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For the reciprocal lattice vector ${\mathbf{G}}_{hkl}=(2\pi /a)(h,k,l)$, sum the atomic scattering factors with the atomic basis phase factors. Taking the positive phase convention gives the structure factorChanging the phase convention complex-conjugates the structure factor and leaves the diffraction wave intensity unchanged. Since the Miller indices are integers,If all three Miller indices have the same integer parity, all pairwise sums are even. Otherwise exactly one pairwise sum is even and two are odd. ConsequentlyThe corresponding structure factors are $f_{\mathrm{Au}}+3f_{\mathrm{Cu}}$ and $f_{\mathrm{Au}}-f_{\mathrm{Cu}}$. Reflections with mixed integer parity are superlattice reflections that reveal chemical ordering in alloys.

Both starting arrangements converge toward the same chemically ordered alloy, exhibiting chemical ordering in alloys. This supports a lower free energy for the ordered unlike-metal neighbour pattern than for the segregated core-shell or disordered precursor at the final temperature. In a simple nearest-neighbour model, ordering is favoured whenwith consistent bond-energy conventions. Bulk mixing and ordering then compensate the loss of configurational entropy on cooling. Annealing alone does not supply numerical bond or surface energies, and elastic mismatch or adsorbed ligands can also matter.

The final morphology does not show a thick single-metal skin: the surface-energy difference is not large enough to overwhelm the bulk preference for ordering throughout the particle. Some atomic-scale surface enrichment is compatible with the drawing, so exact equality of the two surface free energies is not established.

If B has a much lower surface free energy, Surface segregation of B and an A-rich interior become favourable. A nearly continuous B shell is likely if enough B is present and the surface-energy gain exceeds the cost of disrupting favourable A-B bonds; otherwise only partial enrichment occurs.