Tuning the Superspin Dynamics in Inverse Spinel Ferrite Nanoparticle Ensembles via Indirect Cation Substitution

: We used magnetic and synchrotron X-ray diffraction measurements to investigate the possibility of tuning the strength of magnetic interparticle interactions in nanoparticle ensembles via chemical manipulation. Our main result comes from temperature-resolved in-phase ac-susceptibility data collected on 8 nm average-diameter Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25) and Ni 0.5 Zn 0.5 Fe 2 O 4 (Ni50) nanoparticles at different frequencies, χ ′ vs. T| f . We found that the relative peak temperature variation per frequency decade, ϕ = ∆ T T · ∆ log ( f ) —a known measure of interparticle interaction strength—exhibits a four-fold increase, from ϕ = 0.04 in Ni50 to ϕ = 0.16 in Ni25. This corresponds to a fundamental change in the nanoparticles’ superspin dynamics, as proven by the fit of phenomenological models to magnetic relaxation data. Indeed, the Ni25 ensemble exhibits superparamagnetic behavior, where the temperature dependence of the superspin relaxation time, τ , is described in the Dorman–Bessais–Fiorani (DBF) model: τ ( T ) = τ r exp (cid:16) E B + E ad k B T (cid:17) , with parameters τ r = 4 × 10 − 12 s, and (E B + E ad )/k B = 1473 K. On the other hand, the nanoparticles in the Ni50 ensemble freeze collectively upon cooling in a spin-glass fashion according to a critical dynamics law: τ ( T ) =


Introduction
Magnetic nanoparticle ensembles have recently attracted considerable attention due to their existing and potential applications in a broad range of fields, from biomedicine [1][2][3][4][5][6][7] to high-density magnetic recording [8,9], imaging [10][11][12][13][14], and water and soil remediation [15][16][17].Besides their large surface area, which is a characteristic of all nano-systems, important for the behavior of magnetic nanoparticles is the fact that they are made of a single magnetic domain whose giant magnetic moment is called the superspin.At any temperature, the superspin flips across an energy barrier to reverse the magnetization, E B (the process is called superspin relaxation), and the relaxation time τ (i.e., the time needed to complete one full flip) depends on the temperature according to an Arrhenius-type activation law: In Equation ( 1), τ 0 is a characteristic time related to the attempt frequency, f 0 , by τ 0 = 1/2πf 0 , and k B is the Boltzmann constant.This Néel-Brown model [18,19], which was developed for a single, isolated magnetic nanoparticle, is important because it explains the so-called superparamagnetic behavior of an ideal system of monodispersed and noninteracting nanoparticles.Indeed, if a magnetic measurement is carried out on such a system of ferromagnetic nanoparticles at a temperature where the observation time τ obs , (which depends on the measurement details) is shorter than the relaxation time, τ, one would, as expected, measure ferromagnetic properties such as an open hysteresis loop.The system is said to be in its blocked state.On the other hand, if the measurement is carried out at higher temperatures, where the observation time is longer than the relaxation time, one would measure paramagnetic properties, although the material the nanoparticles are made of is ferromagnetic.The system is said to be in its superparamagnetic state.The temperature threshold that separates the two regimes is called the blocking temperature, T B .Obviously, when T = T B , the measurement and relaxation times are equal to one another, i.e., τ = τ obs .
In addition to being an interesting phenomenon from the fundamental standpoint, superparamagnetism has implications that are important for many applications of magnetic nanoparticle ensembles.For example, high-density magnetic recording devices, where each magnetic nanoparticle represents a recording bit, fail catastrophically at temperatures above T B , and there are similar issues related to magnetic imaging applications [20].Moreover, heat generation by magnetic nanoparticle ensembles subjected to an alternating magnetic field is not hysteretic, but essentially depends on the superparamagnetic relaxation details, particularly on the energy barrier to superspin reversal [21].This makes superparamagnetic nanoparticle ensembles ideal agents for cancer hyperthermia therapy, a method that has the potential to kill malignant tumors without the side effects of radiation or chemotherapy [22].Consequently, lots of efforts have been invested in studying the temperature dependence of the dynamic superspin behavior in magnetic nanoparticle systems both at the fundamental and the applied levels [23,24].
It is important to mention that interparticle interactions, particularly magnetic dipolar interactions among the nanoparticles in the ensemble, play a critical role in the collective relaxation of the superspins [25].Yet, the above-described Néel-Brown model does not account for such interactions, and Equation (1) does not realistically describe the temperature behavior of the magnetic relaxation times even for weakly interacting magnetic nanoparticles.Instead, a Vogel-Fulcher (VF) type law [26] has been successfully used in many studies [27,28] to analyze and quantify the strength of interparticle interactions through the T 0 parameter.This is clearly an empirical model, as there is no physical meaning attached to the divergence of the relaxation time as T approaches T 0 .More recently, Dorman, Bessais, and Fiorani (DBF) developed a phenomenological model where the magnetic interparticle interactions are accounted for via a statistical calculation of the dipolar energy [29].The DBF equation, introduces a variable pre-factor τ r and an energy barrier to superspin reversal that has two components: E B = KV (where K is the nanoparticle's magnetic anisotropy constant and V is their average volume) and E ad , additional energy that depends on the strength of the dipolar interactions.The predictions of the DBF model have been confirmed experimentally, particularly for weak interparticle interactions.In a recent study, for example, we used Equation (3) to determine E ad as a function of the average distance using frozen Fe 3 O 4 /hexane ferrofluids of different concentrations [30].Finally, for very strong inter- particle interactions, the temperature dependence of the relaxation time is described by a critical dynamics law [31]: as, in this case, the superspins do not block individually upon cooling below a temperature threshold, but freeze collectively in a spin-glass fashion.In Equation ( 4), T g is the freezing temperature and z and ν are the dynamic scaling and critical exponents, respectively.
Here, we report the results of a study aimed at exploring alternative ways of tuning the strength of the magnetic interparticle interactions in nanoparticle ensembles.Instead of varying the interparticle distances (via ferrofluid dilution), we used chemical manipulation to alter the crystal and magnetic structure of inverse spinel nickel ferrite by replacing a fraction of the magnetic Ni ions with non-magnetic Zn.We carried magnetic and synchrotron X-ray diffraction measurements on ensembles of Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25) and Ni 0.5 Zn 0.5 Fe 2 O 4 (Ni50) nanoparticles of average diameter <D> = 8 nm.Our temperatureresolved ac-susceptibility data collected at different frequencies, χ ′ vs. T| f , show notable quantitative differences between the dynamic behavior of the superspin in the two samples.We found that the relative peak temperature variation per frequency decade, ϕ = ∆T T•∆log(f) , in the Ni50 nanoparticle ensemble is ϕ = 0.04, whereas its Ni25 counterpart value is ϕ = 0.16, almost four times greater.This is highly significant, as ϕ is known to be a good indicator of the dipolar interaction strength in magnetic nanoparticle systems [32].Moreover, by analyzing the temperature variation of the relaxation time in the two nanoparticle ensembles, Ni25 and Ni50, we found fundamental differences between their magnetization dynamics.For the weakly interacting Ni25 ensemble, we found that the superspin relaxation is described by the DBF superparamagnetic model, as demonstrated by the excellent fit of Equation (3) to the data.On the other hand, the τ(T) dependence observed in Ni50 cannot be described by the DBF or VF models.Instead, the critical dynamics law in Equation ( 4) is needed to fit the data collected on this strongly interacting system, indicating that the superspins collectively freeze upon cooling below a temperature threshold.Finally, powder X-ray diffraction measurements on the Ni25 and Ni50 samples uncover the microstructural details that underlie the observed magnetic behavior.In both cases, we found an indirect mechanism by which the magnetic Ni ions are replaced by non-magnetic Zn.Rietveld refinements show that Zn is actually incorporated in the tetrahedral (A-sites) of the inverse spinel where it replaces Fe, which in turn migrates to the octahedral (B-sites) where it displaces Ni.
Our findings are significant because they demonstrate the possibility of tuning the strength of the dipolar magnetic interparticle interactions in magnetic nanoparticle ensembles and, in turn, controlling their superspin relaxation.This is important because understanding and eventually controlling the superspin dynamics in these systems is critical for their operation as functional materials in several applications.

Materials and Methods
Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25) and Ni 0.5 Zn 0.5 Fe 2 O 4 (Ni50) magnetic nanoparticles were synthesized by a simple coprecipitation method.Briefly, suitable amount rations of nickel and zinc nitrate hydrates, Ni(NO 3 ) 2 •6H 2 O and Zn(NO 3 ) 2 •6H 2 O, were mixed with iron chloride hydrate, FeCl 3 •6H 2 O, at room temperature in high-purity H 2 O. NaOH was added to the solution to promote precipitation.The nanophases were crystallized by keeping the precipitate at 90 • C for one hour, followed by washing and drying at 60 • C. X-ray diffraction measurements were performed at the National Synchrotron Light Source (Brookhaven National Laboratory, Upton, NY, USA) using synchrotron radiation of wavelength 0.922 Å. Full profile fits were used to confirm the purity of the synthesized phases and determine the full width at half maximum (FWHM) of the diffraction peaks.Scherrer analysis was subsequently used to determine the average diameter of the nanoparticles in the ensemble.Rietveld refinements were employed to determine the atom positions in the unit cell, as well as the occupancy in the tetrahedral and octahedral sites of the inverse spinel crystal structure of the nanoparticles.Dc-magnetization and ac-susceptibility data were collected using a Quantum Design ® Physical Property Measurement System that allows the temperature of the sample to be varied within the 2 K-350 K range.The temperature dependence of the magnetization, M vs. T, was measured upon heating in an H = 100 Oe external magnetic field using the zero-field-cooled field-cooled (ZFC-FC) protocol.The actual measurements were carried out by accelerating the sample through a set of pickup coils using a dc-motor.These datasets were used to determine the saturation value of the magnetization and the critical temperature that marks the onset of irreversibility upon heating.Temperature-resolved in-phase (χ ′ ) and out-of-phase (χ ′′ ) ac-susceptibility measurements were carried out at different frequencies spanning a three-order-of-magnitude range, from 10 Hz to 10,000 Hz.The ac-susceptibility measurements were carried out on a fixed sample by applying an alternating magnetic field of amplitude H m = 3 Oe.The χ ′ vs. T| f data were used to quantitatively assess the strength of the magnetic dipolar interactions among the superspins and to describe the resulting superspin relaxation behavior of the two nanoparticle ensembles.allows the temperature of the sample to be varied within the 2 K-350 K range.The temperature dependence of the magnetization, M vs. T, was measured upon heating in an H = 100 Oe external magnetic field using the zero-field-cooled field-cooled (ZFC-FC) protocol.The actual measurements were carried out by accelerating the sample through a set of pickup coils using a dc-motor.These datasets were used to determine the saturation value of the magnetization and the critical temperature that marks the onset of irreversibility upon heating.Temperature-resolved in-phase (χ′) and out-of-phase (χ″) ac-susceptibility measurements were carried out at different frequencies spanning a three-order-ofmagnitude range, from 10 Hz to 10,000 Hz.The ac-susceptibility measurements were carried out on a fixed sample by applying an alternating magnetic field of amplitude Hm = 3 Oe.The χ′ vs. T|f data were used to quantitatively assess the strength of the magnetic dipolar interactions among the superspins and to describe the resulting superspin relaxation behavior of the two nanoparticle ensembles.

Results and Discussion
Figure 1a shows the powder X-ray diffraction patterns measured on the Ni0.25Zn0.75Fe2O4(Ni25) and Ni0.5Zn0.5Fe2O4(Ni50) magnetic nanoparticle ensembles used in this study.Synchrotron X-rays of wavelength λ = 0.922 Å were used to record the intensity, I, at different "detector" angles, 2θ, between 15 and 45 degrees.The vertical bars mark the 2θ angular positions of the Bragg reflections from the inverse spinel structure of nickel ferrite.These data demonstrate that there are no impurities Ni25 and Ni50 samples, as all peaks observed in the data correspond to Bragg reflections, and also indicate that no major structural changes (in terms of unit cell and heavy atom positions) occur upon Zn doping, as the two powder diffraction patterns are similar both in terms of diffraction peak positions and integrated intensities.Figure 1b shows a full-profile Le Bail fit [33] to the X-ray powder diffraction pattern collected on the Ni50 nanoparticle ensemble.The filled symbols represent X-ray intensity observed at a given angle, I obs , the solid line is the corresponding best profile (Le Bail) fit, I calc , and the lower trace is the difference curve between the observed and the calculated intensities, I obs − I calc .
Le Bail or full-profile analysis is based on simultaneously fitting all the peaks in a powder diffraction pattern by using variables that describe the unit cell parameters and the peak profiles.Here, we use the program FULLPROF [34] to carry out Le Bail fits with the goal of finding the lattice constants and full width at half maximum (FWHM) of the diffraction peaks.This type of analysis also has the potential to reveal structural modifications induced by the variation in the Ni/Zn ratio.We carried out the full-profile fits starting with the known lattice parameters and space group of NiFe 2 O 4 , and used a pseudo-Voigt function for the peak profiles.The background was automatically calculated and subtracted from the observed powder diffraction pattern.The calculation converges under the simultaneous variation of six parameters, and the best fit yields a cubic unit cell parameter a = 8.414 Å (S.G.F d −3 m) and an FWHM = 0.63 deg.for the peak at 2θ = 36.05deg.We used these values in Scherrer's formula < D >= 0.9•λ FWHM• cos 2θ 2 to determine the average diameter of the nanoparticles in the Ni50 ensemble as <D> = 8 nm ± 3 nm.A similar analysis of the X-ray diffraction pattern recorded on the Ni25 ensemble yields the same value of the average nanoparticle diameter.
The temperature dependence of the dc-magnetization, M vs. T, measured on the Ni50 sample using the zero-field-cooled field-cooled (ZFC-FC) protocol is shown in Figure 2a.Both curves were recorded upon heating from 3 K to 300 K in a small external magnetic field of magnitude H = 100 Oe applied before cooling for the FC branch (open symbols) and after cooling for the ZFC branch (filled symbols).The most significant feature of the data is the temperature at which the two branches start overlapping.For Ni50, we found this onset of irreversibility to occur at T irr ~145 K. Similar ZFC-FC data recorded on the Ni25 nanoparticle ensemble reveal a lower value T irr ~90 K. Figure 2b shows the magnetization curve, M vs. H, recorded on the Ni50 system upon increasing the magnetic field from 0 to 70,000 Oe (filled symbols).
Crystals 2024, 14, x FOR PEER REVIEW 5 of 13 in the Ni25 and Ni50 samples, as all peaks observed in the data correspond to Bragg reflections, and also indicate that no major structural changes (in terms of unit cell and heavy atom positions) occur upon Zn doping, as the two powder diffraction patterns are similar both in terms of diffraction peak positions and integrated intensities.Figure 1b shows a full-profile Le Bail fit [33] to the X-ray powder diffraction pattern collected on the Ni50 nanoparticle ensemble.The filled symbols represent X-ray intensity observed at a given angle, Iobs, the solid line is the corresponding best profile (Le Bail) fit, Icalc, and the lower trace is the difference curve between the observed and the calculated intensities, Iobs − Icalc.Le Bail or full-profile analysis is based on simultaneously fitting all the peaks in a powder diffraction pattern by using variables that describe the unit cell parameters and the peak profiles.Here, we use the program FULLPROF [34] to carry out Le Bail fits with the goal of finding the lattice constants and full width at half maximum (FWHM) of the diffraction peaks.This type of analysis also has the potential to reveal structural modifications induced by the variation in the Ni/Zn ratio.We carried out the full-profile fits starting with the known lattice parameters and space group of NiFe2O4, and used a pseudo-Voigt function for the peak profiles.The background was automatically calculated and subtracted from the observed powder diffraction pattern.The calculation converges under the simultaneous variation of six parameters, and the best fit yields a cubic unit cell parameter a = The temperature dependence of the dc-magnetization, M vs. T, measured on the Ni50 sample using the zero-field-cooled field-cooled (ZFC-FC) protocol is shown in Figure 2a.Both curves were recorded upon heating from 3 K to 300 K in a small external magnetic field of magnitude H = 100 Oe applied before cooling for the FC branch (open symbols) and after cooling for the ZFC branch (filled symbols).The most significant feature of the data is the temperature at which the two branches start overlapping.For Ni50, we found this onset of irreversibility to occur at Tirr~145 K. Similar ZFC-FC data recorded on the Ni25 nanoparticle ensemble reveal a lower value Tirr~90 K. Figure 2b shows the magnetization curve, M vs. H, recorded on the Ni50 system upon increasing the magnetic field from 0 to 70,000 Oe (filled symbols).As expected for a system of magnetic nanoparticles, the magnetization curve does not fully saturate but "asymptotically" approaches M s = 105 emu/g (red dashed line).Notably, this is more than 150% greater than its NiFe 2 O 4 counterpart (green dashed line) in accordance with previous reports of magnetic property enhancement in inverse spinel ferrites when doped with non-magnetic ions [35].It is important to mention, however, that dc-magnetization data alone cannot reveal conclusive information on the superspin relaxation of a magnetic nanoparticle ensemble because of the fundamentally dynamic nature of this behavior.As we show below, ac-susceptibility data and analysis are critical to uncover the microscopic mechanisms and quantitative aspects of the superspin dynamics of these systems.
field-cooled (ZFC-FC) protocol.(b) Field dependence of the dc-magnetization, M vs. H, measured on the Ni50 sample.The saturation magnetization Ms of Ni50 is ~150% higher than its NiFe2O4 counterpart.
As expected for a system of magnetic nanoparticles, the magnetization curve does not fully saturate but "asymptotically" approaches Ms = 105 emu/g (red dashed line).Notably, this is more than 150% greater than its NiFe2O4 counterpart (green dashed line) in accordance with previous reports of magnetic property enhancement in inverse spinel ferrites when doped with non-magnetic ions [35].It is important to mention, however, that dc-magnetization data alone cannot reveal conclusive information on the superspin relaxation of a magnetic nanoparticle ensemble because of the fundamentally dynamic nature of this behavior.As we show below, ac-susceptibility data and analysis are critical to uncover the microscopic mechanisms and quantitative aspects of the superspin dynamics of these systems.All measurements were taken upon heating the sample in 2.5 K steps in alternating magnetic fields of 5 Oe amplitudes.At each temperature, the frequency of the driving field was increased from 10 Hz to 10,000 Hz, and for each of the five measurement frequencies, the values of the in-phase (χ′) and out-of-phase (χ″) ac-susceptibility values were recorded.For Ni50, these χ″ vs. T|f and χ′ vs. T|f data are shown in Figure 3a,b, respectively.In both cases, we note that, at each frequency, the data show a robust peak.First, this All measurements were taken upon heating the sample in 2.5 K steps in alternating magnetic fields of 5 Oe amplitudes.At each temperature, the frequency of the driving field was increased from 10 Hz to 10,000 Hz, and for each of the five measurement frequencies, the values of the in-phase (χ ′ ) and out-of-phase (χ ′′ ) ac-susceptibility values were recorded.For Ni50, these χ ′′ vs. T| f and χ ′ vs. T| f data are shown in Figure 3a,b, respectively.In both cases, we note that, at each frequency, the data show a robust peak.First, this demonstrates the quality of the sample in terms of it containing nanoparticles of nearly the same size with a narrow distribution about the average value.In addition, the peak temperature increases with the increase in frequency.The same qualitative behavior is exhibited by the χ ′ vs. T| f curves collected on the Ni25, as shown in Figure 3c.As explained in more detail in the Introduction, this is due to the dynamic nature of the superspin relaxation and the ability to adjust the observation time in ac-susceptibility measurements through variations in the driving field frequency.
The frequency dependence of the in-phase susceptibility peak temperature is known to contain quantitative information about the strength of the magnetic dipolar interactions between the nanoparticle in the ensemble through the so-called relative peak temperature variation per frequency decade, ϕ = ∆T T•∆log(f) .We used the data in Figure 3b,c to calculate the values of ϕ for the Ni50 and Ni25 systems, respectively.Remarkably, we found values that differ significantly from one another: ϕ = 0.04 in Ni50 and ϕ = 0.16 in Ni25.This shows that the dipolar interactions in the Ni50 nanoparticle ensemble are much stronger than their Ni25 counterparts, and clearly demonstrate the ability to tune the strength of such interactions via chemical manipulations, i.e., without the need to alter mesostructural properties of the ensemble, such as interparticle distance or the average nanoparticle size.This is particularly important because, as we will demonstrate below, the interparticle interaction strength directly affects the superspin dynamics, which, in turn, is responsible for the functionality of these systems in several applications.The next step is to investigate the effect of the observed difference between the interparticle interaction strength in the two nanoparticle ensembles on the microscopic aspects of their superspin dynamics.For Ni25, we found ϕ = 0.16, which falls within the ϕ value range that describes weak to medium interparticle interactions.Clearly, the Néel-Brown formalism would not accurately describe the superparamagnetic relaxation of this system as it was developed for isolated, non-interacting nanoparticles.Attempting to fit Equation (1) to τ(T) data from even weakly interacting magnetic nanoparticles usually yields pre-factor values τ 0 that are unphysically short [23,27].Typically, a Vogel-Fulcher law (Equation ( 2)) is used to describe τ(T) data from magnetic nanoparticle systems where interparticle interactions are present [21,23,27], and the T 0 parameter is a good indicator of the interparticle interaction strength.Yet, the VF law is an empirical model, with no physical basis related to the superspin dynamics.To clarify the microscopic dynamic mechanisms responsible for the superspin relaxation in the Ni25 ensemble, we analyzed its observed temperature dependence of the relaxation time in the framework of the Dorman-Bessais-Fiorani (DBF) phenomenological model.Figure 4 shows these data and their analysis.
The solid symbols represent the measured τ(T) dependence obtained from the acsusceptibility data in Figure 3c, and the dashed line is the best fit to Equation (3).The fit converges with low residuals (χ 2 = 1.23) upon the simultaneous variation of two parameters.The best fit parameters yield a pre-factor τ r = 4 × 10 −12 s and a reduced barrier to magnetization reversal (E B + E ad )/k B = 1473 K.In addition, the high-quality fit to the DBF model unequivocally demonstrates that the blocking-unblocking, superparamagnetic mechanism describes the superspin dynamics upon heating in the Ni25 magnetic nanoparticle ensemble.
The Ni50 ensemble exhibits much stronger magnetic dipolar interparticle interactions, as evidenced by the low value of the relative peak temperature variation per frequency decade, ϕ = 0.04.The DBF model does not apply for such strong interactions, so to investigate the microscopic mechanisms that govern the temperature-dependent superspin dynamics in Ni50, we first attempted to fit the observed temperature dependence of the relaxation time using a Vogel-Fulcher law.This was aimed at checking if the superparamagnetic model (where the nanoparticles' giant magnetic moments block upon cooling below a temperature threshold) still describes the dynamic behavior of this strongly interacting system.Figure 5a shows the results of this analysis.
of the interparticle interaction strength.Yet, the VF law is an empirical model, with no physical basis related to the superspin dynamics.To clarify the microscopic dynamic mechanisms responsible for the superspin relaxation in the Ni25 ensemble, we analyzed its observed temperature dependence of the relaxation time in the framework of the Dorman-Bessais-Fiorani (DBF) phenomenological model.Figure 4 shows these data and their analysis.The Ni50 ensemble exhibits much stronger magnetic dipolar interparticle interactions, as evidenced by the low value of the relative peak temperature variation per frequency decade, ϕ = 0.04.The DBF model does not apply for such strong interactions, so to investigate the microscopic mechanisms that govern the temperature-dependent superspin dynamics in Ni50, we first attempted to fit the observed temperature dependence of the relaxation time using a Vogel-Fulcher law.This was aimed at checking if the superparamagnetic model (where the nanoparticles' giant magnetic moments block upon cooling below a temperature threshold) still describes the dynamic behavior of this strongly interacting system.Figure 5a shows the results of this analysis.The filled symbols here are the measured τ(T) dependence obtained from the acsusceptibility data in Figure 3b, and the dashed line is the best fit to Equation (2).Clearly, the Vogel-Fulcher law describes the data well, but, besides it being an empirical model as discussed above, the best fit value of the interaction parameter, T 0 = 85, appears to be unphysically high (i.e., more than 40% of the peak temperatures of the χ ′ vs. T| f curves).Consequently, we next tried a phenomenological model in which, instead of individually blocking, the superspins collectively freeze in a spin-glass fashion upon cooling below a temperature threshold.This behavior leads to a temperature dependence of the relaxation time described by the critical dynamics law in Equation (4). Figure 5b shows the best fit (dashed line) of Equation ( 4) to the τ(T) data (solid symbols).The fit converges with low residuals and yields the following values for the variable parameters: τ 0 = 1.1 × 10 −8 s, zν = 7.2, and T g = 145 K. Remarkable here is the value of the freezing temperature T g , which turns out to be the same as the onset of irreversibility temperature T irr (Figure 2) observed using a totally different method of measuring the same magnetic event in the Ni50 nanoparticle ensemble.This indicates that the nature of the magnetic event observed around 145 K in the strongly interacting Ni50 sample is indeed collective superspin freezing, fundamentally different from the superparamagnetic blocking observed in the Ni25 ensemble.
Finally, to obtain more insight into the microstructural details associated with the observed magnetic behavior, we carried out Rietveld refinements [36] against synchrotron X-ray diffraction data using the GSAS II software [37].In a Rietveld refinement, one adjusts not only the unit cell parameters and the peak profiles but also the atomic coordinates, thermal parameters, and, if more than one type of atom occupies a given position in the unit cell, the occupancy of that position.Figure 6a shows such a Rietveld refinement of the crystal structure of the Ni25 nanoparticles.The red line here is the best fit, the solid symbols are the observed intensity, I, as a function of the detector angle 2θ, the vertical lines mark the positions of the Bragg reflections, and the lower trace is the difference curve The filled symbols here are the measured τ(T) dependence obtained from the ac-susceptibility data in Figure 3b, and the dashed line is the best fit to Equation (2).Clearly, the Vogel-Fulcher law describes the data well, but, besides it being an empirical model as discussed above, the best fit value of the interaction parameter, T0 = 85, appears to be unphysically high (i.e., more than 40% of the peak temperatures of the χ′ vs. T|f curves).Consequently, we next tried a phenomenological model in which, instead of individually blocking, the superspins collectively freeze in a spin-glass fashion upon cooling below a temperature threshold.This behavior leads to a temperature dependence of the relaxation time described by the critical dynamics law in Equation (4). Figure 5b shows the best fit (dashed line) of Equation ( 4) to the τ(T) data (solid symbols).The fit converges with low residuals and yields the following values for the variable parameters: τ0 = 1.1 × 10 −8 s, zν = 7.2, and Tg = 145 K. Remarkable here is the value of the freezing temperature Tg, which turns out to be the same as the onset of irreversibility temperature Tirr (Figure 2) observed using a totally different method of measuring the same magnetic event in the Ni50 nanoparticle ensemble.This indicates that the nature of the magnetic event observed around 145 K in the strongly interacting Ni50 sample is indeed collective superspin freezing, fundamentally different from the superparamagnetic blocking observed in the Ni25 ensemble.To initialize the peak profile parameters, we started with the same pseudo-Voigt function used in the full-profile fit in Figure 1b.For the unit cell and atom position parameters, we started with the known inverse spinel structure of NiFe 2 O 4 , shown in Figure 6b.We note that the unit cell contains eight tetrahedral (A) sites fully occupied by Fe atoms (light brown) and sixteen octahedral (B) sites, half of which are occupied by Ni atoms (green) and the other half by Fe atoms (light brown).Now, when Zn atoms (blue) are "doped" into this structure to replace 75% of the Ni in order to make Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25), they can, in principle, be incorporated in either the A or the B sites.We explored both scenarios and obtained better fits when the non-magnetic Zn was incorporated in the A (tetrahedral) sites.While this is not unexpected, being consistent with results from previous studies of inverse spinel ferrites doped with non-magnetic atoms [38], it implies that an indirect Ni replacement (by Zn) mechanism is at work here, as the Ni atoms reside in the B (octahedral) sites.Indeed, as shown in Figure 6c, Zn atoms are incorporated in the A sites, where they actually displace Fe atoms, which migrate to the B sites, where they displace Ni.In this particular case-when 75% of the Ni is replaced by Zn-the resulting occupancies in the octahedral (B) sites are one-eighth Ni and seven-eighths Fe, whereas in the tetrahedral (A) sites, they are three-quarters Zn and one-quarter Fe.This is significant because the magnitude of the magnetic moment of each nanoparticle, the

Figure
Figure 1a shows the powder X-ray diffraction patterns measured on the Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25) and Ni 0.5 Zn 0.5 Fe 2 O 4 (Ni50) magnetic nanoparticle ensembles used in this study.Synchrotron X-rays of wavelength λ = 0.922 Å were used to record the intensity, I, at different "detector" angles, 2θ, between 15 and 45 degrees.

Figure 1 .
Figure 1.(a) Powder diffraction patterns collected on Ni0.25Zn0.75Fe2O4(Ni25) and Ni0.5Zn0.5Fe2O4(Ni50) nanoparticles using synchrotron X-rays of wavelength λ = 0.922 Å.(b) Full profile (Le Bail) fit showing the calculated intensity Icalc (solid line) and the observed intensity Iobs on the Ni50 sample (red symbols).The vertical bars show the angular positions of the Bragg reflections, and the lower trace is the difference curve Icalc − Iobs.

Figure 1 .
Figure 1.(a) Powder diffraction patterns collected on Ni 0.25 Zn 0.75 Fe 2 O 4 (Ni25) and Ni 0.5 Zn 0.5 Fe 2 O 4 (Ni50) nanoparticles using synchrotron X-rays of wavelength λ = 0.922 Å.(b) Full profile (Le Bail) fit showing the calculated intensity I calc (solid line) and the observed intensity I obs on the Ni50 sample (red symbols).The vertical bars show the angular positions of the Bragg reflections, and the lower trace is the difference curve I calc − I obs .The vertical bars mark the 2θ angular positions of the Bragg reflections from the inverse spinel structure of nickel ferrite.These data demonstrate that there are no impurities in the 8.414 Å (S.G.F d −3 m) and an FWHM = 0.63 deg.for the peak at 2θ = 36.05deg.We used these values in Scherrer's formula < D >= .• • to determine the average diameter of the nanoparticles in the Ni50 ensemble as <D> = 8 nm ± 3 nm.A similar analysis of the X-ray diffraction pattern recorded on the Ni25 ensemble yields the same value of the average nanoparticle diameter.

Figure 2 .
Figure 2. (a) Temperature dependence of the dc-magnetization, M vs. T, measured on the Ni50 nanoparticle ensemble upon heating in a H = 100 Oe external magnetic field using the zero-field-cooled

Figure 2 .
Figure 2. (a) Temperature dependence of the dc-magnetization, M vs. T, measured on the Ni50 nanoparticle ensemble upon heating in a H = 100 Oe external magnetic field using the zero-fieldcooled field-cooled (ZFC-FC) protocol.(b) Field dependence of the dc-magnetization, M vs. H, measured on the Ni50 sample.The saturation magnetization M s of Ni50 is ~150% higher than its NiFe 2 O 4 counterpart.

Figure 4 .
Figure 4. Temperature dependence of the superspin relaxation time, τ (filled symbols), obtained from ac-susceptibility data collected on the Ni25 sample.The dashed line shows the best fit to the data of the Dorman-Bessais-Fiorani (DBS) model known to describe the superparamagnetic behavior of nanoparticle ensembles.The solid symbols represent the measured τ(T) dependence obtained from the acsusceptibility data in Figure3c, and the dashed line is the best fit to Equation (3).The fit converges with low residuals (χ 2 = 1.23) upon the simultaneous variation of two parameters.The best fit parameters yield a pre-factor τr = 4 × 10 −12 s and a reduced barrier to magnetization reversal (EB + Ead)/kB = 1473 K.In addition, the high-quality fit to the DBF model unequivocally demonstrates that the blocking-unblocking, superparamagnetic mechanism describes the superspin dynamics upon heating in the Ni25 magnetic nanoparticle ensemble.The Ni50 ensemble exhibits much stronger magnetic dipolar interparticle interactions, as evidenced by the low value of the relative peak temperature variation per frequency decade, ϕ = 0.04.The DBF model does not apply for such strong interactions, so to investigate the microscopic mechanisms that govern the temperature-dependent superspin dynamics in Ni50, we first attempted to fit the observed temperature dependence of the relaxation time using a Vogel-Fulcher law.This was aimed at checking if the superparamagnetic model (where the nanoparticles' giant magnetic moments block upon cooling below a temperature threshold) still describes the dynamic behavior of this strongly interacting system.Figure5ashows the results of this analysis.

Figure 4 .
Figure 4. Temperature dependence of the superspin relaxation time, τ (filled symbols), obtained from ac-susceptibility data collected on the Ni25 sample.The dashed line shows the best fit to the data of the Dorman-Bessais-Fiorani (DBS) model known to describe the superparamagnetic behavior of nanoparticle ensembles.

13 Figure 5 .
Figure 5. Temperature dependence of the superspin relaxation time, τ (filled symbols), obtained from ac-susceptibility data collected on the Ni50 sample, and best fits the data (dashed line) of (a) a Vogel-Fulcher empirical model and (b) a critical dynamics law that describes the collective, spinglass-like freezing of magnetic nanoparticle ensembles.

Figure 5 .
Figure 5. Temperature dependence of the superspin relaxation time, τ (filled symbols), obtained from ac-susceptibility data collected on the Ni50 sample, and best fits the data (dashed line) of (a) a Vogel-Fulcher empirical model and (b) a critical dynamics law that describes the collective, spin-glass-like freezing of magnetic nanoparticle ensembles.