COMPUTER-ORIENTED REAL SPHERICAL HARMONICS FOR TEXTURE AND PROPERTIES ANALYSES

An algorithm system to generate symmetric real functional bases for Quantitative Texture Analysis (QTA) is presented. A review of the analytical conditions to be satisfied by the considered functions is given. Suitable two- and three-dimensional bases are proposed. All crystal and sample point groups are analyzed. Computer implementation of the suggested algorithms is straightforward.


INTRODUCTION
Spherical harmonics expansion of texture functions and polycrystal physical properties is a well-established technique. In this context Fourier functional bases provides a through system, although somewhat uncomfortable to use because of its complex character.
Another problem arises from the fact that, as a consequence of traditional interest in metal research (Bunge, 1982;Matthies, 1987), QTA algorithms for highly symmetrical crystal-sample combinations (such as centro symmetric cubic-orthorhombic) have already been studied in detail. However, for low-symmetry systems we have not found a systematic presentation of suitable functional bases.
Here we present a collection of explicit formulae and tables to generate two-and three-dimensional symmetric real functional bases for all crystal and sample point groups. Suggested functions follow all the mathematical and methodical precepts of Bunge's school. Computer implementation of the proposed algorithms is straightforward. THEORETICAL SUMMARY (for further details see Bunge, 1982) We briefly summarise some fundamental relations of QTA, as expressed with the aid of Bunge's spherical harmonics. Their fulfilment is to be considered as compulsory requirements for any alternative functional basis.
Orientation distribution function f(g) may be expanded under symmetry-adapted generalised spherical harmonics T/V(g) The considered functional bases are linear combinations of the matrix elements of the irreducible representation of the three-dimensional rotation group T/mn(g) (Gel'fand, Minlos and Shapiro, 1963). The functions /"V(g) form an orthonormal basis fulfilling the condition REAL TWO-DIMENSIONAL SYMMETRIC FUNCTIONS As described by Bunge_ _(1969 and 1982) and Matthies (1987), it is possible to introduce real functions K (r) (r =-l,-l + 1 l) by means of orthonormalized real combinations of functions (8) If one denotes Aft by with then, one has These functions, like (8), form a complete and orthonormalized basis. Therefore, (6) and (7) may be represented by the new expansion with real coefficients Nffu.

Non-cubic Groups
For these point groups, the At mu may be chosen quite simple: Values of r' are given in Tables I and II, as functions of the parameters and t.
In Table I

Cubic Groups
The characteristic feature of cubic system is their main-diagonal threefold rotation axis.
This symmetry element imposes for the functions (6) or (7)  On the other hand, for these groups m is generally even. Therefore if we set then we may establish the function (6) or (7) in real form. Hence, o_ne can see that the functions K/(r) will be linear combinations of the functions K (r).

Tetrahedral Groups
The tetrahedral group T has twofold symmetries associated to each of the cubic coordinate axes. The following conditions are valid: m 2m'; /-# (-1)l/ n# (20) As a consequence, according to (19) where the coefficients At =# are calculated from the condition (18).
For group Th, besides the conditions (18) and (20), the selection rule 2 1' must be satisfied. So, for even 1, its symmetric functions are the same as those for group T.
The symmetric functions for group Oh are the same as those for O for even I.
The real functions obtained in this paragraph fulfill the orthonormality condition (9). Example Suppose a specimen built-up of quartz (D3) crystallites with triclinic sample symmetry (Bunge and Wenk, 1977). Symmetrized functions (16) corresponding to Table I are  reported in Table III and may be expressed explicitly according to (15).

REAL THREE-DIMENSIONAL SYMMETRIC FUNCTIONS
With the help of coefficients (13) real functions Tlm'n(g) Can also be introduced as orthonormalized real combination of functions Tl (g) [see Bunge (1969 and1982) and Matthies (1987)] 100 O. RAYMOND ET AL.  where the coefficients C are reported in Table IV We obtain that the gl roll coefficients calculated from (26-28) as well the functions reported in Table V, agree with the corresponding entities as given by Bunge (1982), for < 6.