Incommensurate magnetic structure of CrAs at low temperatures and high pressures

The incommensurate magnetic structure of chromium arsenide CrAs below the anti-isostructural phase transition is studied with neutron powder and single-crystal diffraction at high pressures and low temperatures.


S1. Analysis of the neutron powder diffraction data
The neutron powder data obtained from the experiments at HRPT (Figure S1) were refined using Jana2020 (Petricek et al., 2023) in three steps.In the first step, a Le Bail fit of the powder pattern was made, in which the positions of the magnetic reflections were described only by an incommensurate propagation vector k of the form (0, 0, kc).The lattice parameters a, b, and c, the kc component of the propagation vector k, the peak-shape parameters GW and LY, the shift parameter shift and the strain tensor parameter St0022 were refined.The background was refined with about 30 manually set points and 5 refined Legendre polynomial terms.As the data showed elemental chromium and Cr2O3 to be present, this was taken into account as additional phases.In the second step, the nuclear structure was determined by the Rietveld method, for which the magnetic reflections were masked when possible.The refinement was carried out in space group Pnma in accordance with our single crystal data.The powder data showed additionally the presence of CrAs in the paramagnetic high-temperature phase, which was taken into account by refining its lattice parameters and a corresponding phase volume fraction.In the third step, the refined nuclear structure, in combination with the deduced propagation vector, was taken as basis for the subsequent derivation of the magnetic superspace groups using the built-in algorithms from Jana2020 (Petricek et al., 2023).Refinements of the magnetic structure were carried out in all derived superspace groups.As the fit of the magnetic reflections using the magnetic superspace models derived from the Pnma of the nuclear structure did not lead to a satisfactory agreement, it was necessary to test also lower symmetries.For this, the space group in which the nuclear refinement was carried out was reduced to lower-symmetrical space groups.However, in all the lower-symmetrical descriptions the higher Pnma symmetry was retained by fixing the respective atomic coordinates using local symmetry operations.For the refinements of the magnetic structures in all considered magnetic superspace groups, all parameters corresponding to the nuclear structure were fixed and not refined.The refined parameters were in all cases the magnetic modulation amplitudes and 5 background parameters.
For those structures, where multiple Cr positions are present, the magnetic modulation amplitudes were in general not restricted, the only exception being the model corresponding to the double-helical structure reported in the literature (Watanabe et al., 1969).

S2. Analysis of the neutron single-crystal diffraction data
The ω-scans consisted of 31 single frames centred around the set value for ω.The indexing of the reflections was based on the refined orientation matrix.The reflections were then integrated using the method described by Wilkinson et al. (1988).The magnetic reflections were measured on the basis of a magnetic propagation vector of k = (0, 0, 0.353(2)).The measured magnetic reflections were integrated in the same way as the nuclear ones.In a first step, the list of integrated intensities of the nuclear reflections was subjected to the intensity correction for the transmission loss due to the used clamp cell (Figure S2).The corrected intensities were then used as a basis for the refinement of the nuclear structure with Jana2020 (Petricek et al., 2023).The refinements of the nuclear structure at both pressure points were carried out in space group Pnma.Refined parameters were the free atom coordinates, the anisotropic displacement parameters, an isotropic extinction coefficient Giso, and the twin volume fractions twvol2 and twvol3.
The intensity of the magnetic reflections was corrected for the transmission losses in the same way as the nuclear reflections.On the basis of the refined nuclear structure at both pressure points and the propagation vectors, the magnetic models corresponding to different magnetic superspace groups were derived using the built-in algorithms of Jana2020 (Petricek et al., 2023).To determine the magnetic structure, the data were refined in all magnetic superspace groups allowed by the nuclear space group Pnma, as well as all the translationengleiche subgroups of Pnma down to P1.For the refinement of the magnetic structure, the nuclear structure was fixed to the values obtained from the nuclear refinement, and the symmetry restrictions imposed by Pnma were applied also in the cases with lower symmetry.In all the tested magnetic superspace groups, the atoms were treated anisotropically.The twin volume fractions (twvol2 and twvol3) and the magnetic modulation amplitudes not restricted by the symmetry were refined.In cases, where the lower symmetry leads to a splitting of the atom positions, the harmonic displacement parameters were restricted to follow the higher symmetry with the help of local symmetry operations.For those structures where multiple Cr positions are present, the magnetic modulation amplitudes were in general not restricted, the only exception being the model which corresponds to the double-helical structure reported in the literature (Watanabe et al., 1969).

Figure S3
Translationengleiche subgroups of Pnma used for the derivation of the magnetic superspace groups for the refinement of the magnetic structure of CrAs (drawn with the program Subgroupgraph (Ivantchev et. al., 2000, J. Appl. Cryst. 33, 1190-1191).
Table S1 Final agreement factors for the neutron single-crystal data of CrAs at 0.12 GPa and 2 K for the magnetic models derived from nuclear space group Pnma.

Table S2
Final agreement factors for the neutron single-crystal data of CrAs at 0.12 GPa and 2 K for all tested magnetic superspace groups derived from the t-subgroups of Pnma.The superscripts [100], [010] and [001] do not belong to the formal space group symbol, but indicate the direction in which the two-fold screw axis of the respective space group P21 is oriented in order to allow differentiation.A space group symbol in parentheses indicates a subgroup of Pnma.

Figure S1
Figure S1 Neutron powder diffraction patterns measured at 300 K (top) and at 1.5 K (bottom).At 1.5 K, the reflections are indexed in (3+1)-dimensional superspace with the propagation vector k = (0, 0, γ).* marks a reflection belonging to the Cr impurity.** marks a reflection belonging to the stillpresent high-temperature phase.

Figure S2
Figure S2 The geometrical model for the absorption correction of the data measured in the clamp cell.