Simulating magnetic nanoparticle behavior in low-field MRI under transverse rotating fields and imposed fluid flow

In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticle's time constant, τ. As the magnetic field frequency is increased, the nanoparticle's magnetic moment lags the applied magnetic field at a constant angle for a given frequency, Ω, in rad s⁻¹. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluid's temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4 and 7 °C above the ambient in vivo temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluid's temperature in the MRI environment which is characterized by a large DC field, B₀. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B₀. Results are presented for the expected temperature increase in small tumors ( radius) over an appropriate range of magnetic fluid concentrations (0.002-0.01 solid volume fraction) and nanoparticle radii (1-10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful the goal of this work is to examine, by means of analysis and simulation, the concept of interactive fluid magnetization using the dynamic behavior of superparamagnetic iron oxide nanoparticle suspensions in the MRI environment. In addition to the usual magnetic fields associated with MRI, a rotating magnetic field is applied transverse to the main B₀ field of the MRI. Additional or modified magnetic fields have been previously proposed for hyperthermia and targeted drug delivery within MRI. Analytical predictions and numerical simulations of the transverse rotating magnetic field in the presence of B₀ are investigated to demonstrate the effect of Ω, the rotating field frequency, and the magnetic field amplitude on the fluid suspension magnetization. The transverse magnetization due to the rotating transverse field shows strong dependence on the characteristic time constant of the fluid suspension, τ. The analysis shows that as the rotating field frequency increases so that Ωτ approaches unity, the transverse fluid magnetization vector is significantly non-aligned with the applied rotating field and the magnetization's magnitude is a strong function of the field frequency. In this frequency range, the fluid's transverse magnetization is controlled by the applied field which is determined by the operator. The phenomenon, which is due to the physical rotation of the magnetic nanoparticles in the suspension, is demonstrated analytically when the nanoparticles are present in high concentrations (1-3% solid volume fractions) more typical of hyperthermia rather than in clinical imaging applications, and in low MRI field strengths (such as open MRI systems), where the magnetic nanoparticles are not magnetically saturated. The effect of imposed Poiseuille flow in a planar channel geometry and changing nanoparticle concentration is examined. The work represents the first known attempt to analyze the dynamic behavior of magnetic nanoparticles in the MRI environment including the effects of the magnetic nanoparticle spin-velocity. It is shown that the magnitude of the transverse magnetization is a strong function of the rotating transverse field frequency. Interactive fluid magnetization effects are predicted due to non-uniform fluid magnetization in planar Poiseuille flow with high nanoparticle concentrations.     

Probing local magnetic cluster development in (CuZr)93−xAl7Gdx bulk metallic glasses by Al NMR

We report magnetization properties of (CuZr)_93−xAl₇Gd_x bulk metallic glasses from temperature dependent ²⁷Al nuclear magnetic resonance spectroscopy and magnetic susceptibility measurements. Significant non-linear line broadening of ²⁷Al spectra commencing at high temperatures is attributed to the development of a local magnetic susceptibility distribution that prevails over a finite temperature range. Magnetization measurements confirm the linewidth enhancement due to strong frustrated magnetic short-range order. This study provides insight into the nature of magnetic development and frustration in paramagnetic systems.     

Scanning Hall probe microscopy of vortex matter

Scanning Hall probe microscopy (SHPM) is a novel scanned probe magnetic imaging technique whereby the stray fields at the surface of a sample are mapped with a sub-micron semiconductor heterostructure Hall probe. In addition an integrated scanning tunnelling microscope (STM) or atomic force microscope (AFM) tip allows the simultaneous measurement of the sample topography, which can then be correlated with magnetic images. SHPM has several advantages over alternative methods; it is almost completely non-invasive, can be used over a very wide range of temperatures (0.3–300 K) and magnetic fields (0–7 T) and yields quantitative maps of the z-component of magnetic induction. The approach is particularly well suited to low temperature imaging of vortices in type II superconductors with very high signal:noise ratios and relatively high spatial resolution (>100 nm). This paper will introduce the design principles of SHPM including the choice of semiconductor heterostructure for different measurement conditions as well as surface tracking and scanning mechanisms. The full potential of the technique will be illustrated with results of vortex imaging studies of three distinct superconducting systems: (i) vortex chains in the “crossing lattices” regime of highly anisotropic cuprate superconductors, (ii) vortex–antivortex pairs spontaneously nucleated in ferromagnetic-superconductor hybrid structures, and (iii) vortices in the exotic p-wave superconductor Sr₂RuO₄ at milliKelvin temperatures.     

Spin reversal and magnetization jumps in ErMexMn1−xO3 perovskites (Me=Ni,Co)

The erbium-based manganite ErMnO₃ has been partially substituted at the manganese site by the transition-metal elements Ni and Co. The perovskite orthorhombic structure is found from x(Ni)=0.2–0.5 in the nickel-based solid solution ErNi_xMn_1−xO₃, while it can be extended up to x(Co)=0.7 in the case of cobalt, provided that the synthesis is performed under oxygenation conditions to favor the presence of Co³⁺. Presence of different magnetic entities (i.e., Er³⁺, Ni²⁺, Co²⁺, Co³⁺, Mn³⁺, and Mn⁴⁺) leads to quite unusual magnetic properties, characterized by the coexistence of antiferromagnetic and ferromagnetic interactions. In ErNi_xMn_1−xO₃, a critical concentration x_crit(Ni)=1/3 separates two regimes: spin-canted AF interactions predominate at x<x_crit, while the ferromagnetic behavior is enhanced for x>x_crit. Spin reversal phenomena are present both in the nickel- and cobalt-based compounds. A phenomenological model based on two interacting sublattices, coupled by an antiferromagnetic exchange interaction, explains the inversion of the overall magnetic moment at low temperatures. In this model, the ferromagnetic transition-metal lattice, which orders at T_c, creates a strong local field at the erbium site, polarizing the Er moments in a direction opposite to the applied field. At low temperatures, when the contribution of the paramagnetic erbium sublattice, which varies as T⁻¹, gets larger than the ferromagnetic contribution, the total magnetic moment changes its sign, leading to an overall ferrimagnetic state. The half-substituted compound ErCo_0.50Mn_0.50O₃ was studied in detail, since the magnetization loops present two well-identified anomalies: an intersection of the magnetization branches at low fields, and magnetization jumps at high fields. The influence of the oxidizing conditions was studied in other compositions close to the 50/50=Mn/Co substitution rate. These anomalies are clearly connected to the spin inversion phenomena and to the simultaneous presence of Co²⁺ and Co³⁺ magnetic moments. Dynamical aspects should be considered to well identify the high-field anomaly, since it depends on the magnetic field sweep rate.     

Heating in the MRI environment due to superparamagnetic fluid suspensions in a rotating magnetic field

In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticle's time constant, τ. As the magnetic field frequency is increased, the nanoparticle's magnetic moment lags the applied magnetic field at a constant angle for a given frequency, Ω, in rad/s. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluid's temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4 and 7 degree Centigrade above the ambient in vivo temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluid's temperature in the MRI environment which is characterized by a large DC field, B₀. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B₀. Results are presented for the expected temperature increase in small tumors (approximately 1 cm radius) over an appropriate range of magnetic fluid concentrations (0.002-0.01 solid volume fraction) and nanoparticle radii (1-10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful selection of the rotating or sinusoidal field parameters (field frequency and amplitude). The work indicates that it may be feasible to combine low-field MRI with a magnetic hyperthermia system using superparamagnetic iron oxide nanoparticles.     

Anisotropy and relaxation processes of uniaxially oriented CoFe2O4 nanoparticles dispersed in PDMS

When a uniaxial magnetic field is applied to a non-magnetic dispersive medium filled with magnetic nanoparticles, they auto-assemble into thin needles parallel to the field direction, due to the strong dipolar interaction among them. We have prepared in this way magnetically oriented nanocomposites of nanometer-size CoFe₂O₄ particles in a polydimethylsiloxane polymer matrix, with 10% w/w of magnetic particles. We present the characteristic magnetic relaxation curves measured after the application of a magnetic field forming an angle α with respect to the needle direction. We show that the magnetic viscosity (calculated from the logarithmic relaxation curves) as a function of α presents a minimum at α=0, indicating slower relaxation processes associated with this configuration of fields. The results seems to point out that the local magnetic anisotropy of the nanoparticles is oriented along the needles, resulting in the macroscopic magnetic anisotropy observed in our measurements.     

Synthesis of goethite from Fe(OH)2 precipitates: Influence of Fe(II) concentration and stirring speed

Magnetic methods are efficient tools in soil and environmental science. But in such natural environments, several magnetic minerals are generally present. So, synthetic standard samples are necessary for calibration of laboratory techniques. The aim of this study was to synthesise goethite free of magnetic impurities (concentration <∼1 μg kg⁻¹) with different crystal sizes. Goethite was prepared by oxidation of aqueous suspensions of Fe(OH)₂ precipitates. Final products were characterised by X-ray diffraction, infrared spectroscopy, scanning and transmission electron microscopy and magnetic methods. Goethite could be obtained in the absence of any trace of strong magnetic minerals using FeSO₄·7H₂O and NaOH as reactants with the following experimental conditions: temperature=45 °C, [FeSO₄·7H₂O]=0.50 mol L⁻¹, [NaOH]=0.20 mol L⁻¹, stirring speed=760 rpm. The Fe(II) concentration and the stirring speed were varied. It proved possible to modify the size of the goethite crystals by varying the Fe(II) concentration and the stirring speed, but important changes of these parameters induced the formation of other phases, lepidocrocite when the oxidation reaction was drastically accelerated and Fe₃O₄ when the reaction was slowed down. In any case, for weak magnetic fields, a low-coercivity magnetic mineral saturating at weak magnetic fields was observed. It may correspond to traces of δ-FeOOH or to domains structurally similar to δ-FeOOH inside the multidomainic crystals of δ-FeOOH.     

Ultrasmall iron oxide nanoparticles: Magnetic and NMR relaxometric properties

Ultrasmall iron oxide (USPIO) nanoparticles, with diameter mostly less than 3 nm dispersed in an organic carrier fluid were synthesized by polyol route. The evolution of ZFC-FC magnetization curves with temperature, as well as the shift of the ac susceptibility peaks upon changing the frequency, reveal that the nanoparticles in the fluid are non-interacting and superparamagnetic with the blocking temperature T_B ∼10 K. The Mössbauer spectra analysis proposed the core/shell structure of the nanoparticles consisting of stoichiometric γ-Fe₂O₃ core and non-stoichiometric shell. The nanoparticle surface layer has a great influence on their properties which is principally manifested in significant reduction of the magnetization and in a large increase in magnetic anisotropy. Magnetic moments do not saturate in fields up to 5 T, even at the lowest measured temperature, T = 5 K. The average magnetic particle diameter is changed from 1.3 to 1.8 nm with increasing magnetic field from 0 to 5 T which is noticeably smaller than the particle sizes measured by TEM. The estimated effective magnetic anisotropy constant value, K_eff = 2 × 10⁵ J/m³, is two orders of magnitude higher than in the bulk maghemite. Measurements of the longitudinal and transverse NMR relaxivity parameters on water diluted nanoparticle dispersions at 1.5 T gave the values r₁ = 0.028 mmol⁻¹ s⁻¹, r₂ = 0.050 mmol⁻¹ s⁻¹ and their ratio r₂/r₁ = 1.8. Continuous increase of the T₁-weighted MRI signal intensity with increasing Fe concentration in the nanoparticle dispersions was observed which makes this ferrofluid to behave as a positive T₁ contrast agent.     

Production of spin polarized 12N through heavy ion reactions

The structural and magnetic properties of Ho substituted BiFeO₃ (BHFO) have been investigated using ⁵⁷Fe Mössbauer spectroscopy and X-Ray diffraction (XRD) as a function of temperature. The Mössbauer spectrum obtained at room temperature for the as-synthesized BHFO sample exhibits broadened features due to the hyperfine field distributions related to the local variation of the neighbourhood of Fe and the magnetic hyperfine splitting patterns are indicative of magnetic ordering, mostly probably screwed or slightly antiferromagnetic. The spectrum was fitted with two superimposed asymmetric sextets, with similar hyperfine magnetic fields of B_hf1 = 48.0(1) T and B_hf2 = 49.0(1) T, corresponding to rhombohedral BFO. The hyperfine fields of the magnetic components decreased systematically with increasing temperature to a ‘field distribution’ just below the Néel temperature, T_N ~ 600 K. At temperatures above 600 K, the spectral line associated with the Bi₂₅FeO₄₀ impurity phase dominates the spectra. This phase is confirmed by XRD measurements. From the temperature dependence of the site populations of the spectral components an average Debye temperature of θ _D = 240(80) K has been estimated.     

Magnetocaloric effect in La0.67Sr0.33MnO3 manganite above room temperature

The La_0.67Sr_0.33MnO₃ composition prepared by sol-gel synthesis was studied by dc magnetization measurements. A large magnetocaloric effect was inferred over a wide range of temperature around the second-order paramagnetic-ferromagnetic transition. The change of magnetic entropy increases monotonically with increasing magnetic field and reaches the value of 5.15 J/kg K at 370 K for Δμ0H=5 T. The corresponding adiabatic temperature change is 3.3 K. The changes in magnetic entropy and the adiabatic temperature are also significant at moderate magnetic fields. The magnetic field induced change of the specific heat varies with temperature and has maximum variation near the paramagnetic-ferromagnetic transition. The obtained results show that La_0.67Sr_0.33MnO₃ could be considered as a potential candidate for magnetic refrigeration applications above room temperature.     

The role of Mg substitution on the microstructure and magnetic properties of Ba Co Zn W-type hexagonal ferrites

A series of W-type hexagonal ferrites with the composition BaCoZn_1−xMg_xFe₁₆O₂₇ (0?x?0.6) were prepared by the conventional ceramic method to study their structural and magnetic properties as a function of temperature and composition. The characterization using X-ray diffraction indicated that a hexagonal W-type single-phase structure and the effect of composition on the unit cell parameters, density and porosity was studied. The variation of the magnetic susceptibility (χ_M) with temperature for all the investigated samples in the temperature range (300–800 K) shows three regions of behavior that was explained on the basis of the distribution of Zn²⁺ and Mg²⁺ ions in the lattice and leads to the anomalous behavior of the effective magnetic moment μ_eff. The Curie temperature indicated that the critical concentration is at x=0.5. Paramagnetic nature of the samples above the Curie temperature is observed. The Curie Weiss constant θ calculated from the plot of 1/χ_M vs. T (K) is in agreement with the expected value. The effective magnetic moment μ_eff decreases with increasing the intensity of magnetic field. The possible mechanisms contributing to these properties are discussed in the text.     

Large magnetocaloric effect with a wide temperature range in Gd5Si4

Gd₅Si₄ magnets have attracted much attention due to their many appealing properties such as strong ferromagnetism, magnetovolume effect, and large reversal magnetocaloric effect (MCE). However, Gd₅Si₄ exhibits a relatively high Curie temperature (T_C ∼336 K) with a narrow refrigeration temperature span, which limits the refrigeration application at room temperature. Here we show that the T_C of Gd₅Si₄ can be reduced to 330 K and the phase transition temperature range can be effectively expanded by applying a high pressure of 6 GPa to the sample during heat treatment. In addition, the room-temperature magnetic entropy changes are improved and the refrigeration temperature span also becomes wider, which leads to an enhanced relative cooling power (RCP) of 748 Jkg^-1 under a magnetic field change of 5 T. These unique features indicate that the Gd₅Si₄ compound prepared under high pressure can serve as a magnetic refrigerant in a wide temperature range covering room temperature.     

Influence of sample geometry on determination of magnetocaloric effect for Gd60Co30Al10 glassy ribbons using direct and indirect methods

Rare-earth based metallic glasses with high saturation magnetization show a sizeable magnetocaloric effect (MCE) and are subject of extensive research concerning magnetic refrigeration materials. In this work, the magnetic phase transition from paramagnetic to ferromagnetic of Gd₆₀Co₃₀Al₁₀ metallic glass has been characterized and three different methods were applied for the determination of its magnetocaloric specific parameters: (a) direct measurement of the adiabatic temperature change by exposing the material to an adiabatically applied magnetic field; (b) determination of the magnetization M(H,T) and calculation of the temperature dependent magnetic field induced entropy change ΔS_m by application of the Maxwell relation and (c) measuring the total heat capacity C_p(H,T) in zero and non-zero magnetic field. Gd₆₀Co₃₀Al₁₀ glassy ribbons were prepared by melt spinning, a technique that offers very high cooling rates due to the low dimensionality of the sample. Depending on the particular method of measurement, pieces of these glassy ribbons form samples with different appropriate total volume and dimensions. We show that the combination of the pronounced two-dimensionality of the ribbon pieces (aspect ratio ∼100) together with the very high magnetic permeability principally can cause strong internal demagnetizing fields that cannot be neglected when evaluating the intrinsic MCE parameters obtained from different methods.     

Temperature dependence of magnetic anisotropy constant in cobalt ferrite nanoparticles

The temperature dependence of the effective magnetic anisotropy constant K(T) of CoFe₂O₄ nanoparticles is obtained based on the SQUID magnetometry measurements and Mössbauer spectroscopy. The variation of the blocking temperature T_B as a function of particle radius r is first determined by associating the particle size distribution and the anisotropy energy barrier distribution deduced from the hysteresis curve and the magnetization decay curve, respectively. Finally, the magnetic anisotropy constant at each temperature is calculated from the relation between r and T_B. The resultant effective magnetic anisotropy constant K(T) decreases markedly with increasing temperature from 1.1×10⁷ J/m³ at 5 K to 0.6×10⁵ J/m³ at 280 K. The attempt time τ₀ is also determined to be 6.1×10⁻¹² s which together with the K(T) best explains the temperature dependence of superparamagnetic fraction in Mössbauer spectra.     

Local spin reversal and associated magnetic responses in Ga-substituted Pb-hexaferrites

We have studied magnetic structure and properties of Ga-substituted Pb-hexaferrites having the stoichiometry of PbFe_12−xGa_xO₁₉ with x=6 (i.e., Fe:Ga=1:1). According to the neutron diffraction results, this compound is characterized by a collinear spin structure below its Curie temperature (∼325 K). Analysis of the neutron diffraction patterns further indicates that the magnetic-moment direction of Fe³⁺ ions located at the octahedral 2a sublattice is downward while that of the unsubstituted PbFe₁₂O₁₉ is upward at room temperature. With decreasing temperature, the Fe³⁺ magnetic moment at the octahedral 2a sublattice undergoes a reorientation to the upward direction while that of the unsubstituted PbFe₁₂O₁₉ remains upward down to 5 K. This selective local spin reversal at the 2a sublattice of PbFe₆Ga₆O₁₉ was attributed to the weakening of the superexchange interaction between the octahedral 2a site and the tetrahedral 4f_IV site upon the preferential substitution of Ga ions for Fe ions at these two neighboring sites. Comparison of the neutron diffraction results with dc magnetization responses and ac susceptibilities further indicates that the paramagnetic–ferrimagnetic transition at ∼325 K (T_c) is followed by the local spin reversal at lower temperatures.     

Temperature dependence of coercivity and metamagnetism in Tb0.5Y0.5Ni

The temperature dependences of magnetic properties of the FeB structure compound Tb_0.5Y_0.5Ni are examined with the aim to elucidate the origin of intrinsic magnetic hardness in this compound. The easy a axis coercivity of an aligned powder sample is found to be strongly temperature dependent. A domain wall activation model is employed to obtain the parameters H₀(0)=30kOe, V_V(0)=0.24K^?1, and β=0.016K^?1, where H₀(0) is the coercive force at absolute zero, V_T(0) the domain wall activation parameter, and β the coefficient for the linear temperature dependence of the reduced anisotropy. From polycrystalline data, we find a strong temperature dependence of the c axis metamagnetic transition only for the ferromagnetic (F) to antiferromagnetic (AF) direction. From the relative constancy of the AF to F c axis metamagnetic transition compound to the a axis coercivity we conclude that the coercive force is not directly related to the local molecular fields as had been previously suggested.     

Phase separation, ferromagnetism and magnetic irreversibility in La1−xSrxMn1−yFeyO3

Magnetic susceptibility, χ(T), is investigated in ceramic La_1−xSr_xMn_1−yFe_yO₃ (LSMFO) samples with x=0.3 and y=0.15−0.25. A ferromagnetic (FM) transition observed in LSMFO is accompanied with an appreciable decrease of the transition temperature with increasing y, which is connected to breaking of the FM double-exchange interaction by doping with Fe. Strong magnetic irreversibility, observed in low (B=10 G) field, gives evidence for frustration of the magnetic state of LSMFO. The FM transition, which is expanded with increasing B, is more pronounced in the samples with y=0.15-0.20 and broadens considerably at y=0.25, where the irreversibility is increased. Well above the transition, χ(T) exhibits a Curie-Weiss asymptotic behavior, yielding very large values of the effective Bohr magneton number per magnetic ion, incompatible with those of Mn or Fe single ions. At y=0.15 and 0.20 a critical behavior of χ⁻¹(T)∼(T/T_C−1)^γ in the region of the FM transition is characterized by influence of two different magnetic systems, a 3D percolative one with γ₌γ_p≈1.8 and T_C=T_C^(p), and a non-percolative 3D Heisenberg spin system, with γ=γ_H≈1.4 and T_C=T_C^(H), where T_C^(p)<T_C^(H). At y=0.25 the percolative contribution to the critical behavior of χ(T) is not observed. The dependence of χ on T and y gives evidence for phase separation, with onset already near the room temperature, leading to generation of nanosize FM particles in the paramagnetic host matrix of LSMFO. The ferromagnetism of LSMFO is attributable to percolation over the system of such particles and generation of large FM clusters, whereas the frustration is governed presumably by a system of smaller weakly-correlated magnetic units, which do not enter the percolative FM clusters.     

Indium doping effect on the magnetic properties of Y-type hexaferrite Ba0.5Sr1.5Zn2(Fe1-xInx)12O22

The effect of indium doping on structural and magnetic properties of Y-type hexaferrite Ba_0.5Sr_1.5Zn₂(Fe_1-xIn_x)₁₂O₂₂ (x = 0, 0.02, 0.04, 0.06, 0.08 and 0.1) prepared by the solid state reaction method was investigated. The Rietveld refinement method was used to analyze the X-ray diffraction patterns. The magnetic transition temperatures associated with the proper-screw spin phase to the collinear ferrimagnetic spin phase transition can be efficiently modulated by varying indium content. The magnetic transition temperature increases to a maximum with indium content x = 0.04 and then decreases with x, suggesting the possibility that electrically controlled magnetization reversal can be can be effectively tailored by varying indium content. The saturation magnetization at room temperature was decreased as increasing indium content, which can be explained as the metal ions occupation. It is worthy to note that the coercivity of In-doped samples was decreased drastically compared that of undoped sample, which is probably resulted from the reduction in anisotropy field with substitution of In³⁺ for Fe³⁺. The In-doped hexaferrite Ba_0.5Sr_1.5Zn₂(Fe_1-xIn_x)₁₂O₂₂ may be potential candidates for application in magnetoelectric devices.     

Mössbauer studies of PbFe2/3W1/3O3 multiferroics

Mössbauer studies of ceramic samples of the antiferromagnetic perovskite PbFe_2/3W_1/3O₃ have been carried out. It has been established that the temperature of transition to the magnetically ordered state is T _N = 365 K. Iron ions in PbFe_2/3W_1/3O₃ are found to reside in the high-spin Fe³⁺ state. The Fe³⁺ ions occupy inequivalent positions differing in the nearest cation environment, or more precisely, tungsten and iron ions are distributed in a random manner over the sites of the octahedral sublattice. The inequivalent positions arise as a result of the Fe and W ions being statistically distributed over the octahedral sublattice. For T > 0 K, magnetic fields at the nuclei and, hence, the average thermodynamic values of the magnetic moments of Fe³⁺ ions occupying inequivalent positions are different and, at a given temperature, are determined by the number of the nearest magnetic neighbors, with the effective magnetic fields (H _eff) varying differently with temperature. As the temperature is lowered, the fields H _eff level off gradually in response to the effective magnetic fields of iron ions having different numbers of exchange bonds leveling off with decreasing temperature which lowers thermal excitation.