Magnetic moment of rare earth elements in R2Fe14B estimated with {\mu}^+SR

The ferromagnetic (FM) nature of Nd2Fe14B has been investigated with muon spin rotation and relaxation ({\mu}^+SR) measurements on an aligned, sintered plate-shaped sample. A clear muon spin precession frequency (f_{FM}) corresponding to the static internal FM field at the muon site showed an order parameter-like temperature dependence and disappeared above around 582 K (~T_C). This indicated that the implanted muons are static in the Nd2Fe14B lattice even at temperatures above around 600 K. Using the predicted muon site and local spin densities predicted by DFT calculations, the ordered Nd moment (M_{Nd}) was estimated to be 3.31 {\mu}_B at 5 K, when both M_{Fe} and M_{Nd} are parallel to the c-axis and M_{Fe} = 2.1 {\mu}_B. Furthermore, M_R in R2Fe14B with R = Y, Ce, Pr, Sm, Gd, Tb, Dy, Ho, Er, and Tm was estimated from f_{\mu} values reported in earlier {\mu}+SR work, using the FM structure proposed by neutron scattering and the same muon site and local spin density as in Nd2Fe14B. Such estimations yielded M_R values consistent with those obtained by the other methods.


I. INTRODUCTION
Among many permanent magnet materials, Nd 2 Fe 14 B 1 and related intermetallic compounds 2 are known to be very suitable for industrial applications, due to their high saturation magnetization (M s = 16 kG), large energy product (H c M s = 64 MGOe) and relatively low cost compared with that of Sm 2 Fe 17 N x 3 . Furthermore, although the Curie temperature (T C ) is 592 K for Nd 2 Fe 14 B, the Nd 2 Fe 14 B phase does not decompose until 1428 K, resulting in flexibility of its synthesis process. Therefore, Nd 2 Fe 14 B and related compounds are widely used for high performance motors in many devices, electric vehicles and audio speakers.
In the ferromagnetic (FM) phase, past neutron scattering measurements suggested a collinear spin structure at room temperature 4 , in which both Fe and Nd moments (M Nd & M Fe ) are aligned parallel along the [001] direction. The magnitude of the ordered M Fe was almost saturated even at 300 K, i.e. ∼ 2.2 µ B , while M Nd was initially thought to be below 1 µ B 4 . The other neutron work reported that M Fe ∼ 2.32(3) µ B and M Nd ∼ 2.2 µ B 5 , but the recent work revealed that M Fe = 1. 9(1) µ B and M Nd = 1. 5(1) µ B 6 . Then, more detailed magnetization measurements at 4 K on R 2 Fe 14 B with R = La, Y, ... revealed that M Fe = 2.1 µ B 2 , leading to M Nd = 3.2 µ B . In addition, Nd-NMR measurements suggested that M Nd = 2.7 µ B at 4.2 K 7 . An X-ray magnetic circular dichroism (XMCD) study on R 2 Fe 14 B 8,9 implied that the ordered M R s are very close to the values obtained from gJ of 4f electrons, where J is the quantum number of the total angular momentum and g is the Landé factor. This means that M Nd ∼ 3.3 µ B . Furthermore, the FM spin structure in Nd 2 Fe 14 B was found to change at 135 K(= T SRT ) due to a spin reorientation transition from a high-temperature phase with M [001] to a low-temperature phase with M canted along the [110] direction by magnetization measurements [11][12][13][14] . Initially, a collinear FM structure with a canting angle θ = 30.6 • at 4.2 K was proposed based on magnetization measurements on a single crystal sample 15  In order to further elucidate the FM ground state of Nd 2 Fe 14 B, we need another technique sensitive to internal magnetic field(s) ( H int ) in solids. Unfortunately, neutron scattering is unlikely to be useful for investigating the magnetic nature of ferromagnets, because relatively weak magnetic diffraction peaks always overlap with strong nuclear Bragg peaks. Indeed, the estimated M Nd with neutron ranges from 1 to 2.2 µ B 4-6 , which is rather small compared with those obtained with the other techniques. On the other hand, a positive muon spin rotation and relaxation (µ + SR) provides information on the local magnetic environments at the site(s) of the implanted muons, which usually locate at the interstitial site with the minimum electrostatic potential, regardless of magnetic order and/or disorder 19,20 .
In fact, immediately after the discovery of the Nd 2 Fe 14 B system, a µ + SR experiment was performed at the Paul Sherrer Institut 21,22 using powder R 2 Fe 14 B samples with R = Y, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm in the temperature range between 300 and 4.2 K. The µ + SR spectra obtained in zero external field (ZF) exhibited a clear oscillation with one precession frequency for all the samples, indicating both the formation of static FM order and a single muon lattice site. However, since it was very difficult to determine the correct muon site(s) in the lattice, the muon site was assumed to be a tetrahedral site with two Fe and two Nd nearest neighbors, based on the Mössbauer and neutron data of hydrated R 2 Fe 14 B 22,23 . In addition, the lack of information on the local spin density at the muon site made it eventually impossible to estimate the magnitude of M R . As a result, the past µ + SR result is unlikely to be recognized as a crucial work for elucidating the magnetic ground state of Nd 2 Fe 14 B and R 2 Fe 14 B.
We have therefore attempted to measure the µ + SR spectra for Nd 2 Fe 14 B up to above T C to know the variation of H int with temperature and to predict muon site(s) in the lattice with density functional theory (DFT) calculations. Using the predicted muon site and the measured local spin density at the muon site, the magnitude of M Nd was clearly estimated even below T SRT . Furthermore, using the past µ + SR data for R 2 Fe 14 B and the predicted muon site, we have obtained a systematic change in M R with the number of 4f electrons in R.

II. EXPERIMENTAL
Aligned sintered plates of Nd 2 Fe 14 B were prepared from jet-milled fine powder with the composition of 31.8Nd-0.98B-0.10Cu-0.90Co-0.15Al-0.05Ga- +z, +x and −x directions, respectively. The initial muon spin direction Sµ(0) is in the +x direction ( â of the plates) for spin-rotated (SR) mode (a) or in the −z direction ( ĉ) for non-spin-rotated (NSR) mode (b). Thus if the internal magnetic field (Hint) is parallel toĉ, only U and D counters will detect a muon spin oscillation, and that only in SR mode; but if Hint ⊥ĉ, only B and F counters in NSR mode will show an oscillatory signal. Using both configurations, one can estimate the magnetic anisotropy in the sample.
66.02Fe (wt%). The mean particle size of the powder was about 6 µm. The powders were then pressed under a magnetic field of 1.8 T followed by uniaxial pressing with 15 MPa. The pressed powders were sintered at 1293-1353 K for 4 hours in vacuum (< 10 −2 Pa). Finally, the sintered powder 8 × 8 × 8 mm 3 cube was sliced into 1 mm thick plates with the aligned c-axis perpendicular to the plane. The preparation and characterization of the sintered sample are explained in more detail elsewhere 24 .
The µ + SR time spectra were measured on the M20 surface muon beam line using the LAMPF spectrometer of the CMMS facility at TRIUMF in Canada. Four plates with 8 × 8 × 1 mm 3 were arranged onto a sample holder with theirĉ axes parallel to the beam direction (z) as defined in Fig. 2. For measurements in the T range between 1.8 and 300 K, the samples were attached to a low-background sample holder in a liquid-He flow-type cryostat with 0.05 mm thick Al-coated Mylar tape. For measurements in the T range between 300 and 600 K, the samples were fixed onto a silver plate by a 50 µm thick titanium foil, which is sandwiched between a second silver plate with a 16 × 16 mm 2 square aperture through which incoming muons passed. For the former setup, there is essentially no background signal, while for the latter case the µ + SR signal naturally includes a background signal from muons stopped in the surrounding silver plate.
The µ + SR spectra were obtained in either zero applied field (ZF) or transverse field (TF) with four positron detectors [backward (B), forward (F), up (U) and down (D)] arranged as shown in Fig. 2. The initial direction of the muon polarization [S µ (0)] relative to the plane of the plates was set by a Wien filter spin rotator. Here TF means the applied field is perpendicular to S µ (0), i.e. TF y in this study. The experimental techniques are described in more detail elsewhere 19,20 . The resulting µ + SR data were analyzed with musrfit 25 .
The distributions of electrostatic potential and local spin density were predicted by DFT calculations with a generalized gradient approximation (GGA) plus on-site Coulomb interaction (U ), as described in Sec. III B.  Figure 3 shows the ZF-µ + SR time spectra for the Nd 2 Fe 14 B sample recorded at 300 and 2 K. The spectrum obtained with SR mode [S µ ⊥ĉ] exhibits a clear oscillation, while that obtained with NSR mode [S µ ĉ] shows mainly a non-oscillatory relaxation together with an oscillation with a very small amplitude. Since the Fourier transform frequency spectrum of the ZF-µ + SR time spectrum shows the presence of only one component, the two spectra were fitted by a combination of an exponentially relaxing cosine signal and an exponentially relaxing non-oscillatory signal:

III. RESULTS
Here A 0 is the initial asymmetry, P ZF (t) is the muon spin depolarization function in ZF, A FM and A tail are the asymmetries associated with the two signals, λ FM and λ tail are their exponential relaxation rates, f FM (≡ ω FM /2π) is the muon Larmor frequency corresponding to the quasi-static internal FM field, and φ FM is the initial phase. At each temperature, the two spectra were fitted using common λ FM and f FM . (4), and λ tail = 0.0141(6) µs −1 at 300 K. Thus, the deviation from the c-axis of the magnetic field at the muon site, i.e. the magnetic anisotropy at the muon site is estimated to be Θ(300 The same fit to the data at 2 K yielded Θ(2 K) = 6(4) deg. This means that Θ is almost zero below 250 K within the accuracy of µ + SR. Figure 4 shows the temperature dependences of f FM , λ FM , and λ FM /f FM for the Nd 2 Fe 14 B sample. The f FM (T ) curve exhibits an order parameter-like temperature dependence and f FM disappears at temperatures above around 582 K (= T µSR C ), which is slightly lower than T C in literatures, i.e. 592 K 1,2 . Here it should be noted that T µSR C is estimated from the data obtained in ZF, while the other techniques require the application of a large external magnetic field, which naturally enhances FM order. The f FM (T ) curve also shows a sharp local maximum at 135 K (= T SRT ), indicating a change in the local FM environment caused by a spin reorientation transition.
As temperature increases from 2 K, λ FM decreases slightly up to 100 K, then suddenly increases up to 150 K, and then decreases again towards T C with an increasing slope (dλ FM /dT ). However, below the vicinity of T C , λ FM rapidly increases with temperature, and then suddenly drops to zero at T C ; that is, a critical behavior is observed below the vicinity of T C .
It should be noted that λ FM /f FM , which corresponds to the normalized field distribution width, is almost temperature independent at temperatures below 100 K and at temperatures between 150 and 550 K. This means that besides the temperatures around a spin reorientation transition and below the vicinity of T C , H int in the FM phase depends only on the magnitude of the ordered moments. These results suggest that muons are stable in the Nd 2 Fe 14 B lattice until T µSR C . The present result reproduces those in past µ + SR work carried out below room temperature 21,22 .
In order to estimate T C more correctly, Fig. 5 shows the temperature dependence of the weak transverse asymmetry (A TF ) measured with TF=50 Oe in the vicinity of T C , together with that of f FM . Here, "weak" means that the applied TF is very small compared with H int caused by FM order. The wTF-µ + SR spectrum was fitted by a combination of an exponentially relaxing cosine oscillation due to muon spin precession in TF and Eq. (1): (color online) The temperature dependences of fFM and the weak transverse field asymmetry (ATF) with TF=50 Oe. The fFM(T ) curve is the same as that in Fig. 4(a).
Here TC corresponds to the midpoint of a step-like change in the ATF(T ) curve.
in the surrounding silver plate.

B. DFT calculations
First-principles calculations based on a density functional theory (DFT) 26,27 have been performed to determine the muon site in Nd 2 Fe 14 B. A self consistent field (SCF) calculation is carried out using the ultrasoft pseudopotential method 28,29 , where the on-site Coulomb interaction for localized Nd-4f electrons is taken into consideration using the DFT + U method 30,31 . The obtained pseudo SCF charge density is transformed into an all electron form with the projector augmented wave operators 32 , from which the muon occupation site is estimated by the electrostatic potential analysis. The program used for the DFT calculations is an original code developed by one of the authors (K. M.), which has been successfully applied for various materials [33][34][35][36][37] .
The cutoff energies of plane waves are set to be 25 and 200 hatrees for the pseudo wavefunctions and the charge density, respectively. The 4 × 4 × 4 k-point mesh is adopted for the Brillouin zone integration. The generalized gradient approximation 38 is used for the exchangecorrelation functional. The effective Coulomb and exchange parameters for Nd-4f orbitals are assumed to be U = 5 eV 39 and J = 0.5 eV, respectively. Table I shows the result of the structural relaxation in which atomic positions as well as lattice constants are fully optimized. The calculated parameters are in good agreement with the experimental ones 40 . Figure 6(a) depicts the electrostatic potential: The muon site is found to be 8i (0.6745, 0.8838, 0) which is located near the center of a square base of a pyramid composed of Nd-3Fe-B atoms. As shown in Fig. 6(b), the spin density at the muon site is negligibly small, ρ spin = −2 × 10 −3 µ B /bohr 3 , which is eventually zero. It should be noted that the DFT calculations with U = 0 provides very similar muon site and local spin density to those predicted with U = 5 eV. This means that the two significant parameters, i.e. the muon site and ρ spin , are not sensitive to U in the Nd 2 Fe 14 B lattice. On the contrary, the ordered magnetic moment of each element varies with U (Table II). More correctly, the introduction of U = 5 eV reduces M Nd by 10%, while the change in M Fe is about 1%. The magnitude of M Fe at each site is comparable to the reported ones (see Table III). This indicates the importance of the magnitude of U for estimating M Nd by DFT calculations. For non-magnetized ferromagnetic materials in zero applied field, the internal magnetic field at a muon site (H µ ) is represented by 20,41-43 ; This field is connected to the muon-spin precession frequencies through the muon gyromagnetic ratio [f = Hγ µ /(2π) = 0.013553 (MHz/Oe)×H (Oe)] leading to where H dip is the dipolar field, H L is the Lorentz field, H hf is the hyperfine field, and f µ , f L , and f hf are the corresponding muon spin precession frequencies. Furthermore, H L and H hf are connected to the saturated magnetization (M s ) and the local spin density at the muon sites (ρ spin ) as follows: In order to estimate H dip (f dip ), we use the results of neutron diffraction 4 and Mössbauer 44 measurements for the magnitude and direction of the Fe moments. Assuming that the magnitude of the ordered M Fe is 2.1 µ B 2 , H dip at the muon site is easily calculated as a function of the Nd moment using crystal structural data with dipelec 45 .
We start by considering a collinear FM structure along the c-axis, that is, M Fe [001] and M Nd [001]. Since 4πM s = 18.5 kOe at 5 K (see Table IV) 2,22 , H L = (0, 0, 6.2 kOe) from Eq. (5). Moreover, H hf = (0, 0, 0) because of the absence of any local spin density at the muon site. Consequently, we obtain the relationship between |H µ | = H calc µ and the magnitude of the Nd moment (M Nd ), as seen in Fig. 7(a). Here, the measured value of f µ (f exp µ ) is 152.6(2) MHz at 2.2 K, which is very close to the reported value (156 MHz) at 5 K. Thus, in order to explain H exp µ , M Nd is uniquely determined as 3.31 µ B . This is almost equivalent to M Nd estimated from magnetization measurements, i.e. M Nd = 3.2 µ B 2 , confirming the reliability of the predicted muon site from DFT calculations. From the data at room temperature, i.e. 4πM s = 16.0 kOe at 295 K and H exp µ = 151(2) MHz at 300 K, we also obtain that M Nd = 3.01 µ B .
Although we assumed that M Fe = 2.1 µ B , M Nd estimated with the above procedure is found to increase linearly with M Fe (see Fig. 8). On the contrary, Fig. 8 provides an acceptable range for M Fe as 2.0 ≤ M Fe ≤ 2.15 µ B , when M Nd ranges between 3.0 and 3.5 µ B . Furthermore, we assumed that M Fe is identical for all the Fe sites. However, experimental studies and DFT calculations reported that M Fe at each site deviates slightly from 2.1 µ B . In order to know the effect of such deviations on the estimation of M Nd , the relationship between H µ and M Nd is also shown for the two cases in Fig. 7(a) [11][12][13][14] . The corresponding anomaly is clearly seen in the f FM (T ) and λ FM (T ) curves [ Fig. 4]. More correctly, both Fe and Nd moments are thought to be canted towards the [110] direction from the [001] direction, based on both first principles calculations and Fe K-edge x-ray magnetic circular dichroism (XMCD) measurements 17 . The canting angle (θ) was estimated to be 27 • for Fe (θ Fe = 27 • ) and 58 • for Nd (θ Nd = 58 • ) at 4.2 K. Figure 7(b) shows the relationship between H µ and M Nd for several θ Nd values. The µ + SR result clearly excludes a collinear structure, in which θ Fe = θ Nd = 27 • , as an FM ground state. On the other hand, non-collinear structures provide a more plausible M Nd , particularly when θ Nd ∼ 60 • . If we assume that M Nd = 3.2 µ B , θ Nd should be 63 • , which is very close to the value reported by XMCD (58 • ).
Dipole field calculations provide that the magnetic anisotropy at the muon site (Θ) is 16 deg at temperatures below T SRT , while Θ = 0 deg at temperatures above T SRT . Making comparison with the experimental result [Θ(300 K) = 7(4)deg and Θ(2 K) = 6(4)deg], the experimental accuracy of Θ was likely to be overestimated. This is probably due to the fact that S µ (0) for NSR mode is deviated from the z direction by about 10 deg to eliminate the other particles in the muon beam. Nevertheless, we should note that the above estimation for M Nd is based only on the magnitude of f µ , and as a result, the estimated value is not affected by the alignment of the sample.
Another XMCD study at low temperatures 18 proposed the possibility of a non-collinear spin arrangement among the Nd moments. That is, θ Nd ∼ 25 • for the Nd ions at the 4g site, but θ Nd ∼ 80 • for the Nd ions at the 4f site. Figure 7(c) shows the dependence of H µ on M Nd as θ Nd(4f) changes from 73 to 84 • . The calculations also predict that θ Nd = 82 • for M Nd = 3.2 µ B , which looks consistent with the proposed arrangement. However, we should note that there are eight crystallographically equivalent muon sites (8i) in the Nd 2 Fe 14 B lattice. Moreover, such a non-collinear spin arrangement among the Nd moments produces two different H µ s at each 8i site -namely, H µ = 11270 Oe for four of the sites and 11655 Oe for the other four sites. Although the difference of the two H µ s (about 4%) is too small to observe two distinct muon precession frequencies in the ZF-µ + SR spectrum, such a split naturally increases the field distribution width, resulting in an increased relaxation rate λ FM . In reality, λ FM and λ FM /f FM at 2 K are smaller than those at room temperature [ Fig. 4(b)]. This clearly excludes the model of a non-collinear spin arrangement among the Nd moments from the FM ground state for Nd 2 Fe 14 B. Since the λ FM (T ) curve exhibits a broad maximum at around T SRT [see Fig. 4(b)], such a non-collinear spin arrangement among the Nd moments could appear in a limited temperature range particularly below the vicinity of T SRT . Even for this case, the predicted Θ is the same to that for the collinear spin arrangement among the Nd moments, i.e. 16 deg. Therefore, Θ provides no crucial information on the spin arrangement in Nd 2 Fe 14 B within the present accuracy.

B. R2Fe14B
Although we have measured µ + SR spectra only for Nd 2 Fe 14 B, both H µ and M s were reported for the other R 2 Fe 14 B compounds with R = Y, Ce, Pr, Sm, Gd, Tb, Dy, Ho, Er, and Tm (see Table IV  Finally, Fig. 10 shows the relationship between M R and the expected magnetic moment (gJ) derived from Landé g factor and the quantum number of the total angular momentum (J) for free R 3+ ions. M R estimated with the magnetization measurements (M Mag R ) is almost equivalent to gJ 2 , suggesting the presence of stronger exchange field to the 4f moments than the crystal field 2 . On the other hand, the slope of the M µSR R (gJ) curve estimated with µ + SR is steeper than that for the M Mag R (gJ) curve, mainly because | M µSR R |>| M Mag R | for the heavy rare earth elements. Although the reason for this discrepancy is not clear at present, we should note that µ + SR is very sensitive to local magnetic environments. Recently, not only for Nd 2 Fe 14 B but also for Ho 2 Fe 14 B, a non-collinear spin structure for the Ho moment is proposed with neutron using a single crystal sample 59 . This implies the possibility that such non-collinear structure appears in the other R 2 Fe 14 B at low temperatures, which  . It would be thus an interesting subject to reconfirm the magnetic structure in R 2 Fe 14 B at low temperatures using a high quality sample. Finally, this work clearly demonstrates the unique power of a combination of µ + SR and DFT calculations for determining the magnetic moments of rare earth elements through the observation of local H int .

V. SUMMARY
We have studied the internal magnetic field in a sintered Nd 2 Fe 14 B permanent magnet sample with a positive muon spin rotation and relaxation (µ + SR) technique, which provides microscopic magnetic information at the muon site. Combining the µ + SR data with the result of DFT calculations for predicting the muon site in the lattice, the magnitude of the ordered Nd moment was clearly estimated both for a collinear ferromagnetic structure at room temperature and a canted ferromagnetic structure at 2 K. Furthermore, a similar estimation for the ordered moment of the rare earth elements in R 2 Fe 14 B provided reasonable values consistent with those reported by magnetization and Mössbauer measurements. µ + SR has been widely used for investigating a magnetic nature in antiferromagnetic, spinglass, and/or paramagnetic materials, in which both the Lorentz field and hyperfine field are usually zero and, as a result, the dipole field is predominant. On the contrary, the present work demonstrates that a combination of µ + SR and DFT calculations further expands the research field into ferromagnetic materials.

VI. ACKNOWLEDGMENTS
We thank the staff of TRIUMF (especially the CMMS) for help with the µ + SR experiments. This work was supported by Japan Society for the Promotion Science  The internal magnetic field detected with µ + SR 22 , the saturated magnetization 2 , the magnetic moment of R (MR) estimated with µ + SR (M µSR R ), and MR proposed with magnetization measurements at 4 K (M Mag R ) 2 , and gJ, where g is the Landé g-factor and J is the quantum number of the total angular momentum.