trigonal Crystal and Triclinic Specimen Symmetry. Textures Microstruct.
**1994**, 22, 233–260. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Daniels, J.E. Quantitative analysis of domain textures in ferroelectric ceramics from single high-energy synchrotron X-ray diffraction images. J. Appl. Phys.
**2017**, 121, 164102. [Google Scholar] [CrossRef] [Green Version] - Jones, J.L.; Iverson, B.J.; Bowman, K.J. Texture and anisotropy of polycrystalline piezoelectrics. J. Am. Ceram. Soc.
**2007**, 90, 2297–2314. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of the θ-2θ diffraction patterns for a randomly oriented and textured BNT-BT-KNN ceramic.

**Figure 2.**Measured 200 pole density of BNT-BT-KNN were of as measured (

**left**) and background subtracted (

**right**) BNT-BT-KNN (X) overlaid with a March–Dollase functional fit.

**Figure 3.**Comparison of the effect of spherical harmonic order (

**top**) and EWIMV ODF resolution (

**bottom**) on the texture and refinement error of BNT-BT-KNN sintered ceramics. Min and Max of f

_{002}, refinement error Rw are plotted against harmonic cutoff and EWIMV resolution.

**Figure 4.**Comparison of the reconstructed pole figure refined using from neutron diffraction pole figure collected on a 5° spacing with (

**a**) experimental reconstruction, (

**b**) WIMV, (

**c**) E-WIMV, (

**d**) Spherical Harmonic n = 10, and (

**e**) spherical harmonic n = 20. Note that the pole figures are rotated such that the left and right represent the sample normal direction.

**Figure 5.**Measured angular dependent diffraction data for BNT before (

**left**) and after (

**right**) application of strong electric fields (8 kV/mm) for 1 s. Data are plotted vs. the out-of-plan angle χ, with 0 and 90 representing data measured parallel and perpendicular to the applied electric fields, respectively. Electric poling drives substantial ferroelastic domain wall motion with aligns the 111 lattice planes in the field direction (χ of 0).

**Figure 6.**Normalization has a substantial impact on the determined fraction of domains aligned into the field direction. Data normalized using an experimental reference though peak fitting of data before poling (

**left**) estimates a higher degree of domain alignment than a calculated equivalent random intensity (

**right**).

**Table 1.**Summary of Lotgering factor for textured BNT-BT-KNN determined using either an experimental reference (powder diffraction pattern) or theoretical reference (modeled diffraction pattern from the crystallographic model). In addition, estimated errors in the Lotgering factor are reported in the parenthesis.

Lotgering Factor | Experimental Reference | Predicted Reference |
---|---|---|

f | 0.945 (7) | 0.946 (7) |

**Table 2.**Summary of the benefits and limitations of the qualitative and quantitative methods for assessing crystallographic textures in ferroelectric ceramics.

Methods | Benefits | Limitations |
---|---|---|

Lotgering Factor | Qualitative metric to assess the volume fraction of textured material. Helpful in determining the effect of processing on the crystallographic texture. | Only utilizes a single diffraction pattern. Analysis requires data measured using a Bragg-Brentano geometry and does not provide information about angular dependences of the texture. |

March–Dollase | Determines an estimate for the crystallographic texture of a given hkl pole of interest. | The commonly used formulation is limited to the analysis of materials with a fiber texture. Researchers must account for background and absorption contributions. Analysis requires angular-dependent diffraction data. |

ODFs | Rigorous method for determining the strength and symmetry of a materials crystallographic texture. | Determining an ODF through Rietveld texture analysis requires information about the crystallographic structure (space group and atomic positions). Analysis requires angular-dependent diffraction data. |

DOD | Metric to quantify the evolution in the domain alignment in response to a thermomechanical poling process. | Requires precise information about the initial state with a random domain configuration. |

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**MDPI and ACS Style**

Fancher, C.M.
Diffraction Methods for Qualitative and Quantitative Texture Analysis of Ferroelectric Ceramics. *Materials* **2021**, *14*, 5633.
https://doi.org/10.3390/ma14195633

**AMA Style**

Fancher CM.
Diffraction Methods for Qualitative and Quantitative Texture Analysis of Ferroelectric Ceramics. *Materials*. 2021; 14(19):5633.
https://doi.org/10.3390/ma14195633

**Chicago/Turabian Style**

Fancher, Chris M.
2021. "Diffraction Methods for Qualitative and Quantitative Texture Analysis of Ferroelectric Ceramics" *Materials* 14, no. 19: 5633.
https://doi.org/10.3390/ma14195633