Strict limit on in-plane ordered magnetic dipole moment in ${\mathrm{URu}}_2$${\mathrm{Si}}_2$

Neutron diffraction is used to examine the polarization of weak static antiferromagnetism in high quality single crystalline ${\mathrm{URu}}_2$${\mathrm{Si}}_2$. As previously documented, elastic Bragg-like diffraction develops for temperature $T<T_{HO}=17.5$ K at $\mathbf{q}=\left(100\right)$ but not at wave vector transfer $\mathbf{q}=\left(001\right)$. The peak width indicates correlation lengths ${\xi }_c=230\left(12\right)$ Å and ${\xi }_a=240\left(15\right)$ Å. The integrated intensity of the $T$-dependent peaks corresponds to a sample averaged $c$-oriented staggered moment of ${\mu }_c=0.022\left(1\right){\mu }_B$ at $T=1.7$ K. The absence of $T$-dependent diffraction at $\mathbf{q}=\left(001\right)$ places a limit ${\mu }_{\perp }<0.0011{\mu }_B$ on an $f$- or $d$-orbital-based in-plane staggered magnetic dipole moment, which is associated with multipolar orders proposed for ${\mathrm{URu}}_2$${\mathrm{Si}}_2$.

Figure 1

Schematic of magnetic ordering for the (a) known out-of-plane magnetic moments (${\mu }_{\parallel }$) in the SMAF phase below $T_{\text{HO}}$, and (b) proposed in-plane magnetic moments (${\mu }_{\perp }$) from Refs. [9, 10, 11].

Figure 2

Temperature dependence of the peak intensity at several candidate magnetic Bragg peaks. The inset shows the temperature dependence of the peaks which would correspond to in-plane magnetic ordering, on a finer intensity scale. Dotted lines are guides to the eye. The monitor count rate was 6100 counts/s. Error bars represent one standard deviation.

Figure 3

Transverse $\mathbf{q}$ scans at two positions, (a) (001) and (b) (100), which would correspond to in-plane and out-of-plane magnetic ordering, respectively. Only the (100) position shows a magnetic Bragg peak upon cooling through $T_{\text{HO}}$ = 17.5 K. A transverse scan performed at the same position as (001) but without the higher-order filters in the beam [i.e., intentionally including $\lambda /2$ contamination from the (002) nuclear Bragg peak] indicates the correct position of the scan in reciprocal space [black dashed line in (a)], scaled down by a factor of 7200. The horizontal bar in (b) represents the calculated full width at half maximum of the instrumental resolution (see Fig. 5). The monitor count rate was 6100 counts/s. Error bars represent one standard deviation.

Figure 4

Square of the magnetic form factors for several types of spin densities as a function of scattering wave vector $\left|\mathbf{q}\right|$. The first three curves correspond to localized ionic states of uranium [U${}^{4+}$ (5$f^2$) and U${}^{3+}$ (5$f^3$)] and ruthenium (Ru${}^{4+}$), and are described by Eq. (B1). We also roughly approximate the possible spin density that could be associated with Ru- or Si-derived conduction bands, for example as might be appropriate for conduction electrons in the hastatic order theory [11]. The data for Si are those of the x-ray atomic scattering factor squared, which measures the Fourier transform of the total electron density (Ref. [35]).

Figure 5

(a) and (b) Calculated and measured FWHM of Bragg peaks as a function of scattering wave number $q$, for transverse and longitudinal scans. The blue circles ($\lambda$/2) and black squares ($\lambda$) show the measured FWHM for nuclear peaks, and the red triangle shows the measured FWHM for the (100) magnetic peak (error bars represent half of the 95% confidence interval resulting from fits to a Gaussian line shape). The calculated FWHM ($f_{\mathrm{res}}$) are based on resolution function calculations for the following instrumental configuration: incident beam divergence of 60'/$k_i$, PG(002) vertically focused monochromator, 80'collimators before and after the sample, PG(002) flat analyzer, and 33'sample mosaic. For $Q<$ 2 Å${}^{-1}$ the FWHM are measured and calculated (green dashed line) for incident neutrons with $k_i=2\pi /{\lambda }_{PG\left(002\right)}=1.8733$ Å${}^{-1}$. For $Q>$ 2 Å${}^{-1}$, the FWHM are shown (pink dot-dashed line) for $k_i=2\pi /({\lambda }_{PG\left(002\right)}/2)=3.7465$ Å${}^{-1}$. (c) Calculated areas of the resolution function ($A_{\mathrm{res}}$) for transverse scans as a function of $Q$, for the instrumental configuration above (green dashed line is for ${\lambda }_{PG\left(002\right)}$, pink dot-dashed line is for ${\lambda }_{PG\left(002\right)}$/2). (d) Peak height of the normalized two-dimensional resolution function [Eq. (A10)]. The black crosses show the values used for determining the measured cross sections at the specified zone centers.

Figure 6

Blue circles: the temperature dependence of intensity at (001), subtracting the mean value, with a fit to an order parameter curve with fixed $\beta =0.5$, $T_c=17.5\mathrm{K}$ (blue solid line). The range corresponding to one standard deviation in $I_0$ is shown as a gray shaded region. The maximum possible $I_{0_{\max }}=2.50\times {10}^{-6}$ counts/monitor (red circle), is used to determine the upper limit on ${\mu }_{\perp }$. Error bars represent one standard deviation on the measured intensity.

Figure 7

The number of reflections accessible by multiple scattering when the scattering geometry is set to measure (001) (blue dashed line), (003) (red dot-dashed line), and (101) (green line with symbols) for different incident energies($E_i$), with ${\mathbf{k}}_i$ and ${\mathbf{k}}_f$ spanning the $\left(H0L\right)$ scattering plane. At $E_i$ = 4.7 meV (black solid line), no multiple scattering is possible for these configurations within a resolution tolerance of 0.05 Å${}^{-1}$.