Small-angle neutron scattering of nanocrystalline gadolinium and holmium with random paramagnetic susceptibility

A neutron study of nanocrystalline terbium (Balaji G et al 2008 Phys. Rev. Lett. 100 227202) has shown that the randomly oriented anisotropy of the paramagnetic susceptibility tensor may lead to strongly correlated nanoscale spin disorder in the paramagnetic state which can be probed very effectively by magnetic small-angle neutron scattering (SANS). In principle, this scenario is also applicable to other rare-earth metals and the size of the effect is expected to scale with the strength of the anisotropy in the paramagnetic state. Here, we report SANS results (in the paramagnetic state) on nanocrystalline inert-gas condensed samples of Gd and Ho, which represent the cases of low and high anisotropy, respectively.


Introduction
As a consequence of the large volume fraction of grain boundaries, typical average crystallite sizes between 10-50 nm and a random orientation of the crystallographic axes, nanocrystalline inert-gas condensed materials are excellent model systems for investigating the impact of internal interfaces and reduced structural length scales on the magnetic properties of solids [1]. In recent years, the eminent potential of the magnetic small-angle neutron scattering (SANS) technique for studying the magnetic microstructure of nanocrystalline ferromagnetic materials has been demonstrated [2]. In particular, using magnetic SANS, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 4 Present address: Patentverwertungsagentur der saarländischen Hochschulen (PVA), Universität des Saarlandes, Wissens-und Technologietransfer GmbH, Starterzentrum, Gebäude A1 1, D-66123 Saarbrücken, Germany. characteristic parameters such as the correlation length of the spin misalignment, the exchange-stiffness constant, or the mean-square internal magnetostatic stray field have been retrieved.
Due to the position-dependent Ruderman-Kittel-Kasuya-Yosida exchange interaction between the highly localized magnetic moments, the magnetism of the rare-earth metals is particularly prone to structural disorder [3]. Furthermore, the random jump in the direction of the magnetocrystalline anisotropy axes across grain boundaries is expected to be particularly relevant in the rare-earth metals with an orbital angular momentum L = 0. Up to date, only two SANS studies on nanocrystalline rare-earth ferromagnets exist: on Tb (L = S = 3) as a representative of a material with strong magnetocrystalline anisotropy [4] and on Gd (L = 0, S = 7/2) (using the low-capturing isotope 160 Gd) with a comparatively low anisotropy [5] (S: spin angular momentum). Both investigations demonstrate a strong impact of the nanoscale microstructure on the ferromagnetic spin structure. Moreover, the recent study of the paramagnetic SANS from nanocrystalline Tb reported in [6] has revealed a (somewhat counterintuitive) increase of the scattering intensity by almost two orders of magnitude with increasing field (from 0-5 T) at low momentum transfers q, whereas the scattering was found to decrease at high q-values. The latter was ascribed to the suppression of local paramagnetic spin fluctuations. Consequently, a rather unusual crossover of the scattering curves at different fields was observed. A quantitative explanation for the increase of the scattering cross section at low q is based on the well-known anisotropy of the paramagnetic susceptibility tensor of Tb [7]. In particular, for the nanocrystalline material, the random orientation of the crystallographic axes of the individual grains gives rise to a highly nonuniform magnetic response on the nanoscale. In other words, in these samples the scattering contrast between neighboring grains (and thus the total scattering cross section) is strongly increased with the field, in contrast to the usual suppression of magnetic nanoscale disorder, which is commonly associated with a decrease of the SANS signal [2].
An important result of the model given in [6] is the quadratic relation of the magnetic SANS cross section to the (internal) applied magnetic field H i , in agreement with the experimental Tb data. Specifically, for unpolarized neutrons, the azimuthally-averaged differential SANS cross section d d (in the paramagnetic temperature regime) can be expressed as where q is the scattering vector, d nuc d represents the nuclear scattering cross section and S mag denotes the magnetic scattering function, which is related to the size and shape of the grains as well as to the main-axis entries of the susceptibility tensor (see equation (3) in [6]).
It is the purpose of this study to further investigate the predictions of equation (1) for the case of a low-anisotropy rare-earth metal (Gd) and for Ho (L = 6, S = 2), which (similar to Tb) exhibits highly anisotropic magnetic properties.

Experimental
Synthesis of disk-shaped nanocrystalline Gd and Ho samples (diameter: 8 mm; thickness: ∼500 µm) was carried out by the inert-gas condensation technique, as described in detail in [4,5,8]; the Gd samples for the SANS experiments were prepared by employing the low neutron-capturing isotope 160 Gd (enrichment: 98.6%) as a starting material in the evaporation process. The average crystallite sizes of the asprepared nanocrystalline samples were determined by analysis of wide-angle x-ray diffraction data and found to be D = 21 ± 6 nm (Gd) and D = 51 ± 7 nm (Ho). The SANS experiments were performed at the SANS 1 instrument at the Paul Scherrer Institut, Villigen, Switzerland, using unpolarized neutrons with a mean wavelength of λ = 6.0 Å and a wavelength broadening of λ/λ = 10% (FWHM) [9]. A cryomagnet provided control of the external magnetic field (µ 0 H max = 11 T) and temperature; the magnetic field H was applied perpendicular to the wave vector of the incoming neutron beam. The qrange from ∼0.04-1.5 nm −1 was covered with three sampleto-detector distances. The neutron raw data were corrected in the usual way for background scattering, transmission, detector efficiency, detector dead time and solid-angle distortion. Figure 1(a) displays the azimuthally-averaged SANS data obtained from the nanocrystalline 160 Gd sample at T = 300 K, while figure 1(b) shows the corresponding data for nanocrystalline Ho at T = 145 K. When increasing the field from zero to 9 T, the SANS signal of Gd increases by a factor of about 2 at the lowest momentum transfers (i.e. q ∼ = 0.04 nm −1 ), whereas at q ∼ = 1.5 nm −1 a decrease by a  figure 1(b). For Ho, we find an excellent agreement between the data and equation (1) over the whole (q, H i )-range, whereas for Gd the agreement is quantitative only up to fields of ∼0.6 T. This can be seen in figure 2, where the field dependence of the total detector counts of Gd at three different sample-to-detector distances is shown. For field values up to 0.6 T and at intermediate and low q, we find a quadratic increase according to equation (1) (see insert in figure 2), while at higher fields the signal shows a Langevin-like behavior. At high q, the intensity is slightly suppressed with increasing field, which may be attributed to the presence of scattering from critical spin fluctuations [6].

Results and discussion
Although for the case of nanocrystalline Gd it is not necessarily expected to see the described effect due to a vanishing difference between the paramagnetic Curie temperature parallel and perpendicular to the hexagonal cdirection [7,10], θ = θ ⊥ = 317 ± 3 K, the properties of Gd near the Curie point T C ∼ = 293 K [11] may help to understand the situation in this material. In particular, Gd shows a rather broad Curie transition, strong deviations from Curie-Weiss behavior near T C (θ was obtained by extrapolation from temperatures above 400 K [10]) and a nonvanishing anisotropy in the paramagnetic regime up to about 340 K [12]. These properties are in line with the observations reported here, i.e. scattering from critical spin fluctuations and due to random paramagnetic susceptibility, as well as an approach-tosaturation behavior at higher fields. Note also that for a mean grain size of D = 21 nm, a reduced Curie temperature of about 285 K is expected [13]. Furthermore, results from a SANS study in the ferromagnetic temperature range showed that in nanocrystalline Gd the grain boundaries are associated with strong nanoscale spin disorder [5]. Grain-boundary effects may also provide a major contribution to the magnetic-field response observed in the paramagnetic regime. However, since grain boundaries are presumably coupled to fluctuations in the paramagnetic susceptibility on rather short real-space length scales, i.e. only a few nanometers, the corresponding magnetic scattering can be expected at the largest q-values. Thus, if present, the signature of the grain boundaries is likely to be covered by critical scattering in the current data for the case of Gd.
The Ho data were measured at 145 K, well above both θ = 73 K and θ ⊥ = 88 K as well as the antiferromagnetic Néel Temperature T N = 132 K [7]. Accordingly, we find strong field-induced scattering predominantly at the higher fields and a very good agreement with the model proposed in [6]. No saturation effects are seen in the Ho data, although the SANS measurements were conducted up to 9 T. 5 Figures 3(a) and (b) display, respectively, the fit results for d nuc d and S mag for Gd at T = 300 K and for Ho at T = 145 K. Integration of the S mag -data, according to yields a partial invariant Q p of the azimuthally-averaged magnetic scattering cross section (equation (4) in [6]) (b H = 2.91 × 10 8 A −1 m −1 ); Q p is related to a deviatoric component, (χ 11 − χ 33 ) 2 , of the susceptibilty tensor, where χ 11 and χ 33 denote, respectively, the susceptibility along the a-axis and caxis of the hcp lattice. When the integration in equation (2) is carried out, we obtain a lower bound for Q p due to the limited range of experimental scattering vectors (q min q q max ).
The resulting value for Ho, Q p ∼ = 0.060, compares reasonably with the value of |χ 11 − χ 33 | ∼ = 0.046 estimated from single-crystal magnetization measurements [14]. Note that the theory behind equation (1) neglects terms of local demagnetizing field and magnetoelastic coupling, which may result in a reduced value of Q p [6]. Furthermore, our results for nanocrystalline Ho do not provide evidence for grainboundary-induced effects, as can be seen e.g. from the fact that S mag takes on very low values for q-values >1 nm −1 (see figure 3(b)). While this does not necessarily imply that grain boundaries have no relevance for the paramagnetic properties of nanocrystalline Ho, we note that the comparatively large crystallite size of 51 nm of the present Ho sample is associated with a volume fraction of grain boundaries of only about 8% [5].
The results for S mag of nanocrystalline Gd (figure 3(a)) indicate that the random paramagnetic susceptibility of the individual grains leads to significant correlated nanoscale spin disorder at 300 K and moderate fields in accordance with the model proposed in [6]. From equation (2), a value of Q p ∼ = 0.27 can be derived. Note that for q > 0.5 nm −1 , S mag could not be evaluated from the present data due to critical scattering. Since Q p compares directly with the difference |χ 11 − χ 33 | and since Gd is known to exhibit by far the lowest magnetocrystalline anisotropy among the heavy rare-earth metals, one may expect to obtain a much lower Q p -value (for Gd) than for nanocrystalline Ho. However, the experimental temperature value of 300 K differs only by about 15 K from the Curie temperature of nanocrystalline Gd (with D = 21 nm) [13], whereas for the Ho data there is a difference of more than 55 K to θ ⊥ , leading to comparatively low susceptibility values (χ ∝ (T − θ) −1 ) and, thus, to a small value of |χ 11 − χ 33 |.
The observation of a finite magnetic anisotropy of the susceptibility tensor of Gd in the paramagnetic state may be surprising in view of the fact that the orbital angular momentum of the Gd 3+ -ion vanishes. This finding indeed addresses fundamental aspects regarding the origin of magnetic anisotropy of Gd metal, which has recently been investigated in the ferromagnetic temperature regime by means of first-principles density functional theory [15]. The authors of [15] have suggested a mechanism for the magnetic anisotropy energy involving (besides dipolar anisotropy) the spin-orbit interaction of the conduction electrons. In this respect, it would be of interest to investigate in detail the origin of the observed magnetic anisotropy of nanocrystalline Gd in the paramagnetic temperature regime.

Conclusion
In conclusion, we have performed magnetic-field-dependent SANS measurements on nanocrystalline Gd and Ho in the paramagnetic temperature regime. Our data analysis indicates that, when a magnetic field is applied, random paramagnetic susceptibility leads to strong correlated nanoscale spin disorder in these materials. Equation (1), which predicts d d ∝ H 2 i at low q, provides an excellent description of the fielddependent correlations in strongly anisotropic Ho, while for Gd its applicability is limited to fields below about 0.6 T, presumably due to low anisotropy and/or the close proximity to the Curie temperature. However, in view of the pure Sstate nature of the Gd 3+ -ion, it appears to be surprising to find a significant nonzero anisotropy of the susceptibility tensor from the neutron data. Future experiments will address the temperature, grain size and, in particular, the polarization dependence of the paramagnetic SANS in order to determine the temperature dependence of the anisotropy parameters and to investigate the role of the grain boundaries for the paramagnetic state of nanocrystalline rare-earth metals.