Continuous and Discrete Representations of an Iterative Reproduction Method

General formulae are obtained for iterative reproduction of the distribution of 
orientations from both unreduced and reduced pole-figures. The most correct 
representation of them is the continuous one realized by the Bunge–Roe formalism. 
The discrete representations are always connected with the additional assumption 
that the texture function is considered to be constant inside the subsets of orientation 
space. One of the discrete representations contains the elements of the elementary 
pole-figures of the vector method.


INTRODUCTION
We summarize and explain the designations we shall use in this work.
KA: right-handed orthogonal coordinate system fixed to the sample. sample vector with serial number k (k 1,..., K). right-handed orthogonal coordinate system fixed to the crystallite according to a given prescription, taking into account the crystal structure of the crystallite. (1) orthogonal coordinate systems equivalent to Kn Kn, g, .Kn, e 1,..., No. (2) Here we use the upperscript p to emphasize that we mean a passive operator. A passive operator is one that moves the axes of space, all points of the space and hence all vector positions being left unmoved, so that after each operation a vector position is referred to a new set of axes (Bradley and Cracknell, 1972). orthogonal transformation which transforms KA into K n, (Bunge, Esling and Miiller, 1982) Kn, u KA, e 1,..., No.
(3) Generally, the set {u} of all possible orthogonal transformations consists of two separated subsets, G and U, containing only proper and improper rotations, respectively. From any proper rotation g e G the equivalent orthogonal transformations can be got by the equation Ue ge g' e 1,..., No. (4) The matrix elements describing the proper rotation g are f(g)" usually expressed by the Eulerian angles (Pl, (I), (] 02 (Bunge, 1969) or a, fl, y (Matthies, 1982) in the region G" 0< tr <2r, 0<fl <r, 0< 7<2r. (5) orientation density function describing the distribution of proper rotations. The definition is AV 1 --if-(g e G) fa dg f (g) dg (6) where Gy" arbitrary subset of the region G (j= 1,..., J;;), AV/V(g e Gy)" relative volume of all the crystallites with g e Gy. complete set of all the crystal vectors equivalent to h {h} {/,,,/,2,...,/,,} where hi, g,e hi, e 1,..., Nb. (8) Here we use the upperscript a to emphasize that we mean an active operator. An active operator is one that moves the points or position vectors of space, all vectors being referred to a fixed set of axes (Bradley and Cracknell, 1972).
(8) we get Nb 4 vectors/ie equivalent to/i, Figure 2. the set of all particular orientations in the region G (5) at which the vector hie described in Ks is parallel to a fixed sample vector 37k. The subscripts of Tiek refer to that of and 37k, respectively. Example. Let us consider a single crystal with s CEv and its vector set (hi}, Figure 2. 37k is an arbitraril_,v chosen sample vector. We rotate the single crystal till hi1 is parallel with Yk. At this moment let the orientation of Ks with respect to KA be g l. After rotating the crystal around hi, we get all orientations in G at which hi, is parallel to Yk i.e., the set Ti,k. Any set Tiek can be constructed in the same way. The set Tiek can mathematically be described as follows. The coordinates of Yk (Xk, Yk, Zk) are given in KA, while hi. k is an arbitrary sample vector. Let 1 be an orientation of KB at which hi1 is parallel to k. After rotating KB around hi1 we get the set TI k. The sum of the sets Te k (e 1, 2, 3, 4) is denoted by (T/k}. that of hie= (Xie, Yi,, Zie) are given in KB.
Pff,,}(yk)is called the unreduced pole-figure belonging to the crystal vector hg and is in certain cases measurable by anomalous scattering.

REDUCED POLE-FIGURES
All crystallites with orientation g e { ,) give one of their vectors /i ( 1...,//b) parallel to a fixed ,. The distribution of the vectors h in the polycrystalline sample is described by the function 1 f(g) dg.
(16) P(?,,(y is called the reduced pole-figure belonging to the crystal vector hi and can be measured by normal scattering.

REPRODUCTION OF THE ORIENTATION DENSITY FUNCTION FROM POLE-FIGURES
In Figure 4 the orientation of Ks with respect to KA is g. The vectors h, of set {h} are drawn in Ks assuming that Gs C2o. Let the sample vector Yk, be parallel to the vectors t,(e,s= 1,..., Nb). As we already know (see Figure 2) in all the cases when one of the vectors h, is parallel to a fixed )Tg, the orientations of K with respect to KA belong to the set { Tg, Nt,) are containing the orientation g and, consequently, they intersect at g. orientation density function in the set (Tk,}. To the common point g at which the sets (Tk} intersect belong I. Nb values Pt;, (Yk,) informing us about the common value f(g) at the same time. We couple the unknown f(g) value with its information by multiplying them. From Eq. (15) we get After rearranging we get Let us perform the operation prescribed by the right side of Eq.
P{g,)(Yk,) s=l,... Nb. pU.) ()Tk,) means the unreduced pole-figure recalculated from the (hi) approaching function f(S)(g). After taking the average of Eqs (19) we get a formula for approximative determination of f(g) contain-ing all pole-figure data allocated to the orientation g. f(s+)(g) f()(g)" Average t .,(N)" i---1,.',.,l t'(K,}tyk) s=l N (20) A possible average is for example the following (Imhof, 1983a(Imhof, , 1983b: pc ones then f(N)(g)=f(E)(g)=fO)(g), i.e., depending on the choice of f(1)(g) we get an infinite number of solutions. We need therefore to decide for one of them. The reason of the choice of the function fO)(g) 1 from among the other possible solutions is as follows. At the very beginning of the iteration we do not know anything about the function f(g) and therefore we must give the same chance for any orientation to take part in the reproduced f(g).
With fO)(g) 1 it is not the case. The choice of fO)(g) 1 supposes an additional method which leads to the concrete form of fO)(g).
The WIMV-method is therefore based last of all on the choice of fO)(g) _= 1, too. A product-function analogous to (28) was first used by Williams (1968) in the texture analysis.

CONTINUOUS REPRESENTATION
According to the Bunge-Roe formalism (Matthies, 1982)  The pole-figure data are generally given at discrete vectors )7 and so the values F'({hi}) can be calculated from (35) by numerical integration. Equation (32) is still not suitable for determination of all the coefficients C7 ' because the/-odd coefficients are covered by a factor (1 + (-1)/) (Matthies, 1982).
The pole-figures are functions obtained from the distribution of orientations. Equation (22) allows us to follow the same way: from an approaching f(lV)(g) to recalculate the Nth approximation of the pole figures together being the blocks of the next approach f<V+l)(g).
First we construct the set {h} in the coordinate system K which is oriented with respect to KA by g. The vectors 37k (g 1, .., b) in Ka are chosen to be parallel to the vectors of the set {h). The pole-figure values Ptg,)(k) in the numerator of (22) are to be calculated by the Eq. (33).
We start the iteration with f(1)(g)= 1 and by Eq. (27) we get f(2)(g), of course, at discrete orientations we have chosen. The corresponding coefficients (2)cnn can be calculated from Eq. (34) and the approaching pole-figure values P)(k) from (31) using the/-even C n" only. Substituting the recalculated values P(29(h.(k=) into the denominator of (22), with the known f(2)(g) we get')'(3)() at the same discrete orientations. By this procedure we get a solution f(N)(g) with coefficients C7 n coupled to the assumption f(1)(g) _= 1 and to the concrete averaging operator.
On the analogy of Eq. (38) one can define the mean value of the orientation density function in the cell CE (Imhof, 1985) M V(Cjm) f (CE) . , f(C,,,) For constant indices and the vectors h of different orientations g C point always into the same A Y,; in the case of the orientations g CE, however, they point into more than one A Y,, numbered by m'. The set {Tu,} corresponding to the region (AYk)m, is denoted by (Tik}m,. The quantities in (41)  C. Let the cells be chosen to be parallelepipeds around the points (nl, n2, n3) and let in the Eq. (27) g mean the point (nx, n2, n3).
Then the sample vectors figs are parallel to the crystal vectors h (, g 1,...,/) of g. The recalculation of the values occurs by the Eq. (16) with the difference that f(g) inside one cell is considered to be constant. This way is followed by Matthies and Vinel (1982) in the WIMV-method.

SUMMARY
A reproduction method was obtained valid for single fase polycrystalline sample consisting of crystallites with any point symmetry group. In the case of enantiomorphic crystallites only one form of them is assumed to be present in the sample.
The kernel of the method is the Eq. (20) and the one (22). After applying them to the points and to subsets of orientation space we get different representations of the same method. The continuous representation must give the most accurate results because here is not assumed that the orientation density function is constant inside the subsets of the orientation space.
The reproducibility of the orientation density function is directly connected with the length of the integration paths { Tk} or { //k)" Generally some of the subsets T,k or T.,k are equivalent. The larger is the number of non-equivalent subsets in the union the less informative are the pole-figure data about the orientation density function.