Naoto Metoki1,2, 
, Hiroki Yamauchi1, Hiroyuki S. Suzuki3, Hideaki Kitazawa3, Masato Hagihara4, Takatsugu Masuda4, Adam A. Aczel5, Songxue Chi5, Tao Hong5, Masaaki Matsuda5, Daniel Pajerowski5, and Jaime A. Fernandez-Baca5 + Affiliations1Materials Sciences Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan2Institute of Quantum Beam Science, Graduate School of Science and Engineering, Ibaraki University, Tokai, Ibaraki 319-1106, Japan3Research Center for Advanced Measurement and Characterization, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan4Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan5Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, U.S.A.
Received February 15, 2018; Accepted June 21, 2018; Published August 9, 2018
The crystalline electric field (CEF) excitation spectra of PrPd5Al2 have been studied by neutron inelastic scattering in order to reveal the f-electron states, where PrPd5Al2 is a Pr-based isostructural compound of the heavy-fermion superconductor NpPd5Al2. We observed clear CEF excitations up to ∼25 meV. The CEF Hamiltonian \(\mathcal{H}_{\text{CEF}}\) of the Pr3+ ion (3H4) under tetragonal point symmetry was orthogonalized analytically and the CEF parameters were determined from the excitation energies. The f-electron states reproduce the distinctive temperature dependence of the magnetic excitation spectra as well as the macroscopic properties. A point charge model is effective for understanding the systematic change in the f-electron states in RPd5Al2. The Ising anisotropy in CePd5Al2, PrPd5Al2, and NdPd5Al2 originates from the flat orbitals with large Jz due to the CEF potential developed in the crystal structure. We found that the temperature dependence of the magnetic susceptibility in NpPd5Al2 can be explained qualitatively on the basis of the identical charge distribution in RPd5Al2. XY-type anisotropy is expected from the positive Stevens factors αl for the Np3+ ion. Thus, we conclude that the local property is important in NpPd5Al2, whose origin is common in the RPd5Al2 isostructural family.
©2018 The Physical Society of Japan
1. IntroductionThe heavy-fermion superconductivity in NpPd5Al2,1) the first example of a Np-based superconductor, has attracted strong interest in the field of strongly correlated electron systems. Heavy-fermion nature with a large electronic specific heat constant of \(\gamma =200\) mJ/(K2·mol) has been observed at low temperatures. Non-Fermi liquid behavior appears in the temperature dependence of the resistivity, which exhibits superconductivity below the relatively high transition temperature of \(T_{\text{c}} = 4.9\) K. Cylindrical Fermi surfaces as well as pancake- and doughnut-like shapes were revealed by a band calculation,2) where the Fermi level is located at a narrow f band with a large density of states hybridized with the Pd \(4d\) band, as expected from its unique crystal structure. The two-dimensionality is attributed to the single layer of the local RPd3 structure stretched along the c-axis as shown in Fig. 1. The rare-earth element is located at the \(2a\) site with local symmetry \(4/mmm\). There is a very long body-centered-tetragonal unit cell (\(I4/mmm\)) with a very large interlayer distance of \(c/2=7.358\) Å separated by an Al layer, while \(a=4.148\) Å is larger than the Hill limit of ∼3.5 Å.3) Therefore, \(5f\)-electrons are expected to retain some of their localized character.4,5) Our recent inelastic neutron scattering study of UPd5Al2 revealed the existence of a localized \(5f^{2}\) state similar to that in PrPd5Al2.6)
Figure 1. (Color online) Crystal structure of RPd5Al2 with R: actinide or rare-earth element, drawn by VESTA.12)
The f-electron states are very important for discussing the mechanism of the unusual heavy-fermion nature and the superconductivity. Studies on the isostructural family will contribute to the systematic understanding of the unusual properties. Since the discovery of superconductivity in NpPd5Al2, RPd5Al2 (R: Y, Ce, Pr, Nd, Sm, Gd) samples have been synthesized.7,8) Antiferromagnetic transitions were reported for this series of compounds (except for the Y and Pr compounds with the singlet ground-state), where the de Gennes scaling of \(T_{\text{N}}\) was confirmed except for CePd5Al2. RPd5Al2 (R: Tb-Yb) and the new RPt5Al2 (R: Y, Gd-Tm, Lu) have been reported.9) Pressure-induced superconductivity has been discovered in CePd5Al2.10,11)
Very recently, we studied the magnetic structure of NdPd5Al2 by neutron powder diffraction experiments.13) Large Nd moments of \(2.7\pm 0.3\) \(\mu_{\text{B}}\), comparable to \(g_{J}J=3.27\) \(\mu_{\text{B}}\), align parallel to the c-axis and order antiferromagnetically with the propagation \(\mathbf{Q}=(1/2,0,0)\) below the Néel temperature \(T_{\text{N}}=1.2\) K. The magnetic structure consists of the \(+{+}{-}-\) modulation of Nd moments including four Nd layers in a period along the \([1,0,0]\) direction. Interestingly, the in-plane modulation \(\mathbf{Q}=(0.235,0.235,0)\) observed in CePd5Al28) can be described by a similar \(+{+}{-}-\) modulation of a four-layer period along the diagonal \([1,1,0]\) direction, assuming a commensurate modulation with \(\mathbf{Q}=(1/4,1/4,0)\). A sinusoidal modulation of the Ce moment parallel to the c-axis appears below \(T_{\text{N1}}=4.1\) K, while a square modulation was suggested from the third order harmonics observed below \(T_{\text{N2}}=2.9\) K.8) The moment size of 2 \(\mu_{\text{B}}\) is close to the full moment \(g_{J}J=2.14\) \(\mu_{\text{B}}\). These stripelike in-plane modulations of \(|\mathbf{Q}|=0.33a^{\ast}\) for CePd5Al2 and \(0.5a^{\ast}\) for NdPd5Al2 may originate from the nesting properties of the cylindrical Fermi surfaces, which appear as a result of a long unit cell with \(c/a\simeq 3.5\), indicative of a two-dimensional nature. Similar in-plane magnetic correlations are expected in other isostructural compounds including PrPd5Al2.
Nearly full magnetic moments in CePd5Al2 and NdPd5Al2 along the quantization c-axis are the signature of flat f-orbitals with the maximum \(J_{z}\). In light rare-earth elements, the orbital magnetic moment due to the orbital current of the f-electron is much larger than the spin moment, for example, \(L=5\) and \(S=1\) for the Pr3+ ion. Thus, the orbital contribution is dominant.14) A large c-moment indicates the orbital current in the tetragonal c-plane, which originates from wave functions with a large \(J_{z}\). The susceptibility measurements of CePd5Al2 and PrPd5Al2 revealed Ising-type magnetic anisotropy, which can be reproduced by a mean-field calculation using the crystalline electric field (CEF) Hamiltonian \(\mathcal{H}_{\text{CEF}}\) with a large negative CEF parameter \(B_{20}\). These features with Ising-type anisotropy and large c-moments are common, and were also revealed in a recent magnetic study on high-quality single crystals of NdPd5Al2.15) These features can be understood from the flat f-orbitals due to the unique crystal structure in Fig. 1. The f-orbital sandwiched by Pd and Al ligand layers is expanded through hybridization2) with the adjacent Pd1 atoms on the same c-plane, and also Pd2.
On the other hand, XY-type anisotropy in NpPd5Al2 has been reported to be different from that of the rare-earth-based compounds, which may be related to the unusual physical properties of NpPd5Al2.11) Previous NMR studies reported d-wave superconductivity mediated by this XY-type magnetic fluctuation.4,5) Physical properties of this isostructural family RPd5Al2 should be understood in a systematic manner. For this purpose, the f-electron states in PrPd5Al2 have been studied by neutron inelastic scattering spectroscopy. The six excitations of non-Kramers Pr3+ (\(4f^{2}\)) with \(J=4\), the same as in Np3+ but with \(5f^{4}\), are enough to determine the CEF parameters uniquely. The analysis may be easier than that of Nd3+ with \(4f^{3}\). Pr with small neutron absorption and weak incoherent scattering is favorable for neutron scattering experiments, while Gd and Sm are extremely strong neutron absorbers.
2. Experimental ProcedurePolycrystalline samples of PrPd5Al2 were grown by arc-melting stoichiometric amounts of Pr (3N), Pd (3N5), and Al (5N) in a pure Ar atmosphere. As-grown samples (20 g) were mechanically cut into small blocks and sealed in an aluminum can filled with 4He thermal exchange gas. The sample was checked by X-ray powder diffraction. No impurity phase was detected within the experimental sensitivity. The chemical composition examined by electron microprobe analysis was in good agreement with the stoichiometry. A single phase was confirmed except for a very thin oxidized surface region, which was mechanically removed.
Neutron inelastic scattering experiments were carried out using the cold and thermal triple-axis spectrometers CTAX (CG-4C) and TAX (HB-3), respectively, installed at the High Flux Isotope Reactor (HFIR) in the Oak Ridge National Laboratory (ORNL), USA. We measured with the final neutron energy \(E_{f}\) fixed at \(E_{f}=3\) and 14.7 meV, giving the energy resolutions of about \(\Delta E=0.1\) and 1 meV at the incoherent elastic position for CTAX and TAX, respectively. A vertically focused PG(002) monochromator was used for the incident beam of both spectrometers. A double bent PG(002) analyzer was used in the CTAX experiments, while a vertically bent and horizontally flat analyzer was used in the TAX experiments. The sample was cooled to 4 K using a closed-cycle refrigerator.
3. ResultsFigure 2 shows the neutron inelastic scattering spectra of PrPd5Al2. The scattering vector \(\mathbf{Q}\) was fixed at \(|\mathbf{Q}|=1\) and 1.8 Å−1 for the CTAX and TAX experiments, respectively. We observed clear peaks at 4 K, which are excitations from the ground-state. Transitions between excited levels were observed with elevating temperature. The linewidths are almost resolution-limited, indicating nondispersive on-site excitations between the well-localized f-electron states, namely CEF excitations. Actually, the dispersion is expected to be very small according to the virtual magnetic transition temperature, which is predicted to be less than 0.5 K by extrapolation from the de Gennes scaling.7) This is consistent with our observation of sharp excitation peaks in the powder sample. We fitted the data with multiple Gaussian peaks to obtain the peak positions. The excitation energies of the six observed transitions were 1.455, 1.707, 8.039, 16.43, 18.99, and 24.17 meV. See also Table A·II. Since seven levels are expected under the tetragonal CEF of the Pr3+ ion with the \(4f^{2}\) configuration (\(^{3}H_{4}\)), we concluded that we successfully observed all six transitions related to the f-electron levels. The CEF parameters \(B_{lm}\) were determined from the excitation energies using the analytical procedure described in the Appendix. The derived CEF parameters, energy levels, and wave functions are summarized in Table I. The CEF parameters \(B_{20}\) and \(B_{40}\) are qualitatively consistent with those in a previous study,11) but the higher-order terms \(B_{44}\), \(B_{60}\), and \(B_{64}\) may be different. The wave functions are almost identical to those in a previous study11) of macroscopic properties. These differences and the similarity can be seen by comparing the data in Table I in this paper and Table III in Ref. 11. Consequently, the energy levels are somehow different and the CEF spectra cannot be reproduced from the CEF parameters \(B_{lm}\) published previously.
Figure 2. (Color online) Neutron inelastic scattering spectra of PrPd5Al2 measured with (a) cold neutron spectrometer CTAX and (b) thermal neutron spectrometer TAX. The solid lines are the model calculation based on \(\mathcal{H}_{\text{CEF}}\). (c) Excitation regime.
 »View table | Table I. CEF parameters \(B_{lm}\), energy levels, and wave functions of f-electrons in PrPd5Al2 derived from the neutron inelastic scattering spectra shown in Fig. 2. |
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The CEF spectra calculated from the derived \(B_{lm}\) are shown by solid lines in Fig. 2. The calculated results are in good agreement with the experimental data. The scattering cross section is given as16) \begin{align} \frac{d\sigma}{d\Omega\,d\omega} &= N\left(\frac{\gamma r_{0}g_{j}}{2}\right)^{2}\frac{k_{f}}{k_{i}}f^{2}(\mathbf{Q})\exp[-2W]\sum_{\alpha}(1-\kappa_{\alpha}^{2})\notag\\ &\quad\times \sum_{i,j}n_{i}|\langle j|J_{\alpha}|i\rangle|^{2}\delta(E_{i}-E_{j}-h\omega). \end{align} (1) Here, γ, \(r_{0}\), and \(g_{j}\) are the gyromagnetic ratio, classical electron radius, and g-factor, respectively. \(k_{i}\) and \(k_{f}\) are the incident and final wave number of the neutron, respectively. The magnetic form factor f was ignored in our calculation, since measurements were performed at a single Q position on each instrument. The Debye–Waller factor \(\exp[-2W]\) was assumed to be unity. The term (\(1-\kappa_{\alpha}^{2}\)) gives the component of the angular moment \(J_{\alpha}\) (\(\alpha=x,y,z\)) perpendicular to \(\mathbf{Q}\). It becomes a constant after powder averaging.
The intensity of the CEF excitation depends on the occupation of the initial state \(n_{i}\) and the matrix element \(\langle j|J_{\alpha}|i\rangle\), where i and j denote the initial and final states, respectively. \begin{equation} n_{i} = \exp[-\beta E_{i}]\biggm/\sum_{i} \exp[-\beta E_{i}] \end{equation} (2) The temperature dependence of the spectra is given by \(n_{i}\) in Eq. (2). Note that we used only two scaling factors of the calculated intensities, corresponding to the different beam fluxes of the two instruments CTAX and TAX. The temperature dependence of the spectrum shown in Fig. 2 can be well understood from the occupation of the initial states \(n_{i}\). This eliminates the possibility of a phonon origin of the observed peaks. The possible broad phonon background is expected to be negligibly small. The phonon intensity is proportional to \(n(\omega,T)+1\), where \(n(\omega,T)\) and ω are the Bose factor and phonon frequency, respectively.17) The existing phonon should be observed from the lowest temperature, and the increase in the intensities should be very small; only 50, 20, and 5% at 8, 16, and 25 meV, respectively, for \(T=100\) K. These values are inconsistent with the experimental data.
Figure 3 shows the magnetic susceptibility χ (upper panel) and magnetization (lower panel) published previously11) in comparison with the results of the same mean-field calculation based on the CEF parameters determined in this neutron spectroscopic study. The magnetic specific heat divided by the temperature \(C_{\text{mag}}/T\) and the magnetic entropy \(S_{\text{mag}}\) are shown in Fig. 4. Without the mean-field parameter and Pauli paramagnetic contribution \(\chi^{0}\), the calculated susceptibility deviates only slightly from the experimental data for \(T<30\) K and \(T>150\) K, as denoted by dashed lines in Fig. 3. On the other hand, our calculations (solid lines) assuming the mean-field parameter of \(\lambda_{c}=-0.2\) mol/emu, \(\chi_{c}^{0}=-9\times 10^{-4}\), and \(\chi_{a}^{0}=-2\times 10^{-4}\) emu/mol perfectly agree with the experimental data as shown in Fig. 3. The calculated magnetization curves in the present study are identical to those in the previous study and explain the experimental data very well. The differences in our calculated results from the experimental data of \(C_{\text{mag}}/T\) and \(S_{\text{mag}}\) are also very small.
Figure 3. (Color online) Temperature dependence of the magnetic susceptibility χ and \(1/\chi\) of PrPd5Al2 (upper panel) and the magnetization curve as a function of the applied field H (lower panel). The marks are the results of a previous study,11) while the curves are the calculation based on \(\mathcal{H}_{\text{CEF}}\) determined in the present study with (solid) and without (dashed) λ and \(\chi^{0}\).
Figure 4. (Color online) Temperature dependence of the magnetic specific heat divided by the temperature \(C_{\text{mag}}/T\) and the magnetic entropy \(S_{\text{mag}}\) of PrPd5Al2. The marks are the result of a previous study11) while the blue and red curves are the calculation assuming \(\mathcal{H}_{\text{CEF}}\) determined from the present neutron scattering study.
We could not identify the broad peak at 24 meV as a CEF excitation, and the peak may be spurious. We considered it as corresponding to the \(E_{1}\)–\(E_{8}\) transition between excited states, which should be observed with elevating temperature. Note that the absence of the peak for the \(E_{0}\)–\(E_{8}\) transition at 4 K is consistent with the selection rule; \(\Delta J_{z}=0\) or \(\pm 1\) is not satisfied for the \(E_{0}\)–\(E_{8}\) but is satisfied for \(E_{1}\)–\(E_{8}\). This peak assignment was ruled out after calculating the CEF level scheme from the excitation energies, as described in the Appendix. The \(E_{1}\)–\(E_{8}\) splitting of 24 meV leads to large CEF parameters \(B_{l0}\) (e.g., \(B_{20}\simeq 5\) K). It results in a very large splitting between \(\Gamma_{t5}\) doublets, which is given by the second term (square root) in Eq. (A·6). Then the \(\Gamma_{t5}\) doublet becomes the ground-state, instead of the \(\Gamma_{t1}\) singlet, even when \(B_{44}=B_{64}=0\). As a consequence, the CEF excitation spectra completely changed because of the differences in the wave functions; thus, the model calculation could not reproduce the experimental data. This situation is unavoidable when the \(E_{1}\)–\(E_{8}\) transition exceeds some critical energy. Therefore, we choose \(E_{8}{\text{--}}E_{1}\simeq 19.34\) meV, which reproduces the entire CEF level scheme. Furthermore, we found that the obtained CEF parameters \(B_{lm}\) can reproduce the magnetic excitation spectra and the temperature dependences, as shown in Fig. 2, except for the 24 meV peak. This means that the wave functions of the f-electron states are also in good agreement with the experimental data. The \(E_{1}\)–\(E_{8}\) transition is superposed on the \(E_{0}\)–\(E_{6,7}\) transition as shown in Table A·II; thus, the former is not visible as an isolated peak. Nevertheless, the existence of the \(E_{1}\)–\(E_{8}\) transition in this energy range is convincing; the observed spectra can be reproduced with the contribution of the \(E_{1}\)–\(E_{8}\) transition. Without this contribution, the calculation is inconsistent with the experimental result.
The existence of the \(E_{1}\)–\(E_{8}\) transition was confirmed from the model calculation. Figure 5 shows the calculated spectral weight at 19 meV compared with the experimental data. At \(T=4\) K only the excitation from the ground-state \(E_{0}\)–\(E_{6,7}\) contributes to the scattering intensity, while the contribution of \(E_{1}\)–\(E_{8}\) between excited levels increases with the temperature, reaching almost half of the total intensity at \(T=100\) K. The total intensity is in good agreement with the experimental data. This means that the data can be reproduced by the model calculation when the contribution from the \(E_{1}\)–\(E_{8}\) transition is included. We have confirmed that the absence of this peak results in a significant deviation of the model calculation from the experimental data. In addition, the expected integrated intensity of the \(E_{1}\)–\(E_{8}\) transition was only about half of the peak observed at 24 meV. Thus, this peak cannot be explained with the \(E_{1}\)–\(E_{8}\) transition, even if we assume an unrealistic shift of the CEF level inconsistent with the CEF Hamiltonian \(\mathcal{H}_{\text{CEF}}\). We conclude that the 24 meV peak cannot be the \(E_{1}\)–\(E_{8}\) transition, which may be somehow related to the CEF level scheme. The phonon origin is definitely ruled out from the temperature dependence.
Figure 5. (Color online) Temperature dependence of the calculated spectral weight for the \(E_{0}\)–\(E_{6,7}\) and \(E_{1}\)–\(E_{8}\) transitions denoted by triangles and squares, respectively. The closed circles are the total. The open circles are the experimental data.
The two transitions at 19 meV can be distinguished by measuring the \(\mathbf{Q}\) dependence on a single crystal, or by polarization analysis, which give a definitive CEF level scheme. The matrix elements are anisotropic with a finite \(\langle E_{1}|J_{z}|E_{8}\rangle^{2}\) of 1.15, while \(\langle E_{1}|J_{x,y}|E_{8}\rangle=0\). On the other hand, the \(E_{0}\)–\(E_{6,7}\) transition shows the opposite trend, namely, \(\langle E_{0}|J_{z}|E_{6,7}\rangle=0\) and \(\langle E_{0}|J_{x,y}|E_{6,7}\rangle^{2}=0.894\). Therefore, when \(\mathbf{Q}\) is parallel to the c-axis, the signal of the \(E_{1}\)–\(E_{8}\) transition disappears, while the \(E_{0}\)–\(E_{6,7}\) transition is observed to have the maximum intensity. The polarization analysis provides an ideal experimental condition to separate these transitions. When the c-axis is perpendicular to the scattering plane, only the \(E_{0}\)–\(E_{6,7}\) transition is detected in the VF-SF channel, while the \(E_{1}\)–\(E_{8}\) transition can be separated by measuring the VF-NSF channel; possible phonons can be measured in the HF-NSF channel and subtracted. Here VF, HF, SF, and NSF mean the vertical guide field (\(H\parallel\mathrm{z}\)), horizontal guide field (\(H\parallel\mathrm{Q}\)) for the neutron spin, and the spin-flip and non-spin-flip processes, respectively.
Figure 6 shows the integrated intensity of the 24 meV peak obtained by a simple Gaussian fitting. The temperature dependence of the intensity indicated by open circles is in good agreement with the occupation number \(n_{i}\) of the second excited level (\(i=2\)) denoted by a solid line, rather than that for the first excited level (\(i=1\)) shown by a broken line. The inset of Fig. 6 shows a plot of the integrated intensity as a function of the occupation number at the first excited level (open triangles) and the second excited level (closed circles). The linearity of the data from the second excited level is better than that from the first excited level. Reflecting the small energy difference between the first (1.455 meV) and the second (1.707 meV) excited levels, the difference in the temperature dependence of the occupation number \(n_{i}\) is very small. Nevertheless, the second excited level is considered more appropriate for the initial state of the 24 meV peak than the first excited level. This initial state of the second excited level may be inconsistent with the \(E_{1}\)–\(E_{8}\) transition. The matrix element square of the \(E_{2,3}\)–\(E_{8}\) transition is not dominant and corresponding to ∼37% of the total intensity.
Figure 6. (Color online) Temperature dependence of the integrated intensities of the 24 meV peak observed in PrPd5Al2. The solid and broken lines show the occupation of the second and first excited levels, respectively. The inset is a plot of the integrated intensities as a function of the occupation of the second (closed circles) and first (open triangles) excited levels, respectively. The lines are linear fits, which are shown as a function of temperature in the figure.
We also tried some different peak assignments to explain the 24 meV peak, but they were unsuccessful. For example, we assigned the 24 meV peak to \(\Gamma_{t3}\) and the 16 meV peak to \(\Gamma_{t1}\). Then we found that the energy levels were not reproduced, exhibiting a serious inconsistency between the \(B_{lm}\) parameter set and the peak assignment.
The origin of the 24 meV peak still remains an open question. A possible scenario would be a spurious peak of the \(E_{2,3}\)–\(E_{4}\) transition, because the initial state would be the second excited level and these peaks are isolated from the others. However, we have not succeeded in finding out a reasonable explanation, for example, based on higher harmonics. An experiment using a chopper spectrometer might be helpful for removing a possible spurious peak related to higher harmonics using the TAX spectrometer. The broad excitations for \(\Delta E=3{\text{--}}6\) and 10–15 meV with no temperature dependence are due to phonon or background scattering.
4. DiscussionThe CEF potential18) \(v(\mathbf{r})\) of the electron density ρ may be expanded in spherical harmonics as \begin{equation} v(\mathbf{r}) = \int \frac{e\rho(\mathbf{R})}{|\mathbf{r}-\mathbf{R}|}\,d\mathbf{R}= \sum_{lm}A_{l}^{m}r^{l}Y_{lm}, \end{equation} (3) where \begin{equation} A_{l}^{m} = (-1)^{m}\frac{4\pi}{2l+1}\int \frac{\rho(\mathbf{R})}{\mathbf{R}^{l+1}}Y_{l-m}\,d\mathbf{R}. \end{equation} (4) The matrix elements of \(v(\mathbf{r})\) are proportional to the operator equivalents \begin{equation} \mathcal{H}_{\text{CEF}} = \sum_{lm}A_{l}^{m}\alpha_{l}\langle r^{l}\rangle \left(\frac{2l+1}{4\pi}\right)^{1/2}O_{lm}, \end{equation} (5) described with CEF parameters \(B_{lm}\) as \begin{equation} \mathcal{H}_{\text{CEF}} = \sum_{lm}B_{lm}O_{lm},\quad B_{lm} = A_{l}^{m}\alpha_{l}\langle r^{l}\rangle. \end{equation} (6) \(A_{l}^{m}\) is given by the charge distribution \(\rho(\mathbf{R})\) as in Eq. (4). When \(\rho(\mathbf{R})\) is identical in isostructural compounds, \(B_{lm}\) can be estimated from one another using Eq. (6). Here \(\alpha_{l}\) is known as the Stevens factor and \(\langle r^{l}\rangle\) is the distribution of the f-electrons. Table II shows the list of Stevens factors of light rare-earth R3+ ions obtained with the Russel–Saunders coupling scheme, which was used in the calculation with Eq. (6). We can notice a sign change of the Stevens factors from negative for Nd3+ (\(4f^{3}\)) to positive for the Pm3+ (\(4f^{4}\)) ion. This crossover may not be widely recognized owing to the lack of systematic studies beyond this boundary. Pm is hard to get, Sm and Eu ions show valence instability. The CEF scheme of Tb, Dy, Ho, and so forth, with large J are too complicated, while Gd (\(L=0\)) shows no CEF splitting. Most U3+ are itinerant, and the following actinide elements Np, Pu, and so on, are difficult to use under the strong regulation.
»View table | Table II. Stevens factors of light rare-earth R3+ ions.18) |
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By using \(B_{lm}\) for PrPd5Al2 obtained in the present neutron scattering study, we estimated \(B_{lm}\) for the series of RPd5Al2 compounds, as summarized in Table III and denoted by “Eq. (6)”. The estimated values for CePd5Al2 are in good agreement with those in previous neutron and magnetic susceptibility studies for CePd5Al2,8,11) except for the small negative \(B_{44}\). The list of \(B_{lm}\) denoted by “point charge”19) in Table III is the values obtained from a model constructed from three point charges of +0.66, +0.21, and +0.4 at the Pd1, Pd2, and Al sites in a unit cell, respectively. These three charges are determined so as to reproduce \(B_{20}\), \(B_{40}\), and \(B_{44}\) for CePd5Al2 to give a simple representation. The values of \(B_{lm}\) for PrPd5Al2 obtained from this point charge model are similar to the present experimental result. Actually the calculated susceptibility is qualitatively consistent with the experimental data. There is a local RPd3 layer stretched along the c-axis between two Al layers in the crystal structure as shown in Fig. 1. The + charges at the nearest-neighbor Pd1 site on the same plane as the R atoms and at Pd2 attract f-electrons in the in-plane direction. We can expect a flat orbital with a large \(J_{z}\), which exhibits Ising-type anisotropy. This point charge model is implicated in the band picture of NpPd5Al22) or itinerant CePd5Al2,11) consisting of two-dimensional f bands due to the hybridization with \(4d\) bands of adjacent Pd atoms. These band calculations may correspond to an extreme limit for itinerant f-electrons. Note that the CEF potential is not the actual dielectric potential of the atomic charge, but it is the effective charge potential.
»View table | Table III. List of CEF parameters for RPd5Al2. |
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The Stevens factors of Pm3+ (\(4f^{4}\)) are identical to those of Np3+ in the Russell–Saunders coupling scheme.19) The positive sign does not change in the intermediate coupling scheme, whereas the values of the Stevens factors may be slightly different.20) The magnetic anisotropy of NpPd5Al2 should be XY-type due to the positive \(B_{20}\) originating from the positive Stevens factor. Either the Ising or XY-type anisotropy is determined by the character of the J multiplet, which is relevant to the Stevens factors.
Figure 7 shows the calculated magnetic anisotropy of NpPd5Al2 based on the point charge model21) compared with the previously published experimental data.1) The XY-type anisotropy and the overall temperature dependence are qualitatively consistent with the experimental data. Therefore, we can consider that the localized character of \(5f\)-electrons in NpPd5Al2 is very important.
Figure 7. (Color online) Magnetic susceptibility of NpPd5Al2 and the result of the mean-field calculation based on the point charge model. The published data on the susceptibility of NpPd5Al2 are reproduced with permission of Professor Aoki.1)
NdPd5Al2 exhibits uniaxial anisotropy with negative \(B_{20}\). However, the anisotropy becomes smaller than CePd5Al2 and PrPd5Al2. Actually \(\chi_{a}\) in NdPd5Al2 increases at low temperatures. A significant increase in \(\chi_{a}\) has not been observed in CePd5Al2 and PrPd5Al2. The uniaxial anisotropy based on the \(f^{3}\) configuration of the Nd3+ ion should be observed in NpPd5Al2 when the valence state is Np4+ with the \(5f^{3}\) configuration. In other words, the XY-type anisotropy in NpPd5Al2 is evidence for the Np3+ state. This valence state is consistent with the band calculation, which predicts the number of \(5f\) to be 3.7, roughly corresponding to Np3.3+. This ionic state slightly deviates from the trivalent state.2) The isomer shift reported in a recent Mössbauer study is empirically consistent with the trivalent state.22)
The study of the CEF level scheme in NpPd5Al2 and NdPd5Al2 is highly interesting. Our charge model provides only a rough estimation of the CEF potential. The details in the CEF level scheme, wave functions, and so forth, are different from those in real systems. The CEF parameters determined by the mean-field calculation of the susceptibility data are helpful for understanding the importance of the local properties for the f-electron state in the RPd5Al2 family. The magnetic susceptibility data of NdPd5Al2 have been published already15) and neutron inelastic scattering experiments have been carried out.23)
5. ConclusionsThe f-electron states of PrPd5Al2 were revealed by neutron inelastic scattering. The flat f-orbitals with large \(J_{z}\) become stable under the CEF potential of the unique crystal structure, which is the origin of the Ising anisotropy. A systematic understanding of RPd5Al2 based on this CEF potential is possible. In particular, the XY-type anisotropy in NpPd5Al2 can be qualitatively understood on the basis of the positive Stevens factors. This means that the local property is important for the physical properties of RPd5Al2, including the heavy-fermion superconductivity in NpPd5Al2.
Acknowledgments
We thank the US-Japan Cooperative Program on Neutron Scattering using resources at HFIR, operated by ORNL and sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US DOE. We thank D. Aoki, S. Araki, F. Honda, T. Matsumura, and T. Takeuchi for the stimulating discussions and the permission to use their data and software for the CEF calculations.
AppendixPr3+ ions in PrPd5Al2 with the space group \(I4/mmm\) are located at the \(2a\) site with tetragonal point symmetry \(4/mmm\). The J multiplet (\(^{3}H_{4}\)) splits into five singlets and two doublets,24) \begin{equation} \Gamma_{J=4} = 2\Gamma_{t1} \oplus \Gamma_{t2} \oplus \Gamma_{t3} \oplus \Gamma_{t4} \oplus 2\Gamma_{t5}. \end{equation} (A·1) The CEF Hamiltonian \(\mathcal{H}_{\text{CEF}}=\sum_{l,m}B_{lm}O_{lm}\) is described by the (\(9\times 9\)) matrix \begin{equation} \begin{bmatrix} a &&&&f &&&&0\\ &b &&&&g\\ &&c &&&&h\\ &&&d &&0 &&g\\ f &&&&e &&&&f\\ &g &&0 &&d\\ &&h &&&&c\\ &&&g &&&&b\\ 0 &&&&f &&&&a \end{bmatrix}, \end{equation} (A·2) where the diagonal and off-diagonal elements are the sum of \(B_{lm}O_{lm}\) with \(m=0\) and 4, respectively. The off-diagonal elements provide the mixing of orbitals \(|J_{z}\rangle\) with \(|J_{z}\pm 4\rangle\). The coefficients for \(B_{nm}\) are given in Table A·I.
»View table | Table A·I. Coefficients for CEF parameters \(B_{lm}\) in the \(\mathcal{H}_{\text{CEF}}\) matrix elements. |
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This (\(9\times 9\)) CEF matrix can be reduced to a (\(3\times 3\)) and three (\(2\times 2\)) matrices, in which two of the (\(2\times 2\)) matrices are identical, giving two doublet states [the middle matrix in Eq. (A·3)]. \begin{equation} \begin{bmatrix} a &f &0\\ f &e &f\\ 0 &f &a \end{bmatrix},\quad \begin{bmatrix} b &g\\ g &d \end{bmatrix},\quad \text{and}\quad \begin{bmatrix} c &h\\ h &c \end{bmatrix} \end{equation} (A·3)
The (\(3\times 3\)) matrix gives the energy levels \(E_{0}\), \(E_{1}\), and \(E_{8}\) as in Eqs. (A·4) and (A·5) \begin{align} \Gamma_{t1}:E_{0}, E_{8} = \frac{a+e \pm \sqrt{(a-e)^{2}+8f^{2}}}{2}, \end{align} (A·4) \begin{align} \Gamma_{t2}:E_{1} = a. \end{align} (A·5) Similarly, we obtain \begin{align} \Gamma_{t5}:E_{2,3},E_{6,7} = \frac{b+d \pm \sqrt{(b-d)^{2}+4g^{2}}}{2}, \end{align} (A·6) \begin{align} \Gamma_{t3,t4}:E_{4},E_{5} = c \pm h. \end{align} (A·7)
The energy levels can be assigned easily and almost uniquely. We can expect wave functions that contain orbitals with large \(J_{z}\) at lower energies from the strong Ising-type anisotropy. It is easily shown mathematically that \(\Gamma_{t2}\) cannot be the ground-state. The \(\Gamma_{t5}\) doublet can be ruled out because of the absence of the magnetic order and no quasi-elastic response at low temperatures. The doublet \(\Gamma_{t5}\) ground-state having non-zero expectation value of \(J_{z}\) results in the divergence of \(\chi_{c}\) in the Curie term, which is not the case. The scenario with either \(\Gamma_{t3}\) or \(\Gamma_{t4}\) at the ground-state is also impossible. When \(\Gamma_{t3}(\Gamma_{t4})\) is the ground-state, \(\Gamma_{t4}(\Gamma_{t3})\) and \(\Gamma_{t5}\) are low-lying levels. Thus, there is no way to explain the large \(\chi_{c}\) with these orbitals having a small \(J_{z}\). The observed large \(\chi_{c}\) can only be explained with the largest \(\langle i|J_{z}|j\rangle^{2}\leq 16\) in the low-lying levels. Therefore we can assign \(E_{0}\) (\(\Gamma_{t1}\)) to the main component of \(|{\pm}4\rangle\) and minor \(|0\rangle\) and \(E_{1}\) (\(\Gamma_{t2}\)) only with \(|{\pm}4\rangle\). The lower energy of \(\Gamma_{t1}\) than that of \(\Gamma_{t2}\) is mathematically true and physically not surprising because \(\Gamma_{t1}\) is interpreted to be stabilized with the mixing of \(|0\rangle\) due to the off-diagonal term of \(\mathcal{H}_{\text{CEF}}\) under tetragonal symmetry.
We obtain three equations by subtracting Eq. (A·5) from Eqs. (A·4), (A·6), and (A·7), \begin{equation} \left\{ \begin{array}{l} E_{8}-E_{1}-(E_{1}-E_{0}) = e-a\\ E_{2,3}-E_{1}+E_{6,7}-E_{1} = b+d-2a\\ E_{4}-E_{1}+(E_{5}-E_{1}) = 2(c-a) \end{array}\right.. \end{equation} (A·8) The left-hand sides of Eq. (A·8) can be determined experimentally from the excitation energies. The right-hand sides can be given by linear equations of three diagonal CEF parameters, \(B_{20}\), \(B_{40}\), and \(B_{60}\). Therefore, we can obtain these parameters by solving Eq. (A·8). Similarly, we obtain off-diagonal CEF parameters \(B_{44}\) and \(B_{64}\) from the following equations. \begin{equation} \left\{ \begin{array}{l} (E_{8}-E_{0})^{2} = (a-e)^{2}+8f^{2}\\ (E_{2,3}-E_{6,7})^{2} = (b+d)^{2}+4g^{2}\\ E_{5}-E_{4} = 2|h| \end{array}\right. \end{equation} (A·9) All three equations in Eq. (A·9) should be satisfied by two parameters \(B_{44}\) and \(B_{64}\). The consistency of the calculated energies and experimental data was confirmed. The ground-state energy \(E_{0}\) can be determined using Eq. (A·4). The negative sign of \(B_{44}\times B_{64}\) reproduced the energy level but the positive sign could not. In the latter there is inconsistency between the \(B_{lm}'s\) and the level assignment. The signs of \(B_{44}\) and \(B_{64}\) could not be determined from the present study. The calculated spectra, magnetic susceptibility, and specific heat are identical irrespective of the sign.
In our analysis, the CEF parameters are determined only from the observed excitation energies as summarized in Table A·II. The relevant f-electron states shown in Table I are obtained from the eigenstates of \(\mathcal{H}_{\text{CEF}}\) in Eq. (A·2). The consistency of the f-electron states with the experimental results has been confirmed from the magnetic excitation spectra [Fig. 2 using Eq. (1)] and macroscopic properties (Figs. 3 and 4); the agreement of the calculation and the experimental data is satisfactory.
»View table | Table A·II. Energy differences in the CEF spectra. \(E_{8}\)–\(E_{1}\) is determined to reproduce the entire CEF level scheme, and the other values are results from a simple Gaussian fit of the experimental data shown in Fig. 2. |
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