Intercrystalline structure distribution in polycrystalline materials
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Materials", Metall. Trans. A, 1986, vol. 17A, pp. 21992207. Irreducible representations of SO(3), the group of proper rotations in R 3, have been given in terms of functions of the Euler angles by Wigner, tll Margenau and Murphy," [21 Rose, TM Gel'fand, Minlos, and Shapiro, t4] and Roe, TM among others. These representations involve generalized spherical harmonics containing the augmented (normalized) Jacobi polynomials. Margenau and Murphy and Gel'fand et al. restrict the second Euler angle to the range 0, 7r. Roe, while citing Margenau and Murphy with reference to his choice of Euler angles, does not explicitly limit the range of the second angle. Gel'fand et al. note that, if the orientation g is given by ~1, 0, ~02, its inverse g - 1 is given by 7r - ~2~0, 7r - ~Pl.Rose (p. 53) points out that d~,m ( - f l ) = ( - 1 ) '- dJm,m (fl). With this relation, the representations -~o2, - 0 , -~p, and 7r - ~P2, 0, 7r - ~p are equivalent. In Roe's representation, the augmented Jacobi polynomials are expressed in terms of the cosine of the second Euler angle, so that the effect on the polynomials of a change of sign of the angle is not apparent. Failure to properly consider this effect is responsible for the following errors in Adams' paper: Equation [19] is incorrect. The non-sidedness property requires only that r be even. In Table I, the equivalent entries under L~tmmfor the misorientation inverse property should be ( - 1 ) m+nL,t-~-~ and l* rs Lan m. In Table II, the entry under S~m, for the non-sidedness property should require only that r be even. By following a procedure analogous to that used to obtain Eq. [18] it is possible to show that ifL~ (to, a , f l , 7 ) = j* L~* (to, a, fl, 7) (real function), then L~lmn = ( - - ll'~m+nl ) ~tff'~" This relation was used in constructing Table I, but not given explicitly in the paper. In addition to the above, the sentence in the first full paragraph of the second column on page 2201 should read, "The comers of the triangle, Co, Cl, and c2, are ordered according to increasing projected length on coordinate ya.,, In Figure 4, c~ and c2 should be interchanged.
(b) Fig. 10--Fractographs in the F69 specimen. (a) Scanning electron micrograph and (b) optical micrograph.
We wish to thank S.-M. Kang for experimental assistance. REFERENCES 1. Y. Tomita, S. Oki, and K. Okabayashi: Tetsu-to-Hagan~, 1978, vol. 64, pp. 459-68. 2. Y. Tomita, S. Oki, and K. Okabayashi: Tetsu-to-Hagan~, 1978, vol. 64, pp. 78-87. 3. S.-M. Kang and H. Kwon: Metall. Trans. A, 1987, vol. 18A, pp. 1587-92.
Intercrystalline Structure Distribution in Polycrystalline Materials P. R. MORRIS, JUNWU ZHAO, and B.L. ADAMS The following concerns B. L. Adams, "Description of the Intercrystalline Structure Distribution in Polycrystalline METALLURGICALTRANSACTIONSA
The authors wish to acknowledge the support of the United States Department of Energy, Office of Basic Energy Sciences, under grant #DE-FG02-85ER45203. REFERENCES 1. Eugen Wigner: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Fr
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