Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.6, p. 296

Section Antiphase domain boundaries

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Antiphase domain boundaries

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In the diffraction pattern of an ordered alloy, a dissimilar broadening can often be observed for structure and superstructure peaks (with the former being present in both the ordered and disordered states). The superstructure peaks, in fact, bear microstructural information on the interface between the ordered regions in the material: broadening occurs when domains meet out of phase, creating an antiphase domain boundary (APB or APDB). A general formula for APDB-related broadening does not exist: for a given ordered structure, the Fourier coefficients correspond to the normalized value of [A_{{\rm APDB},hkl}(L) = \overline {F(0)F^*(L)}], where F(0) is the structure factor of a cell positioned at L = 0 and [F^*(L)] is the complex conjugate of the structure factor of a cell at a distance L along the direction [hkl]. Being the result of a combination of probabilities, the peak is always expected to be Lorentzian.

Explicit formulae have been derived for the Cu3Au ordered alloy (L12 phase; Wilson, 1943[link]; Wilson & Zsoldos, 1966[link]; Scardi & Leoni, 2005[link]). Several types of boundaries can form, depending on the way that the domains meet: the broadening depends both on the boundary plane and on the local arrangement of Au atoms leading to conservative (no Au atoms in contact) or nonconservative (Au atoms in contact) boundaries. By arranging the indices in such a way that hkl and that l is always the unpaired index, the broadening of the superstructure reflections can be described as (Scardi & Leoni, 2005[link])[A^{\rm APDB}(L) = \exp [- 2L\delta f(h,k,l)]. \eqno (3.6.51)]In this formula, δ = γAPDB/a0 is the probability of occurrence of an APDB, a0 is the unit-cell parameter and f(h, k, l) is a function of hkl defined in Table 3.6.2[link], obtained from the results of Wilson (1943[link]) and Wilson & Zsoldos (1966[link]).

Table 3.6.2| top | pdf |
Models for antiphase domain boundaries for the Cu3Au case

N = h2 + k2 + l2.

IDModelf(h, k, l)
1 Random 2/3
2 {100} planes [\displaystyle {2 \over 3}{{h + k + l} \over {\sqrt N}}]
2.I {100} planes, no Au–Au contacts [(k + l)/{\sqrt N}] if h is the unpaired index
[(h + l)/{\sqrt N}] if k is the unpaired index
[(h + k)/{\sqrt N}] if l is the unpaired index
2.II {100} planes, only Au–Au contacts [\displaystyle {{2h + k + l} \over {2\sqrt N }}] if h is the unpaired index
[\displaystyle {{h + 2k + l} \over {2\sqrt N }}] if k is the unpaired index
[\displaystyle {{h + k + 2l} \over {2\sqrt N }}] if l is the unpaired index
3 {110} planes [\displaystyle {2 \over 3}{{4h + 2k} \over {\sqrt {2N} }}]
3.I {110} planes, Au displacement parallel or perpendicular to plane normal [\displaystyle {{4h} \over {\sqrt {2N} }}] if h is the unpaired index
[\displaystyle {{2h + 2k} \over {\sqrt {2N} }}] otherwise
3.II {110} planes, Au displacement at 60° to plane normal [\displaystyle {{2h + 2k} \over {\sqrt {2N} }}] if h is the unpaired index
[\displaystyle {{3h + k} \over {\sqrt {2N} }}] otherwise
4 {111} planes [\displaystyle {{8h} \over {3\sqrt {3N} }}] if h ≥ (k + l)
[\displaystyle {{4(h + k + l)} \over {3\sqrt {3N} }}] otherwise

The average distance between two APDBs is given by 1/δ. For a random distribution of faults, the broadening is Lorentzian and AAPDB = exp(−4Lδ/3).


Scardi, P. & Leoni, M. (2005). Diffraction whole-pattern modelling study of anti-phase domains in Cu3Au. Acta Mater. 53, 5229–5239.Google Scholar
Wilson, A. J. C. (1943). The reflexion of X-rays from the `anti-phase nuclei' of AuCu3. Proc. R. Soc. Lond. Ser. A, 181, 360–368.Google Scholar
Wilson, A. J. C. & Zsoldos, L. (1966). The reflexion of X-rays from the `anti-phase nuclei' of AuCu3. II. Proc. R. Soc. Lond. Ser. A, 290, 508–514.Google Scholar

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