Cubic nvariant Spherical Surface Harmonics in Conjunction with Diffraction Strain Pole-Figures

Four kinds of cubic invariant spherical surface harmonics are introduced. It has been shown previously that these harmonics occur in the equations relating measured diffraction (line-shift) elastic strain and macro-stresses generating these strains for the case of textured cubic materials. As a consequence, these harmonics are important for the determination of unknown macro-stress tensor components if the o.d.f. expansion coefficients are known. On the other hand, they play a role in the determination of unknown o.d.f. expansion coefficients if the macro-stresses are known using for instance, a tensile test device on the diffractometer. Then, even the odd-order o.d.f. expansion coefficients can be obtained. In this paper, special attention is given to the mathematical construction of the cubic harmonics, the physical symmetry requirements they are subject to and some examples are given exhibiting both even and odd order harmonics.


INTRODUCTION
It has been recently shown (van Baal, 1983;Brakman, 1983;Brakman 1984a) that the "classical" sin2ff-law (D611e, 1979) used for residual macro-stress determination may be adapted for the case of textured materials using relevant orientation distribution function (o.d.f.) theory.It is common practice in residual stress analysis to denote the deviation angle from the specimen's surface normal by the angle k whereas the angle b measures the angular distance in the specimen's surface (Fig. 1).For only trll and 0"22 : 0 and texture-free cubic materials the FIGURE Experimental arrangement.equation relating the measured diffraction (line-shift) elastic strain and these stresses reads: where: <e'z(O, dp)> s (hkl)[Crlt + r22] +1/2.s2(hkl)r,sin2p le) Here the Reuss-model of poly-crystalline elastic crystal coupling has been used and s11, S1122 and S1212 are the well-known single (cubic) crystal elastic compliances.Note the predicted linear behaviour between (e') and sin2O.For the case of a specimen exhibiting crystallographic texture and residual macro-stresses (for instance, cold-rolled steel sheet samples with orthorhombic specimen symmetry) it has been shown (ld) (Brakman, 1984a) that Eq. ( 1) may be modified using o.d.f, theory yielding: (e'zz(b, ) ) s1(hkl)[0.11+ 0"22] + 1/2s2(hkl)0"osin2O So 2 Phu(/, ) (0" where the series-expansion method of the o.d.f, on the rotation group in R3 has been used cf. the theory given by Bunge (1982, pp. 47-118).
Note that it follows using Eq. ( 2) that for the {hoo} and {hhh) reflec- function F2 FIGURE 3 Function F2 for j 12 and # 1.
tions the relevant expressions for (e'z) do not exhibit any texture- dependency at all if the Reuss-model is taken to be valid.
Note that Eq. ( 2) exhibits in principle, both even and odd j.Note also that function F1 and F3 are invariant w.r.t, any permutation of h, k and whereas F2 and F4 develop a negative sign for 24 of the 48 possible permutations of h, k and I. See for the latter case Figs. 3 and 5 whereas the former is demonstrated in Figs. 2 and 4. function F3 FIGURE 4 Function F3 for j 13 and # 1.

PHYSICAL SYMMETRY CONSIDERATIONS
Centro-symmetry of the diffraction strain pole-figure ('z(, ) Since the diffraction strain (e'zz) is a physical quantity depending on the angles ff and tk it can be represented for a given (macro)  The iso-density lines in Figures 2 thr.1983) and also by Brakman (1984a).From Friedel's law it follows that, similar to "conventional" diffraction intensity polefigures, these pole-figures should exhibit centro-symmetry, that is the following should hold: ('zz(rC-,rc + 4')) (5) Using Eqs.(I-1) through (1-4) (see Appendix), it follows from Eq. (2) that this is not the case.Analysis reveals that just like the case for the "conventional" diffrac- tion intensity pole-figures (Bunge, 1982, p. 106) the contributions due to crystals exhibiting a [h"kT] vector in the , tk direction should be taken into account in addition to [hkl-I.This problem has been discussed extensively elsewhere (Brakman, 1984b) where it was found that treating the 2 orientations exhibiting a [_hkl] and a I-h-kTJ vector in the q, b- direction respectively, is sufficient.Note that a [h'kTJ does not imply an inversion of cubic crystal reference axes: the choice of [hk-/'J as a vector parallel to the q, b direction generates a physically different crystallite orientation within the context of this problem (Brakman, 1984a).
Then it follows, assuming equal structure factors of these two orientations, that the functions F2 and F4 vanish from Eq. ( 2).
The correct expression is immediately obtained by then omitting all terms exhibiting F2 and F, in Eq. ( 2).
It may be inferred (see the Appendix) that centro-symmetry of (e'z (Eq.( 5)) is fulfilled then.Since (e'z is a physical quantity it is also re- quired that the eventual equation be invariant w.r.t, any permutation ofh, k and I. Since only the functions F1 and F3 remain this is indeed the case.Note that functions F2 and F, vanish from Eq. ( 2) for both even and oddj whereas functions F1 and F3 persist for both even and oddj values.Hence, it can be argued that the "selection-rule" obtained from the requirement cf.Eq. ( 5) is the vanishing of F2 and F4 from the eventual equation rather than imposing a restriction upon the permissible values of j as is the case with the diffraction intensity pole-figures (Bunge,  1982, p. 106).
Note that for the {hhk} and {hko} reflections the functions F2 and F, may be shown to be equal to zero (as was also the case for the {hoo}, (hho} and {hhh} reflections).This may also be seen from Figs. 3 and 5.In addition to this it can be shown (Brakman, 1984b) that for the complete treatment of Eq. ( 2) for these reflections the consideration of the crystallite orientations associated with [h-h-k-I, or [ho] is superfluous due to the "symmetrical" nature of these directions.It can be shown that they yield exactly the same result as follows from [hhk] and [_hko] respectively."Consequently, for the determination of (e'zz (if these reflections are used) the treatment of only [hkk-I or [_hko] respectively is sufficient omitting the terms exhibiting F2 and F, in Eq. ( 2).

Cubic crystal symmetry
The cubic symmetry coefficients ,ucf.Eqs.(3a) through (3d) are defined as a consequence of point-group 432: A 4-fold rotation axis parallel to [001] yields that rn in Eqs.(3a) through (3d) should be equal to a multiple of 4, the 3-fold rotation axis parall,el to [111] leads to the well-known relationship that the coeffici- ents A'j 'u must satisfy i.e.
AT'"P7 cos (8) -j for both even and odd j and a 2-fold rotation axis parallel to [100]   yields: A -''u (-1)JA'fl u (9) The other symmetry elements of 432 do not yield additional useful relationships w.r.t, the context of this paper.
As such, rotations only are needed which comply with the series- development method of the o.d.f, on the rotation group in R3 as has been introduced by Bunge (1982, pp. 47-118).
However, upon considering point-group m3m a centre of inversion occurs which cannot be accounted for using rotations only.It has however, been shown (Brakman, 1984b) that due to this centre of inversion all physically relevant crystallite-orientations can be "reached" using rotations only.Consequently, Eq. (2) (minus the F2and F,-contributions) should represent physical reality (the Reuss- model taken to be valid) for textured materials consisting of cubic crystals exhibiting symmetries of.point-groups 432 or m3m.
One can distinguish now between 3 cases: (i) crystal symmetry of.point-group m3m and specimen symmetry of.mmm.In this case the right-handed and left-handed o.d.f.'s are both equal and (equivalent) to the o.d.f.(Bunge, 1982, p. 105).The treat- ment given here concerns this o.d.f, and use of rotations only (as has been used in the derivation of Eq. ( 2)) is sufficient.
Note that specimen symmetry cf.point-group 222 can not occur if the crystals satisfy m3m.(ii) crystal symmetry of.point-group 432 and specimen symmetry cf.
222.In this case the right-and left-handed o.d.f.'s are different from each other (possibly in every orientation g) and they have to be treated separately.However, right-and left-handedness has to be defined with respect to some physical property.For polycrystalline diffraction experiments and measurement of elements of 2nd order (strain) tensors (as is the case here) it can be expected that measure- ments obtain contributions from both right-and left-handed o.d.f.'s at the same time.This case is treated in Eq. (4.323) from Bunge (1982) for polycrystalline diffraction experiments.For the case of diffrac- tion strain pole figures the o.d.f, coefficients C as they occur in ]L(' LEv M R Eq. (2) have to be substituted by: MgCj 'v d- where and M L denote the fractions of the right-and left-handed crystals respectively.The odd order C do not drop from Eq. (2).Con- sequently, the virtual o.d.f, giving rise to the measured diffraction strain pole-figure is different from the one yielding the measured intensity pole-figure.Eq. ( 2) is still adequate.

DISCUSSION
It can be seen from Figures 3 and 5 that functions F2 and F, exhibit a symmetry cf.point-group 432 for both even and odd j.
On the other hand, functions Ft and F3 (Figures 2 and 4 depict examples of them for both even and odd j) satisfy m3m for both even and odd j.
As such their behaviour resembles the "normal" harmonics cf.Eq. ( 10) for even j.
It has been shown by Bunge (1982, pp.100-107) that for the case of diffraction intensity pole figures the treatment of the crystallite- orientations following from taking [hkl] and [h-F/] parallel to the k, b direction leads to the occurrence in the eventual equations for the pole figure intensity of a factor equal to: [1 + (-1)] (11) As a consequence, the o.d.f, expansion coefficients C cannot be determined for odd j from those pole-figures.
It is argued now that for the case of diffraction strain pole figures a different situation develops.It has been obtained (Brakman, 1984b)  that instead of Eq. ( 11) the following factors then occur in the eventual equations: [1 + (-1)zs] for Ft and F3 (12a) [1 + (-1) 2j+ ] for F2 and F4 (12b) which complies with the vanishing of F2 and F, as mentioned previously.Hence, it follows using Eqs.(I-1) thr.(1-4) that the o.d.f.expansion coefficients C" are not undeterminable for oddj from diffrac- tion (line-shift) elastic strain pole-figures.
Note that the "selection rule": C" undeterminable for odd j for the case of diffraction intensity pole-figures is replaced here by the vanishing of functions F2 and F4 for both even and odd j.
Note that for all hkl: F2 F4 0 for j 4 and F2 0 for j 8. CONCLUSIONS (i) Four kinds ofcubic invariant spherical surface harmonics have been introduced.Two of them (functions F1 and Fa) satisfy for even and odd order the symmetry-requirements of point-group m3m whereas the other two (functions Fz and F4) comply with point-group 432 for even and odd j.For the treatment of diffraction strain pole- figures the functions F. and F, are not necessary (except for the case of anomalous scattering).
(ii) the functions derived may serve as a tool for the determination of unknown macro-stresses in textured cubic materials exhibiting residual or applied stress if the o.d.f, expansion coefficients are known.On the other hand, if the macro-stresses are known these functions may be used to obtain, the o.d.f, expansion coefficients C for both even and odd j provided enough diffraction strain measurements are available.
of a diffraction strain pole-figure as introduced by Hoffman et al. (