Codex @codex  0

For a stoichiometric $A_{3}B$ alloy with one corner sublattice and three equivalent face sublattices, let $p_c$ and $p_f$ be the respective occupation probabilities of species $B$. Define $\rho =(4p_c-1)/3$. Conservation of composition, $p_c+3p_f=1$, gives $p_c=(1+3\rho )/4$ and $p_f=(1-\rho )/4$. Thus the random alloy has $\rho =0$ and the fully ordered corner-$B$ pattern has $\rho =1$. The coherent superlattice reflection amplitude is $\rho (f_B-f_A)$, so its wave intensity scales as ${\rho }^2$, while the equal-parity fundamental amplitude $f_B+3f_A$ is composition-fixed. The definition assumes equivalent face sublattices; endpoint normalization alone does not uniquely specify an arbitrary order coordinate.

An ordered cubic $A_{3}B$ structure with B at cube corners and A at face centres. The chemically labelled Bravais lattice is primitive cubic; its superlattice reflections distinguish it from a chemically disordered face-centered cubic lattice.

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.

Replace every coloured site in the drawing by an identical site carrying $\bar{f}$. The face-centering lattice translations now preserve the average crystal structure, whose Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors isA coherent Bragg diffraction peak requires $\mathbf{G}\cdot {\mathbf{a}}_j\in 2\pi \mathbb{Z}$ for all three primitive lattice translation vectors. Thus $k+l$, $h+l$, and $h+k$ must all be even, equivalent to all three Miller indices having the same integer parity. This is the face-centred cubic selection rule: $(110)$ is forbidden and $(111)$ is allowed. The restoration of these lattice translations explains the disappearance of the ordered superlattice reflections.

Use the same atomic basis phases as in the ordered structure factor, replacing each atomic scattering factor by its site average. For the reflection with equal integer parity,Conservation of the total composition removes the alloy sublattice order parameter from this amplitude. For the reflection with mixed integer parity,The coherent powder diffraction wave intensities consequently satisfyHere the lattice constant, atomic scattering factors, powder diffraction multiplicities, and omitted experimental factors are held fixed. The ratio of a superlattice reflection to a fundamental reflection can therefore measure the magnitude of the alloy sublattice order parameter.

For mixed parity, one of the three pair sums is even and two are odd. Thus their phase factors sum to $-1$, givingThese superlattice reflections reveal chemical ordering. Their kinematic intensity is proportional to $|f_{\mathrm{Sc}}-f_{\mathrm{Al}}{|}^2$, and vanishes in the hypothetical limit of equal species scattering factors.

Keep the matrix spots and add precipitate reflections at all integer $(h,k,0)$ with $0<h^2+k^2\le 8$. Even-even spots are fundamental; every other allowed position in this zone is a superlattice reflection. The added positions are $100,010,110,210,120$ and their symmetry-related signs. Each position actually denotes the corresponding signed index pair, not one radial ring. The precipitate grid spacing is $1/a_p$; the nearest fundamental separation is $2/a_p=0.4878{\mathring{\mathrm{A}}}^{-1}$. If matrix and precipitates are coherent and similarly oriented, their fundamental spots approximately coincide, with finite mismatch potentially giving splitting or broadening. The sketch assumes common orientation, small strain, appreciable sampled precipitate volume and predominantly single scattering; electron diffraction can have dynamical intensity effects.

Choose an objective aperture around a precipitate superlattice reflection, such as $100$, which is absent from the disordered aluminium matrix. The selected beam then makes precipitates bright in dark-field transmission electron microscopy, while the matrix gives little intensity in that beam. Tilt the grain to place the chosen reflection close to its Bragg diffraction condition and preferably approach a two-beam geometry. This reduces excitation error and can increase precipitate brightness; the exact maximum also depends on foil thickness, strain and dynamical scattering. Selecting a shared fundamental reflection would give much poorer chemical-order selectivity.