Going beyond the dipole approximation to improve the refinement of magnetic structures by neutron diffraction

We have made an analysis of neutron-diffraction data for a number of rare-earth compounds to assess the use of the dipole approximation in magnetic structure refinements. We present two examples. In the case of ${\text{CePd}}_2{\text{Si}}_2$, our analysis confirms the published magnetic structure, but we find that use of the “exact” magnetic form factor of ${\text{Ce}}^{3+}$ gives a significantly improved description of the data. For ${\text{NdBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$, however, we find that incorrect conclusions have previously been drawn about the magnetic structure through the use of the dipole approximation. We have extended the magnetic modeling suite MCPHASE to enable a straightforward calculation of the higher-order anisotropic terms in the form factor of rare-earth-based systems.

Figure 1

(Color online) Magnetic Bragg-peak intensities for ${\text{CePd}}_2{\text{Si}}_2$ calculated (i) in the dipole approximation and (ii) going beyond the dipole approximation (upper figure). In both cases the moments were assumed to be aligned parallel to the [110] direction. The lower figure shows deviations between the exact calculation and dipole approximation expressed as $R=100\left({\left|F_M^{\text{exact}}\right|}^2-{\left|F_M^{\text{dip}}\right|}^2\right)/{\left|F_M^{\text{dip}}\right|}^2$ (bars) and comparison with experimental data obtained on a single crystal at $T=1.3\text{K}$ (Ref. 17) plotted as $R=100\left({\left|F_M^{\text{exp}}\right|}^2-{\left|F_M^{\text{dip}}\right|}^2\right)/{\left|F_M^{\text{dip}}\right|}^2$ (symbols).

Figure 2

Models for the magnetic structure of ${\text{NdBa}}_2{\text{Cu}}_3{\text{O}}_{6.97}$. Only the Nd moments are shown. A magnetic structure refinement with the dipole form factor suggests a tilt of the spin axis from the $c$ direction of $\theta =12°$ (right). With the exact anisotropic form factor a better fit is achieved with moments parallel to $c$ (left).

Figure 3

(Color online) $4f$ charge density of ${\text{Nd}}^{3+}$ calculated from the action of the crystal field in ${\text{NdBa}}_2{\text{Cu}}_3{\text{O}}_{6.97}$ at $T=2\text{K}$.