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. 2021 Apr 15;12:2272. doi: 10.1038/s41467-021-22515-3

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 4 — a Temperature-dependent population of CdC_nN_4–n coordination environments as determined by our MC pseudospin simulations (small symbols; error bars denote the standard error measured from five independent MC simulations), neutron PDF measurements (open squares) and ¹¹³Cd MAS NMR spectroscopic measurements (open circles), compared with the non-interacting analytical result (solid lines, ref. ³⁴). The sublattice magnetisation (red symbols, red line)—or polarisation, in the specific context of Cd(CN)₂—is an order parameter for the transition between the disordered ( $P n \bar{3} m$ ) spin-ice state and the low-temperature ( $I 4_{1} / a m d$ ) antiferroelectric state. b Variable-temperature ¹¹³Cd MAS NMR spectra (black lines) and corresponding fits (red lines) used to extract the values given in a. c Variable-temperature neutron PDF data (lower curves) and calculated PDFs from $γ = 0$ and $\frac{2}{3}$ QC configurations (upper curves) used to determine the PDF-derived values in a. d Single-crystal X-ray diffuse scattering pattern ((hk0) plane) measured at 298 K and calculated from a coupled CN-orientation/Cd-displacement model based on the 298 K pseudospin configuration. e Effective magnetic diffuse scattering pattern ((hhl) plane) extracted from our 300 K MC simulations by interpreting CN^– orientations from a single pyrochlore sublattice as classical spin vectors with the Dy³⁺ magnetic form factor (top panel). Calculated magnetic scattering for the pyrochlore Ising ferromagnet for the same relative temperature $T / J_{eff}$ = 1.5 to which the Cd(CN)₂ data correspond (bottom panel). In both cases, fluctuations away from the spin-ice ground state broaden the pinch-point features observed in experimental magnetic diffuse scattering pattern measured, for e.g., spin-ice Dy₂Ti₂O₇ at 1.3 K⁶⁸. Instead the scattering feature resemble more strongly the inelastic magnetic neutron scattering for strongly fluctuating spin-ice system such as Pr₂Zr₂O₇⁵⁰.