Site-selective NMR measurements in single crystal PrNb2Al20

We report the result and analysis of the nuclear magnetic resonance (NMR) measurements at the magnetic field of 8.57 T and temperature of 80 K using a single crystal PrNb2Al20. Combining the field angle dependence of NMR spectra, numerical simulation, and band calculation, we deconvoluted the NMR signals from the Nb, Al 48 f, and Al 96g sites. Unfortunately, the overlapping of NMR lines prevents us to extract the NMR signals from the Al 16c site. However, the obtained nuclear quadrupole resonance (NQR) parameters for the Nb and Al 48f sites are consistent with the previous reports [T. Kubo et al., 2014 JPS Conf. Proc. 3 012031; 2015 J. Phys.: Conf. Ser. 592 012093; 2016 J. Phys. Conf.: Ser. 683 012015] and for the Al 96g site are refined to be vQ = 0.98 MHz and η = 0.46.


Introduction
PrT 2 X 20 system (T = Ti, V, Nb, Rh, Ir, X = Al, Zn) which crystallizes in a cubic CeCr 2 Al 20 -type structure with the space group Fd3m has been studied intensively because of their intriguing behaviors at low temperatures such as superconductivity, quadrupole ordering, and non-Fermi liquid (NFL) behaviors [1,2,3,4,5]. In this system, the 4 f electronic ground state of Pr 3+ ion is considered to be the non-Kramers' Γ 3 doublet which has no magnetic dipole but has two electric quadrupoles O 20 and O 22 , and one magnetic octupole T xyz . These multipole degrees of freedom are believed to play an important role in exotic ground states in these materials at low temperatures. Among them, PrNb 2 Al 20 shows NFL behaviors such as T -linear resistivity below 2 K and logarithmic divergent specific heat divided by temperature below 10 K, in which the relationship between these behaviors and quadrupole degrees of freedom has been discussed [1].
To clarify the importance of multipole degrees of freedom, site-selective nuclear magnetic resonance (NMR) measurements are desirable since NMR can obtain the microscopic electronic information from static and dynamical viewpoints based on the local symmetry at the ligand sites of Pr ions. The NMR spectrum of PrNb 2 Al 20 becomes complex because of its crystal structure. Thus, it is important to deconvolute the complex NMR lines since the microscopic electronic state at the ligand sites directly reflects the essential information of coupling between conduction electrons and multipoles through the bond symmetry between the Pr and ligand sites [6,7,8,9,10]. In this paper, we report the results of 93 Nb-and 27 Al-NMR measurements using a single crystal and the analysis of the NMR lines.  directions at 8.57 T, and at 80 K. Colored dash-dotted, solid, and dashed lines are the results of the simulation for Nb, Al 48 f , and Al 96g sites, respectively. As shown in the inset of Fig. 1(b), θ and ϕ denote the field angle from the crystal c-axis and in the ab-plane, respectively. Both the ideal and actual angles are presented. The details of the simulation are described in the text.

Experimental Procedure
The single crystal sample of PrNb 2 Al 20 was prepared by the Al-flux method. Details of the sample preparation method were reported in the previous paper [1]. The field angle dependence of NMR spectra is measured by using a 17 T superconducting magnet and a phase-coherent pulsed spectrometer. The angle between the applied field and crystal axes is controlled by a uniaxial goniometer. 93 Nb-(nuclear spin I = 9/2, gyromagnetic ratio γ/2π = 10.421 MHz / T, nuclear quadrupole moment Q = −0.32 × 10 −28 m 2 ) and 27 Al-(I = 5/2, γ/2π = 11.094 MHz / T, Q = 0.1466 × 10 −28 m 2 ) [11,12] NMR lines are obtained by summing up the fast Fourier transformed spin echo signals obtained with a step of 5 kHz at a fixed magnetic field µ 0 H ≈ 8.57 T. Figure 1 shows 93 Nb-and 27 Al-NMR spectra obtained at the fixed magnetic field 8.57 T applied parallel to one of the (a) ⟨111⟩ and (b) ⟨110⟩ axes, and at a temperature of 80 K. Each of the resonance lines are as sharp as about 30 kHz. There are one Nb site and three crystallographically inequivalent Al sites (16c, 48 f , and 96g) in PrNb 2 Al 20 . Due to the local symmetry at the ligand sites which is lower than cubic as described below, these sites split further under the magnetic field. In order to deconvolute the obtained lines into the above sites, we carried out the simulation numerically solving the (2I + 1) × (2I + 1) nuclear spin hamiltonian H by an exact diagonalization method. Such a hamiltonian can be written as follows:

Results and Discussion
where h is the Planck constant (ℏ ≡ h/2π) and e is the elementary charge. The first term in Eq. (1) is a Zeeman interaction between the nuclear spin I and applied field H, and is expressed in Eq. (2). The second term is a nuclear quadrupole interaction between the nuclear spin and electric field gradient (EFG) created by the surrounding charge distribution at the nuclear site, and is expressed in Eq. (3) . EFG is characterized by the second rank EFG tensorṼ In the principal coordinate system,Ṽ is diagonalized and Eq. (3) can be rewritten as Eq. (4) is the asymmetry parameter, and V XX , V YY , and V ZZ are the principal values ofṼ which satisfy the relation |V ZZ | ≥ |V YY | ≥ |V XX | and V XX +V YY +V ZZ = 0. The third term is a hyperfine interaction between the nuclear spin and electron spins, which is represented by the second rank classical dipole field tensorD and Knight shift tensorK, and is expressed in Eq. (5).D describes the classical dipole interaction between the nuclear spin and Pr 4 f magnetic dipoles.K has the information of c-f hybridization and is decomposed into the isotropic part K iso and the anisotropic part K ani .D andK ani are traceless and are designated by their principal values (D 1 , D 2 , D 3 ) and (K 1 , K 2 , K 3 ) that satisfy the relation of |D 3 | ≥ |D 2 | ≥ |D 1 | and |K 3 | ≥ |K 2 | ≥ |K 1 |.
In order to calculate NMR lines, we have to know the principal values and the principal axis directions ofṼ,D, andK. The principal values ofD are calculated by the lattice sum of the classical dipole field , where µ i = g J µ B J i is the 4 f electronic magnetic dipole in which g J , µ B , and r i are the Landé g-factor, the Bohr magneton, and the distance between the site considered and ith Pr site, respectively. The magnitude of magnetic dipole is estimated to be ≈ 0.2 µ B at 80 K from the crystalline electric field analysis [10]. ν Q and η for Nb, Al 48 f , and Al 96g sites are previously determined [13,14,15]. The principal values ofK are treated as fitting parameters.
Next, we consider the principal axis directions. Generally, these directions do not coincide with the crystal axes or with each other due to the local symmetry at the ligand sites. In such a case, the analysis taking into account of the local symmetry is previously reported [9,15,16,17,18,19]. Shown in Fig. 2 is the geometry of neighboring Nb, Al 48 f , and Al 96g atoms around a Pr atom. The local symmetry axis directions at these ligand sites are also represented. For the Nb site with local symmetry of .3m, the maximum principal axis direction coincides with the local threefold rotation axis. As shown in Fig. 2(a) with the crystal axis directions, this axis is parallel to [111] direction for the Nb site at (1/2, 1/2, 1/2).Ṽ, D, andK are axially symmetric around this axis. The Al 16c site has the local symmetry of .3m, thus the principal axis directions are same with those for the Nb site. For the Al 48 f site at which local symmetry is 2.mm, there are two mirror planes. For example, the Al 48 f site at (1/8, 1/8, z) have the mirror planes (110) and (110). In this case, the principal axis directions are constrained in these mirror planes but the maximum principal axis direction cannot be determined beforehand. For the Al 96g site at which local symmetry is ..m, there are only one mirror plane which is parallel to one of the {110} plane. In this case, one of the three principal axes are perpendicular to the plane and the other two axes are in the plane. Since the principal axis directions in the plane cannot be predetermined, we calculatedṼ for each sites by a band calculation in local density approximation for the isostructural LaNb 2 Al 20 . The details of the calculation method are described in the previous paper [20]. As shown in Fig. 2   Using the principal values and principal axis directions, we can obtain each of the tensor components in the crystal coordinate by an appropriate frame transformationT c = RT p R −1 , whereT p andT c are the representations of a tensor in its principal and crystal coordinate, respectively, and R is a frame transformation matrix between these coordinates. In addition, tensor components are transformed from one site to another under symmetry operations such as 120 • rotation around one of the ⟨111⟩ axes and mirror operation for one of the {110} planes.
Based on the above considerations, we analyzed the NMR lines. In Table 1, we summarized the number of the NMR sites and corresponding number of NMR lines for two magnetic field directions. As shown in Fig. 1, the number of the observed NMR lines are larger than expected number of NMR lines for each of the field directions. This is due to a misalignment of the sample, which causes further splitting of the NMR sites. Such a misalignment is inevitable since we use a uniaxial goniometer in the present measurement. For general field direction, we have four Nb, six Al 48 f , and twelve Al 96g sites.
First, we determined the exact field angle by calculating the 93 Nb-NMR lines because the principal axis directions ofṼ,D, andK at Nb sites are uniquely determined as discussed above. We defined the    Fig. 1, we could assigned 93 Nb-NMR lines to the four Nb sites. From the fitting of these lines, we obtained (θ exp , ϕ exp = 57 • , 39 • ) for Fig. 1(a) and (θ exp , ϕ exp = 89 • , 37 • ) for Fig. 1(b). Obtained NQR and Knight shift parameters are summarized in Table 2 with the result of the band calculation. The obtained values are consistent with the previous reports [13,14,15]. Therefore, we could determined the field angle for two magnetic field direcitons. Next, we analyzed the 27 Al-NMR lines by using the obtained field angles. NMR lines from Al 16c and 96g sites overlap strongly each other so that, in general, it is difficult to deconvolute them. Nevertheless, we could assigned NMR lines from one of the Al 96g site since these lines could be clearly separated from another Al lines as shown by the dashed lines in Fig. 1. The obtained NQR parameters (ν Q = 0.98 MHz, η = 0.46) are consistent with the previous NMR and NQR result [15] and the V ZZ direction corresponds to the predetermined one by the band calculation. Fore more precise deconvolution of the Al 16c and 96g lines, the NMR measurement using a double-axis goniometer and the NQR measurement below 1.5 MHz are needed.
Finally, we discuss about the 27 Al-NMR lines from the Al 48 f site. As shown by the solid lines in Fig. 1, all of the 27 Al-NMR lines from the Al 48 f site are well reproduced by the simulation and the obtained NQR parameters (ν Q = 2.28 MHz, η = 0.17) are consistent with the previous reports [14,15]. The EFG principal axis directions at the Al 48 f site correspond to the calculated ones, thus the validity of the band calculation is confirmed. In addition, we could determinedK at the Al 48 f site as shown in Table 2.
By using these assignment of the ligand sites in PrNb 2 Al 20 , we may bring out the information of multipole fluctuations. Measurements of the temperature and field dependences of the Knight shift and NMR relaxation rate using single crystal PrNb 2 Al 20 are ongoing and the results will be presented elsewhere.

Summary
We measured the field-angle dependence of 93 Nb-and 27 Al-NMR signals for single crystal PrNb 2 Al 20 . Together with previously reported NQR parameters and band calculation, we succeeded to deconvolute the NMR signals from the Nb, Al 48 f , and one of Al 96g sites. The experimental results are well reproduced by using the numerically obtained EFG tensors, thus confirms the availability of the band calculation. Unfortunately, the strong overlapping of the NMR lines prevents us to obtain the precise NMR and NQR parameters for Al 16c site in the present stage. Nevertheless, these results help us to investigate the importance of multipole degrees of freedom for the anomalous properties of PrNb 2 Al 20 .