INTRODUCTION

Most materials used in industry (metals and alloys, ceramics, and minerals) have a polycrystalline structure. Many properties of polycrystals depend on how the individual grains that make up the material are oriented relative to a fixed external coordinate system. If the grains in the material have random orientations, this state is called textureless. In the case of preferential orientations, it is called crystallographic texture [1, 2].

The most complete information about the texture of a material is provided by the orientation distribution function (ODF): a probability density distribution function normalized to the orientation space volume. The main task of quantitative texture analysis is the reconstruction of the ODF from its two-dimensional projections: pole figures (PFs) obtained using X-ray or neutron diffraction.

The ODF can be reconstructed using so-called direct methods (WIMV method, arbitrarily defined cells method: ADC, vector method) based on the fact that the discrete half-sphere of the PF corresponds to families of projection tubes in the ODF space [3–7]. The advantage of direct methods is the strict fulfillment of the nonnegativity condition of the ODF, but a disadvantage is the occurrence of false maxima.

Component methods are also used, where the ODF is described by a set of standard components with different weights and scattering parameters [8, 9]. Its advantages include a small number of texture components, direct determination of the volume fractions of the obtained components, fulfillment of the nonnegativity condition of the ODF, and absence of false maxima. The main drawback is the difficulty in determining the initial values of parameters (texture components, their weights, and scattering) from PF analysis.

Analogs of the component method are the automated component method, the method of ODF reconstruction using the superposition of normal distributions, and the MTEX method [4, 10–12]. In these methods, the ODF is determined as a superposition of a large number of standard distributions with fixed scattering.

The method of ODF reconstruction by expanding it in a Fourier series of spherical harmonics (harmonic method) is characterized by simplicity, universality, and stability of the solution found [1, 3, 13, 14]. Its disadvantages include the determination of only the even part of the ODF owing to the peculiarities of the diffraction experiment, which leads to the appearance of negative values and false maxima [14–16], as well as its inapplicability to sharp textures.

The harmonic method allows for the reconstruction of the ODF for any material classes and any sample symmetries, primarily for cubic or hexagonal sample symmetries and orthorhombic lattice symmetries, which are characteristic of the textures of rolled metals. To fulfill the nonnegativity condition of the ODF and eliminate false maxima in the harmonic method, approaches such as regularization of the solution and the positivity method are used [17, 18].

The aim of this work is to reconstruct the ODF for materials with low symmetry of the lattice and the sample using the harmonic method.

EXPERIMENTAL

The primary processing of experimental PFs included correction for the defocusing effect arising from the sample tilt in the process of PF measurement, symmetrization, and pseudonormalization.

Correction factors for the defocusing effect were determined according to the accepted method [19, 20]. The following equation was used for pseudonormalization factors \(N_{i}^{'}\) [21]:

$$N_{i}^{'} = \frac{{\int\limits_0^{{{\Phi }_{{\max }}}} {\int\limits_0^{2\pi } {\sin (\Phi ){\text{d}}\Phi {\text{d}}\beta } } }}{{\int\limits_0^{{{\Phi }_{{\max }}}} {\int\limits_0^{2\pi } {I(\Phi ,\beta )\sin (\Phi ){\text{d}}\Phi {\text{d}}\beta } } }},$$
(1)

where Φ and β are the radial and the azimuthal angles on PF, Φmax is the maximum tilt angle when measuring PF, and I(Φ, β) are the measured intensities.

Symmetrization consists of equalizing the intensity values at symmetrical points on the PF:

$$\begin{gathered} {{I}^{{{\text{sym}}}}}({{\Phi }_{1}},{{\beta }_{1}}) = {{I}^{{{\text{sym}}}}}({{\Phi }_{2}},{{\beta }_{2}}) \\ = \left[ {I({{\Phi }_{1}},{{\beta }_{1}}) + I({{\Phi }_{2}},{{\beta }_{2}})} \right]{\text{/}}2, \\ \end{gathered} $$
(2)

where (Φ1, β1) and (Φ2, β2) are the symmetrical points on PF, and I(Φi, βi) and Isym(Φi, βi) are the actual measured and symmetrized values of the intensity.

Symmetrization is necessary to reduce statistical errors in experimental PFs, the level of which can be estimated from differences in intensity at symmetric points [17]. Note that, in the case of triclinic symmetry of the sample, a plane is selected relative to which the reflections on the PF are located symmetrically, and symmetrization is carried out relative to this plane.

The pole density Ph(y) = NhIh(y) is related to the ODF f(g) by the following relations [1, 16]:

$$\begin{gathered} {{P}_{h}}(y) = \left[ {Rf(h,y) + Rf( - h,y)} \right]{\text{/}}2, \\ Rf(h,y) = \frac{1}{2}\int {f(g){\text{d}}g} , \\ \end{gathered} $$
(3)

where h is the crystallographic direction, y is the direction in the sample, and g is the orientation in the space of Euler angles (integration is carried out over all orientations that transform the direction h into the direction y: gh = r).

The harmonic method for reconstructing the ODF is based on its expansion in a Fourier series in three-dimensional symmetric spherical functions [1]:

(4)

where g is the orientation in Euler space and \(C_{l}^{{\mu \vartheta }}\) are the expansion coefficients (l = 0, 2, …, lmax is the order of spherical functions; μ = 1, 2, …, M(l) and ϑ = 1, 2, …, N(l) are summation indices; and M(l) and N(l) are the number of linearly independent harmonics depending on the symmetry of the lattice and the sample, respectively).

Symmetric spherical functions are a linear combination of generalized harmonic functions \(T_{l}^{{mn}}\) [1]:

(5)

where \(\ddot {A}_{l}^{{m\mu }}\) and \(\dot {A}_{l}^{{n\vartheta }}\) are coefficients that ensure the fulfillment of the symmetry conditions of the lattice and the sample, and \(P_{l}^{{mn}}\)(cosΦ) are the generalized associated Legendre functions.

The ODF described by Eq. (2) corresponds to the pole density on the direct pole figure (DPF) [1]:

$${{P}_{h}}(y) = \sum\limits_{l = 0}^{{{l}_{{\max }}}} {\sum\limits_\mu ^{M(l)} {\sum\limits_\vartheta ^{N(l)} {\frac{{4\pi }}{{2l + 1}}C_{l}^{{\mu \vartheta }}\ddot {k}_{l}^{{*\mu }}(h)\dot {k}_{l}^{\vartheta }(y)} } } ,$$
(6)

where \(\ddot {k}_{l}^{{*\mu }}\)(h) and \(\dot {k}_{l}^{\vartheta }\)(y) are surface spherical harmonics, which have symmetry between the lattice and the sample (the asterisk denotes complex conjugation).

The reverse pole figure is described in a similar way.

Symmetric surface harmonics \(\dot {k}_{l}^{\vartheta }\) in much the same way as spherical functions are a linear combination of normalized spherical surface harmonics \(k_{l}^{n}\) [1]:

$$\begin{gathered} \dot {k}_{l}^{\vartheta }(\Phi ,\beta ) = \sum\limits_{n = - l}^l {\dot {A}_{l}^{{n\vartheta }}k_{l}^{n}(\Phi ,\beta )} , \\ k_{l}^{n}(\Phi ,\beta ) = \frac{1}{{\sqrt {2\pi } }}{{e}^{{in\beta }}}\bar {P}_{l}^{n}(\cos {\kern 1pt} \Phi ), \\ \end{gathered} $$
(7)

where \(\dot {A}_{l}^{{n\vartheta }}\) are the symmetry coefficients and \(\bar {P}_{l}^{n}(\cos {\kern 1pt} \Phi )\) are the normalized associated Legendre functions.

Spherical harmonics \(k_{l}^{n}\)(Φ, β) and \(T_{l}^{{mn}}\) are generally complex, and as a result, the coefficients \(C_{l}^{{\mu \vartheta }}\) are also complex. However, one can introduce real functions \(\bar {k}_{l}^{n}\)(Φ, β) and \(\bar {T}_{l}^{{mn}}\) that are linear combinations of the corresponding complex functions. Symmetric real harmonics are defined in much the same way as symmetric complex harmonics according to Eqs. (5) and (7). Real harmonics also form an orthogonal basis and can be used for decomposing the ODF into a Fourier series [22].

The unknown coefficients of the expansion \(C_{l}^{{\mu \vartheta }}\) are determined by minimizing the square of the residual r2:

$${{r}^{2}} = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{{\omega }_{{i,j}}}{{{\left\{ {{{N}_{i}}I_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}}) - P_{{{{h}_{i}}}}^{{{\text{model}}}}({{y}_{j}})} \right\}}}^{2}} \to \min ,} } $$
(8)

where I and J are the amount of PFs and the number of measured points in a sample, \(I_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}})\) are the measured intensities for the ith PF at the jth point, Ni are the unknown normalizing factors for the ith PF, \(P_{{{{h}_{i}}}}^{{{\text{model}}}}({{y}_{j}})\) is the model value of polar density determined by Eq. (6), and wi,j is the weight determining the significance of the measurement.

The least squares method is often used to solve Eq. (3). Writing the unknown values \(C_{l}^{{\mu \vartheta }}\) as vector c, the measured pole densities \(P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}})\) as vector p, and introducing the matrix

we obtain the following equation [23]:

$$C = {{({{A}^{{\text{T}}}}A)}^{{ - 1}}}{{A}^{{\text{T}}}}p.$$
(9)

In view of the poor conditionality of the matrix A caused by statistical errors in the initial data and the nonoptimality of the diffraction experiment, the solution may contain large errors, for reduction of which regularization of the solution was used [17]. After using ridge regression, the solution takes the following form [17, 24]:

$$C = {{({{A}^{{\text{T}}}}A + \lambda Q)}^{{ - 1}}}{{A}^{{\text{T}}}}p,$$
(10)

where λ is the regularization parameter and Q is the smoothness functional (in this case chosen as Q = l(l + 1)/(2l + 1), l = 0, 2, …, L).

A correctly selected regularization parameter λ makes it possible to reduce the probability of the appearance of false maxima and large negative ODF values without significantly underestimating the intensity. The inverse matrix in Eqs. (9) and (10) is determined using singular value decomposition.

The error in determining the ith coefficient is [17]

$$\Delta {{c}_{i}} = \sqrt {D[{{c}_{i}}]} = \sqrt {{{\sigma }^{2}}{{{\left[ {{{{({{A}^{{\text{T}}}}A + \lambda Q)}}^{{ - 1}}}} \right]}}_{{ii}}}} ,$$
(11)

where D[ci] is the dispersion, σ2 = r2/(m – n), r2 is the sum of squared residuals obtained as a result of the solution; and m and n are the numbers of rows and columns of matrix A.

In order to find the unknown normalization factors Ni, it is necessary to use an iterative algorithm, since an additional condition is imposed on Ni: Ni > 0. In our case, the normalization factors were found using the TRF method [25].

The error in calculation of the ODF is [17]

$$\Delta f = \sqrt {\frac{1}{{2l + 1}}\sum\limits_{l,\mu ,\vartheta } {{{{\left( {\Delta C_{l}^{{\mu \vartheta }}} \right)}}^{2}}} } .$$
(12)

To check the adequacy of the solution, the RP factor was used. The RP factor for the ith PF and the average RP factor were calculated as [8, 26]

$$R{{P}_{{{{h}_{i}}}}}(x) = \frac{1}{J}\sum\limits_{j = 1}^J {\frac{{\left| {P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}}) - P_{{{{h}_{i}}}}^{{{\text{calc}}}}({{y}_{j}})} \right|}}{{P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}})}}} 0\left[ {x,P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}})} \right],$$
(13)
$$0\left[ {x,P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}})} \right] = \left\{ \begin{gathered} 1,P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}}) > x \hfill \\ 0,P_{{{{h}_{i}}}}^{{\exp }}({{y}_{j}}) \leqslant x, \hfill \\ \end{gathered} \right.$$
(14)
$$RP(x) = \frac{1}{I}\sum\limits_{i = 1}^I {R{{P}_{{{{h}_{i}}}}}(x).} $$
(15)

It should be mentioned that the ODF coefficients calculated by the described method correspond only to the even part of the ODF. To approximately determine the odd part, one can use the iterative positivity approach [18].

RESULTS AND DISCUSSION

The harmonic method of ODF reconstruction was applied to a model single-component base texture and the texture of a Mg–4.5% Nd alloy subjected to equal-channel angular pressing. The lattice symmetry in both cases corresponds to the symmetry of magnesium and is hexagonal with point group 6/mmm, while the sample symmetry is triclinic. The maximum degree of expansion in the series is lmax = 24 (experimental PFs [27]).

The synthetic base texture is defined by a single component with Euler angles Φ1 = 20° and Φ = φ2 = 0° and a weight of 0.1. Six DPFs were used as input data: {001}, {100}, {101}, {102}, {103}, {110}.

Figure 1 shows the original and calculated DPFs, and Fig. 2 shows sections of the true (model) and calculated ODF at φ2 = 0. The RP factors and calculated ODF errors for the model and real textures are given in Table 1.

Fig. 1.
figure 1

Initial (a) and calculated (b) direct pole figures for model single-component texture.

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Fig. 2.
figure 2

Sections of the initial (a) and calculated (b) ODF for model single-component texture (φ2 = 0).

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Table 1.   RP factors and ODF calculation errors for the model texture and texture of Mg–4.5% Nd alloy
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The RP factors and Δf show good agreement between the calculated and original data, but the calculated ODF contains false peaks and its maximum density is lower than the original. The degree of uncertainty in ODF lies in the range 0 ≤ f(g) ≤ 2 [1]; therefore, the false maxima in this reconstructed OFD can be attributed to a nontextured state. Thus, the main difference between the calculated and model ODF is the difference in maximum density.

For the reconstruction of the ODF of the alloy sample, the same set of DPFs was used as for the model texture. Figure 3 shows the original and calculated DPFs, and Fig. 4 shows a section of the reconstructed ODF at φ2 = 0.

Fig. 3.
figure 3

Initial incomplete (a), symmetrized incomplete (b), and calculated complete (c) direct pole figures for Mg–4.5% Nd alloy.

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Fig. 4.
figure 4

Section of the reconstructed ODF for Mg–4.5% Nd alloy (φ2 = 0).

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The RP factors and Δf for the alloy texture are higher than for the model texture, but they also show good agreement between the calculated and original data. However, negative values are present in the ODF, which are due to limitations of the harmonic method. The absolute magnitude of the negative ODF values is significantly smaller than the maximum intensity. The ODF contains two texture components g1 = {95, 25, 0} and g2 = {270, 30, 0} with maximum intensities of 13.81 and 2.23, respectively. The Miller–Bravais indices for these components are (\(\bar {1}\)2\(\bar {1}\)6)[1\(\bar {2}\)11] and (\(\bar {1}\)2\(\bar {1}\)6)[\(\bar {1}\)2\(\bar {1}\)1].

CONCLUSIONS

The model single-component texture and the texture of the Mg–4.5% Nd alloy sample subjected to equal-channel angular pressing are characterized by hexagonal lattice symmetry and triclinic sample symmetry. The conducted research using the harmonic method showed that, in both cases, the RP factors and the error of calculation of the ODF exhibit good agreement between the calculated and original data. Additionally, the ODF of the magnesium alloy sample contains two texture components (\(\bar {1}\)2\(\bar {1}\)6)[1\(\bar {2}\)11] and (\(\bar {1}\)2\(\bar {1}\)6)[\(\bar {1}\)2\(\bar {1}\)1] with maximum intensities of 13.81 and 2.23.

Thus, the applied harmonic method for ODF reconstruction is an efficient tool for texture studies of various materials. It allows for the reconstruction of ODFs and full PFs, expanding the range of materials for texture studies to include ceramics, rocks, and other nonmetallic materials with low symmetry.