Modelling the magnetization data of Fe3O4 nanoparticles above blocking temperature

Nanoparticles of Fe3O4 are prepared by simple co-precipitation method. The sample is characterized using an x-ray diffractometer, transmission electron microscope, and vibrating sample magnetometer. The x-ray diffraction pattern of the sample clearly shows that it is a single-phase magnetite. The transmission electron micrograph shows that the sample has a narrow distribution in particle size with average particle size of 9.9 nm. The SAED pattern only shows the diffraction planes correspond to magnetite and no other phase impurity is detected. The calculated thickness of the magnetic disordered shell due to the reduction in particle size is found to be 1.7 nm. The magnetization of the sample is measured as a function of temperature and applied magnetic field. The zero-field cooled and field cooled curves of the sample are measured in the presence of 250 Oe applied magnetic field and both the curves bifurcate at 170 K. The peak in the zero-field curve indicates that the sample has a blocking temperature of around 100 K. The magnetization as a function of applied magnetic field data at 200, 225, 250, 275 and 300 K are measured (up to ±20 kOe). These magnetization data are used for the fitting to analyze the magnetic behavior of Fe3O4 nanoparticles. . The magnetization of nanoparticles systems is influenced by several factors such as particle size distribution, disordered surface, magnetocrystalline anisotropy, magnetic moment distribution and magnetic interactions. The ignorance of such factors while analyzing the magnetization data leads to discrepancies in the results. The surface effects are sensitive to the reduction in particle size leading to the spin frustrations on the surface suggesting a magnetic disordered layer which affect the magnetic behavior of nanoparticles. This work presents the analysis of the magnetization data in an appropriate magnetization expression which takes into consideration the effect of magnetic moment distribution. This distribution in the magnetic moment is found to be significantly influenced the magnetization analysis and affected by the magnetic disordered surface which accounts for the presence of magnetic anisotropy and magnetic interactions on the particles surface. The results and observations are discussed in detail.


Introduction
Magnetic behavior of ferro, ferri or antiferromagnetic nanoparticles systems is studied widely because these systems are trending in the development of nanotechnology.These systems show exciting and unique properties due to the reduction in particle size such as superparamagnetism [1].Superparamagnetism is the phenomenon that occurred when the thermal energy becomes more than the energy keeping the particles oriented in a direction and the particles start flipping their direction randomly.There are different models and approaches used in literature to investigate the magnetic behavior in these systems by modelling their magnetization data [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].The modelling of magnetization data gives the estimation of different magnetic parameters that are responsible for controlling the magnetic dynamics in a system.The Langevin model is mostly used to study the magnetic behavior of nanoparticles systems in superparamagnetic region and found to yield unphysical results [7][8][9][10][11][12].This model assumes that the particles in a system are of same size and there is no directional dependence of magnetization vector present inside the crystal.However, the magnetization of the real nanoparticles system depends on many factors such as particle size distribution, magnetic anisotropy, particle interactions, magnetic moment distribution, particle shape etc.Several authors attempted to study the magnetization of ferro, ferri or antiferromagnetic systems in superparamagnetic region by considering the effect of magnetic moment distribution [4,9,11,12].This distribution in magnetic moment arises due to the particle size distribution present in the system.In a study of magnetic nanoparticles system of ferritin with narrow distribution in particle size, the magnetic moment distribution is found to be wide and the presence of the intraparticle magnetic disorder is reported to be the cause of this wide distribution in magnetic moment [9].A report on antiferromagnetic nanoparticles of maghemite claimed that in systems with narrow particles size distribution, the effect of magnetic moment or particle size distribution in the analysis of magnetization data can be ignored [12].In a recent study, the combined effect of particle size distribution and magnetic anisotropy is considered to study the magnetization in antiferromagnetic nanoparticles of ferritin [21].It is concluded that the effect of magnetic anisotropy on magnetization process is weak in systems with narrow particle size distribution and significantly affects the systems with broad particle size distribution.Another study on magnetic nanoparticle systems with broad distribution in particle size shows that the consideration of only particle size distribution in magnetic analysis is not sufficient [22].The consideration of combined effect of several factors due to finite size and surface effects that influence the magnetization in the nanoparticles systems increases the complexity of the analysis.There is still exact information on magnetic behavior of nanoparticles that needs to be investigated to understand their behavior properly.
Fe 3 O 4 is a well-studied magnetic system because of its applications in biomedicine, wastewater treatment, magnetic resonance imaging etc. as it is a non-toxic, efficient, scalable, and economic material.The analysis of magnetization data of this system above blocking temperature has been studied by many researchers [2,16,[23][24][25][26][27][28][29].There is rarely any study that discusses the contribution of magnetic disordered particle surface in the estimation of magnetic moment distribution in Fe 3 O 4 nanoparticles.This motivated the idea of the present study.In this work, the magnetic analysis of 10 nm Fe 3 O 4 particles synthesized by co-precipitation method is investigated.The magnetization data of this system is analyzed using the role of log-normal distribution in particle magnetic moment at different temperatures above blocking temperature.Also, the obtained magnetic moment distribution is compared with the particle size distribution obtained from transmission electron micrograph.

Experimental details
Fe 3 O 4 nanoparticles are prepared by following a simple co-precipitation method.For this, 0.1 M aqueous solution of FeCl 2 .4H 2 O and 0.2 M of FeCl 3 .6H 2 O mixed .The ratio of Fe 2+ to Fe 3+ is strictly maintained at 1:2 [30,31].The pH of this solution maintained is less than 2. In this case, the pH of this mixed solution is 1.6.After that, the 8 M aqueous solution of NaOH is added to this mixed solution of ferrous and ferric salts at a faster rate, while continuously stirring at room temperature.The black precipitate are seen to be formed and the pH of the final solution is maintained at 12.These precipitate are washed several times with deionized water and the last two washings are performed with ethanol.These washed precipitate are then dried at a temperature of 100 o C. The dried sample is then ground to get fine powder of Fe 3 O 4 .The reaction takes place as follows: Structural characterizations of samples are done with x-ray diffractometer (PANalytical X'Pert PRO MPD) using Cu-K α radiation and transmission electron microscope (FEI Technician).Magnetization measurements are done with a vibrating sample magnetometer (Quantum Design, PPMS).

Structural characterization
The x-ray diffraction pattern of the prepared sample is shown in figure 1 after correcting for the baseline.The diffraction peaks are broader that depicts the nanocrystalline nature of the sample.The interplanar spacing 'd' and corresponding (hkl) values are also mentioned in the graph.It shows that the sample prepared is a single phase Fe 3 O 4 .The crystallite size is calculated using Scherrer formula as shown in equation (2).The average crystallite size is calculated to be 9 nm from x-ray diffraction graph.
Here B M is the width where intensity is half of its peak value, t is the crystallite size, λ is the wavelength of the Cu-kα radiation used which is equal to 1.54 Å and θ is the Bragg angle.The average value of lattice constant 'a' is found to be 0.836 nm.The x-ray diffraction patterns of Fe 3 O 4 and γ Fe 2 O 3 look similar.One cannot easily distinguish between the x-ray diffraction pattern of these two compounds.However, it is observed that the characteristics peaks of γ Fe 2 O 3 with (hkl) values (210) and (211) are not present in this sample [32].The calculated value of lattice constant 'a' from x-ray diffraction pattern is close to the expected value for Fe 3 O 4 i.e., 0.838 nm [10,33,34].Most of the literature reported the value of lattice constant 'a' to between 0.836 to 0.842 nm [31,35].The lattice constant for maghemite is found to be 0.833 to 0.835nm [10,36,37].These evidences lead to the conclusion that the present synthesized sample is Fe 3 O 4 .Figure 2 shows the transmission electron micrograph and selected area electron diffraction pattern (SAED) for Fe 3 O 4 nanoparticles.It is confirmed from the SAED pattern that there is no other phase impurity present in the sample.The calculated average value of lattice constant 'a' from SAED pattern is 0.837 nm.It again confirms the formation of Fe 3 O 4 nanoparticles.The histogram showing the particle size distribution is plotted from the micrograph after measuring the size of 105 particles as shown in figure 3.This histogram is seen to peak at 10 nm.The histogram is fitted using log-normal distribution function in particle size f (D) and is given as ( ) . Here s and n are the parameters characterizing the log-normal distribution.The value of s = 0.24 and n = 9.6 nm.The mean and standard deviation can be found out from this function as n e s 2 and ( ) , respectively.From this fitting, the mean particle size turns to be 9.9 nm with a standard deviation of 2.4 nm.This particle size distribution is narrow in nature.

Magnetization
Zero field cooling (ZFC) and field cooling (FC) susceptibility χ is measured as a function of temperature T as shown in figure 4.These curves are measured in the presence of 250 Oe external applied magnetic field.The value of blocking temperature is dependent on the applied magnetic field.The peak in the zero-field curve shows the blocking temperature T B of the system at 100 K.The temperature where both the curves bifurcate is called bifurcation temperature T bf which is observed at 170 K. Above T bf , the thermal energy of the particles becomes dominant and overcomes the energy barrier.The particle magnetic moment gets unblocked above blocking temperature.Figure 5 shows M-H loops for Fe 3 O 4 nanoparticles at 10 and 300 K.At low temperature, the M-H curves show hysteresis because the particle magnetization vector relaxes slowly, and it appears that the magnetic moments of the particles are blocked.
In order to study the magnetic behavior in the given system, the magnetization M as a function of applied magnetic field H data are measured at 200, 225, 250, 275 and 300 K as shown in figure 6.The magnetization is seen to be increasing with increase in the strength of applied magnetic field because the magnetic moments tend to align with the direction of the magnetic field.There is observed a decrease in magnetization with temperature which is due to the random flipping of magnetic moments with an increase in thermal energy.The magnetization of Fe 3 O 4 nanoparticles is less than the bulk value due to the reduction in particle size [38,39].It is considerably smaller than the bulk Fe 3 O 4 i.e., 92 emu/g [10,40].It can be attributed to the fact that due to reduction in particle size, the surface spins become more disordered.This magnetic disordered surface layer around the ordered magnetic core suggests a core-shell morphology.The contribution to the magnetization due to this disordered surface spin is assumed to be negligible as compared to the core of the particle.One can easily find this thickness of magnetic disordered surface layer of a nanoparticle system by the given formula [38] where M B is the bulk magnetization of the sample at a given temperature, D is the particle size and h is the thickness of the disordered layer.The magnetization curves in figure 6 are used to study the magnetic behavior of Fe 3 O 4 nanoparticles by fitting procedure in an appropriate magnetization expression.A non-linear least square method is used for the fitting of the magnetization data.The computer codes for the magnetization expressions are written in PYTHON.

Modelling
The modified Langevin equation is used to describe the magnetization process above blocking temperature and is given by [21] Here, μ is the particle magnetic moment, k is the Boltzmann constant.M 0 is the saturation magnetization and is equal to Nμ, where N is the number density of particles and where χ is the susceptibility.The magnetization curves are fitted using this equation as shown in figure 7. It is seen that the experimental and the fitted data deviate from each other.The quality of the fits is determined by R 2 called as coefficient of determination.If the value of the coefficient of determination is remarkably close to 1 then the quality of fits is assumed to be good.A similar kind of deviation in magnetization data in other magnetic nanoparticles system is also found in others work [7][8][9][10][11].The estimated fit parameters obtained using this equation are shown in table 1. Figure 8 shows the temperature dependence of magnetic  moment and saturation magnetization.The value of magnetic moment is found to be linearly increasing and the saturation magnetization is decreasing with increasing temperature.This is not the expected behavior of particle magnetic moment in a system of nanoparticles.The value of magnetic moment per particle is given as M 0 ρV, where V is the particle volume and ρ is the density of magnetite.The extrapolation of M 0 in figure 8 gives 25.7 emu/g at T = 0 K. if one assumes the particles of spherical shape then the volume is given as , D is the diameter of the particle.Assuming the core-shell morphology of the sample then using equation (3) and the bulk value M B for Fe 3 O 4 at room temperature is 92 emu/g [10], it is found that the thickness of this spin disordered layer around the particle is about 1.7 nm which gives the magnetic size of 8.3 nm.The density of Fe 3 O 4 is 5.50 g/cm 3 , then the particle magnetic moment turns out to be 4500 μ B per particle.This value of magnetic moment per particle is very small as compared to the obtained in the fitting results.These yielded results are not according to the expected behavior and can attributed to the ignorance of consideration of several factors in magnetization expression.The most important factor that affects magnetization is particle magnetic moment distribution [9].Now, if one considers a log-normal distribution in magnetic moments in equation (4) it becomes [12] 0 The function f (μ) represents the log-normal distribution in particle magnetic moment.Now, the magnetization data in figure 6 is fitted using this equation.The fitted magnetization data are seen to be well passed through the experimental data as shown in figure 9.The values of magnetic fit parameters are shown in table 2. The value of  coefficient of determination shows that the quality of the fits is very good.The experimental and fitted data are passing through each other.The values of mean magnetic moment is also found to be close to the expected value.The particle size found from these values is very close to the magnetic size.Figure 10 shows the variation of mean magnetic moment 〈μ〉 and the saturation magnetization N〈μ〉 as a function of temperature T and as per the expected behavior.The value of the width parameter s in this case is found to be large which shows the magnetic moment distribution is broad and is overestimating the magnetic moment distribution.It is known that the   distribution in magnetic moment arises due to the presence of particle size distribution.From TEM, it is shown that the sample has a narrow distribution in particle size.In a study of maghemite nanoparticles coated with oleic acid and having narrow distribution in particle size, it is observed that the equation (4) and equation (5) gives the comparable results [12].Because the particles surface is modified with oleic acid.The oleic acid reduces the interactions of particles with other particles and results in the suppression of surface effects by covalently bonded to the surface of the particle [34,41].In the present ferrimagnetic system of Fe 3 O 4 nanoparticles the results   yielded for both the equations are different.In a study of antiferromagnetic ferritin nanoparticles similar kind of observations are reported, the width parameter 's' is found to be wide when magnetization curves are fitted at temperature data 30 and 65 K, although the sample has narrow particle size distribution [9].The fitted magnetization data of antiferromagnetic NiO which considers the role of magnetic moment distribution at temperature 320 and 350 K also results in a wide 's' parameter [11].This is due to the inner or surface magnetic disorder causing the presence of uncompensated magnetic moment results in the broadness of magnetic moment distribution.The magnetic moment distribution in the system arises due to the dependence of particles size distribution but it is contradicting the value of 's' that is indicating the wide magnetic moment distribution.It means there is another possibility that is contributing to this magnetic moment distribution.It can be explained on the basis of surface effects that accounts due to the disordered surface spins.The broken bonds and the missing neighbor atoms on the surface of the particle are the cause of disordered surface spins [42].These disordered spins cause spin frustrations on the surface of the particle.The increased effect of this magnetic disordered surface becomes more prominent with the reduction in particle size, roughness of the surface, low crystallinity and due to the synthesis method [34,41].The effects like spin canting, uncompensated magnetic moment, spin glass behavior etc. arise on the surface of the particles [43,44].
In the present analysis, it is found that the magnetic moment distribution is overestimated.If K is the effective magnetic anisotropy constant, then it is given by = K kT V 25 B .Assuming the spherical shape of nanoparticles, the value of K for this system is obtained is 6.59X10 4 J/m 3 .The value of magnetic anisotropy constant for bulk Fe 3 O 4 is K bulk is 1.35X10 4 J/m 3 [27,33].The effective magnetic anisotropy constant is the combination of bulk and surface anisotropy and given by = + K K bulk K D 6 surface , where K surface is the surface magnetic anisotropy constant [45].From this, it is concluded that the surface anisotropy is large in this system.This large value of surface anisotropy is associated with spin canting [46].The increase in surface anisotropy also suggests the increased particle interactions which modifies the magnetic behavior.Therefore, because of the magnetic disordered surface the magnetic anisotropy and particle interactions becomes prominent on the surface of nanoparticles [47].These factors significantly affect the estimation of magnetic moment distribution in the present analysis.The spin frustrations on the surface causes the overestimation of magnetic moment distribution.The consideration of magnetic anisotropy and particles interactions in the magnetization analysis can enhance the quality of fits and yield more better meaningful magnetic fit parameters but it makes the analysis more complicated and tedious.Further the investigations of these factors on magnetization of this system will be reported in future studies.

Conclusion
This work presents a study on the magnetization of Fe 3 O 4 nanoparticles system.The particle size is found to be 10 nm from transmission electron micrograph.The size of the spins disordered layer is about 1.7 nm.The blocking temperature of the sample is found to be 100 K.The magnetization data as a function of temperature above blocking temperature is analyzed by considering the effect of magnetic moment distribution.The magnetic moment distribution is found to be important when analyzing the magnetization data and significantly improves the results.It is found that this distribution of magnetic moment in the sample is not only dependent on the distribution of particles size.The surface effects i.e., surface magnetic anisotropy and particle interactions that come in to play due to the magnetic disordered shell of the particles cause the overestimation of magnetic moment distribution in the present analysis.The role of surface effects in the analysis of magnetization data are very important and should always be considered while analyzing the magnetization data.

Figure 1 .
Figure 1.Room temperature x-ray diffraction pattern of Fe 3 O 4 nanoparticles.The labeling are miller indices (hkl) and interplanar spacing 'd' correspond to Fe 3 O 4 planes .

Figure 3 .
Figure 3. Fitting of histogram obtained from transmission electron micrograph with log-normal distribution in particle size.

Figure 4 .
Figure 4. ZFC (solid symbol) and FC (open symbol) susceptibility χ as a function of temperature T for Fe 3 O 4 nanoparticles in 250 Oe applied magnetic field.

Figure 5 .
Figure 5. M versus H loops of Fe 3 O 4 nanoparticles at 10 and 300 K.

Figure 8 .
Figure 8.The variation of magnetic moment μ and saturation magnetization M 0 as a function of temperature T.

Figure 10 .
Figure 10.The variation of mean magnetic moment 〈μ〉 and saturation magnetization N〈μ〉 as a function of temperature T.

Table 1 .
Values of fit parameters M 0 , μ, χ and R 2 obtained by fitting the magnetization data to equation (4) for Fe 3 O 4 sample.

Table 2 .
Values of fit parameters N, s, n, χ and R 2 to equation (5) for Fe 3 O 4 samples.The values of 〈 μ〉 and N〈μ〉 are calculated.