Inelastic neutron scattering study of a nonmagnetic degenerated ground state in a cubic compound PrMgNi$_{4}$

Cubic praseodymium-based intermetallic compounds with nonmagnetic $\Gamma_{3}$ doublet ground states of 4$f^{2}$ electrons have received much interests. They show a variety of phenomena such as multipole order and heavy fermion superconductivity, which originate from quadrupolar degrees of freedom due to the interaction between 4$f$ and conduction electrons ($c-f$ hybridization). In case a crystalline electric field (CEF) ground state of the Pr$^{3+}$ ion with a cubic point group is a nonmagnetic doublet, physical properties are governed by electric quadrupoles and/or magnetic octupoles active in the doublet. For example, PrPb$_{3}$ with a $\Gamma_{3}$ doublet ground state shows an antiferroquadrupole order at $T_{Q}$ = 0.4 K [1] below which the electric quadrupoles are modulated sinusoidally [2]. On the contrary, PrAg$_{2}$In [3] and PrMg$_{3}$ [4] with the $\Gamma_{3}$ doublet ground states do not show any long-range orders.
In the present work, we have focused on PrMgNi$_{4}$ crystallizing in the cubic MgSnCu$_{4}$-type structure [5]. We have grown single crystals by the Mg self-flux method. Single-crystal x-ray structural analysis revealed that Mg atoms are substituted for the Pr atoms at the 4$a$ site by 4.5 %. The analysis of the Schottky-type anomaly in the specific heat $C$($T$) and van-Vleck paramagnetic behavior of the magnetic susceptibility suggest the CEF ground state of Pr$^{3+}$ to be the $\Gamma_{3}$ doublet. Nevertheless, no long-range quadrupole order has been observed down to 0.1 K. However, a shoulder-type anomaly of $C$($T$) was seen around 0.7 K. The absence of the long-range quadrupole order may be attributed to symmetry lowering owing to the atomic disorder, since the shoulder of $C$($T$) becomes more pronounced when the atomic composition is close to the stoichiometric one.
To determine the CEF level scheme of Pr$^{3+}$ ion in PrMgNi$_{4}$, we have performed inelastic neutron scattering experiments of the powdered sample using a time-of-flight neutron spectrometer MARI. Magnetic excitation peaks were observed at $E$ = 1.1 and 2.5 meV as labeled (1) and (2) in Fig. (a), respectively, with incident neutron energy of $E_{i}$ = 11.8 meV. Moreover, with $E_{i}$ = 30 meV, as shown in Fig. (b), excitation peaks were observed at 10.7 and 11.7 meV labeled (3) and (4), respectively, and a shoulder at around 5.9 meV labeled (5). As a reference for the phonon contribution, the inelastic spectra of a nonmagnetic LaMgNi$_{4}$ are shown with the black circles in the figures. Because the intensities of the excitations at 1.1, 10.7, and 11.7 meV are nearly $Q$-independent for $|Q| <$ 5 Å$^{-1}$, they are ascribed to the magnetic excitations between the CEF levels. The peak intensity at 1.1 meV becomes half with increasing temperatures from 5 K to 20 K, indicating the transition from a low-lying state to an excited state. To reproduce the temperature dependence of the excitation peaks and the selected excitations between the CEF levels, we have chosen the CEF parameters as $W$ = -3.3 K and $x =$ 0.8 for the cubic $T_{d}$ point group. Thereby, the CEF scheme is expressed as the $\Gamma_{3}$ doublet ground state and excited states $\Gamma_{4}$ (12.5), $\Gamma_{1}$ (30.0), and $\Gamma_{5}$ (134 K). This level scheme, however, does not reproduce the broad peaks at 2.5 and 5.9 meV labeled (2) and (5), respectively. To reproduce these excitations, we need another CEF parameter $B_{2}^{0}$, which is the dominant parameter for a tetragonal point group. Thereby, the symmetry lowering of the Pr ions by the atomic disorder splits the $\Gamma_{3}$ doublet, to hinder the long-range quadrupole orders.

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