Three-Dimensional Texture Analysis after Bunge and Roe " Correspondence Between the Respective Mathematical Techniques

Bunge’s and Roe’s three-dimensional texture analysis methods, although both founded on harmonic analysis, show some differences between the various mathematical techniques used. This paper establishes the correspondence relation between the respective mathematical techniques allowing one to compare works done in either variant. Taking the latest developments in three dimensional texture analysis into account, the correspondence relations hold for the odd degrees as well as for the even ones. Finally numerical tables give the extension of the symmetry coefficients B’" (after Bunge) and R[,, (after Roe) to all the degrees of the series expansion, even and odd, including 34.


INTRODUCTION
In 1965, Bunge on the one hand and Roe (1965) on the other hand have simultaneously published a theory about three-dimensional analysis of crystallographic textures.Their theories are founded on the same principle: using pole figures which are bi-dimensional projections for the purpose of calculating the three-dimensional texture function f(7).Although the base of their methods is similar, the formalism is slightly different, essentially because of: indifferent definition of the Euler angles rathe utilization of polynomials having different phase-and norm- conventions 96 C. ESLING, E. BECHLER-FERRY AND H. J. BUNGE --the way of taking the symmetries into account.
As the scientific publications in the field of texture analysis use either terminologies, the comparison between the various results published requires some correlation relations that we intend to establish.

THE EULER ANGLES
Definition of the Euler angles after Bunge and Roe The texture functionf(g) describes the orientation distribution of the crystallites in the sample.These orientations are generally described by means of Euler angles making the coinciding of two orthogonal reference systems possible: Ka bound to the sample and Kn bound to the crystallographic lattice of grains.This coincidence is generally achieved by three successive rotations around the K n reference axes.FIGURE la Euler angles after Bunge (1969).FIGURE lb Euler angles after Roe (1965).The variables are different only because of the second rotation, whose axes (X' and Y') are orthogonal, which introduces a phase difference of amplitude H . Thus, both triplets described the same orientation of the KB reference if they are linked by the relations:

0=0
(1) The invariant measure d/7 The O.D.F.depends on the Euler angles: This density function has to be positive and to verify the normalization condition: (3) The integral is extended over the whole of the Euler space: The Euler angles q91, O, (/92 constitute a system of curvilinear coordinates.In this system the differential integration element reads: dg-I(q91, 0, q92).dqgl.dO.dq92.
It should be noticed that this differential volume element is not the simple product of the differentials of each variable, as in the case of Cartesian coordinates, for which: dv dx dy dz (5) The differential integration element (also named "invariant measure") used by Bunge: 1 dg n 2 sinO.dO.dq91" dtp2 (6) is such as: This selfsame factor is consequently found again between the functions of orientation density: 1 W(, 0, (./9)----2jq21, (I), (P2) (10) used by Bunge and Roe respectively.

THE THREE DIMENSIONAL FUNCTIONS
Relation between the polynomials P"()) and Ztm,,() A convenient representation of the texture function f(g) is its development into series on the basis of the generalized spherical harmonics.According to H. J. Bunge, this development reads" f g Z C'f"" T'fl "(g   (11)   or f(o, , q2)= Z C "e'm2 P"() einel (12) In R. J. Roe's terminology the corresponding development reads: w(@, , )= W,e-'Zt,().e -'" (13) The expression of the generalized spherical harmonics contains polynomials which are associated to Jacobi's polynomials.
The above norm relations immediately lead to the norms of random dis- tributions: and frand."dg -"frand.dg 1 fia.d. (20) Wrand" sin0.dO" dO.dq w,a,d, sin0"dff" dO'dq 1 w,,.a. (21) Imposing the norm condition to the serial development off(g) gives: (E C'" TTfl"(g))dg EC'" T "(g) 1 ( 22) The integral of the generalized spherical harmonics may be considered as a particular case of the orthonormalization relations: I (g) dg l(g) .ag (g).ro(g) .d ,o o o (23)   Therefore the norm condition of f(g) determines the value of the first coefficient of rank 0 in the development: Co=1 (24) and it follows from a similar calculation: ooo "/ Relation between the coefficients C;"" and W;"" of the O.D.F.sThe determination of the set of coefficients {C7'"}/< l,,,,,x of thef(g) develop- ment constitutes an important stage in studying the texture of a polycrystal.Indeed, this set of coefficients contains the whole information about the orientation distribution in a strongly digested form, and enables the later computation off(g) as well as that of the anisotropic physical properties.
As Euler angles used by Bunge and Roe differ, the plottings of the O.D.F.
sections are generally not identical.
The two kinds of graphs can be deduced from one another by geometrical transformations which arise from the correspondence relations Eq. ( 1).
These geometrical transformations are particularly simple when the sample and the crystal lattice have symmetry elements.In the following we are going to describe these transformations in the major case of an orthorhombic sample symmetry and a cubic lattice symmetry.
As a matter of fact, the application of these symmetries allows us to restrict the variation range of all three Euler angles to the interval I-O, -].Especially,   for the Euler angles {q91, , q92}, the following equivalence relation holds: ,, ,, The combination of the correspondence relations (1) and the symmetry relation (31) leads to: 0 These relations show that the @ and oz variables are connected with one another by a symmetry plane at qgz 45; likewise qg q92 45.The plottings of the O.D.F.sections in the Euler space, in the terminology of respectively Bunge and Roe, correspond to one another by means of these symmetry planes, as shown in Figure 2.

THE TWO DIMENSIONAL FUNCTIONS
The mathematical relation which makes it possible to go from O.D.F. to pole figure is an integral (Bunge, 1969; Bunge, 1982).Thus the various functions necessary to study pole figures are only bi-dimensional.The correspondence relations are then particular cases of the relations which have just been established for the three-dimensional functions.
Spherical coordinates on the pole sphere The pole figure P,i is the distribution density of the normals h to a family of crystallographic planes {hi, ki, li} in the sample reference frame.
The orientation of the normals to the considered planes is described with its polar coordinates.These polar coordinates make up an orthogonal curvi- linear system of coordinates on the sphere; they are defined identically by both Bunge and Roe (Figure 3).(Bunge, 1969;1982) b) Roe's definition (Roe, 1965).
Relations between the polynomials P; (cos) and P'() A pole figure can be developed into a series on the spherical surface harmonics.These functions constitute indeed a complete orthogonal basis on the sphere.

CONSIDERING THE SYMMETRIES OF THE TEXTURE FUNCTION
The dual origin of symmetries: crystal and sample symmetry The f(9) texture function satisfies two invariance types that result from the crystal symmetry on the one hand, from the sample syro.metry on the other hand.The symmetries of a crystal lattice can be defined by a symmetry point group where GB is the rotation subgroup: GB { g' } j 1,NB Due to these symmetries, the f(9) function is invariant for the corresponding orientations which could not physically be distinguished from one another: f(g?.g) =f(g) (55) In the same way the crystallites distribution in the polycrystal may have statistical symmetries gA which generally are induced by the shaping processes of the material: f(9"g) =f(g) (56) Both invariance conditions Eq. (55-56) can be taken into account indepen- dently from one another; indeed one of them operates on the right whereas the other one operates on the left.The treatment of these symmetries being otherwise identical, only either type is to be examined.
Bunge creates functions adapted to the symmetries by means of linear combinations of the generalized spherical harmonics TT't "(g) (Bunge, 1965b;   1982).Roe expresses the Wtm, coefficients of the series as being a function of only a smaller number of these, which are linearly independent (Roe, 1966).
The "lower" symmetry (non-cubic): selection rules We will take the trigonal crystal symmetry and the orthorhombic sample symmetry as examples and investigate these two cases.
The trigonal crystal symmetry Let us recall that the rotations subgroup of the trigonal holohedrism is characterized by one threefold axis and per- pendicular to it, three binary axes.The threefold axis (parallel to the Z axis of the K system) selects the generalized spherical harmonics T""(g) whose left index is a multiple of 3. Likewise it selects the Wt,,3,, the other coefficients being necessarily zero.

Bunge and Roe
The description of the symmetries is simplified if the Z axis of the crystal system is parallel to the four-fold axis which has the highest multiplicity.
Considering moreover a binary axis orthogonal to the four-fold axis implies describing the tetragonal symmetry which corresponds to a subgroup of the cubic symmetry.
Due to the particular value of this angle, the functions adapted to the cubic symmetry cannot be obtained as above using only selection rules.The functions adapted to the cubic symmetry are linear combinations of spherical harmonics: +l fin(g) E mu Tmn(g)   (66) 4m--!As the real ./i 4' coefficients verify the relation: F4m/z (__ 1)/ 2zi/4m/z (67) the above mentioned linear combination also reads: Tf"(g) A?" "(g) + A" T"(g) + (-1)' T*(g) ( 68) The index takes on the values 1, 2, 3 M(l) and so numbers the elements of the orthogonal basis: &u, , 6,,, f() '"'()d 2N% In the case of lower symmetry, the number of functions adapted to the symmetries is obtained simply by counting from their definition.Thus in the case of the tetragonal symmetry, the number of functions is an immediate result of the selection rules expressed in Eq. (63-64): even odd (70) FIGURE 5 M(l) number of linear independent symmetry--adapted functions corresponding to tetragonal symmetry (dotted curve) and cubic symmetry (solid curve).

FIGURE 2
FIGURE 2 Correspondence between O.D.F.sections represented in the terminology of, respectively, Bunge and Roe a) Sections at constant (192 (i.e.constant b) Sections at constant (91 (i.e.constant k --go1).