Qualitative Evaluation of the Magnetocrystalline Anisotropy in Spinel Ferrite Nanoparticles Using Polarized Neutron Powder Diffraction

ABSTRACT

Magnetocrystalline anisotropy is a key parameter governing the performance of magnetic nanoparticles in many applications. However, disentangling its intrinsic contribution from other sources of effective anisotropy, such as surface effects, dipolar interactions or shape anisotropy, remains highly challenging. Here, we report a novel approach to qualitatively estimate the magnetocrystalline anisotropy of two Co_xFe_3−xO₄ nanoparticles with different Co contents (x = 0.11 and 0.61) using polarized neutron powder diffraction (PNPD). The off‐diagonal elements of the susceptibility tensors and degree of asymmetry of the magnetization ellipsoids obtained from the PNPD refinements reveal that the sample with x = 0.61 presents a larger magnetocrystalline anisotropy than the sample with x = 0.11, which is consistent with the effective anisotropy derived from magnetometry. Moreover, comparison of the PNPD‐derived magnetization ellipsoids across materials with varying anisotropies confirms the direct relationship between the ellipsoid asymmetry and magnetocrystalline anisotropy. These findings establish PNPD as a powerful tool for qualitatively probing intrinsic anisotropies in nanoparticle systems, paving the way for the rational design and optimization of magnetic nanoparticles for advanced applications.

Polarized neutron powder diffraction is used to qualitatively probe the magnetocrystalline anisotropy of cobalt‐doped ferrite nanoparticles. The analysis of local magnetic susceptibility tensors and magnetization ellipsoids reveals how cobalt incorporation modifies both the magnitude and orientation of anisotropy, providing a new approach to compare intrinsic magnetic anisotropies in nanoparticle systems beyond conventional magnetometry.

1. Introduction

Magnetic nanoparticles have maintained a sustained interest during decades both due to their interesting fundamental properties and their use in widespread fields [1, 2, 3, 4]. One of the key properties of magnetic materials is the magnetic anisotropy where the magnetic properties depend on the direction along which they are measured. In particular, it is easier to magnetize the material in certain directions (easy axis) than in others (hard axis). The additional energy required to magnetize a specimen in a particular direction compared to the one required to magnetize it along the easy direction defines the anisotropy constant, K. This has important implications for the magnetization reversal, dictating, for example, the shape of the hysteresis loop or how difficult it is to reverse the magnetization, i.e., the coercivity, H_C [5]. Importantly, magnetic anisotropy is a crucial parameter in many applications, like permanent magnets, spintronics or magnetic hyperthermia [6, 7, 8, 9]. During the last decade, the development of new rare earth‐free high anisotropy materials, like Mn‐based alloys, carbides or nitrides, has surged [10, 11, 12, 13, 14]. Additionally, traditional materials, like different types of ferrites, are being reevaluated for moderate anisotropy applications [15, 16, 17, 18]. In this sense, cobalt ferrite (CoFe₂O₄) and non‐stoichiometric cobalt ferrite (Co_xFe_3−xO₄) nanoparticles have attracted considerable attention due to their appealing properties, including moderate saturation magnetization, high magnetic anisotropy, and cost‐effective, straightforward synthesis [9, 15, 19]. Recently, the potential of Co_xFe_3−xO₄ nanoparticles for high anisotropy applications has been demonstrated by optimizing the particle size, shape, and composition [20, 21]. For example, 20 nm Co_0.7Fe_2.3O₄ nanoparticles have been shown to exhibit energy products, (BH)_max, in excess of 20 kJ/m³ [9]. However, despite the intense research in hard magnetic materials, there are many aspects of these materials that are still not fully understood, which precludes the efficient development of optimized nanostructured magnets.

In particular, determining K in assemblies of nanoparticles is not straightforward. Perhaps the simplest approaches are the ones based on magnetization measurements, like from the law of approach to saturation [22] or using simple relations between the anisotropy field or the blocking temperature with K [23, 24]. In addition, direct or indirect information about K can also be obtained from other measurements, like ferromagnetic resonance [25, 26], torque magnetometry [27], small angle neutron scattering [28, 29], Mössbauer [30], ac‐susceptibility [31, 32], or nuclear magnetic resonance [33, 34]. Nevertheless, the magnetic behavior of nanoparticles is usually determined by diverse extrinsic magnetic factors [8], like dipolar interactions [35, 36], surface effects [37, 38], size effects [39, 40, 41], exchange coupling [42, 43], or shape anisotropy [44, 45, 46]. Thus, most of the approaches to obtain K of nanoparticles only provide an effective anisotropy, K_eff, where all the different contributions to the anisotropy are convoluted into a single parameter.

In order to tune and optimize the performance of magnetic nanoparticles it is critical to comprehend the intrinsic properties of the material, like the magnetocrystalline anisotropy. It is important to highlight that the magnetocrystalline anisotropy is related to the type of ions comprising the magnetic material and their position in the crystal lattice. Consequently, this is a unique characteristic of each magnetic material [5]. Unfortunately, in most of the experimental approaches it is not possible to separate between the intrinsic and extrinsic contributions. Thus, a reliable approach for estimating the intrinsic magnetocrystalline anisotropy is needed.

This study presents a novel approach to qualitatively evaluate the magnetocrystalline anisotropy in nanoparticles using polarized neutron powder diffraction (PNPD). The results show that PNPD suitably accounts for the differences in magnetocrystalline anisotropy between samples and at different temperatures.

2. Results and Discussion

Two types of Co_xFe_3−xO₄ nanoparticles with different Co content [Co_0.61Fe_2.39O₄ and Co_0.11Fe_2.89O₄, as determined by ICP (inductively coupled plasma); Table S1] were prepared using a thermal decomposition process (see Experimental Section). These two samples are labeled as H‐Co and L‐Co, respectively.

The nanoparticles displayed very similar particle sizes, as determined from vertex‐to‐vertex distances in transmission electron microscopy images (Figure 1a,b), with values of 33(6) and 31(9) nm for H‐Co and L‐Co, respectively, and a polydispersity of approximately 20%. Their shapes range from cubic to cuboctahedral (Figure 1c,d) as expected for large particle sizes [9]. Note that energy‐dispersive X‐ray spectroscopy mapping on the same samples has shown a homogeneous distribution of the Co and Fe ions, ruling out any possible core/shell structure [47]. In addition, the particle size and shape were further quantified by calculating the circular equivalent diameter (D_CIRC) and the circularity parameter from the projected particle area [47, 48]. This approach reduces the overestimation associated with vertex‐to‐vertex size measurements and allows a more robust comparison of particle morphology across the series. The resulting D_CIRC values are 28(4) nm and 26(6) nm for H‐Co and L‐Co, respectively (Figure S1). The circularity values remain nearly constant across all compositions, with average values around 0.75, confirming the morphological uniformity of the nanoparticle series and their faceted, non‐spherical geometry. Moreover, X‐ray diffraction analysis confirms the expected cubic spinel structure ($Fd\overline{3}m$) in both samples, with crystallite sizes of 19.5(6) nm and 18.0(2) nm for the H‐Co and L‐Co samples, respectively (Figure 1d). Although these values are slightly lower than the ones obtained by transmission electron microscopy, they remain within the expected range.

FIGURE 1.

Low‐ (a,b) and high‐resolution (c,d) TEM images and corresponding particle size histograms for samples H‐Co (a,c) and L‐Co (b,d), respectively. Scale bars: 200 nm (low‐resolution) and 20 nm (high‐resolution). (e) XRD patterns for H‐Co and L‐Co showing the experimental data (red symbols), calculated profiles (black line), and their difference (blue line). The vertical markers indicate the expected Bragg reflection positions. The corresponding χ²‐residuals for H‐Co and L‐Co patterns are 1.81 and 11.2 respectively.

From the magnetic viewpoint, at low temperatures the samples present a high field magnetization (M_5T) in the range expected for Fe₃O₄‐CoFe₂O₄ nanoparticles of this size [49, 50, 51], with M_5T for H‐Co (89(2) Am²/kg) being slightly larger than that of L‐Co (82(2) Am²/kg) (Figure 2a and Table 1). The difference in M_5T arises not only from replacing different amounts of Fe²⁺ cations with Co²⁺, but probably also due to the presence of oxygen vacancies in the structure [47]. In addition, both samples present a moderately high coercivity, μ₀H_C = 1.23(3) and 0.91(3) T for H‐Co and L‐Co, respectively. The difference between the μ₀H_C values of the two samples is expected from the dissimilar Co content [52]. Note that no exchange bias was observed after field cooling, indicating magnetic homogeneity and negligible surface‐related effects [53]. Finally, the squareness of the hysteresis loops (R = M_R/M_S, where M_R and M_S are the remanent and saturation magnetizations; Table 1) is within the expected values for systems with cubic anisotropy (R ∼ 0.83‐0.87) [54]. Importantly, this hysteretic behavior persists up to room temperature, demonstrating that both samples remain in the blocked state (Figure 2b,c), i.e., below the superparamagnetic state, where the thermal energy induces fluctuations of the magnetic moments, leading to a vanishing remanent magnetization [24]. Indeed, the estimated blocking temperatures (T_B), obtained from fitting the temperature dependence of H_C with the Kneller model (Figure 2d, see Experimental), are 370(20) K and 320(20) K for H‐Co and L‐Co, respectively (Table 1). These values are consistent with a higher magnetic anisotropy of the H‐Co sample, since T_B is proportional to KV, and the particle volume (V) is nearly identical for both samples. Moreover, the fitted Kneller exponents, n = 0.87(1) (H‐Co) and 0.70(1) (L‐Co), further support the predominance of cubic anisotropy in both systems [9, 55, 56, 57].

FIGURE 2.

(a) Magnetization versus applied field (µ₀H) hysteresis loops at 5 K for H‐Co and L‐Co. Hysteresis loops recorded as a function of temperature (5–300 K) for (b) H‐Co and (c) L‐Co. (d) Temperature dependence of the coercive field (µ₀H_C) for the H‐Co and L‐Co samples, where the symbols are the experimental data and the solid lines correspond to the Kneller fits.

TABLE 1.

Summary of the data obtained from magnetic measurements at low temperature (5 K) and their analysis. M_5T refers the magnetization at 5T, μ₀H_C is the coercive field, R is the reduced remanence, T_B is the blocking temperature, n the Kneller exponent and K_eff,SW and K_eff,LAS the effective anisotropy constants calculated from Stoner‐Wohlfarth and the law of approach to saturation, respectively.

Sample	5 K			Kneller law		Keff,SW (105 J/m3)	Keff,LAS (105 J/m3)
	M5T (Am2/kg)	μ0HC (T)	R	TB (K)	n
H‐Co	89(2)	1.23(3)	0.83(2)	370(20)	0.87(1)	1.82(6)	2.0(2)
L‐Co	82(2)	0.91(3)	0.80(2)	320(20)	0.70(1)	1.28(5)	1.21(3)

To evaluate K_eff for each sample, two different approaches were used, the law of approach to saturation (LAS), K_eff,LAS, and the Stoner‐Wohlfarth model for cubic magnetocrystalline anisotropy, K_eff,SW (see Experimental) [58]. Both approaches yield similar K_eff (1.82(6)–2.0(2) × 10⁵ J/m³ for H‐Co and 1.28(5)–1.21(3) × 10⁵ J/m³ for L‐Co; Table 1), consistent with Co_xFe_3−xO₄ nanoparticles of similar composition [9, 21, 59, 60, 61]. This indicates that the magnetic properties are predominantly governed by intrinsic effects rather than extrinsic factors such as finite size or surface contributions, as expected for high anisotropy particles of this size [62, 63, 64, 65]. Additionally, the analyses showed that K_eff(H‐Co) is 40%–60% (depending on the model) larger than K_eff(L‐Co), as expected from the higher Co content [52].

To confirm the Co²⁺ ions as the origin of the observed differences in K_eff, an X‐ray absorption spectroscopy (XAS) and X‐ray magnetic circular dichroism (XMCD) study was carried out (Figure 3a,b). Note that the unique electronic structure of the Co²⁺ ions, leading to large orbital moments, and its strong spin‐orbit coupling, usually results in a strong anisotropy when this ion is present in the crystal structure, particularly for spinel structures [52]. The Fe‐edge spectra corroborate the presence of iron in different oxidation states, with Fe atoms occupying octahedral (O_h) sites as Fe²⁺ and Fe³⁺, and tetrahedral (T_d) sites as Fe³⁺. The interpretation of the cobalt spectra is more challenging, but the negative intensity of the main peak in the XMCD spectra suggests the predominance of Co²⁺ cations in O_h positions. To quantify the cation distribution, the experimental XAS and XMCD L₃‐edge spectra were fitted using a linear combination of ligand‐field multiplet calculations for each site, assuming that the entire contribution at the Co‐edge can be attributed to Co²⁺ in O_h sites (Figure S2a–d). Although other oxidation states or site occupations could, in principle, be present, their contributions usually fall within the fitting error. Based on this assumption, the calculated compositions for the samples were (Fe³⁺ _0.88(5))[Fe²⁺ _0.48(3)Fe³⁺ _1.03(3)Co²⁺ _0.61(3)]O_3.91(7) and (Fe³⁺ _0.91(8))[Fe²⁺ _0.79(4)Fe³⁺ _1.19(4)Co²⁺ _0.11(4)]O_4.00(8) for H‐Co and L‐Co, respectively (Table S2; where () and [] indicate tetrahedral and octahedral sites, respectively). From this analysis, it can be observed that, to maintain charge neutrality, the sample with higher cobalt content must exhibit a reduction in the number of O²⁻ anions, whereas sample L‐Co does not show such oxygen vacancies. This electroneutrality could also be stabilized by partial oxidation of Co²⁺ to Co³⁺. However, previous studies have demonstrated that the most plausible hypothesis is the formation of oxygen vacancies [47].

FIGURE 3.

XAS and XMCD spectra at the Fe and Co L_2,3 edges for samples H‐Co and L‐Co, recorded at 15 K under an applied magnetic field of 5 T.

From the sum rules analysis, we extracted the separate contributions for the spin (m_S) and orbital (m_L) average magnetic moments for each edge and calculated the total magnetic moment per formula unit (m_T). It can be observed that the m_T, although it differs slightly from the M_5T value obtained by magnetometry (as it is commonly observed in this type of analysis for non‐metallic magnetic materials) [66], shows a similar trend than M_5T, with a higher moment for sample H‐Co. On the other hand, we observed that m_L for the Fe‐edge is practically negligible compared to that calculated for Co, confirming that cobalt cations are primarily responsible for the large anisotropy observed in these nanoparticles. Interestingly, m_L for both samples is virtually identical. Thus, the higher anisotropy of the H‐Co sample must arise from the larger amount of Co²⁺ in the structure. It is worth noting that although anisotropy generally increases with cobalt content within a certain range, the dependence is not strictly linear, and it may deviate for larger nanoparticles [47]. Finally, in order to gain further insight into the origin of the differences in magnetic anisotropy between the two samples and to estimate the magnetocrystalline anisotropy of both samples, PNPD measurements were carried out, where the spin‐dependent scattering cross‐sections are measured using an incident neutron beam polarized either parallel or antiparallel to the applied magnetic field.

First, the compositions obtained from PNPD, using the Co/Fe ratios determined by ICP, showed a clear discrepancy with those derived from the XAS‐XMCD analysis (Figure 4; Figure S3), which suggested a higher fraction of cobalt atoms occupying tetrahedral sites. Notably, in the PNPD refinements, the occupancies of Co in the T_d and O_h sites were refined independently. The results indicate a comparable distribution of Co between both sublattices, without evidence of a strong site preference within the experimental uncertainty. The PNPD refinements, assuming no oxygen vacancies, yielded (Co_0.20(5)Fe_0.80(5))[Co_0.40(5)Fe_1.60(5)]O₄ and (Co_0.05(1)Fe_0.95(1))[Co_0.06(1)Fe_1.94(1)]O₄ for H‐Co and L‐Co, respectively (cell parameters depicted in Table S3). Note that unpolarized neutron powder diffraction is unable to resolve such differences in the Co composition due to the smaller scattering length of Co (0.2490 fm) compared to Fe (0.9450 fm). In contrast, polarized neutrons overcome this limitation, even when only low‐resolution data are available, since high‐resolution data are generally unattainable for nanoparticles due to peak broadening caused by size effects. The difference patterns (spin‐up minus spin‐down intensities) calculated for two models, with and without Co in the T_d sites, are markedly different, whereas the corresponding sum patterns (spin‐up plus spin‐down intensities) are nearly indistinguishable (Figure 4; Figures S3 and S4). This distinction arises because the difference pattern is free from parasitic peaks originating from the cryostat or possible non‐magnetic impurities and is dominated by magnetic scattering, which is primarily determined by the magnetic moments of Co²⁺ (3 μ_B) and Fe³⁺ (5 μ_B). Therefore, considering the input provided by the PNPD analysis, the XAS‐XMCD spectra were re‐examined assuming the presence of Co²⁺ also in the T_d sites. The inclusion of this cation moderately improved the agreement between experimental and simulated spectra, as reflected by a reduction of the normalized residual (Res; see Experimental) from 0.42 to 0.31, and, without altering the oxygen vacancy scenario previously identified. The analysis yielded compositions that are much closer to those obtained from PNPD (Figure S2e,f). The resulting compositions from XAS‐XMCD are (Fe³⁺ _0.88(5)Co²⁺ _0.18(5))[Fe²⁺ _0.48(3)Fe³⁺ _1.03(3)Co²⁺ _0.43(3)]O_3.91(7) and (Fe³⁺ _0.91(8)Co²⁺ _0.03(8))[Fe²⁺ _0.79(4)Fe³⁺ _1.19(4)Co²⁺ _0.08(4)]O_4.00(8) for H‐Co and L‐Co, respectively (Table S1).

FIGURE 4.

PNPD difference patterns (spin‐up minus spin‐down) for H‐Co (2 and 300 K, 2 T) and L‐Co (2 K, 2 T), showing the experimental data (red symbols), calculated profiles (black line), and their residuals (blue line). The vertical markers indicate the expected Bragg reflection positions. The corresponding χ²‐residuals for H‐Co and L‐Co patterns are 1.58 (sample H‐Co, 2 K); 1.92; (sample H‐Co, 300 K) and 1.51 (sample L‐Co, 2 K), respectively.

As a diffraction‐based method, PNPD yields insights into the intrinsic magnetocrystalline anisotropy, unaffected by extrinsic factors such as particle interactions or surface effects. When analyzed within the local susceptibility framework [67], this technique also provides site‐resolved magnetic information through the determination of local susceptibility tensors [68]. These tensors reveal how each crystallographic site contributes to the overall magnetic anisotropy of the system and are typically visualized as ellipsoids, where the longest and shortest axes correspond to the easy and hard local anisotropy directions, respectively.

When the PNPD data for both samples, measured at 2 K under a magnetic field of 2 T, were initially refined assuming the cubic symmetry ($Fd\overline{3}m$) expected for cobalt‐doped ferrites, the model failed to reproduce the experimentally observed anisotropy. For the cubic space group, symmetry constraints forbid any local distortions in the T_d sites, while they are allowed in the O_h sites. Remarkably, our analysis shows that the refined off‐diagonal terms (which contain the information about anisotropy K) of the local magnetic susceptibility tensor for O_h sites are zero within experimental error, indicating an absence of detectable local anisotropy. This result, however, appears inconsistent with the magnetometry and XMCD data and reports in the literature [20, 47, 52]. To address this discrepancy, we reduced the symmetry in our refinement to the tetragonal space group I4₁/amd [69, 70], the maximal subgroup of the cubic space groups. Although refinements were performed with both $Fd\overline{3}m$ and I4₁/amd yielded nearly identical goodness‐of‐fit, the reduced symmetry is the only approach that allows for a deviation from the spherical shape in the local magnetic susceptibility of the O_h sites, i.e., the existence of anisotropy. Note that the refinement using the tetragonal lattice shows the ratio c/ $\sqrt{2·}$ a = 0.997(2), i.e., the cubic and tetragonal lattices cannot be distinguished within our experimental accuracy.

As shown in Figure 5, the I4₁/amd space group produces a clear asymmetry of the ellipsoids at the O_h positions, which is also reflected in the non‐zero off‐diagonal elements of the susceptibility tensor (χ), confirming the presence of magnetic anisotropy. This observation demonstrates that the octahedral coordination plays the dominant role in governing the anisotropy of the samples. Combined with the sum rules analysis, which revealed that Co is the element with a non‐quenched orbital moment, these findings indicate that Co²⁺ cations at O_h sites are primarily responsible for the significant enhancement of magnetic anisotropy observed in these cobalt‐doped ferrites.

$$\begin{array}{c}\hfill \chi \left(\mathrm{H}-\mathrm{Co},2\mathrm{K}\right)=\left(\begin{array}{ccc}4.35(7)& 0& 0\\ 0& 4.35(7)& -1.21(4)\\ 0& -1.21(4)& 4.60\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=3.26;\mathrm{B}=4.35;\mathrm{C}=5.69\end{array}$$

$$\begin{array}{c}\hfill \chi \left(\mathrm{L}-\mathrm{Co},2\mathrm{K}\right)=\left(\begin{array}{ccc}4.04\left(4\right)& 0& 0\\ 0& 4.04\left(4\right)& -0.34\left(2\right)\\ 0& -0.34\left(2\right)& 4.04\left(4\right)\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=3.93;\mathrm{B}=4.04;\mathrm{C}=5.11\end{array}$$

FIGURE 5.

Magnetization ellipsoids of the H‐Co samples (2 K and 300 K, 2 T) and the L‐Co sample (2 K, 2 T) as viewed from different directions. The O_h sites are shown in red and the T_d sites in green. The arrows indicate the directions of the crystallographic axes.

Moreover, it can be observed that the ellipsoids for H‐Co appeared more elongated than for L‐Co, which is consistent with the larger χ₂₃ component of the susceptibility tensor for H‐Co compared to L‐Co. This observation indicates that, at 2 K, the magnetic anisotropy of H‐Co is higher than that of L‐Co. However, to compare between different conditions or different samples, it is more convenient to use the ellipticity factor of the ellipsoid. For simplicity, we used the ratio of the long axis to the shortest axis of the ellipsoid, as e = C/A. Here, the values A, B, and C are the eigenvalues of the susceptibility tensor. The comparison between the parameter e of the two samples yields e(H‐Co, 2 K)/e(L‐Co, 2 K) = 1.74/1.3 = 1.34, which is consistent with the K_eff values obtained from SW, where K_eff decreases from 1.82 × 10⁵ J/m³ (H‐Co) to 1.28 × 10⁵ J/m³ (L‐Co), i.e., a ratio of anisotropies of 1.42.

Given the clear differences observed between H‐Co and L‐Co at low temperature, additional PNPD measurements were conducted for H‐Co at 300 K under a 2 T field to evaluate how temperature affects its magnetic anisotropy (Figure 4). The high temperature susceptibility tensor also shows non‐zero off‐diagonal elements, confirming the presence of anisotropy (Figure 5). However, in this case, χ₂₃(H‐Co, 300 K) is smaller than χ₂₃(H‐Co, 2 K), indicating a weaker anisotropy, as expected. When comparing the ellipsoid eigenvalues for the two temperatures, e(H‐Co, 2 K)/e(H‐Co, 300 K), a ratio of 1.18 is obtained, suggesting that the anisotropy at 300 K is somewhat smaller than at 2 K. Notably, in this case, the ratio of effective anisotropies is K_eff(2 K)/K_eff(300 K) = 1.82(6) × 10⁵/1.4(4) × 10⁴ = 13, indicating a much stronger reduction of the effective anisotropy with temperature compared to the e‐ratio. This indicates that the K_eff values obtained from magnetometry were strongly affected by superparamagnetic effects due to the high characteristic measuring time, τ_m [24], of magnetometry compared to neutron diffraction, and the proximity of 300 K to T_B.

$$\begin{array}{c}\hfill \chi \left(\mathrm{H}-\mathrm{Co},300\mathrm{K}\right)=\left(\begin{array}{ccc}4.85\left(7\right)& 0& 0\\ 0& 4.85\left(7\right)& -0.69\left(4\right)\\ 0& -0.69\left(4\right)& 4.60\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=3.86;\mathrm{B}=4.95;\mathrm{C}=5.69\end{array}$$

Interestingly, whereas the directions of the ellipsoids corresponding to the O_h sites for the L‐Co and H‐Co samples are different, the ones for the low temperature and room temperature for the H‐Co nanoparticles are the same (Figure 5). This indicates that the L‐Co and H‐Co samples have different magnetocrystalline easy axes, while there is no reorientation with temperature in the H‐Co sample, suggesting that the easy axis is sensitive to cobalt content and reflects a complex anisotropy landscape in these non‐stoichiometric ferrites. This is consistent with the change in the sign of the cubic anisotropy constant when moving from Fe₃O₄ to CoFe₂O₄. In addition, in non‐stoichiometric Co‐ferrite, it has been shown that the relative strength of the K₁ and K₂ components of the cubic anisotropy depends strongly on the Co content [71, 72], further complicating the discussion of the easy axis evolution with Co content.

To evaluate the sensitivity of the local susceptibility ellipsoids to changes in anisotropy, we performed preliminary PNPD experiments on 30 nm Fe₃O₄ nanocubes [49] and 10 nm Mn₃O₄ nanoparticles [73] (see Supporting Information and Figure S5). Note that for Fe₃O₄ nanoparticles with similar dimensions to those studied here, the magnetic anisotropy is expected to be significantly smaller than in Co‐doped ferrites [52, 74], while Mn₃O₄ is characterized by a much higher anisotropy than CoFe₂O₄ [75, 76]. As can be seen in the susceptibility tensor and the ellipsoids of the Fe₃O₄ nanocubes (Figure S5a), the off‐diagonal elements are virtually zero and the ellipsoids are almost spherical, i.e., e ∼ 1, thus confirming the much smaller anisotropy of this type of particles. In contrast, for Mn₃O₄, the magnetic structure and, thus, the susceptibility tensor and the ellipsoids are more complex, where both T_h and O_h sites exhibit anisotropy [77, 78, 79]. Indeed, the e value of the O_h sites, e = 2.37, is considerably larger than for the cobalt‐doped ferrite nanoparticles, in concordance with the expected larger anisotropy of Mn₃O₄.

$$\begin{array}{c}\hfill \chi \left(\mathrm{F}{\mathrm{e}}_3{\mathrm{O}}_4,150\mathrm{K},5\mathrm{T}\right)=\left(\begin{array}{ccc}3.79\left(2\right)& 0& 0\\ 0& 3.79\left(2\right)& 0.05\left(2\right)\\ 0& 0.05\left(2\right)& 3.79\left(2\right)\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=3.73;\mathrm{B}=3.73;\mathrm{C}=3.89\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{Td}\left(\mathrm{M}{\mathrm{n}}_3{\mathrm{O}}_4,5\mathrm{K},3\mathrm{T}\right)=\left(\begin{array}{ccc}4.66\left(8\right)& 0& 0\\ 0& 4.66\left(8\right)& 0\\ 0& 0& 2.2\left(1\right)\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=2.23;\mathrm{B}=4.66;\mathrm{C}=4.66\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{Oh}\left(\mathrm{M}{\mathrm{n}}_3{\mathrm{O}}_4,5\mathrm{K},3\mathrm{T}\right)=\left(\begin{array}{ccc}-1.57\left(8\right)& 0& 0\\ 0& -1.75\left(7\right)& 2.59\left(2\right)\\ 0& 2.59\left(2\right)& -0.31\left(7\right)\end{array}\right)\\ \hfill \mathrm{Tensor}\mathrm{eigenvalues}\mathrm{A}=-1.57;\mathrm{B}=1.66;\mathrm{C}=-3.72\end{array}$$

However, note that although relating the e parameters of the magnetization ellipsoids of similar samples measured in the same conditions should give a good qualitative comparison between their anisotropies, comparing e of different materials should be taken as indicative, particularly if the type of anisotropy is different.

These results demonstrate that PNPD is a powerful tool to investigate the intrinsic magnetic properties of ferrimagnetic nanoparticles, even in the presence of extrinsic effects, like interactions or surface effects. This is crucial to qualitatively evaluate the magnetocrystalline anisotropy, allowing direct correlation between different materials. This type of comparison will allow to optimize the design of diverse types of materials, ranging from rare‐earth‐free permanent magnets to materials with THz resonances.

3. Conclusions

This study demonstrates that PNPD enables a qualitative and comparative estimation of magnetocrystalline anisotropy in nanoparticles, independent of extrinsic contributions such as surface or dipolar effects. Through the analysis of the susceptibility tensors and local magnetic ellipsoids, we found that Co_0.61Fe_2.39O₄ nanoparticles exhibit an anisotropy approximately 1.3 times higher than Co_0.11Fe_2.89O₄ (i.e., K(Co_0.61Fe_2.39O₄)/K(Co_0.11Fe_2.89O₄) = 1.3), in full agreement with magnetic measurements. Furthermore, PNPD accurately reproduces the temperature evolution of the anisotropy, evidencing its high sensitivity to intrinsic magnetic properties. Altogether, these results highlight the potential of PNPD as a powerful tool to investigate and compare magnetic anisotropy across nanoparticle systems, offering valuable guidance for the development of optimized materials for technological applications.

4. Experimental Section

4.1. Chemicals Used

Cobalt (II) chloride (CoCl₂, ≥ 99%), iron(III) acetylacetonate (Fe(acac)₃, 99.99%), manganese (II) chloride (MnCl₂, ≥ 97%), iron (III) chloride (FeCl₃, ≥ 97%), sodium oleate (NaOl, > 97.0%), oleic acid (OA, 90%), oleylamine (ONH₂, 70%), benzyl ether (Bz₂O, 98%), 1‐octadecene (ODE, 90%), 1‐decanol (Dol, ≥ 98%) and hexane (Hx, > 95%) were purchased from Merck KGaA (Sigma–Aldrich), except sodium oleate, which was acquired from TCI America, and ethanol, which was purchased from Panreac S.A. The solvents were used as received without further purification.

4.2. Synthesis of Cobalt‐Doped Ferrites

The cobalt‐doped ferrite nanoparticles were synthesized following a slightly modified procedure previously reported for the synthesis of iron‐based cubic spinels (MFe₂O₄) through thermal decomposition of organometallic compounds in high‐boiling solvents containing OA and ONH₂ as stabilizing surfactants [9, 47]. In a typical synthesis, 12 mmol of metal precursors (Fe(acac)₃ and CoCl₂) were dissolved in a solution containing 1.8 mmol of OA, 2.4 mmol of ONH₂, and 20 mL of Bz₂O in a 100 mL three‐neck round‐bottomed flask. Initially, the mixture was degassed by bubbling N₂ at 100°C for 20 min, and then it was heated up to the desired decomposition temperature (270°C), setting a heating rate (1°C/min) and nucleation and growth time of 15 min to control the final particle size. During the heating and growth processes, the mixture was exposed to a N₂ flow. Finally, the flask was removed from the heating mantle and was allowed to cool under an inert atmosphere. All nanoparticles were washed by several cycles of ethanol‐induced sedimentation, centrifugation at 5000 rpm, disposal of supernatant solution, and redispersion in chloroform.

Two different samples were synthesized by varying the Co/Fe precursor ratio. The sample labeled L‐Co (low cobalt content) was prepared with a Co/Fe ratio of 0.07, while the sample labeled H‐Co (high cobalt content) was obtained with a Co/Fe ratio of 1. To better control cobalt incorporation and approach the desired stoichiometry, the total precursor concentration in H‐Co was reduced to 6 mmol, compared to the commonly used 12 mmol [9]. Due to the lower compatibility of the CoCl₂ precursor with the nonpolar reaction medium, the incorporation efficiency of Co ions into the spinel lattice is lower than that of Fe ions [9, 47]. Therefore, the actual composition of the nanoparticles was determined by ICP analysis.

4.3. Synthesis of Fe₃O₄ and Mn₃O₄ Nanoparticles

The Mn₃O₄ nanoparticles were synthesized by thermal decomposition of Mn(II) oleate (synthesized by refluxing Mn(II) chloride and sodium oleate in a mixture of water, hexane, and ethanol) in ODE in the presence of Dol at high temperatures under Ar flow [80].

Similarly, the Fe₃O₄ nanocubes were synthesized by thermal decomposition of Fe(III) oleate (synthesized by refluxing Fe(III) chloride and sodium oleate in a mixture of water, hexane, and ethanol) in a mixture of OA and ODE [81].

4.4. Structural, Compositional, and Morphological Characterization

The structural characterization of the nanoparticles was performed by X‐ray powder diffraction (XRD) using a Bruker D8 ADVANCE ECO diffractometer equipped with Cu K_α radiation. The obtained patterns were analyzed by the Rietveld refinement using the FullProf package [82].

The CIF structural models were taken from the Crystallography Open Database (COD 5910063) for CoFe₂O₄.

The cell parameters obtained from XRD are shown in Table S3.

Note that from the XRD fits rather small microstrains, in the order of ε ≈ 6 × 10⁻⁴, were obtained for both samples.

The relative concentrations of cobalt and iron in the nanoparticles were measured by inductively coupled plasma atomic emission spectroscopy (ICP‐OES) using an Agilent Vista‐MPX instrument. The analysis was performed using 238.204 and 228.616 nm emission lines for Fe and Co, respectively. Calibration curves were constructed using multi‐element standard solutions with concentrations ranging from 0.5 to 50 mg/L, ensuring high linearity. Approximately 1.5 mg of each sample was dissolved in 0.6 mL of freshly prepared aqua regia (1:3 nitric acid to hydrochloric acid ratio). The resulting solution was diluted to 25 mL with Milli‐Q water and subsequently used for ICP analysis.

The morphology and size distribution of the nanoparticles were analyzed by transmission electron microscopy (TEM) using a TALOS F200X G2 microscope (Thermo‐Fisher Scientific). This microscope is equipped with a high‐brightness field emission gun (X‐FEG, 80–200 keV) and operated at an accelerating voltage of 200 kV, employing a high‐speed CETA camera with a resolution of 4096 × 4096 pixels. For each sample, the mean size and particle size distribution were obtained from a statistical analysis of over 200 nanoparticles and fitted to a log‐normal distribution function. Two complementary approaches were used to characterize particle dimensions: (i) the direct measurement of the longest vertex‐to‐vertex distances (D_TEM) and (ii) the calculation of the circular equivalent diameter (D_CIRC) derived from the projected particle area [47, 48]. Additionally, the circularity parameter was determined for each nanoparticle to evaluate morphological uniformity and faceting. Only particles with clear boundaries were considered to ensure accuracy in the perimeter and area measurements.

X‐ray absorption spectroscopy (XAS) and X‐ray magnetic circular dichroism (XMCD) measurements were carried out at the I10‐1 beamline of the Diamond Light Source on dried nanoparticle samples deposited on carbon tape. Spectra at the Fe and Co L_2,3 edges were recorded in total electron yield mode at 15 K and 5 T. The XMCD signals were normalized to the integrated area of the corresponding XAS spectra after linear background subtraction. The XAS and XMCD spectra were simulated using ligand field multiplet theory implemented in the CTM4XAS 5.5 program [83]. Slater integrals F(dd), F(pd), and G(pd) were reduced to 80%. The crystal field splitting parameters were set to 1.5 eV for Fe³⁺‐O_h, 1.1 eV for Fe²⁺‐O_h, −0.6 eV for Fe³⁺‐T_d, and 1.0 eV for Co²⁺‐O_h and 0.3 eV for Co²⁺‐T_d. To assess the agreement between experimental and simulated spectra, the residual difference between both curves (Res) was evaluated by integrating the absolute difference over the full energy range and normalizing it to the total area of the experimental spectrum.

$$Res=\frac{\int I_{exp}\left(E\right)-I_{calc}\left(E\right)dE}{\int I_{exp}\left(E\right)dE},$$

where I_exp(E) and I_calc(E) correspond to the experimental and simulated spectra, respectively. Element‐specific effective spin (m_S,eff) and orbital (m_L) magnetic moments were extracted from the XMCD data by a sum‐rule‐based analysis according to:

$$m_S^{eff}=-n_h\left(\frac{6p-4q}{r}\right)$$

$$m_L=-\frac{4}{3}{n}_h\left(\frac{q}{r}\right),$$

where n_h is the effective number of 3d‐band holes, assumed to be 4.3 for Fe²⁺/Fe³⁺ and 3 for Co²⁺. The parameters p and q represent the integrals of the XMCD signal over the L₃ and L_2,3 absorption edges, respectively, while r is the integral of the XAS signal over the L_2,3 edges.

4.5. Magnetometric Characterization

The magnetic properties of the nanoparticles were measured on tightly and randomly packed powder samples using a 9 T Quantum Design PPMS magnetometer. The powder was sealed in a small Teflon tape capsule and manually compressed to obtain a compact, randomly oriented pellet. Hysteresis loop measurements were recorded from 5 to 300 K following a 5 T field‐cooling (FC) protocol from 300 K. The uncertainty in the high‐field magnetization values (M_5T) is mainly determined by mass normalization, whereas the uncertainty in the coercive field (H_C) was estimated from the magnetic field step used near M = 0.

Note that no correction to account for the residual organic content of our samples (expected to be around 10% weight) was applied to the magnetometry data [84, 85].

The temperature dependence of coercivity, H_C(T) was analyzed using Kneller's law [86]:

$$H_c\left(T\right)=H_c\left(0\right)\left(1-{\left(\frac{T}{T_B}\right)}^n\right)$$

where H_C(T) is the coercive field at temperature T, T_B is the superparamagnetic blocking temperature, H_C(0) is the coercivity extrapolated at 0 K, and n is an exponent with typical values of 0.5 or 0.77.

The effective anisotropy constant (K_eff) was calculated following the Stoner–Wohlfarth (SW) model [9], assuming a cubic magnetocrystalline anisotropy, and by applying the Law of Approach to Saturation (LAS) [58].

According to the SW model, K_eff is given by:

$$K_{eff,SW}=\frac{H_C\left(0\right)M_S\left(0\right)}{0.64}$$

where H_C(0) and M_S(0) are the coercive field and saturation magnetization at 0 K extracted from the Kneller and Bloch fits [86], respectively.

Additionally, using the LAS, the magnetization approaching saturation (from 1 to 5 T) was fitted with the following relation:

$$M\left(H\right)=M_S\left(1-\frac{a}{H}-\frac{b}{H^2}\right)+\chi H$$

with the parameter b related to K_eff as:

$$b=\frac{8K_{eff,LSA}^2}{105M_S^2}$$

In this equation, M(H) is the magnetization as a function of applied magnetic field H, M_S is the saturation magnetization, and a, b, c, and χ are fitting constants.

The fits of the Kneller, Block and LAS laws were carried out using the non‐linear fit option of OriginPro and the errors of the different obtained parameters are the standard deviations obtained from the fits.

4.6. Polarized Neutron Powder Diffraction

The PNPD experiments were performed at the D1B diffractometer (λ = 2.524 Å) of the Institut Laue–Langevin (ILL, Grenoble, France) equipped with a 128°‐range one‐dimensional detector, and at the 6T2 diffractometer (λ = 2.4 Å) of the Laboratoire Léon Brillouin (LLB, Saclay, France) equipped with a two‐dimensional detector. At D1B, the beam polarization was achieved with a ³He spin filter [87, 88], whereas at 6T2 a pyrolytic graphite monochromator combined with a supermirror bender was employed. Measurements for the H‐Co sample were conducted at 2 and 300 K under a magnetic field of 2 T, and for the L‐Co sample at 2 K and 2 T, both using the one‐dimensional detector. The Mn₃O₄ sample was also measured with the one‐dimensional detector at 5 K and 3 T. In contrast, the Fe₃O₄ sample was measured above the Verwey transition, at 150 K and 5 T, using the two‐dimensional detector. To avoid any rotation of the nanoparticles in the applied field, the powder samples were tightly compacted. Note that the refinements did not show any preferred orientation.

The polarized neutrons patterns were fitted using the software package CrysPy [89], specially developed for polarized neutron diffraction data analysis [67, 90]. The definition of different parameters refined by CrysPy are given in Ref [89]. Note that in PNPD experiments the sum and the difference of the spin‐up and spin‐down patterns are refined simultaneously. It is important to highlight that the difference pattern is free from the contribution of any parasitic reflections (e.g., due to the cryostat). The occupancies of Co and Fe in the tetrahedral (T_d) and octahedral (O_h) sites were refined independently, allowing the cation distribution to be determined without imposing a fixed site preference. However, due to the limited statistical sensitivity of the diffraction data to oxygen, the oxygen stoichiometry was fixed to the nominal value (Fe,Co)₃O₄. Attempts to refine the oxygen content resulted in large uncertainties and strong parameter correlations, preventing a reliable determination.

To reduce the fitting parameters, microstrains were not taken into account in the fits of the PNPD data.

In the refinements, a Néel‐type ferrimagnetic configuration was used, where the spins of Co and Fe in the T_d sites are aligned parallel, as in magnetite. This configuration contrasts with the magnetic structure of pure Co₃O₄, in which the T_d sublattice becomes antiferromagnetic below approximately 40 K, producing a characteristic (200) reflection [91]. In our case, only a very weak diffuse feature is observed near 35°, likely indicating residual antiferromagnetic correlations at the T_d sites (Figure S3).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: smll73648‐sup‐0001‐SuppMat.pdf.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.