COMMENTS TO A PUBLICATION OF T . I . SAVYOLOVA CONCERNING DOMAINS OF DEPENDENCE IN POLE FIGURES

A questionable publication of T.I. Savyolova is analysed. The paper claims, that it is possible to determine Ph(y) pole figure values for y E Y(h; ho, Y0) using only data from Y E Y0 of a single pole figure P0 (Y)The contradiction to the common knowledge can be resolved well understanding that there is a general difference between the term solutions (or continuation of solutions) of an ultrahyperbolic equation (satisfied by the axis distribution function A(h, y)) and the term pole figure. Pole figures considered in texture analysis are two-dimensional projections of a three-dimensional object (the ODF f(g)). For limited data sets the equation AA(h, y) AyA(h, y) bears only a necessary, but not sufficient character in order to get solutions of interest, i.e. it cannot be guaranteed that a h-specific continuation of a starting solution P0 (Y) will be a "h-projection" of the same ODF, which belongs to a concrete sample and possesses the "h0-projection" P0 (Y). Consequently, to name such a h-specific continuation "pole figure" is incorrect. The consideration of formal (unambiguous only by artificial conditions) continuations of solutions may be mathematically interesting, but is ofno practical importance for texture analysis. Examples (already considered by the authors about twenty years ago) are given, how to construct in a much more simple way continuations of solutions of the same useless type like in the paper under discussion.


INTRODUCTION
In her publication "Calculation of domains of dependence for pole figures with an ultrahyperbolic equation" (Savyolova, 1995; see also Savyolova, 1996) the author comes to the spectacular conclusion that it is possible, using the (arbitrary) intensity distribution given in a yregion Y0 of a single pole figure P0 (Y), to determine the intensities for some other pole figures Ph,(y) in (h0, Y0, hi)-specific regions Yi.
We definitely disagree with this statement. The reason for the dubious conclusion by T.I. Savyolova is finally a misunderstanding and not appropriate use of the (well defined in texture analysis) term "pole figure", considering solutions of an equation of only necessary but not sufficient character. The mathematically formulated reply can therefore be relatively short and is given in the appendix.
There are some motives for an enlarged form of the present publication: The "fairy-tale" on ODF-determinations, using only one (and even incomplete) pole figure still appears from time to time in literature. We would like to recommend potentially interested readers of Savyolova's paper not to waste time in a, at first sight, new interesting direction (cf. Nikolayev and Schaeben, 1997).

And at last:
It is not simple to read and to analyse Savyolovas paper (doubtless containing interesting valuable mathematical aspects) and to find out the critical moments related to texture analysis. Therefore (unlike the request of the appendix) in the main part we try to give some necessary background information, more or less popular (or even simplified) explanations of the nontrivial interesting problem, and will demonstrate several weak points of the discussed paper.
Also, because Savyolova's idea, if it works, must be applicable to the case of "ideal pole figures" too (triclinic (i.e. "no") crystal symmetry and no +hi-reduction by Friedels law), at the beginning we will not consider these complicating features, but unessential for the basic problem under consideration. We have already seen in the past, that there may appear some difficulties between abstract mathematical considerations of a problem and its realization in a "language" prescribed by the experimental possibilities, i.e. adapted to the practical situation in texture analysis. So, at the very beginning of the 1980s, we derived, for instance, the mathematically exact solution of the problem ofpole figure inversion in a closed analytical form using two completely different methods (Matthies, 1979;1980;Muller et al., 1981). For the above mentioned ideal case Cideal pole figures" P) this solution reads: Therefore, the aim of any method of pole figure inversion is to extract maximum information about the ODF from the deficient 210 S. MATTHIES AND C. ESLING volume of experimental data. In this respect, reasonable ODF approximations ought to be constructed, and the restricted nature of the pole figure-related input data should concern only the resolution power of these approximations.
From a mathematical point of view, there is a perfect symmetry between the direct Ph(Y) and the inverse Ry(h) pole figures which can both be considered as specific representations of the so-called biaxial function A(h, y) (Bunge, 1982): Disregarding problems of experimental feasibility and considering A(h, y) as a purely mathematical object with continuously varying y and h, abstract coordinate transformations mixing h and y are also imaginable.
In 1982, Savyolova published a rather elaborate study from a mathematical point of view, that was indeed based on such mixed variables. We could infer from this paper that the problem of pole figure inversion in such a representation can be solved using the socalled Radon transformation and that there is an infinite number of possible formulations of the solution of the problem. However, up to now there has been published not even one variant from this multiplicity of solutions, which refers to experimentally accessible quantities in terms ofh and y and which can be directly compared with our simple, analytically closed and transparent solution (Eqs. (1), (2)).

ANALYTICAL DEPENDENCIES BETWEEN POLE FIGURES
A quite simple fact immediately leads to acknowledging that there must be some kind of dependency between y and h in the biaxial function A(h, y) (of. Eq. (3)). As is well known, the orientation space G has a volume of 8r2: where r (0, o), and Ar is the angular part of the Laplace operator in spherical co-ordinates: Ar (1 z 2) 02 z z2-2z + o z 1 g 2 0 2' z cos . (7) The harmonic method is based on the eigenfunctions El, re(r) of this operator (Edmonds, 1957) with the eigenvalues l(1 + 1): mr Yl,m (r) Z(l + 1) Yl,m (r).
The series representation of pole figures by these eigenfunctions is absolutely model-independent: A(h'y):47rZ E 21+1 1=o m,n=-I Using (8), Eq. (6) can immediately be deduced from (9). The relation (6) has already been observed and discussed by us during international methodological seminars in Metz in the early 1980s (Baro et al., 1982).
It was analysed at that time as a possibility to reduce the necessary volume of experimental data for ODF reconstruction. But finally it was rejected.
The reason why we have concluded that this path is an impasse is the following. Equation (6) obviously connects the y and h spaces in a specific way. Thus, as a matter of principle, it should be possible to transform y-related information into a h-related one. Since Eq. (6) (using the most abstract characterization) is a partial second-order differential equation for a function of four variables, an unique solution requires information on the values of the asked function (as well as on its first derivatives in h) on a finite h-depending three-dimensioaal boundary, i.e. in any case, these requirements will be linked with continuous h variations, which are mathematically conceivable, but cannot be realized experimentally (not even in a bad approximation, especially for the derivatives, due to the discrete and accidental character of the available reflections hi).
At least from this formal view, it seems apparent to us, that one single pole figure (a single h, i.e. one point in the h space) cannot serve as a basis for reasonable estimations of distributions in a continuous h domain.
Solutions of Eq. (6) of common (ambiguous) character, or formally constructed in an unique way, but using artificial model conditions are not of practical interest. Moreover, up to now we even have not considered the much more important question, whether Eq. (6) will be able at all to deliver solutions of interest for texture analysis resting on a limited set of information only. Some decisive circumstances have to be underlined yet. Independently from the fact whether at all and how an information can be transmitted from Y0 to Y1, at least one thing is quite sure: the domain Y1 for Ph (y) is determined through .4 purely geometrically by h0, hi and Y0. Concerning its determination, equations of type (6) are absolutely unimportant. On the contrary, possible formal continuations of solutions into a domain Yf, which are deduced from an abstract treatment of Eq. (6), must be complemented with additional conditions such as Y f3 Y. This is the minimum level at which the necessary information, according to which the P(y) are projections of a common object defined in a space of higher dimension, has to be taken into account. This common property of true pole figures in no way follows from Eq. (6) or any transformed variants of it without additional information.
Indeed, only possessing the data of A(h, y) for all h and y, where all are satisfying (6), the projection character in the form (3) can be concluded for this data set. Equation (6) does not know anything whatsoever about projections.
In the most general solution Eq. (9) arbitrary C/"'" values are admissible. For a C'" set, that exactly explains P0(Y) > 0 (in the common case the C 'n cannot be determined unambiguously from one given pole figure, see also Bunge and Esling, 1979), even the fulfillment of the simple condition P (y) > 0 cannot be guaranteed for a "hi pole figure" formally calculated via Eq. (9), although it is an exact solution of Eq. (6).
In other words, Eq. (6)  Eq. (6) is not sufficient to deliver true pole figure data P (y) by any kind of "continuation of its solutions", resting only on P0 (y)-data.
Therefore, the contradiction under discussion can be resolved well understanding that there is a general difference between the terms "solutions or continuations of solutions" of Eq. (6) (that might be of some abstract mathematical interest) and the term "pole figure" in the sense (3), the only object of interest for texture analysis. There is nothing new about the determination of the Yi either. Since the mid 1970s (beginning with the vector method (Ruer and Baro, 1977)) its inverse formulation (which A cf. In spite of the possibility to demonstrate the main idea considering first the simplest case of ideal pole figures, Eq. (3) (without symmetries and loss of information in the normal diffraction experiment (Friedel's law)), her paper is burdened with all these elements of secondary importance for the problem under consideration. As a consequence it is rather difficult to follow it up and to analyse it in detail. The basic program of the author is as follows. By transforming variables h and y (10), it is possible to bring Eq. (6) (.4 -, F) into the form of an ultrahyperbolic Eq. (11): Xl x (h, y), x2 x2(h, y), Zl Zl (h, y) and z2 z2(h, y), Then x and X 2 are transformed into polar coordinates (r, qo) and z and z2 into (p, b), respectively. Considering F as an abstract function of these variables it can be decomposed into a series of Bessel functions similar to Eq. (9). In this respect, a model parameter a > 0 is required and can either be finite or infinite. For known solutions in given r and p domains there are in both cases possibilities to continue the solution into new regions of the p domain, correspondingly lowering the size of the r domain. Some prescriptions for the determination of the Bessel function series coefficients are also provided (S. (11)). Analysing these connections in more detail the necessity of the above-mentioned hderivations, when we analysed in Section 3 Eq. (6) as a partial secondorder differential equation, can be seen.

S. MATTHIES AND C. ESLING
Then an attempt is made to apply the results of the abstract case to the case of pole figures (i.e. F= A). Concretely, the case of a single pole figure and only for special h with 0h-7r/2 is considered.
However, as opposed to the above abstract case, the description of the ways to get new solutions is relatively fuzzy and difficult to follow. For instance, the series coefficients have to be derived from a system of algebraic equations (see below S. (17)) which is apparently underdetermined in the general case (this is for example in agreement with the difficulties mentioned in connection with the analysis of Fig. 1).
The consequence of the forced coupling of h y in the Bessel functions (instead of being clearly separated as in the exact expression (9)) in order to get a resolvable scheme are assumptions of kind F(p a) O, or cOF/c3p],= O. After this, the ambiguity Of a continued solution may be overcome. However, these conditions having been imposed in an artificial way, they may lead to satisfactory approximations for an infinite space, but they may also have consequences in the finite pole figure space of a magnitude that can hardly be evaluated.
Furthermore, an independence of F from qo is mentioned. This does not correspond to the reality (apart from a trivial particular case see below), because crystal lattices with cylindrical symmetry do not exist (which additionally was supposed for any h). Trying to confirm the results obtained by the author, at least for some numerical examples, we already failed with the very first continuation domain. Using the relations at the bottom of S. p. 190 from a-2(2 + x/), it follows Xt-60 .F or the h0 of the pole figure measured in the domain Y0(0 < 0 < 60, 120 < tgY0 < 180) we select 0 90o, We have some difficulties to imagine how the author intends to generate a peak in Yi' from the zero intensity in Y0. Using the terminology of Fig. 1, the present case obviously belongs to a situation for which Yi' is located outside Y1 and possesses projection threads nowhere intersecting the G-space re#on (A + B + C) determined by Y0. Figures 4 and 5 give an additional example in connection with the discussion of Fig. 1. If instead of the texture component at the above g we have one at g*= {351.85 , 60.47 , 1.12} the h0 pole figure will be identically the same. However, the corresponding h figure ("determinable" by these h0 pole figure data) given in Fig. 5 differs from that of Though commonly one counter-example is sufficient in order to question the merit of a certain concept, it seems reasonable additionally to consider some other issues of Savyolova's paper.

THE BERYLLIUM EXAMPLE
The author tries to confirm the derived relationships on the basis of some experimental data. Unfortunately, this example only represents a very particular and irrelevant case. On the basis of the assumptions 220 S. MATTHIES AND C. ESLING made, according to which F= A(hi, y) should not depend on oh, and Oy, it can immediately be inferred that this is possible only for a cyclic fibre component, with N =H =(0 , 0) (Matthies et al., 1990). Consequently, already the starting assumptions determine an ODF, not only a pole figure type. In other words, in Eq. (9), all C ''n with m, n 0 have to be identical to 0. The consequence is, that all coefficients C/' completely describing the ODF in this case can be determined on the basis of only one pole figure. Thus, any single pole figure already represents the whole ODF of such a sample. But, as it is well known, any other pole figure can then be determined in the entire y domain from the ODF.
In this sense, a statement with a similar content by the author is only a confirmation of a necessary condition, without any additional information based on Eq. (11).
Finally, trying quantitatively to understand the example, we again met with difficulties to follow the "ultra-hyperbolic calculating mode". Because r2-p 2 =-4 (for any X), we obtain in the Eq. (S. (19)) for cos on S. page 194 a zero, located in denominator, and were thus unable to continue the calculations in order to confirm the results of the author.

SYMMETRIES
Symmetries play a very important role in quantitative texture analysis and have therefore to be considered with utmost care. The author has, to some extent, taken this into consideration; it is reflected in the 4-hi solution variants and leads to a reduction of the domain of solution continuation via the condition G G + tO G-(S.p. 190). But some basic errors appear even at this relatively elementary level.
Over half of the work is devoted to the case h0 (0 90 , ) for which the statement "F+= F-" is made (S.p. 189, bottom). This is simply not true as it can be seen, for instance, with Eq. (S. (2)) and (S. (3)) on S. page 186. The cause of such errors is the somewhat lax determination of azimuthal quantities (0 < < 2r) only via tan (see the top of S. page 188). In reality, + anddiffer. Therefore, statements like "for every kind of lattice symmetry" (S.p. 190) about the validity of the derived relations are even more surprising, the more COMMENTS TO A PUBLICATION OF T.I. SAVYOLOVA 221 so as nowhere in the paper can any consideration be found concerning the consequences of the crystal symmetry for the problem described.
Moreover, because "every kind of lattice symmetry" means also the triclinic case, from this view our simplifying (but not changing the validity of the general statements) limitation to this symmetry case in the appendix can be considered as justified.
Up to now in this review, we have deliberately considered only the triclinic case (and ideal pole figures), without the +h symmetry appearing due to normal scattering. But the crystal symmetry (point group fin) has to be treated in an absolutely equivalent way to the +h symmetry. This means, that the exact relation for the experimental ("reduced") pole figures reads: where hi, on the lehand side, is only a name for the whole set of equivalent hi. (Matthies et al., 1988).
Independently of the fact that, numerically, the Ph,. (y) values for additional points of input, which should first be shown before statements with so far-reaching consequences can be made as "every kind of...'.

SINGLE POLE FIGURES
In order to avoid any risk of misunderstanding with the reader, we would like to underline that it might be possible, in specific cases (very sharp textures, high crystal symmetry, and satisfaction of the necessary MPDS-conditions (Helming, 1992)), to determine an ODF from a single pole figure with a fair degree of approximation. For instance, if only a few nonoverlapping sharp peaks (surrounded by large regions with very small intensities) are present in a cubic (1 1) pole figure.
Consequently, using this ODF f(g) it is possible to calculate any other pole figure with the same accuracy. However, this is feasible only by using a "detour via the ODF'. It does not require either Eq. (6) or its equivalent formulations as in Eq. (11). But, on the contrary, it needs Eqs. (13) and (3), which are the only relevant ones, and special cases of the structure of the given pole figure.
T.I. Savyolova considers a problem of a totally another type: to calculate in Y1 a distribution (designed by her "hi pole figure"), directly from a pole figure /30(y) measured in Y0, as a continuation of a solution of Eq. (11). Such a "hi solution" determined by the proposed procedure is not of any practical interest for texture analysis. To name it "hi pole figure" is incorrect. How to construct quite simply (without complicated and rather nontransparent ultra-hyperbolic and h ymixing activities) "continuations of solutions" of the same useless type (but also satisfying Eq. (6) and resting only on P0 (Y) data in Y0), was already mentioned at the end of Section 4 and is described in some more detail in the appendix. Even only trying to go into this direction, there will be difficulties in connection with the discrete character of measurable projection directions hi, with additional restrictions to the formal domain for solution continuations as consequences of a proper trettment of the crystal symmetry as well as of the projection character of pole figures and their definite positivity. It is quite questionable whether a proper treatment of crystal symmetry by h, y-mixed variables exists at all, these being the basic elements of the author's idea.
In a publication with such spectacular statements, a certain degree of numerical accuracy should be warranted. Moreover, before implementing highly sophisticated mathematical tools, some attention should be paid to simple analogic pictures of the problem, like Fig. 1 that shows fight from the beginning the dubious nature ofthe statements made in the paper in its present form.
In this connection we would like to point out that the computer programs for standard functions have been available to the texture community since 1987 (Matthies et al., 1987), including the scientific environment of the author. They have been specially developed for proper testing procedures with high numerical accuracy and can answer such questions as those treated here (e.g. Figs. 2-5) in a matter of seconds.