Effect of lattice distortion on uranium magnetic moments in ${\mathrm{U}}_4{\mathrm{Ru}}_7{\mathrm{Ge}}_6$ studied by polarized neutron diffraction

In a cubic ferromagnet, small spontaneous lattice distortions are expected below the Curie temperature, but the phenomenon is usually neglected. This study focuses on such an effect in the ${\mathrm{U}}_4{\mathrm{Ru}}_7{\mathrm{Ge}}_6$ compound. Based on DFT calculations, we propose a lattice distortion from the cubic $Im\text{−}3m$ space group to a lower, rhombohedral, symmetry described by the $R\text{−}3m$ space group. The strong spin-orbit coupling of the uranium ions plays an essential role in lowering the symmetry, giving rise to two different U sites (U1 and U2). Using polarized neutron diffraction in applied magnetic fields of 1 and $9\mathrm{T}$ in the ordered state ($1.9\mathrm{K}$) and in the paramagnetic state ($20\mathrm{K}$), we bring convincing experimental evidence of this splitting of the U sites, with different magnetic moments. The data have been analyzed both by maximum entropy calculations and by a direct fit in the dipolar approximation. In the ordered phase, the ${\mu }_{\mathrm{L}}/{\mu }_{\mathrm{S}}$ ratio of the orbital and spin moments on the U2 site is remarkably lower than for the free ${\mathrm{U}}^{3+}$ or ${\mathrm{U}}^{4+}$ ion, which points to a strong hybridization of the U $5f$ wave functions with the $4d$ wave functions of the surrounding Ru. On the U1 site, the ${\mu }_{\mathrm{L}}/{\mu }_{\mathrm{S}}$ ratio exhibits an unexpectedly low value: the orbital moment is almost quenched, like in metallic $\alpha$-uranium. As a further evidence of the $5f\text{−}4d$ hybridization in the ${\mathrm{U}}_4{\mathrm{Ru}}_7{\mathrm{Ge}}_6$ system, we observe the absence of a magnetic moment on the Ru1 site, but a rather large induced moment on the Ru2 site, which is in closer coordination with both U positions. Very similar results are obtained at $20\mathrm{K}$ in the ferromagnetic regime induced by the magnetic field of $9\mathrm{T}$. This shows that applying a strong magnetic field above the Curie temperature also leads to the splitting of the uranium sites, which further demonstrates the intimate coupling of the magnetic ordering and structural distortion. We propose that the difference between the magnetic moment on the U1 and U2 sites results from the strong spin-orbit interaction with different local point symmetries.

Figure 1

Mutual orientation of the cubic $Im\text{−}3m$ unit cell and the rhombohedral $R\text{−}3m$ in the hexagonal axes. U1 sites are marked red, U2 green, Ru1 and Ru2 gray, and Ge are yellow.

Figure 2

(a) Indicated section in the $R\text{−}3m$ space group representation that is perpendicular to the ${\left[001\right]}_{\mathrm{hex}}$ axis, and (b) the corresponding MAXENT magnetization density map measured in $9\mathrm{T}$ and $1.9\mathrm{K}$. Contour lines are at $0.004{\mu }_{\mathrm{B}}{\AA }^{-3}$ .

Figure 3

Flipping ratios measured at $1.9\mathrm{K}$ and $9\mathrm{T}$ compared with the calculated values using the dipolar approximation.