crystals. The intensity of this diffuse scattering was zero for l = 0 (in the
planes passing through the centre of reciprocal space), and increased
with the magnitude of l. This is a typical signature of diffuse scatter-
ing from displacive correlations, with displacement polarized along
the c direction. Interestingly, in the Mn[Co]′ and Cd[Co] samples the
diffuse scattering intensities follow tetragonal symmetry, instead of
the cubic symmetry of the Bragg peaks. Note that because displacive
diffuse scattering is absent in the hk0, h 0 l and 0kl sections it has no
contribution to the projected diffuse scattering tiles from Figs. 2 and
3 and thus does not influence the analysis of vacancy distributions.
Data availability
The raw data on which this manuscript is based are openly available for
download from https://doi.org/10.5287/bodleian:8JB5XgybE. These
data include the scattering images given in Fig. 2 and the Monte Carlo
configurations from which Figs. 3 and 4 are derived.
Code availability
All custom code used in this study was developed using widely available
algorithms. Copies of the code used can be obtained upon request.
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Acknowledgements A.S. and A.L.G. gratefully acknowledge financial support from the
Leverhulme Trust UK (grant no. RPG-2015-292), and T.D.B. acknowledges F.W.O.–Vlaanderen
(Research Foundation Flanders) for a Postdoctoral Fellowship. A.S. thanks the Swiss National
Science Foundation for Ambizione and PostDoc Mobility Fellowships (PZ00P2_180035,
P2EZP2_155608) and Diamond Light Source for the provision of beamtime (MT13639,
MT20876, CY22610). M.L.R.G. thanks the Consejo Nacional de Ciencia y Tecnología (Mexico)
for a scholarship. A.L.G. thanks the European Research Council for funding (grant nos 279705
and 788144), P. D. Battle (Oxford) and A. R. Overy (Oxford) for valuable discussions, N. P.
Funnell (ISIS), J. A. Hill (UCL) and C. S. Coates (Oxford) for assistance with single-crystal
growth, and A. L. Thompson for assistance with synchrotron measurements.Author contributions A.S., T.D.B., H.-B.B. and A.L.G. designed the research. H.L.B.B., M.L.R.G.,
H.J.G. and T.D.B. synthesized the materials. A.S., D.C., A.B. and H.-B.B. measured the single-
crystal diffuse scattering patterns. A.S. performed the 3D-ΔPDF refinement. A.S. and A.L.G.
developed and implemented the Monte Carlo model. A.S. and T.D.B. calculated percolation
properties. A.S. and A.L.G. wrote the manuscript, with input from all authors.Competing interests The authors declare no competing interests.Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/s41586-020-
1980-y.
Correspondence and requests for materials should be addressed to A.L.G.
Peer review information Nature thanks Simon Billinge, T. Richard Welberry and the other,
anonymous, reviewer(s) for their contribution to the peer review of this work.
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