Observation of Modulated Quadrupolar Structures in ${\mathrm{PrPb}}_3$

Neutron diffraction measurements have been performed on the cubic compound ${\mathrm{PrPb}}_3$ in a [001] magnetic field to examine the quadrupolar ordering. Antiferromagnetic components with $q=(\frac{1}{2}\pm \delta \frac{1}{2}0)$, ($\frac{1}{2}\frac{1}{2}\pm \delta 0$) ($\delta \approx \frac{1}{8}$) are observed below the transition temperature $T_{\mathrm{Q}}$ (0.4 K at $H=0$) whose amplitudes vary linear with $H$ and vanish at zero field, providing the first evidence for a modulated quadrupolar phase. For $H<1\mathrm{T}$, a nonsquare modulated state persists even below 100 mK suggesting quadrupole moments associated with a ${\Gamma }_3$ doublet ground state to be partially quenched by hybridization with conduction electrons.

Figure 1

Evolution of the magnetic scattering in a field of $H=4\mathrm{T}$ applied along the [001] direction, obtained in a temperature interval of $0.125\mathrm{K}<T<0.81\mathrm{K}$. The $\mathit{Q}$ scans were performed along the line $(h\frac{1}{2}0)$, as indicated by a red arrow in the $(hk0)$ reciprocal plane (inset), where open and closed circles represent the nuclear and the magnetic reflections observed at $T=0.125\mathrm{K}$, respectively.

Figure 2

Field dependence of the square root of the $(\frac{3}{8}\frac{1}{2}0)$ reflection intensity (circles) at $T=0.32\mathrm{K}$. The background was subtracted from the data. The intensity of the $(\frac{1}{8}\frac{1}{2}0)$ peak (triangles) is also shown as a function of $H$. The small arrows indicate the direction of the field scan.

Figure 3

$H\mathrm{\text{−}}T$ phase diagram of ${\mathrm{PrPb}}_3$ in a [001] magnetic field. Open and closed triangles denote the transition points determined from the $T$- and $H$-scan measurements, respectively. For both cases, the upward and downward triangles represent the increasing and decreasing $T(H)$ scans, respectively. AFQ I denotes the possible incommensurate structure with the wave vectors $\mathit{q}=(\frac{1}{2}\pm \delta \frac{1}{2}0)$ and/or $(\frac{1}{2}\frac{1}{2}\pm \delta 0)$, ($\delta \sim \frac{1}{8}$). AFQ II denotes the antiphase structure as discussed in the text. The inset shows the temperature dependence of the center position of the magnetic reflection at around $\mathit{q}=(\frac{1}{2}-\delta \frac{1}{2}0)$ in a field of $H=4\mathrm{T}$, obtained on cooling below 0.75 K as shown by the arrow.

Figure 4

Field-induced antiferromagnetic moments and the possible quadrupole structures along the [100] axis ($y=0$) in a [001] field. (a) Sinusoidal structure with $\mathit{q}=(\frac{1}{2}\pm \delta \frac{1}{2}0)$, and (b) antiphase structure with $\mathit{q}=(\frac{1}{2}\pm \delta \frac{1}{2}0)$ and $(\frac{1}{2}\pm 3\delta \frac{1}{2}0)$. We assume that $\delta =\frac{1}{8}$ for simplicity. Uniform components are not included. The solid lines denote the amplitude of the staggered components, whose phase factor is arbitrarily chosen.