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Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013, Article ID 609819, 10 pageshttp://dx.doi.org/10.1155/2013/609819

Research ArticleSize Effects on Magnetic Properties of Ni0.5Zn0.5Fe2O4Prepared by Sol-Gel Method

Min Zhang,1 Zhenfa Zi,1 Qiangchun Liu,1,2 Peng Zhang,1,2

Xianwu Tang,1 Jie Yang,1 Xuebin Zhu,1 Yuping Sun,1 and Jianming Dai1

1 Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China2 School of Physics and Electronics Information, Huaibei Normal University, Huaibei 235000, China

Correspondence should be addressed to Jianming Dai; [email protected]

Received 8 May 2013; Revised 28 June 2013; Accepted 28 June 2013

Academic Editor: Yong Ding

Copyright © 2013 Min Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ni0.5Zn0.5Fe2O4 particles with different particle sizes have been synthesized by sol-gelmethod. X-ray diffraction results show that allthe samples are pure cubic spinel structure with their sizes ranging from 9 to 96 nm.The lattice constant significantly decreases withfurther increasing annealing temperature. The magnetic measurements show superparamagnetic nature below the particle size of30 nm, while others show ferrimagnetic nature above the corresponding blocking temperature.The blocking temperature increaseswith the increase in particle size, which can be explained by Stoner-Wohlfarth theory.The saturationmagnetization increases as theparticle size increases, which can be explained by the cation redistribution on tetrahedral A and octahedral B sites and the domainwall motion. The variation of coercivity as a function of particle size is based on the domain structure.

1. Introduction

Spinel ferrites have attracted more and more attention due totheir various technological applications in some fields, suchas microwave absorption, high-speed digital tape, ferrofluid,magnetic recording, and photomagnetic materials [1–5].Among the spinel ferrites, nickel zinc ferrite is one of themost versatilemagneticmaterials as they have high saturationmagnetization, high Curie temperature, excellent chemicalstability, low coercivity, and biodegradability [6]. It is a mixedspinel structure based on a face-centered cubic lattice of oxy-gen ions, with functional units of (Zn

𝑥Fe1−𝑥)[Ni1−𝑥

Fe1+𝑥]O4.

Zn2+ andNi2+ ions are known to have very strong preferencesfor the tetrahedral A and octahedral B sites as depictedby curled and square brackets [2], respectively, while Fe3+ions partially occupy the A and B sites. In the case ofZn1−𝑥

Ni𝑥Fe2O4ferrite, it was found that for 𝑥 greater than

0.5, Fe3+ moments in A and B sites have collinear arrange-ment, whereas for 𝑥 less than 0.5, Fe3+ moments in the Bsite have noncollinear arrangement [7]. The compositionalvariation can result in the redistribution of metal ions in theA and B sites, which can modify the properties of nickel

zinc ferrites.The nickel concentration effect on structure andmagnetic of Ni

𝑥Zn1−𝑥

Fe2O4has been reported [8].The result

showed the superparamagnetic nature of the samples for 𝑥 =0.1 and 𝑥 = 0.3 whereas the material showed ferromagneticfor 𝑥 = 0.5, but the crystallite size increased unobviouslyfrom 12 to 17 nm corresponding 𝑥 = 0.1 to 𝑥 = 0.5.Therefore,the nickel concentration played an important role in deter-mining the magnetic properties of Ni

𝑥Zn1−𝑥

Fe2O4. In our

previous studies [9], it is found that Ni0.5Zn0.5Fe2O4(NZFO)

presents the best magnetic and microwave absorption abilityinNi𝑥Zn1−𝑥

Fe2O4system.Therefore, it is necessary to further

study the magnetic properties of NZFO nanoparticles.George et al. [10] have studied finite size effects on the

structural andmagnetic properties ofNiFe2O4powders.They

found the specific saturation magnetization decreased withdecreasing grain size, whichmay be due to noncollinearmag-netic structure and surface effects. The coercivity reached amaximumwhen the grain size was 15 nm and then decreasedas the grain size increased further, which can be explainedon the basis of domain structure. Chen and Zhang [11] havereported size effects on magnetic of MgFe

2O4spinel ferrite

nanocrystallites. The MgFe2O4nanoparticles showed typical

2 Advances in Materials Science and Engineering

superparamagnetism, which unambiguously correlated withthe particle size from 6 to 18 nm. However, few groups haveinvestigated size effects on magnetic properties of NZFOferrite prepared by sol-gel method.

In the present work, the sol-gel method has been used toprepareNZFOwith different heat treatment temperatures. Tothe best of our knowledge, it is the first time to systematicallydemonstrate size effects of nanocrystallite ferrite on themagnetic behavior. The possible mechanism is discussedhere.

2. Experimental

In order to synthesize NZFO, stoichiometric amounts ofnickel nitrate, zinc nitrate, and iron nitrate were dissolved indeionized water under heating and magnetic stirring. Afterstirring for 30min, citric acid was slowly added to the mixednitrates solution. The mole ratio of citric acid and total metalions was controlled to be 1.5 : 1. Urea was added to adjustthe pH value to 7. The mixed solution was stirred at 80∘Cuntil forming viscous brown gel. Then, the viscous brown gelwas placed in the oven at 80∘C for 1-2 days to obtain a drygel. The as-burnt powders were obtained when the dry gelwas calcinated at 350∘C for 3 h. Finally, the as-burnt powderswere annealed in the muffle furnace at different temperaturesin the range 400–1100∘C in steps of 100∘C for 2 h with aheating rate of 5∘C/min in air. The as-burnt powders withdifferent annealing temperatures were named as NZFO-350,NZFO-400, NZFO-500, NZFO-600, NZFO-700, NZFO-800,NZFO-900, NZFO-1000, and NZFO-1100, respectively.

Phase analysis of the products was performed by PhilipsX’pert PRO X-ray diffractometer with Cu K

𝛼radiation. TEM

(JEM-2010) was used to show the morphology and particlesize distribution. The magnetic properties of the NZFOferrite powders were measured by using a superconductingquantum interference device magnetometer measurementsystem (SQUID, MPMS-5T). Zero-field-cooling (ZFC) andfield-cooling (FC) magnetization curves were performed inthe temperature range between 5 and 350K under an appliedmagnetic field of 100Oe.

3. Results and Discussion

3.1. Structure and Morphology. Figure 1 shows XRD patternssamples treated under different annealing temperatures. TheXRD patterns have a good agreement with the standardJCPDS cards for nickel zinc ferrite (card no. 08-0234), whichconfirms single phase cubic spinel structure (space group𝐹𝑑3𝑚) of ferrite samples. Figure 1(a) shows that the as-burntsample appears diffraction peak of spinel ferrite, but thecrystallinity is still relatively low, with less defined diffractionpeaks. Figure 1 shows that the corresponding diffractionpeaks become narrower and sharper with increasing anneal-ing temperature, which indicates the growth in crystallitesize [12] and much better crystallinity. It is expected that ifone introduces annealing temperature in the system muchhigher, the molecular concentration at the crystal surface willincrease and hence the crystal growth will be promoted [13].

In addition, a higher temperature can enhance the atomicmobility and make grains get more energy to grow up.

The lattice constant 𝑎 for the samples is shown in Table 1.The 𝑎 value obviously decreases as the annealing temperatureincreases from 400 to 700∘C. The sample calcined at lowertemperature is partially crystallization. So, surface defects canoccur within the lattice, but the crystallization will enhancewith the increase of annealing temperature, which can resultin lattice contraction.The 𝑎 is also observed to increase as theannealing temperature increases from 700 to 800∘C but againdecrease for samples annealed at above 800∘C. This increaseand decrease of 𝑎 could be attributed to redistribution ofcations between tetrahedral and octahedral sites and zincloss from the sample [14], respectively. In addition, theredistribution of cations between tetrahedral and octahedralsites is also supported by the magnetic measurement, asdiscussed later in this paper. Figure 1(b) shows refinementvalue of XRD pattern for NZFO-1000.The similarity betweenthe experimental and simulated pattern confirms single phasecubic spinel structure of nanoparticles.The average crystallitesize for all NZFO nanoparticles is calculated from intensity(220), (311), (511), and (440) peaks by the Debye-Scherrerequation

𝐷 =0.9𝜆

𝛽 cos 𝜃, (1)

where 𝐷 is the crystallite size, 𝜆 is the wavelength of CuK𝛼(1.540598 A), 𝜃 is the angle of Bragg diffraction, and 𝛽

is the full width at half maxima (FWHM) broadening. Theobtained crystallite size at different annealing temperature islisted in Table 1. The calculation results show that crystallitesize increases from 9 nm to 96 nm with increasing annealingtemperature, whichmay be due to the increasing crystallinity.

In addition, theWilliamson andHall (W-H) plots [15] arealso used to calculate the crystallite size. The equation is asfollows:

𝛽 cos 𝜃 = 𝜀 (4 sin 𝜃) + 𝜆𝐷, (2)

where 𝛽 (FWHM in radian) is measured for different XRDlines corresponding to different planes, 𝜃 is the Bragg angle,𝜀 is the strain, and 𝐷 is the crystallite size. Equation (2)represents a straight line between 4 sin 𝜃 (x-axis) and 𝛽 cos 𝜃(y-axis). The values of 𝜀 and 𝐷 are obtained by the slope (𝜀)and intercept (𝜆/D) of line, respectively. The strain rapidlyincreases for smaller crystallite which can be due to theincreasing defect density. The value of 𝜀 obviously decreaseswith increasing annealing temperature (listed in Table 1),which are consistent with those calculated from 𝑎 values.Figure 2 shows the linear fitting W-H plots of NZFO-500,NZFO-600, NZFO-800, and NZFO-1000. From the param-eters of linear fitting, the 𝜆/D values are 0.01528, 0.00475,0.00432, and 0.00206, respectively, corresponding to NZFO-500, NZFO-600, NZFO-800, and NZFO-1000.Therefore, thecalculation 𝐷 values are 10 nm, 32 nm, 36 nm, and 75 nm,respectively, which are consistent with the previous calcu-lated crystallite size by the Debye-Scherrer equation. So theannealing temperatures play an important role in controllingthe crystallite size of the nanocrystallites.

Advances in Materials Science and Engineering 3

(a) (b)Rp = 2.312% Exp

NZFO-400 CalRwp = 2.944% Exp-Cal

𝜒2 = 0.859 Mar

NZFO-350 NZFO-1000

(220

) (311

)

Inte

nsity

(a.u

.)

Inte

nsity

(a.u

.)

Inte

nsity

(a.u

.)

(222

)

(400

)

(422

)

(511

)

(440

)

(620

)

(533

)2𝜃 (deg)2𝜃 (deg)

NZFO-1100

NZFO-1000

NZFO-900

NZFO-800

NZFO-700

NZFO-600

NZFO-500

2𝜃 (deg)20 30 5040 60 70 80

20 30 60 70 8020 30 5040 504060 70 80

Figure 1: XRD patterns of Ni0.5Zn0.5Fe2O4at different annealing temperatures. Inset (a) shows XRD patterns of as-burnt powders and

annealed at 400∘C. Inset (b) is refinement result of XRD patterns for NZFO-1000.

Table 1: Structural and magnetic parameters of Ni0.5Zn0.5Fe2O4 ferrites from 350 to 1100∘C.

𝑇 (∘C) 𝐷 (nm) 𝑎 (A) (𝜀2)1/2

× 10−4𝑀𝑠(𝜇B/f.u.) 𝐻

𝑐(Oe) 𝑀

𝑟(𝜇B/f.u.)

300K 300K 10K 300K350 9 — — 0.26 0 478 0400 10 8.3965 — 0.79 3.6 687 0.0017500 13 8.3922 76.7 1.53 5.6 530 0.0085600 30 8.3775 13.7 2.03 58.2 114 0.21700 33 8.3750 13.5 2.77 58.0 — 0.19800 46 8.3792 10.7 2.87 50.8 117 0.26900 81 8.3776 1.57 3.12 32.3 — 0.171000 88 8.3766 2.15 3.09 25.4 44 0.121100 96 8.3774 2.14 3.14 15.1 — 0.06

Figure 3 shows the TEM morphology of NZFO-600 andNZFO-800.Theparticles are similar spherical and polyhedralshapes. For NZFO-800 sample, classical polygonal grain andgrain boundary morphologies are present, which shows ahigher degree of crystallinity than that of NZFO-600 sample.The insets in Figures 3(a) and 3(c) are particle size distribu-tion graph by counting 200 nanoparticles. The histogram ofthe size distribution is characterized by a Gaussian function

(solid line). It is found that average particle size of NZFO-600and NZFO-800 is obtained as 23 nm and 46 nm, respectively,which are in agreement with those of the XRD patterns.Therefore, the average crystallite size obtained from XRDanalysis is taken as the average particle size. High-resolutionTEM (HRTEM) analysis is employed to determine the crystalfacets and orientation, as shown in Figures 3(b) and 3(d).In Figure 3(d), two sets of lattices are present and they are


NZFO-500

1.0 1.5 2.04 sin 𝜃

𝛽cos𝜃

0.04

0.02

0.00

(a)

NZFO-600

1.0 1.5 2.04 sin 𝜃

𝛽cos𝜃

0.04

0.02

0.00

(b)

NZFO-800

1.0 1.5 2.04 sin 𝜃

𝛽cos𝜃

0.02

0.01

0.00

−0.01

−0.02

(c)

NZFO-1000

1.0 1.5 2.04 sin 𝜃

𝛽cos𝜃

0.02

0.01

0.00

−0.01

−0.02

(d)

Figure 2: Williamson and Hall plot graphs for NZFO-500, NZFO-600, NZFO-800, and NZFO-1000.

oriented at a certain angle with the interfringe spacing of0.24 nm and 0.25 nm, corresponding to spinel (222) and (311)lattice planes of NZFO-800 ferrite.

3.2. Magnetic Properties. Figure 4 shows the room temper-ature M-H curves for Ni-Zn ferrites particles with differentparticle sizes. A similar room temperatureM-H curve resultwas also observed by Jiang et al. [16]. The inset in Figure 4shows the magnified view of theM-H curves at lower appliedfield, which shows that hysteresis appears obviously whencalcinated at 600∘C. In Figure 4, the NZFO-350, NZFO-400,and NZFO-500 samples exhibit nonsaturated magnetizationeven at the maximum applied field of 10 kOe, and thecoercivity (𝐻

𝑐) and the remanent magnetization (𝑀

𝑟) are

almost zero, which indicate the superparamagnetic nature.The magnetic moment is obtained using nonlinear curve fitof Langevin function. The function is expressed as [17]

𝑀 = 𝑀𝑠(coth(

𝜇𝐻

𝑘𝐵𝑇) −𝑘𝐵𝑇

𝜇𝐻) , (3)

where 𝜇 = 𝑀𝑠𝜋𝐷3/6 is the true magnetic moment of

each particle, 𝑀𝑠is the saturation magnetization, 𝑘

𝐵is the

Boltzmann constant, and 𝑇 is the absolute temperature. Thefit results are displayed in Figure 5.

Figure 6 shows M-H loops of the nanoparticles withdifferent particle sizes at 10 K. NZFO nanoparticles showtypical hysteresis behaviors. The values of 𝐻

𝑐are listed in

Table 1.The variation of 𝑀

𝑠and 𝐻

𝑐with different particle size

is shown in Figure 7. The magnetic moment for formulaunit in Bohr magneton is calculated and the obtained dataare displayed in Table 1. It is seen that 𝑀

𝑠decreases as the

particle size decreases. Kumar et al. [18] had reported that theexistence of spin canting, cation distribution, and disorderedsurface layer could result in decreased𝑀

𝑠. The surface effects

become significant as the particle size decreases, which canlead to the decrease of 𝑀

𝑠. Another possible factor is the

redistribution of cations between A and B sites, which growsthe net magnetic moment. According to Neel’s two sublatticemodel of ferrimagnetism, (Zn2+

0.5Fe3+0.5)[Ni2+

0.5Fe3+1.5]O4


NZFO-600200 particles counted

Num

ber o

f par

ticle

s

0

10

20

30

40

50

60

10 20 30 40 50 60

Particles diameter (nm)

50nm

(a)

5nm

(b)

NZFO-800200 particles

Num

ber o

f par

ticle

s

0

10

20

30

40

20 30 40 50 60 70 80Particles diameter (nm)

50nm

counted

(c)

5nm

(d)

Figure 3: (a) TEM image of NZFO-600 ferrite: the inset is particle size distribution graph, (b) high magnification TEM image for NZFO-600ferrite, (c) TEM image of NZFO-800 ferrite: the inset is particle size distribution graph, and (d) highmagnification TEM image for NZFO-800ferrite.

configuration has 6 𝜇𝐵for formula unit. The value is higher

than our 𝑀𝑠value, which confirms cation disorder and

redistribution. Sreeja et al. [19] had confirmed an abnor-mal cation distribution of Ni

0.5Zn0.5Fe2O4with different

annealing temperature by Mossbauer spectroscopic study.At lower sintering temperatures, weaker magnetic superex-change interaction and lattice defects can also lead to thesmaller value of 𝑀

𝑠[20]. Figure 7 shows that 𝐻

𝑐increases

rapidly as particle size increases with a maximum value of58.2Oe at 600∘C (30 nm) and then decreases with furtherincreases in particle size. The same observation of𝐻

𝑐change

with particle size inNi-Zn ferritewas reported in earlier study[16]. From the inset of Figure 7, 𝐻

𝑐increases as the particle

size increases, reaches a maximum value, and then decreasesat 10 K as well as at 300K. The values of 𝐻

𝑐decrease as the

temperature ofmeasurement increases. A critical particle sizeof 10 nm is obtained at 10 K.The critical particle size decreasesas the temperature of measurement decreases from 300K to10K. Thus, in Figure 7, the peak value of 𝐻

𝑐has shifted to

the lower particle size when the temperature decreases from300K to 10K. A similar result had been reported by Georgeet al. [10]. The values of 𝐻

𝑐and𝑀

𝑟near to zero for NZFO-

350, NZF-400, and NZFO-500 display superparamagnetic

nature at 300K. Generally, the𝐻𝑐for magnetic nanoparticles

is closely related to their particle size. Smaller particle sizescorrespond to a lower𝐻

𝑐.

This variation of the𝐻𝑐with particle size can be explained

on the basis of domain structure, critical size, and the surfaceand interface anisotropy of the crystal. A crystallite willspontaneously break up into a number of domains in order toreduce the large magnetization energy if it is a single domain.The ratio of the energy before and after division into domainsvaried as√𝐷 [10], where𝐷 is the particle size. So, the energyreduces as 𝐷 decreases, which suggests that the crystalliteprefers to remain single domain behavior for quite small𝐷.

In the single domain region, the variation of coercivity asa function of particle size is expressed as [17]

𝐻𝑐= 𝑔 −

ℎ

𝐷3/2, (4)

where𝑔 and ℎ are constants and𝐷 and𝐻𝑐are particle size and

coercivity. Therefore, 𝐻𝑐increases as 𝐷 increases in below a

critical particle size.


0.25

0.00

−0.25

−0.15 0.00 0.15

1.0

0.5

0.0

−0.5

−1.0−0.1 0.0 0.1

3

2

1

0

−1

−2

−3

−10 −5 0 5 10H (kOe) H (kOe)

H (kOe)

H (kOe)

M(𝜇

B/f.

u.)

M(𝜇

B/f.

u.)

M(𝜇

B/f.

u.)

M(𝜇

B/f.

u.)

NZFO-350

NZFO-400

NZFO-500

NZFO-1100NZFO-1000NZFO-900NZFO-800NZFO-700

NZFO-600

NZFO-1100

NZFO-1000

NZFO-900

NZFO-800

NZFO-700

NZFO-600

NZFO-500NZFO-400NZFO-350

NZFO-1100NZFO-1000NZFO-900NZFO-800NZFO-700

NZFO-600NZFO-500NZFO-400NZFO-350

T = 300K

Figure 4: Room-temperatureM-H loops forNi-Zn ferrites with different particle sizes.The inset shows themagnified view of theM-H curvesat lower applied field for a series of sample.

2

1

0

−1

−2−10 −5 0 5 10

H (kOe)

M(𝜇

B/f.

u.)

NZFO-350NZFO-400NZFO-500

T = 300K

Figure 5: The nonlinear curve fit of Langevin function for NZFO-350, NZFO-400, and NZFO-600.

6

4

2

0

−2

−4

−6

−10 −5 0 5 10H (kOe)

M(𝜇

B/f.

u.)

NZFO-350NZFO-400NZFO-500NZFO-1000

NZFO-800NZFO-600

T = 10K

Figure 6: The M-H loops for Ni-Zn ferrites with different particlesizes at 10 K.


3

2

1

0

−1

Saturation magnetizationCoercivity

Coe

rciv

ity (O

e)

Coe

rciv

ity (O

e)

Satu

ratio

n m

agne

tizat

ion

(𝜇B

/f.u.

)T = 10K

0

150

300

450

600

750

0 20 40 60 80 100

−200 20 40 60

0

20

40

60

80 100 120Particle size (nm)

Particle size (nm)

T = 300K

Figure 7: Saturation magnetization and coercivity at 300K versus particle size of the nanosized Ni-Zn ferrites. The inset shows particle sizedependence of the coercivity at 10 K.

In multidomain region, the particle size dependence ofthe coercivity is expressed as

𝐻𝑐= 𝑎 +𝑏

𝐷, (5)

where 𝑎 and 𝑏 are constants and 𝐷 is particle size. So,the coercivity decreases as particle size increases above acritical particle size. These equations’ analysis results arecorresponding to our experimental results.

As a result, it indicates a critical particle size for thetransition from single domain tomultidomain behavior closeto 30 nm at 300K. In the spherical particle model, the criticalsize from single domain to multidomain can be calculatedwith the following formula [21]:

𝐷𝑚=9𝜎𝑤

2𝜋𝑀2𝑠

, (6)

where 𝜎𝑤= (2𝑘

𝐵𝑇𝐶|𝐾1|/𝑎)1/2 is the domain wall energy,

𝑘𝐵is Boltzmann constant, 𝑇c is Curie temperature, 𝐾

1is

magnetocrystalline anisotropy constant, a is the lattice con-stant, and𝑀

𝑠is the saturation magnetization. The particle is

considered to be single domain below𝐷𝑚, while the particles

are multidomain above 𝐷𝑚. For NZFO, 𝑇c = 538K, 𝑎 =

8.39 × 10−8 cm, |𝐾1| = 1.7 × 104 erg/cm3, and𝑀

𝑠= 310Gs.

From (6), the calculation value of𝐷𝑚is about 25.9 nm, which

is almost consistent with the experimental result (30 nm).So, the average particle size of NZFO-600 is close to thecritical size. For 𝐷 < 𝐷

𝑚, these nanocrystallites are single

domain, and the surface effect becomes important. In theprepared NZFO powders, the low magnetization value canbe attributed to noncollinear surface spins that present in thesurface of nanoparticles. As a result, the𝑀

𝑠decreases, which

is confirmed by the experimental curve of𝑀𝑠that decreases

more rapidly as the values of 𝐷 decrease. For 𝐷 > 𝐷𝑚,

the magnetic domain structure appears. Compared with thesingle domain, multidomain particles require fewermagneticfields to switch for domain wall motion, which improvessaturation magnetization [22]. Moreover, the surface effectsbecome weak for larger particle size and the sample reachesthe saturation of bulk ferrite. Thus, the experimental curveof 𝑀𝑠versus 𝐷 in Figure 7 is almost horizontal for higher

value of 𝐷. This can also be due to the low strain value. So,the effects of strain on the magnetic of NZFO-900, NZFO-1000 and NZFO-1100 can be ignored.

Figure 8 shows the variation of magnetization with tem-perature (5K < 𝑇 < 350K) curves in an external fieldof 100Oe recorded in ZFC and FC modes. The ZFC-FCcurves separation at lower temperature can be speculatedas a high field irreversibility (𝑀FC > 𝑀ZFC) behaviorbelow a certain temperature, irreversibility temperature, 𝑇irr.The irreversibility behavior also indicates that there is anonequilibrium magnetization state. The difference between𝑀ZFC and 𝑀FC values become much larger at a certaintemperature with increasing particle size for all examples,which may have some reasons as follows.𝑀FC and𝑀ZFC ofdifferent magnetic systems are found to be related throughthe expression [23]

𝑀ZFC ≈ 𝑀FC𝐻app

𝐻app + 𝐻𝑐, (7)

𝐻app + 𝐻c ≈ {𝐻app, 𝐻app ≫ 𝐻𝑐,

𝐻𝑐, 𝐻

𝑐≫ 𝐻app,

(8)

where 𝐻app and 𝐻𝑐are the applied field and coercivity,

respectively. Here, 𝐻app = 100Oe, and 𝐻𝑐varies with

particle size.𝑀ZFC value is calculated by using the previousexpression from the measured 𝑀FC at different annealing


NZFO-350M

(𝜇B

/f.u.

)

0.05

0.04

0.03

0.02

0.01

FC

ZFC

TB

0 150 300T (K)

(a)

NZFO-400

M(𝜇

B/f.

u.)

FC

ZFC

TB

0 150 300T (K)

0.12

0.08

0.04

(b)

NZFO-500

M(𝜇

B/f.

u.)

0.24

0.18

0.12

0.06

FC

ZFC

TB

0 150 300T (K)

(c)

NZFO-1000

M(𝜇

B/f.

u.)

FC

ZFC

0 150 300T (K)

0.7

0.6

0.5

0.4

(d)

Figure 8: Temperature dependence of magnetization for field cooled (FC) and zero-field cooled (ZFC) Ni0.5Zn0.5Fe2O4nanoparticles at

applied field of 100Oe.

temperatures. For example, according to (7), the value of𝑀ZFC and 𝑀FC will be almost identical at 300K because𝐻app = 100Oe compared to 𝐻

𝑐= 3.6Oe for NZFO-400,

which can be further proven ZFC and FC curve shapes ofFigures 8(a), 8(b), and 8(c).

The ZFC magnetization curves appear maximum at theblocking temperature 𝑇

𝐵at which the relaxation time equals

the time scale of the magnetization measurements. FromFigures 8(a), 8(b), and 8(c) curves, the ZFC and FC are almostoverlapped above 𝑇

𝐵, indicating the presence of the small-

sized particles [24]. The 𝑇𝐵value of NZFO-350, NZFO-400,

NZFO-500 are 121 K, 123 K, and 208K, respectively, indicat-ing that the different particle size is characterized by differentaverage energy barrier. Note also that the measured 𝑇

𝐵value

of NZFO above annealing temperature of 600∘C is higherthan 350K, as shownFigure 8(d).TheobtainedM-T curves ofNZFO above annealing temperature of 600∘C are similar, soonly theM-T curves of NZFO-800 is shown. According to theStoner-Wohlfarth theory, the magnetocrystalline anisotropy

𝐸𝐴of a single domain particle can be approximated as follows

[25]:

𝐸𝐴= 𝐾𝑉sin2𝜃, (9)

where 𝐾 is the magnetocrystalline anisotropy constant, 𝑉 isthe volume of the nanoparticle, and 𝜃 is the angle betweenthe magnetic direction and the easy axis of the nanoparticle.When 𝐸

𝐴is comparable with thermal activation energy,

𝑘𝐵𝑇 with 𝑘

𝐵as the Boltzmann constant, the magnetization

direction of the nanoparticle starts to fluctuate and goesthrough rapid superparamagnetic relaxation. The 𝑇

𝐵is the

threshold point of thermal activation. Above 𝑇𝐵, thermal

activation can overcome the anisotropy energy barrier andthe nanoparticles become superparamagnetic with the mag-netization direction randomly flipping.Therefore, a variationof magnetization with applied field at 300K (𝑇 > 𝑇

𝐵) is

shown in Figure 4, the particles have adequate thermal energy𝑘𝐵𝑇 to attain complete thermal equilibrium with the applied

field during the measurement time, and, hence, hysteresis


disappears. In single domain, larger particles possess a higher𝐸𝐴and require a larger 𝑘

𝐵𝑇 to become superparamagnetic.

Therefore, 𝑇𝐵increases as particle size increases. Below 𝑇

𝐵,

the thermal energy is no longer able to overcome the magne-tization anisotropy energy barrier, remanent magnetizationand coercivity appear and then exhibit a hysteretic feature,just as shown in Figure 6. According to previous analysis, allthe analysis results of ZFC and FC curves are in agreementwith magnetization curves.

4. Conclusion

XRD analysis reveals that all samples are the single phasecubic spinel structure, and higher annealing temperaturecould lead to lattice shrinkage and grain growth.The strain isalso induced during the annealing process. The particle sizefrom TEM morphology is in close agreement with the crys-tallite size by W-H plots. The room magnetic measurementshows superparamagnetic nature for NZFO-350, NZFO-400, and NZFO-500 ferrites, and others show ferrimag-netic nature.The room temperature saturationmagnetizationincreases as particle size increases, with a maximum valueof 3.14 𝜇

𝐵/f.u. corresponding the particle size of 96 nm. The

coercivity increases with increasing particle size and reachesa maximum when the particle size reaches a critical size andthen decreases as the particle size increases further. This isdue to the transition from single domain to multidomainstructure. ZFC and FCmagnetization behaviors confirm sys-tematically the effect of surface effects on magnetic behavior.

Acknowledgments

This work was financially supported by the NationalNature Science Foundation of China (Grants nos. U1232210,11274314, 51002156, and 11104098) and the Natural ScienceMajor Foundation of Anhui Provincial Education Depart-ment (Grant no. KJ2012ZD14).

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