APPLICATION OF NORMAL DISTRIBUTIONS ON SO(3) AND S FOR ORIENTATION DISTRIBUTION FUNCTION APPROXIMATION

The orientation distribution function (ODF) in a polycrystalline sample is of special interest in texture analysis. Its determination from pole figures leads to an ill-posed problem, the solution of which is non-unique. In the present paper the properties of normal distributions on the rotation group SO(3) proposed by Parthasarathy (1964), Savyolova (1984) are discussed. A method for ODF determination based on the superposition of the normal distributions is proposed. The parameters of normal distributions are determined from the experimental pole figures. The application of this method is demonstrated for a rolling texture of beryllium.


INTRODUCTION
The central problem of the quantitative texture analysis is the determination of the orientation distribution function of texturized samples from pole figures measured by X-ray or neutron diffraction techniques. The most widely applied method to solve this problem is the series expansion formalism proposed by Bunge (1969). It is not capable of avoiding regions of negative values and ghost effects in the ODF. The reason for these phenomena was revealed by Matthies (1979). He showed that ghost effects are caused by the lack of information on the odd part of the ODF and, consequently, it is in principle not possible to determine the "true" ODF from pole figure data. This discovery stimulated the development of numerous approaches for the ODF determination from pole figures by using additional information on ODF. One such approach consists in using special model functions. Pospiech (Cracow) and Lucke (Aachen) groups got valuable results in ODF representation by a finite number N of normalized bell-shaped distributions on SO(3). In their papers (1981,1986) the authors have used a Gaussian type function suggested by Bunge (1969) for the bell-shaped distribution. However, this function posseses several disadvantages. One of them is that it cannot adequately describe the whole width of the spectrum for bell-shaped curves beginning with very sharp peaks up to the random distribution. Moreover, there is no explicit analytical expression for the corresponding pole figures, and therefore the "Gaussian" component fit cannot be directly 161 achieved from the pole figures. A new method for ODF determination from experimental pole figures was proposed in (Nicolaev, Savyolova, Feldmann, 1992). In this method, the ODF is represented by a superposition of a number of normal distributions, suggested by Savyolova (1984). Basing on the known analytical relationships between ODF's and pole figures formed by such type of distributions, the parameters of normal distributions are found from experimental data for cubic lattice symmetry.
In the present paper we take the normal distribution on the rotation group which satisfies the central limit theorem in Parthasarathy K. P. (1964). The properties of these normal distributions are discussed. It is possible to obtain explicit expressions for the density of normal distribution only for specific cases. One of such cases is the central normal distribution. The connection of the central normal distribution on SO(3) group with fundamental solution of the corresponding diffusion equation is shown (see Perrin, 1928, Roberts, Winch, 1984, Heyer, 1987. Some projections of the normal distribution are obtained. These projections coincide with the normal distribution on the unit circle (Mardia, 1972) and with the Perrin distribution on the two dimensional unit sphere (Perrin, 1928).
In the paper (Matthies, Muller and Vinel, 1988) the central normal distribution are only analysed.
The analogues of the normal distribution for rotation group SO(3) and for hyperspheres S are discussed in (Schaeben, 1992, see also the references in this article).
In the present paper the central normal distributions are used for approximation of ODF. The calculations are illustrated for hexagonal lattice symmetry. The application of the proposed method is demonstrated for the example of the rolling texture of beryllium.
The canonical normal distributions are used for approximation of ODF in (Savyolova, 1989 (Bunge, 1969;Nikolaev, Savyolova, Feldmann, 1992;Savyolova, 1984;Matthies, Miller and Vinel, 1988). The first model of a central normal type distribution was proposed by Bunge where E is the unit matrix of the same order as Tg, gi(t) are the parameters of group whose tangent vectors at unity are mutually orthogonal. A distribution /, on a group G is said to be infinitely divisible if for every where * denotes convolution.
integer n exists a distribution/,n such that/, =/,, A distribution/, on a group G is said to be an idempotent factor if/z*2--/.
DEFNmON 1 A distribution /z is said to be normal if it is infinitely divisible without idempotent factors and admits representation of the type fs Tgdlt(g)=exp { aA'+ :a} where Ai, i-1, 2, 3 are the matrices (2), (a) is a real positive definite or semidefinite matrix, and ti, 1, 2, 3, are constants.
The central limit theorem for the rotation group SO(3) gives the necessary and sufficient conditions under which the limit of a sequence of distributions of the type #* may be normal (Parthasarathy, 1964;Savyolova, 1984). THEOREM If n(1 det (g)) < C, C constant, lim__.oo n(e gn) F (),), then the limit of/* exists and is a normal distribution whose parameters are given by where g,, so(3)g d,,(g), e is the unit in S0(3).

The Properties of Normal Distribution on SO(3)
The normal distribution lz(g), g SO(3), (3) may be represented by a series where C7 are the corresponding Fourier-type coefficients, Tg'n(g) are spherical functions on the group SO(3). Let the three Eulerian angles g { aG/3, ),}, 0 < t, ), < 2if, 0 </3 < at, be the three rotations with the help of these the coordinate system KA can be oriented parallel to Ks with dg sin [3 dcr dfl d),, where Ks is the coordinate system of a given crystal, and KA is the coordinate system of the sample. In the present paper everywhere except for part 2.3 the choice of the-Eulerian angles is the same as defined by Bunge (1969)  where PT'n(cos fl) are Jacobi polynomials (Vilenkin, 1965). Let a 2= b 2= e 2 in (6). Then we obtain the central normal distribution where cos (t/2) cos (fl/2) cos ((a + ),)/2) (Vilenkin, 1965).
The central normal distribution (9) has been obtained by Perrin (1928) for the 2 3 Brownian motion on the sphere S in the space R. In Appendix A we give the definition of normal distribution on the sphere S n-1 in the space Rn, n > 2, as a solution of the parabolique differential equation.  (Bunge, 1969) with /Sh) (4r)-' | f(g) dp, f'(g, go, e)= 1 NANn i=l'=E f(g, gn]gogA,, e),  (Matthies, 1979). Additional informationfor example, additional model type suppositionsis necessary. In the present paper it is assumed that the true texture function is a superposition of the central normal distributions (7). If the centre of the central normal distribution is go =/= {0, 0, 0}, go SO (3), where cos t=[Tr(gog-1) 1]/2, has to be used instead of the (7). The corresponding PF has the form P, go, e)= P(O)= (4/+ 1)exp {-21(2/+ 1)e2}P2t(cos 0), where cos 0 (h, go) is the scalar product of h and go. The derivation of the formula (12) is given in the Appendix B.

Method of Calculation of ODF for Polycrystalline Sample of Hexagonal
Lattice Symmetry Consider the two steps of solving the problem (21): Step 1. Determination of the number of components N of texture and their coordinates g,.
Step 2. Determination of the parameters e., M., 1, 2,..., N. In case of hexagonal lattice symmetry with crystal symmetry D6h the texture components can be distinguished if the positions of the maxima in the pole figures do not coincide (Bucharova and Savyolova, 1989). Then we can find the coordinates of the centres of normal distributions by the method described elsewhere (Bucharova and Savyolova, 1985).
where 1 MA NB a=M,S n E E P(rA,g;'gn,,,r,,,g;'ga,n,f,)), k=l n,n'=l m,m'=l bi Pg, ) N ((rA.g;lgnk, )) d. If the positions of components of texture are not isolated, additional conditions, such as the minimum number of normal distributions, or other considerations, have to be taken into account.

Numerical Results
An application of the proposed method is demonstrated by the example of a rolling texture of beryllium produced by rolling of the fiber texture with axis n {0 , 0}, go { a, 90 , 0}, a e [0 , 360). In this example the sample coordinate system K, and the crystal coordinate system Kn are fixed as follows: Z, is parallel to the rolling direction (RD), Xt is parallel to the transverse direction (TD), and Y, is parallel to the normal direction (ND); Xn, Yn, Za are parallel to the crystal directions [10i0], [i2i0] and [0001] respectively. In this part of the paper the Eulerian angles are chosen as in _(Roy, 1965_) g= {t, fl, y}-={qJ, O, }. The experimental PF {0001}, {1010} and {1011}, obtained from X-ray measurements, are shown in Figures 5a, 6a, 7a respectively. For this texture it is difficult to find the number of components N and the coordinates of their centres from the positions of the maxima of function/5P07) (Bucharova and Savyolova, 1989). We have used the method of minimization of the functional (21) with the two PF {0001} and {10i0}, taking into account the minimum numlr of normal distributions. Choosing different N (N > 1) and g, we find the parameters of ODF as a superposition of normal distributions f (g) Mtf(g, g,, e,, n-',) + M2f2(g, g2, e2, at2) + M3f3(g, g3, e3, tit3) + M4.f4(g, g4, e4, t4) "" Msfs(g, gs, es, t5) + M6f6(g, g6, 86) + M7fT,  Figures 5b, 6b and 7b respectively. The errors of even part of ODF of the proposed method can be estimated from the following relations (Roe, 1965;Matthies, 1986) 0.2 f exp ntheor/-*x12 d;,  Figure 10 (/max 16). We see that the representation of the ODF as a superposition of normal distributions provides the following characteristics" the ODF satisfies the positivity condition and is free of ghost effects.
We use all information in pole figures for approximating the ODF by  superposition of a number of normal distributions. Certainly, we can make a mistake if we define the odd part of ODF by the odd part of superposition of central normal distributions. The check of the odd part of the ODF can be made by other methods. We can find ODF as the sum of 6-functions on group SO(3) Savyolova, 1985, 1989). Also, we can define the orientations of grains in polycrystalline sample (about 1000 grains) (Bunge, 1969), and then we must use a conventional zE-test.