APPROXIMATION OF THE POLE FIGURES AND THE ORIENTATION OF DISTRIBUTION OF GRAINS IN POLYCRYSTALLINE SAMPLES BY MEANS OF CANONICAL NORMAL DISTRIBUTIONS

The orientation distribution function (ODF) as determined from experimental pole figures (PF) in a polycrystalline sample by classical spherical harmonics analysis can have ghost effects and regions of negative values. The regions of negative values and the ghosts are a consequence of the loss of information on the "odd" part of ODF. In the present paper the canonical normal distributions (CND) on the rotation group SO(3) and on the sphere S in R used in texture analysis are discussed. The examples ofCND on SO(3), S and their PF calculated for hexagonal lattice symmetry and for a rolling texture of beryllium are demonstrated.


INTRODUCTION
The most widely applied method to solve the determination problem of the ODF from experimental pole figures is the series expansion formalism proposed by Bunge (1969, 1982).This method has ghost effects and regions of negative values (Matthies, 1979).The ghosts are a consequence of the inversion symmetry of experimental pole figures which leads to the loss of information on the "odd" part of ODF.
Let ODF have the harmonic representation through the generalised spherical functions c,..r.(g), !=0 m,n =--! where Cmn are determined Cm,, are Fourier-type coefficients of f(g).Coefficients from experimental pole figures Ph,(Y) using the Eq. ( 1), where PF are represented as a series in spherical functions (Bunge, 1969).
In the present paper the canonical normal distributions are discussed (Savy- olova, 1989).The canonical normal distributions are used for approximation of the pole figures and orientation distributions of grains in polycrystalline samples.

SO(3) AND S 2
Let us consider the normal distributions on the rotation group SO(3) as defined by Parthasarathy (1964), Savyolova (1984).DEFINITION 1 A distribution # is said to be normal (Gaussian) if it admits representation of the type T d#(g) exp oijAiA + oi Ai   (3) where T are the irreducible representations, A are the infinitesimal operators of representations T, (cij), i, j 1, 2, 3, is a real positive definite or semidefinite matrix, and i, 1, 2, 3, are constants.
Using the definition exp B we can calculate the right-hand part of expression (3) for CND as the sum of series where El is the unit matrix of order (2/+ 1), l 1, 2,....If two matrices A (ao), B (b0) of order n, i, j 1, 2,..., n, have a 0 b 0 0 with odd (i +j), then for their product C =AB (@) we have the same property c 0 0 with odd (i + j).
For the unitary matrix Bi we obtain Bt (Ol)-lAiDl,   (5)   where D is the orthogonal matrix, A is the diagonal matrix.Thus we have exp Bt (Dr) -1 exp A(D).
In that case the CND's reduce to the Brownian motion distributions on SO(3) (Roberts, Winch, 1984).
The other normal distributions on SO(3) we can get from formula (3) using the method of calculation (5).
The normal distributions on sphere S 2 in R 3 can be obtained as "projections" from normal distributions on group SO(3) (Nikolayev, Sav),olova, 1986; Bucha- rova, Savyolova, 1993).We can get the functions on S if we consider the functions on SO(3) that are constant on right-hand contiquity classes for supgroup of rotation with respect to a certain axis.We take the angle of rotation 0_< y<2.
THE CALCULATION OF THE CANONICAL NORMAL DISTRIBUTIONS ON SO(3) AND ON S 2 AND THEIR POLE FIGURES Consider the calculation of pole figures for canonical normal distribution on SO(3) for hexagonal lattice symmetry.
PF from CND in SO(3) with /1'33 --0 is the even part of CND on S 2 (Nikolayev,  Savyolova, 1986).The proof of this property follows from definition of CND on $2(8), if we take hi {0, 0} coincident with axis OZ with angle of rotation 0-< y < 2st.Then for ever,ty pole figure without any crystal and sample symmetry we have the coefficients C,,,n from Eq. (1) for ODF (2) with even only.
For ODF approximation by CND we can find the coordinates of the centres of normal distributions for hexagonal lattice symmetry by the method described in Bucharova, Savyolova (1985); for cubic lattice symmetry see Nikolayev et al.  (1992).
For ODF approximation by the canonical normal distributions the determina- tion of the parameters of CND from pole figures is discussed in Savyolova (1989).The parameters of the CND (al, trEE, t'33) are determined from their coefficients C when l= 2 (7).
C,, of CND for l>2 in that case It is easily to calculate the coefficients because the matrices B(4) are diagonal.
Using the canonical normal distributions for ODF approximation we a priori suppose that ODF is the sum of several CND's.The unknown parameters in cases 2, 3 can be calculated by selection method or from coefficients Cmn with l=4.
In Figure 5 the experimental pole figure (0001} is shown for rolling texture of beryllium (continuous line) and the even part of CND on S 2 with parameters (1/16, 1/32) (dotted line).
The error of the approximation of ODF by the CND was estimated by the Figure 5 PF {0001} for beryllium experimental data (continuous line); CND on S 2 with parameters /16, 32 dotted line).
On the left-hand side of Figures 1, 2 the CND's on S 2 are displayed with different parameters (try1, trEE) on the upper hemisphere S2.They are close to zero on the lower hemisphere S2.The even parts of CND's on S 2 with the same parameters (trl, trEE) are shown on the fight-hand side of Figures 1, 2. They coincide with PF {0001} for hexagonal lattice symmetry (or also for a polycry- stalline sample without any symmerty).We see that the maximum of pole figure is one half the maximum of CND on S2.
Figure 4 shows that if instead of parameter tr we put the parameter tr22, and inversely, instead of parameter tr22 we put tr, then the pole figure {0001} rotates by an angle of r/=-.
Figure 5, 6 display examples of approximation of experimental PF {0001} by of CND on $.Using the coefficients Cn with l 2 of harmonic the even part 2 representation of experimental PF {0001} as a series in spherical functions the parameters (trl, tr22) were found by the method described in case 2 with tr33 0.
Figure la; ld the same as in Figure lb.

Figure 2
Figure 2 CND on S with parameters 2a