Generalization of the Concept of Sample Symmetry- Fuzzy Symmetry,Symmetroids, Similarity

The crystal orientation distribution function of polycrystalline materials, i.e. the texture, 
may exhibit internal symmetries due to symmetries of the production steps, or more generally, 
to the whole materials history. The “sharpness” of such symmetries can be quantified 
in terms of various symmetry parameters. If the symmetries of subsequent production 
processes are different, e.g. of sheet rolling and deep drawing, then these symmetries may 
still be recognized in the final texture. In the same way also similarities of textures and 
properties of different materials can be quantified. Symmetry parameters have many 
practical applications. Examples of that are the determination of rolling direction corrections, 
determination of the “plastic spin”, estimation of coarse-grained materials, or 
finding the “correct” (symmetry adapted) axis system in a material.


INTRODUCTION
The sample symmetry of an orientation distribution function (ODF) is illustrated in Fig. 1. The volume fraction of crystals in the orientation g is equal to that of crystals in the symmetrically equivalent orientations g.gS. Hence, sample symmetry is defined by the relationship f(g.gS)=f(g); gSEGs, where g is an element of the (rotational) point symmetry group Gs. Contrary to crystal symmetry which is fulfilled in each volume element individually, sample symmetry is defined only for the totality of all volume elements of crystallites in a statistical sense. Hence, it may be expected that it is less "sharp" than crystal symmetry. Furthermore, sample symmetry originates from the symmetry (or the symmetries) of all texture forming processes which have taken place in the history of the considered material, e.g. the orthorhombic symmetry of the sheetrolling process or the axial symmetry of wire drawing or many others.
The symmetries of these processes can be more or less "sharp" so that the same must be assumed for the sample symmetry of a texture func, tion. Also different symmetries of different such processes may have been superposed, e.g. the orthorhombic symmetry of the sheet-rolling process and a monoclinic symmetry of a subsequent deep-drawing process. Hence, Eq. (1) will in general, only be an approximation.

DEFINITION OF A SYMMETRY DEGREE-FUZZY SYMMETRY
The "degree" of symmetry according to the symmetry element gS can be judged by the difference function A f (g, gS) =f(g) f (g. gS).
(2) It may, however, be convenient to judge the degree of symmetry by a single-valued parameter. Such parameters can be defined in many different ways by various functionals F of the difference function Eq. (2) written formally as F(g s) F[Af(g, gS)]. (3) As an example we may define the symmetry degree by the least-squares parameter cr (i.e. a quadratic symmetry parameter) cr(g s) (4) which can also be expressed in the form cr(g s) ff(g)"f(g" gS)dg f[,f(g)]2dg (5) i.e. by the auto-correlation function-of the texture function f(g). This parameter may vary between 0 < r(g s) < 1, where exact symmetry r(g s) < 1 "fuzzy" symmetry 0 no symmetry symmetry degree or. (7) Values of r < correspond to some "unsharp" or "fuzzy" symmetry.
If the symmetry described by gS is unsharp then gS itself is also unsharp. Hence, it is necessary to consider the symmetry parameter r, Eq. (4), also for other rotations Ag instead of only the fixed symmetry elements gS. Hence, we define analogous to Eq. (4) ff (g) which is the auto-correlation function of the texture function considered for variable Ag. Symmetry is then indicated by Max[cr(Ag)] at Ag g.
Eq. (9) allows to find the symmetry elements g and (he symmetry degree cr(g ) of these elements.

SYMMETRY PARAMETERS FOR POLE FIGURES
The symmetry parameters can also be defined for individual pole figures instead for the texture as a whole. In analogy to Eq. (8)  The direction Ag. f is that obtained from f by the rotation Ag. The symmetry in this pole figure can be .judged analogous as in the case of the whole texture by Eq. (9).

DEFINITION OF PARTIAL SYMMETRY SYMMETROIDS
A texture function may be composed of partial functions./)(g) with the volume fractions V; where each partial function may have its own sample symmetry group Gi v; ..l)(g) a; i--I In this case the function a(Ag), Eq. (8), will have local maxima Max[cr(Ag)] at Ag g/S E G/s (13) in orientations g/S which are the sum (unification set) of all symmetry elements of all the different symmetry groups G of all partial texture functions f.(g). It must be mentioned that the values of a belonging to the symmetry elements g of one of the partial functions need not necessarily be identical tT(gsl) : s2 sl s2 tr(g); g,g EG.
This is due to the contributions of the other partial functions j(g) with c which will, in general, be different in these orientations. It may even be that a particular symmetry element gSX also belongs to the symmetry group G/ of another partial function f(g) which thus also contributes to the local maximum in this particular orientation. Hence, it seems consequent to consider all local maxima ofthe function a(Ag) as symmetry elements of some (hypothetical) partial texture function (g) in the sense of Eq. (9) independent of whether or not it is possible to actually split the function f(g) into such partial function f.(g) as is assumed in Eq. (12). The orientations g/S of these local maxima may be called symmetroids of the texture function f(g). In the general case the symmetroids do not form a group. Rather they may be considered as the sum (or unification set) of the elements of several groups g GUG U...ua m,

TEXTURE SIMILARITY
We may compare two different textures f(g) and f2(g). We may call them "similar" if the difference Af(g) =f2(g) -fl(g) is small. As in the case of symmetry it may be convenient to have a single-valued parameter by which to judge "similarity". This may be a functional F of the difference function Af(g) which we may write formally as F F[Af(g)]. (17) As an example we may define a least squares parameter 0-1,2 analogous to Eq. (4) (18) with this parameter similarity may be judged in the form identity 0-1,2 < similarity --+ 0 no similarity similarity degree 0-.
Rotated Similarity If two texturesfa(g) andf2(g) are not identical then some similarity may still exist if we consider either of them in its own particular sample coordinate system. Hence, we may define "rotated similarity" by o'l'2(Ag) 1-f[f2(g Ag) - The function 0 "1'2 (Ag) may have several local maxima. Then the two textures fa(g) and f2(g) are similar (possibly with different degree) in different relative orientations Ag i. In analogy to Eq. (8) the similarity function al,2(Ag) is the intercorrelation function of the two texture functions fl(g) and f2(g). The similarity parameter a1,2, Eq. (18), referred to identical sample coordinate systems KA1 KA2 is, of course, contained in cr 1'2 (Ag) 0 "1'2 ffl'2(Ag) for Ag {0, 0,0}.
It is obvious that symmetry or fuzzy symmetry in the sense of Eqs. (8), (9) can be considered as "self-similarity" of a texture function f(g).

Similarity of Pole Figures
Also the concept of similarity can be applied to individual pole figures.
In analogy to Eq. (20) we define the similarity parameter for the (hkl) o'(ha,h,:,,, Similarity of these two pole figures is then judged in analogy to Eq. (19). This parameter is important for ODF analysis from pole figures. If some of the "input pole figures" used in ODF analysis are "too similar" then they do not increase the independent input information needed for ODF-calculation.

SYMMETRY PARAMETERS BASED ON SERIES EXPANSION
We represent the texture function f(g) by a series expansion in the Using the addition theorem of the functions Tn(g) the symmetry function a(Ag), Eq. (8) Eq. (28) shows again that a(Ag) is a quadratic symmetry parameter.

Rotation Axes in z-Direction
Symmetry axes of rth order in z-direction of the sample coordinate system KA are represented in terms ofthe "selection rule" (see e.g. Bunge, 1982) C n --0 for n r. n'.
It must be mentioned that in the sample symmetry rotation axes of all orders r may occur, particularly also those for r , i.e. axial symmetry.
In fact, these latter ones are particularly interesting.
The parameter r, Eq. (30), can also be defined in any rotated sample coordinate system according to Eq. (22). In the sample coordinate system rotated through Ag the texture function f(g) may be written in the form with the "transformed" texture coefficients (see e.g. Bunge, 1982) +A Almn ms Z C Tn(/kg) Then the position of axes of order r can be seen directly in the pole figure of the original sample coordinate system KA.
Local minima of the function r(Ag) indicate symmetries and more generally symmetroids according to Eq. (15).
The criterion #r(/'kg), Eq. (34), is defined by absolute values of the coefficients C n in contrast to the criterion r(Ag), Eq. (8), which is defined by the square of the difference function. Hence, the criterion r(Ag) exhibits apexes in the symmetry positions whereas cr(Ag) goes through these positions with horizontal tangent.
Mirror Plane Perpendicular to the y-Axis Taking crystal symmetry into account the series expansion Eq. (26) can be written in the form (see e.g. Bunge, 1982) This function possesses mirror symmetry perpendicular to the y-axis of the sample coordinate system KA if the express.ion in brackets is real. If we assume (for simplicity) that the functions k (hkl) are real (as it is usual in cubic crystal symmetry) then this requires c;n=c; #n (real).
Hence, a symmetry parameter indicating mirror symmetry in all pole figures perpendicular to the sample direction y can be defined by A=O n=-A with F' related to F according to Eq. (54). The existence of a mirror plane or two-fold axis is then estinated according to Eq. (48).
The criterion qahk tM is particularly obvious because it contains the terms sin n/3 in the series expansion of a pole figure. These terms change their sign when going through/3 0. Hence, these are the terms which break the mirror symmetry in/3 0.

SIMILARITY PARAMETERS BASED ON SERIES EXPANSION
In complete analogy to Eqs. (27), (28) may be helpful. According to this parameter it can be estimated how much "new" information the pole figure (h'k'l') contributes to ODF analysis additional to the pole figure (hkl).

SYMMETROIDS DUE TO TEXTURE FORMATION PROCESSES
Textures are formed by solid state processes of any kind. These processes modify an initial texture into the final texture. If several such processes are acting one after the other, the final texture is the result of the superposition of all these processes fend(g) Mn (R)"" (R) M2 (R) M1 (R)finitial(g).
Each of these processes may have its own symmetry which is the symmetry of the "modifyer operators" Mi. Strictly speaking, the final texture can only have a sample symmetry which is the lowest subsymmetry of all processes, i.e. of all modifyers Mi. The individual symmetries may, however, still be seen in the final texture in the form of symmetroids. This is particularly interesting in geological sciences, where a final texture is measured in order to get information about a sequence of "modifyers" which have contributed to it during the materials history.

SOME EXAMPLES
In the following some examples are considered which illustrate the practical application of the defined parameters and the conclusions which can be drawn from them. The first example is the rolling texture of a cold rolled copper sheet.   Fig. 2(b) as a function of the angle fl 90 + AI,Eq. (50). This curve shows a very sharp minimum at fl0 16 which represents the alignment correction of the rolling direction. The accuracy of the value/0 is much higher than with the conventionally used alignment corrections.
The curve Fig. 2(b) also shows a second minimum at/0 + 90 which corresponds to the transverse direction. This shows that the sample symmetry is orthorhombic. The value of U(TD) is higher than M(RD). This illustrates that the two mirror planes in RD and TD have different symmetry degrees. The one in TD is more "fuzzy" than the one in RD. In Fig. 3 the analogous situation is illustrated for a recrystallized copper sheet. The pole figure (200) is shown in this case. The parameter (fl) is plotted in Fig. 3(b). Again the alignment correction angle fl0 59 can be obtained from this diagram with good accuracy. Besides the minimum in RD there are now other ones in RD+90 and RD 4-45. This time the values at RD and TD are very close to each other, there is, however, a slightly higher deviation in RD 4-45. This shows that the sample symmetry is now close to tetragonal but a slight right-left asymmetry from the rolling direction can be seen (Nielsen and Bunge, 1996).
A third example concerns a coarse grained material. The samples were A1 99.5% purity, deformed respectively 50% and 70%. After that they were annealed 24 h at 400. The resulting texture is a weak cube texture.
As a result of that the pole figures are rather "spiky". Nevertheless, Fig. 4(a), i.e. for the 70% material, shows definitely the four minima similar to Fig. 3 indicating the tetragonal sample symmetry characteristic for the cube texture. The situation in Fig. 4(b) is even more "fuzzy" but still the four minima can be recognized. Another example is shown in Fig. 5. The material is a ferritic steel Stl4Q5. Texture measurement, this time, was done with the oblique section method (see e.g. Welch, 1980). In this technique oblique sections through the sheet are used which have ideally their normal direction under equal angles (i.e. 54.7) to the sheet directions ND, RD, TD. These sample cutting angles as well as the alignment ofthe sample in the sample holder may, however, deviate from their ideal values. In this case the "back-transformation" of the measured pole figures to the sheet coordinate system ND, TD, RD may be incorrect. Hence, ODF analysis was carried out in the "measuring coordinates" assuming triclinic sample symmetry. Then the symmetry parameter or=2(Ag), Eq. (34), was determined, this time in the whole two-dimensional area, Eq. (38). The result is plotted in the middle-left in Fig. 5. One sees very clearly the minima corresponding to the three orthorhombic axes i.e. ND, TD, RD.
It is also seen that one of them is much less "fuzzy" than the other two. This is the normal direction. Also a broad "fuzzy-band" can be seen in the rolling plane. From this diagram the transformation angles for transformation into ND, TD, RD can easily be obtained. The transformation was carried out in the middle-right diagram and also in the pole figures (lower line in Fig. 5). This example shows, on the one hand, the most reliable evaluation method for oblique section measurement. On the other hand it shows the much sharper nature ofthe symmetry axis in ND and ofthe mirror plane in the rolling plane compared with the symmetry elements in the rolling plane. lower part of this figure shows the pole figures represented in the conventional coordinate system ND, TD, RD. As a further example, a series of aluminum samples was considered.
They were recrystallized with the starting texture being essentially the cube texture. Then samples were cut at different angles r to the original rolling direction and were stretched 20% in the indicated direction SD in Fig. 7. The (111) pole figures of the stretched samples are shown in Fig. 8. Besides (111) also (200)   texture coefficients C n the texture similarity parameter pl'2(Ag), Eq. (62), was determined whereby any ofthe textures was compared with the one deformed in rolling direction r=0. Since the deformation process does not violate the mirror symmetry in the sheet plane, the actual symmetry of the deformed samples is monoclinic (rather than triclinic). Hence, it is sufficient to plot the similarity parameter only as a function of a rotation/3 about the sheet normal direction. The result is shown in Fig. 9 for two selected cases, i.e. r 0 and r 20. The curve for r 0 compares a texture with itself (rotated through the angle/3).
It is seen that the minimum at fl 0 goes down to zero (identity) but there are more "fuzzy" similarities at fl 90 and fl 4-45 corresponding to the orthorhombic and tetragonal components in the sample symmetry already mentioned in Fig. 3. These similarities can still be seen in the sample deformed at 7-= 20 , though much more "fuzzy". The angle Aft at which the highest similarity is reached, i.e. the minimum of l,2(fl) is plotted in Fig. 10. This minimum value M qOmin(7" is plotted in Fig. 11. It is seen that the symmetries of the deformed textures are more and more fuzzy the more the angle r deviates from the original symmetry axes in r =0 and r 90. But also stretching the sample at r=45 conserves the symmetry (Bunge and Nielsen). The final example concerns a magnetite ore. From a big piece of this material two samples were taken but the ralative orientation of the sample axes was not specified. The (111) pole figures of them are shown in Fig. 12. Only later on it turned out to be necessary to refer the two textures to the same sample coordinate system. This relationship could be retrieved under the assumption that the two textures from nearby locations of the original big piece were similar enough. Hence, the similarity parameter 991'2(Ag), Eq. (62), was calculated in the whole range Ag {qo 4qo2 }. The result is shown in Fig. 13. It is seen that the similarity parameter has a well-developed minimum which gives the most probable relative orientation of the two-sample coordinate systems.

CONCLUSIONS
The orientation distribution function ODF of the crystallites of polycrystalline materials may exhibit (statistical) symmetries, i.e. sample symmetries, which are the result of corresponding symmetries of the production process. It turns out that these symmetries are often rather "fuzzy" compared with crystal symmetry. It is then necessary to characterize them by a "symmetry degree" which may vary in a wide range from very sharp to rather diffuse symmetries. Also there may be remnants of symmetries from several production steps superposed which do not form a symmetry group. Nevertheless, such generalized "symmetroids" may be successfully used to judge the influences of subsequent production steps which the material has undergone. Several different symmetry parameters can be defined. If texture analysis is done by the series expansion method (harmonic method) then the most convenient way is to use symmetry parameters based on the texture coefficients C which are already provided by routine texture analysis programs.
In several examples it was shown that the symmetry parameters exhibit very sharp minima (cusps) which allow to find the position of symmetry elements in a sample with high precision, e.g. 0.5. This was successfully applied to determine a misalignment correction for the rolling direction in texture measurement with much higher precision than it was possible without this method.
In a similar way the alignment angles can also be found for the oblique section method of texture measurement. Symmetry parameters are integrals over the whole orientation distribution function. As such they are rather insensitive to the particular features of the texture. This is particularly helpful in the case of coarse grained materials in which symmetries can still be distinguished even if the texture has great "statistical noise", i.e. it looks "spiky" as in Fig. 4.
The high sensitivity with which symmetry elements can be localized allows to identify small changes of their positions also with high sensitivity. As an example the rotation of the mirror plane after extension of sheet samples in oblique directions was considered. This has been called the "plastic spin" in terms of plastomechanics (see e.g. Dafalias, 1984;Raniecki and Mr6z, 1989). At the same time it is also seen that the "'plastic spin" describes only part of the texture changes after oblique extension. Also the degree of symmetry is increasingly modified the more the extension direction deviates from the original symmetry directions. This is important, for instance, in deep drawing where the direction of extension varies continuously around the circumference of a drawn cup, thus leading to cup-earing.
Besides the symmetry parameters also similarity parameters can be defined which compare two textures with each other. Also similarity parameters are a useful tool. This was shown, for instance, in two geological samples taken from neighbouring positions. On the one hand, it showed that neighbouring samples have similar textures, and, on the other hand, it allowed to find the corresponding sample axes with high accuracy. With the help ofthe so defined symmetry parameters as well as similarity parameters, formerly only qualitatively considered features of textures, e.g. "fuzzy symmetries" or "similarity" can now be defined on a quantitative basis. This may have many practical applications as was shown here only in a few selected examples.