THE INFLUENCE OF CRYSTAL SYMMETRY ON THE DETERMINATION OF THE ORIENTATION OF ISOLATED TEXTURE COMPONENTS FROM POLE FIGURES

The explicit relationship between the orientation of isolated texture components and peaks in pole figures is given. With this relationship the influence of triclinic and monoclinic crystal symmetry on the determination of the of isolated texture components is considered. In the special case that the problem is not ill-posed the method of the determination of the orientation of isolated texture components from pole figures is proposed.


INTRODUCTION
Sharp textures, which only consist of few non-overlapping components, reflected correspondingly sharp in the pole figures (PF) too, are considered.The problem of the determination of the orientation of these components from pole figure data is discussed in the present paper.
The properties of symmetry of the diffraction mechanism (Friedel's law) as well the properties of crystal symmetry can lead to non-uniqueness of the solution of this problem even though the single texture component is available.All possible 32 crystal classes can be subdivided into three types (Bunge, Esling and Muller, 1980; Esling, Bunge   and Muller, 1980): 1) groups containing rotations only; 2) groups containing the inversion centre; 3) groups containing inversion axes but not the inversion centre itself.In the cases of types 1) and 2) the orientation of an isolated texture component can be determined from PF wheares for the type 3) it cannot, in principle, be unambiguously determined from PF (Matthies and Helming, 1982).
The problem of the determination of the orientation of isolated texture components from pole figures has been dealt with in the works (Matthies., 1981; Helming., 1993;  Helming.et al., 1994).Another way of looking at this problem for the cases of hexagonal (point group D6h), cubic (point group Oh) and trigonal (point group D3) crystal symmetry was proposed in the papers (Bukharova, Savyolova, 1985; Nikolaev, Savyolova, 1987;   Ivanova, Savyolova, 1993).In the present paper it has been developed for the cases of triclinic and monoclinic crystal symmetry.

DESCRIPTION OF THE ORIENTATION OF A CRYSTALLITE
Let KA be the sample coordinate system and Ks be the crystal coordinate system.The orientation g of the crystallite is defined as the rotation which transforms the sample coordinate system into the crystal one.In particular, it is expressed in terms of the Eulerian angles g {a, fl, ?'}.In the present paper the sequence of rotations is chosen as follows (Korn, and Korn, 1968): 1) rotation of Ka around the axis ZA by the angle 2) rotation of K around the axis Y by the angle fl, 3) rotation of K] around the axis Z by the angle ,; 0< o, '< 2r, 0<<r.
Denote the matrix representation of the orientation g by T(g).T(g) is the transformation matrix for the coordinates y (yl, yZ, y3) of an arbitrary vector in the sample coordinate system KA expressed by means of the coordinates h (h1, h2, h3) of this vector in the crystal coordinate system K (Bunge, 1982): where y'= y2 h'= h y3 h The matrix T(g) is an orthogonal matrix, detT(g) 1.

RELATIONSHIP BETWEEN THE ORIENTATION OF ISOLATED TEXTURE COMPONENT AND PEAKS IN 'UNREDUCED' POLE FIGURES IN THE CASE OF TRICLINIC CRYSTAL SYMMETRY
Let 'unreduced' pole figure Ph(Y) be pole figure determined with the help of anomalous scattering (Matthies and Helming, 1982).In the case of the triclinic crystal symmetry (group C,, only one sort of enantiomorphic crystals exists in the sample) a peak at g go in the orientation space provides peaks in the 'unreduced' PF Ph (Y) and Ph2 (y) at y and Y2, respectively: y' T(g )h' y' T(go)h2 (2) The relation between the vector h (see Eq. 1) [h, hE] and the vector Y3 [Yl, Y2] is well-known y T(go)h . (3) The Eq. ( 2), (3) can be rewritten as the system of vector equations where Ti is the /-column vector of the orientation matrix T(go).The determinant of the system of Eg. (4) isn't equal to zero.Hence, Eq.( 4) has a unique solution.It is given by the following expression where hi= (h, hi 2, h), [hi, h2] is the /-coordinate of the vector product of hi, h, (hi, h2) is their scalar product.As an example, assume that hi (0, 0, 1) and h (1, 0, 0).Then the orientation matrix is equal to Y [Yl, ya]l y ) By comparing the matrix elements of the orientation matrix T(g0), expressed in terms of the Eulerian angles (Korn, and Korn, 1968), with the corresponding ones of Eq.
(5) one obtains the relation between the Eulerian angles of the texture component and peaks in two 'unreduced' PF (Bukharova, 1990).For example, the angle r0 is given by + y3 (h3_ h (hi, h2))} }. is valid.In the case of triclinic crystal symmetry (po_int groups C1, Ci) a peak at go in the orientation space provides two peaks in the PF Phi (Y) at +Yi, where Yi T(g0)h'i.
The statement is proved easily by going over from the crystal coordinate system KB to crystal coordinate system/B in which the coordinates of the vectors hi and h,. are equal to 61 (1, 0, 0), fil (cos q, sin q, 0). (10) Let us denote the rotation which transforms KA into a by g0.The connections between go and 0 and between T(go) and T(g0) are given by (Bunge, 1982)   go lg0, T(go) T-I(gB)T(g0)T(a), where B is the rotation which transforms/ into K.It can be straightforward verified that only in two cases of (a) (d) the solution of the system is an orthogonal matrix with determinant equal to 1.These matrices yield two orientations, namely, the correct orientation g, 0 {O0,/0, '0} and an uncorrect orientation , which is expressed in terms of correct orientation as follows gd 2zg6 {O0,j0, '0 + (12 where z is a rotation is the second order around the axis Z. It should be mentioned the situation when the vectors hi and h2 are perpendicular to each other.As it can be shown in this case two PF yield four orientations which are related to each other by the elements of the rotation group D2 Now let us consider three PF /5l(y), /52(y), /53(y).
Statement 2. If (hi, [h2, h3]) * 0 and each of the vectors h l, h2, h3 is not parallel to the vector product of the remaining ones, then the correct solution go ,130, } can be obtained.
INFLUENCE OF MONOCLINIC CRYSTAL SYMMETRY ON THE DETERMINATION OF THE ORIENTATION OF ISOLATED TEXTURE COMPONENTS FROM POLE FIGURES Monoclinic crystal system contains three crystal classes: C {z}, C3 {m}, C2h {z, rn}.Let go be a peak in the orientation space.The rotational part of the groups C2, Czh is the group C2.Hence, two equivalent peaks at go and at zgo exist in the orientation space.The rotational part of the group Ca is the group C. Hence, a single peak at go exists in the orientation space.Because the groups C2, Ca and CEh give rise to the common Laue group C2h the PF are identical for all these groups.The orientation g0 can be reproduced from PF within the multiplication from the left to the element 2z of the rotation group C2.For the groups C2 and CEh both go and zgo are solutions of the problem.For the group Ca this is not the case.Consequently, in the case of the group Ca the orientation of an isolated texture component is not determined unambiguously from PF.
For the groups C2 and C_2 the orientation of an isolated texture component can be determined from two PF Ph and Phi with h (1, 0, 0) and h (cos (p, sin tp, 0).This statement is based on the fact that peaks at go and at g0 in the orientation space provide two peaks in any of these PF.Consequently, Eq. ( 12) is valid.It gives the solution of the problem.

CONCLUSION
With the help of the explicit relationship between the orientation of isolated texture components and peaks in pole figures the influence of crystal symmetry on the determination of this orientation from different PF can be considered.In the special case that the problem is not ill-posed the number of PF sufficient to determine the orientation of isolated texture components unambiguously depends on crystal symmetry and on the _symmetry of measured PF.In the case of triclinic crystal symmetry three PF /Sh(y), Ph2(Y), eh3(Y) provide an unambiguous result on the condition that ((h, [h2,   h3]   0) and each of the vectors hi, h2, h3 is not parallel to the vector product of remaining ones.In the case of monoclinic crystal symmetry (point groups C2, C2h) the orientation of an isolated texture component is determined unambiguously from two PF /6hl(y), /6h2(Y with h (h, h21, 0), h (hE , hE 2, 0).For the group C it is not determined unambiguously from PF.
ORIENTATION OF ISOLATED TEXTURE COMPONENTS FROM POLE FIGURES IN THE CASE OF TRICLINIC CRYSTAL SYMMETRYNow we assume that the pole figures would be obtained by 'normal' diffraction experiments.For PF/'h(Y) obtained by'normal' diffraction experiments and 'unreduced'