The Use of a Quadratic Form the Determination of Non-negative Texture Functions for

The classical analysis of measured pole figures of textured polycrystals by the series expansion method does not necessarily produce a non-negative texture function. The main reason for this is, that the method is unable to find the terms of odd rank of the series expansion. A new method is proposed, which introduces the non-negativity condition into the series expansion method by the use of quadratic forms. The method is found to be successful when treating sharp textures, which have a considerable zero range in Euler space. The preliminary determination of this zero range by experimental methods is however not necessary.


INTRODUCTION
The well-known series expansion methods allows the calculation of the crystal orientation distribution function of a polycrystal from measured pole figures (Bunge, 1969).
The O.D.F. is written as follows" The C coefficients are to be obtained from measured data such as pole figures.
An orientation distribution function cannot be negative because of 2 P. VAN  i) Small errors in the measured pole figures. ii) Truncation of the series expansion at some maximal value L of l. iii) Omission of the terms with odd values of I. As was indeed first pointed out by Matthies (1979), the classical method for analysis of measured pole figures only produces the so-called "reduced O.D.F.", which corresponds to the terms of even rank l: The true O.D.F. is then given by: f(g) =.if(g) + f(g) (3) where f(g) corresponds to the terms of odd rank l.
The true O.D.F. is usually zero in a large part of Euler space ("zero range") when the texture is sharp. In such case, the reduced O.D.F. can have rather important negative peaks. False positive peaks ("ghost maxima") may appear as well. Moreover, it will usually underestimate the value of the true maxima. Bunge and Esling (1979) have suggested that the non-negativity condition could be used in order to estimate the odd partf(g) of the texture function. They have proposed a method which requires the preliminary determination of the zero range. Results have recently been presented (Bechler-Ferry, . The method tries to satisfy Eq. 3 in the zero range. It does not necessarily satisfy the non-negativity condition outside the predetermined zero range. Truncation effects might still cause minor negative values in the zero range. Liicke, Pospiech, Virnich and Jura (1981) (Pospiech and Jura, 1974;Morris, 1975;Humbert and Bergmann, 1980;Van Houtte, 1980;Lian, Xu and Wang, 1981). In case complete pole figures are available, it is better to use version (b), which tries to fit the pole figures recalculated from the quadratic form directly with the measured pole figures. Version (a) and (b) will each have their own error criterion. Let f(g) be an estimation of the true texture function f(g). f(g) can be written as a series expansion: The coefficients C can be found by numerical integration when the estimation f(g) is known (Bunge, 1969). Both criteria define a sum of squares A to be minimized. The method for adjusting the function h(#) (Eq. (4)) has the character of a least squares minimization method.

ELABORATION OF THE QUADRATIC FORM
Equation (4) gives an estimation of the true texture function by a quadratic form h2(g). We will use a form for h2 ( The function f(g) is always normalized f(g)dg 1 because of (9) and (11) It will now be attempted to find the coefficients H by a least squares method. To that purpose, the partial derivatives dH must be calculated (As from Eq. (6) or Eq. (7)). Useful quantities are (see Eq. (5)): As is defined by Eq. (6).
Assume that these are known at a certain iteration step.
The following procedure will now be adopted in order to find a better estimation for Hf .
First, the function h(#) itself (Eq. (10)) will be computed at a grid pattern in Euler space. A mesh length of 3 was used in the present work.
Equation (9) then allows the calculation of fs(g) for the points of the grid pattern.
This finally produces a better value for H by the steepest descent method: (H'*)"w Hv + aOH" ( must be negative).
It is strongly advised to select a value for a which minimizes the new value of A. This can be done by methods which require no additional computer time (see Appendix I). This procedure should be repeated until it is seen that no significant decrease of A is obtained any more. A minimum for A as a function of the H is found by then.
At the present state of the analysis, it cannot be decided whether this minimum is a local minimum or "the" minimum.
Experience gained with case studies shows, that the results usually are very satisfactory.
The iteration scheme described above requires a first estimation of the H'V-coefficients. Appendix II explains how this first estimation can be obtained.
The end result of the iteration is a set of H'-coefficients which can produce a non-negative O.D.F. (Eqs (9-11)). These coefficients can easily be normalized:  Bunge (1969)) is 7.
The Cfv coefficients of this texture are known beforehand both for even and odd I. The even coefficients for up to 22 were used as input data for the test. Version (a) of the method was used.
The resulting quadratic form obtained after 2 iteration steps produced an O.D.F. without traces of ghost maxima. Figure 1  ,. 0 , @ 45 section plane. sion L 22 with both even and odd terms. Tests on more complex (but still sharp) textures have shown that this is a general tendency, but that the power of resolution of the method is nevertheless poorer than that of a classical series expansion with L 34 which would also include correct odd terms (Figures 2-3). A subsequent test has been carried out on a complex texture of which both even and odd terms were known beforehand. This texture is a copper-type rolling texture simulated by the lath-version of the relaxed Taylor theory, as reported earlier (Van Houtte, 1981). The result of such a Taylor simulation is a set of 294 discrete crystallite orientations, each with its own weight. Such a texture is normally transformed into a continuous texture function f(#) by putting a Gaussian distribution upon each of the 294 orientations. The superposition of these 294 Gaussian distributions then produces the continuous texture function.
Both the even and odd C 'v coefficients can easily be calculated for such textures. In Figure 5 given by Van Houtte (1981), Gaussian dis-   Figure 4 of the present paper shows the result of the same Taylor simulation (lath model) but with o 7 and L 34.
Of this texture, the even C coefficients for up to 22 were used as input data for a test of version (a) of the quadratic form method.
After 3 iterations, the result of Figure 5 was found. Figure 6 shows the values of the texture functions of Figures 4-5 along the skeleton line.
The results of the classical series expansion for L 22 (even + odd and even only) are also shown. It is seen that the quadratic form comes very close to the values obtained by the classical series expansion for L 34, the latter being undistinguishable from the correct value of the O.D.F.t It can be seen in Figures 4 and 5 that this excellent fitting in the vicinity of the maximum of the texture function is not always reached in the parts of Euler space where the function is weaker. There is however no trace left of ghost maxima.
The method has finally be applied on several textures obtained from incomplete pole figures measured by X-ray diffraction. The method made the ghost maxima disappear and enhanced the value of the principal maxima, especially in the case of sharp textures. One example will be given here: the deformation texture of a technically-pure copper specimen after 95o cold rolling. The texture has already been reported elsewhere (Van Houtte, 1981). Four incomplete pole figures have been analyzed producing the Cfv coefficients of even rank up to 22. Figure 7 gives the reduced O.D.F. which contains ghost maxima. Its highest value is 16.33. Figure 8 shows the quadratic form obtained after 6 iteration steps. The highest value is 26.74. It is interesting to note, that the pole figures recalculated from the quadratic form showed no excessive negative oscillations in their nonmeasured parts.
One iteration step of the quadratic form method with L 22 requires approximately the same amount of computer time as the usual calculation of the sections through Euler space of an O.D.F. by the series expansion method for L 34 (making extensive use of precalculated function libraries). In the case of very simple textures (Figure 1), as few iteration steps as 2 are sufficient; 4 steps are recommended for more complex textures.
-1" Gaussian distributions with bo 7 can nearly exactly be fitted by a classical series expansion with L 34.
Reduced O.D.F. (L 22) of a cold rolled copper specimen (95 reduction).  Esling and Bunge (1981), this is a common feature of all methods which try to find the odd part of an O.D.F. by using the non-negativity condition. Some textures which do not have a zero range have an isotropical background. In that case, the present method can also be used, provided this isotropical component is known. It could simply be subtracted from the texture, thus producing a new texture which has a zero range and upon which the non-negativity conditions can be applied.
Although the present method requires the existence of a zero range, its preliminar determination is not needed. It is capable of finding the zero range itself in the case of sufficiently sharp textures for which the variation range off(g) is strongly restricted by the positivity condition .
Another advantage of the method is that the quadratic form gives a far better estimation of peak heights of sharp textures than the classical series expansion (even + odd terms) does at the same degree of expansion (L 22). This result is nevertheless not so good as a classical series expansion (even + odd terms) up to l= 34. The problem there is to find the odd C 'v coefficients up to 33. So far, only a method which models an O.D.F. by a superposition of various Gaussian distributions (L/icke et al., 1981) has been able to produce odd C coefficients up to such a high rank l. Such a method has the disadvantage that its success strongly depends upon good modelling of a particular O.D.F., i.e., it is not an automatic method.
The present method seems to be less accurate in the weaker regions of the O.D.F. than in the regions of high intensity. This problem might be related to the quadratic nature of the error criterion used. At the present time, no computer programmes have been written for version (b) of the method which tries to fit complete pole figures directly. It can be expected on theoretical grounds that this version might even work if less than M(L) pole figures have been measured (Bunge (1969), Figure 4.3), since the preliminar knowledge of the even C coefficients is not required in this case. For this reason, this version of the method