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arXiv:0809.0203v2 [cond-mat.mtrl-sci] 3 Sep 2008

Surface anisotropy broadening of the energy barrier

distribution in magnetic nanoparticles

N P´ erez1, P Guardia1, A G Roca2, M P Morales2, C J Serna

2, O Iglesias1, F Bartolom´ e3, L M Garc´ ıa3, X Batlle1and A

Labarta1

1Departament de F´ ısica Fonamental and Institut de Nanoci` encia i Nanotecnologia

IN2UB, Universitat de Barcelona, Mart´ ı i Franqu´ es 1, 08028 Barcelona, Catalonia,

Spain

2Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana In´ es de la Cruz 3,

Cantoblanco 28049, Madrid, Spain

3Instituto de Ciencia de Materiales de Arag´ on, CSIC-Universidad de Zaragoza,

Dpto. F´ ısica de la Materia Condensada, Pedro Cerbuna 12, 50009 Zaragoza, Spain

E-mail: nicolas@ffn.ub.es

Abstract.

in magnetic fine particles of nanometer size is discussed within the framework

of the T ln(t/τ0) scaling approach.The comparison between the distributions of

the anisotropy energy of the particle cores, calculated by multiplying the volume

distribution by the core anisotropy, and of the total anisotropy energy, deduced by

deriving the master curve of the magnetic relaxation with respect to the scaling

variable T ln(t/τ0), enables the determination of the surface anisotropy as a function

of the particle size.We show that the contribution of the particle surface to the

total anisotropy energy can be well described by a size–independent value of the

surface energy per unit area which permits the superimposition of the distributions

corresponding to the particle core and effective anisotropy energies. The method is

applied to a ferrofluid composed of non-interacting Fe3−xO4 particles of 4.9 nm in

average size and x about 0.07. Even though the size distribution is quite narrow

in this system, a relatively small value of the effective surface anisotropy constant

Ks = 2.9 × 10−2erg cm−2gives rise to a dramatic broadening of the total energy

distribution. The reliability of the average value of the effective anisotropy constant,

deduced from magnetic relaxation data, is verified by comparing it to that obtained

from the analysis of the shift of the ac susceptibility peaks as a function of the frequency.

The effect of surface anisotropy on the distribution of energy barriers

Submitted to: Nanotechnology

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1. Introduction

Nowadays magnetic fine particles [1] are routinely used in many technological

applications such as magnetic recording [2] and magnetic resonance imaging [3, 4], and

they are considered promising materials for various biomedical applications [5] and non–

linear optics [6]. Consequently, the study of their magnetic properties have attracted

many efforts along the last decades, in particular because considerable deviations from

bulk behavior have been widely reported for particle sizes below about 100 nanometers.

This is because of finite-size effects and the increasing fraction of atoms lying at the

surface with lower atomic coordination than in the core as the size of the particle

decreases.In fact, magnetic properties at the particle surface are governed by the

breaking of the lattice symmetry associated with several chemical and physical effects

leading to a site-specific surface energy, usually taken as a local uniaxial anisotropy

normal to the surface.Generally, it is assumed that this local surface anisotropy

averages over the whole particle surface giving rise to an effective uniaxial anisotropy

acting on the net magnetization of the particle. In the nanometer range of sizes, the

contribution of the surface anisotropy to the total effective anisotropy of the particle

may be larger than that of the core, a fact which highly enlarges the characteristic

switching time of the particle magnetization as a result of the increase in the effective

energy barrier. Actually, determining the characteristic switching time by studying

the non-equilibrium dynamical response of the magnetization has been one of the most

used methods to get an estimation of the effective value of the particle anisotropy per

unit volume, which in many cases has been found to be one or two orders of magnitude

greater than that corresponding to bulk counterpart and not proportional to the particle

volume [7, 8, 9, 10, 11].

Many experimental results [9, 12, 7, 11] have been interpreted in terms of the

phenomenological, ad–hoc, equation originally proposed in Ref. [10] for the effective

anisotropy per volume unit of a spherical particle of diameter D

Keff= Kv+6Ks

D,

where Kvis the core anisotropy energy per unit volume and Ksis the effective surface

anisotropy per unit of surface area which, in general, is assumed to be particle–size

independent. The usually adopted assumption of radial surface anisotropy is not in

contradiction with equation 1 since, in real samples, departures form ideal spherical

shape and surface roughness result in an effective uniaxial contribution to Ks.

It is worth noting that for spheroidal particles, Kvcontains the contributions coming

from magnetocrystalline and shape anisotropy energies. Besides Keff is an effective

uniaxial anisotropy which represents the height of the energy barrier per unit volume

blocking the swithching of the particle magnetization. Kvand Ksare also treated as

effective uniaxial anisotropies.

In Eq. 1, the contributions of the core and surface to the total effective anisotropy

are assumed to be solely additive excluding cross–linked effects. In spite of the simplicity

(1)

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of this assumption and the fact that there is not theoretical justification for it, Eq.

1 has been succesfully applied to show that experimental values of Keff determined

from ac susceptibility measurements scale with 1/D in some fine particle systems [12].

Moreover, magnetic resonance experiments in maghemite nanoparticles [7] have revealed

an anisotropic contribution to the internal field associated with a positive uniaxial

anisotropy originating at the particle surface which dominates over the cubic anisotropy

contribution of the maghemite core and scales [7] with 1/D.

The ad hoc assumption of a surface anisotropy normal to the particle surface

and described by a uniform surface density Ks has also been applied to the study

of the magnetization and switching processes of a single particle in many numerical

calculations based on atomistic Monte Carlo simulations [13, 14, 15, 16], Landau-

Lifshitz-Gilbert equation [13, 17] and micromagnetics models [18]. Many of the results

of these calculations reproduce most of the anomalous properties associated with

surface-anisotropy effects observed in fine particle systems.

phenomenological model based on this assumption [18] has been used to calculate

the astroids corresponding to the phase diagrams for ellipsoidal particles which are

in agreement with recent micro–SQUID experiments on isolated particles [19].

At the moment, Ref. [20] constitutes the only attempt to assess the validity of Eq.

1 using an atomistic model for the surface anisotropy, namely the N´ eel model [21, 22].

It has been shown that the surface energy of a particle with a cubic lattice can be

effectively represented by a first order uniaxial contribution due to particle elongation,

which is proportional to Ksand scales with the surface; a second order contribution

which is cubic in the net magnetization components, is proportional to K2

with the volume; and a core–surface mixing contribution which is smaller than the other

two contributions. Correspondingly, the effective energy barrier of a particle could be

consistent with Eq. 1 only for elongated particles but not for spherical or truncated

octahedral ones. However, these conclusions have been drawn in the framework of a

simple atomistic model for which there is not a general justification derived from more

realistic modelizations of the crystal field, spin–orbit coupling, and disorder taking place

at the surface atoms.

In this work, we show that Eq. 1 may also account for the effective energy barriers

of a size distribution of non-interacting spheroidal magnetic particles.

a method to evaluate the effective contribution of the surface and core anisotropies

based on the comparison between the distributions of particle volumes, obtained

from transmission electron microscopy (TEM), and energy barriers calculated from

thermoremanent magnetization measurements. This method is similar to that applied in

Ref. [23] to study the power law dependence of the energy barrier on the particle volume

in antiferromagnetic ferrihydrite nanoparticles. We show that the effective contribution

of the particle surface to the total anisotropy energy can be well described by a size–

independent value of the surface anisotropy density in accordance with Eq. 1, which

permits the superimposition of the two energy distributions corresponding to the particle

core and total effective anisotropies. It is worth noting that relatively small values of

In particular, a simple

sand scales

We propose

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the effective surface anisotropy density give rise to a dramatic broadening of the energy

barrier distribution even for a narrow distribution of particle volumes.

2. Sample and structural characterization

Monodispersed iron oxide Fe3−xO4nanoparticles were synthesized by high temperature

decomposition Fe(III)-acetylacetonate, coated by oleic acid and dispersed in hexane

with extra oleic acid added as stabilizer [24, 25]. In fact, it was the sample called M5

in Ref. [26], where the full structural characterization can be found.

X-ray diffraction evidenced very good crystallinity, inverse spinel structure with

lattice parameter a = 0.838(2) nm, similar to that of bulk magnetite, and average

particle diameter 5.8 ± 1.0 nm [26].

The phase of the iron oxide particles and their stoichiometry were identified by

M¨ ossbauer spectroscopy [26].The spectrum recorded at 16 K was very similar to

those reported for magnetite nanoparticles that have already undergone the Verwey

transition [27] (see Fig. 9 in Ref. [26]). This spectrum was fitted to five discrete sextets

following a fitting model previously proposed by other authors [28, 29] (see Table 4 in

Ref. [26]). Three of the five components of the spectrum, amounting 73% of the total

spectra area, showed values of the isomer shift less than 0.5 mm/s that may be attributed

to Fe3+ions in the octahedral and tetrahedral sites of the inverse spinel structure. The

other two components showed values of the isomer shift greater than 0.6 mm/s and were

attributed to Fe2+ions lying in octahedral sites. Therefore, the Fe2+atomic fraction

was 0.27 and the average stoichiometry of the particles was estimated to be Fe2.93O4.

This result could be compatible with the presence of up to 21% of maghemite phase in

the form of an overoxidized shell surrounding the particle core.

Fig. 1 shows TEM micrographs of the sample. Particles were spheroidal in shape

and very uniform in size, with polydispersity lower than 20 % of the mean size.

The size distribution was determined measuring the internal diameter of about 3500

particles and the resultant hystogram was fitted to a Poisson–like distribution function,

f(D) = f0Daexp(−D/b), with f0 = 0.65, a = 12.5, and b = 0.36 (in appropriate

units for D to be in nm). From the fitted function an average diameter ?D? = 4.9

nm with standard deviation σ = 1.3 nm was estimated. Assuming that the particles

are ellipsoidal in shape with equal minor axes, we evaluated the average value of the

aspect ratio of about 100 particles to be 1.1 with standard deviation σ = 0.1. We only

considered prolate shape, due to the difficulties in distinguishing prolate and oblate

shapes in TEM micorgraphs.

3. Magnetic characterization

Magnetic measurements were carried out in a commercial Superconducting Quantum

Interference Device (SQUID) magnetometer in the temperature range within 1.8 and 200

K and in magnetic fields up to 5 T. The ac susceptibility was measured by appliying an

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Figure 1. (a) Histogram of the particle diameters obtained from TEM images. (b)

High resolution image of a nanoparticle showing lattice interference fringes. (c) Bright

field image showing the particles. (d) Z-contrast image of the same particles as in (c).

ac field of 2 Oe of amplitude and frequency within 0.1 and 1320 Hz. Magnetic relaxation

at zero field and several temperatures were recorded after field cooling the sample under

50 Oe from rooom temperature down to the measuring temperature, switching off the

field and then recording the magnetization decay as a function of time. All the magnetic

measurements were performed using a especial container for liquid suspensions.

Magnetization measurements revealed bulk–like saturation magnetization at 5 K,

Ms= 78 emu g−1[25, 30].

The temperature dependence of the magnetization was measured increasing the

temperature under an applied field of 50 Oe after zero field cooling and field cooling

(ZFC–FC experiment) the sample from room temperature to 1.8 K. Fig. 2 shows the

ZFC–FC curves of the magnetization which join togheter at Tirr≈ 40 K indicating that

all the particles were superparamagnetic above this temperature. The maximum of the

ZFC curve was located at a mean blocking temperature of about 15 K. The relatively

small irreversibility between the ZFC and FC curves above 15 K and the abrupt increase

of the FC curve below this temperature were indicative of a non-interacting or very weak

interacting superparamagnetic system of nanoparticles.

The Weiss temperature obtained extrapolating the reciprocal FC susceptibility in

the superparamagnetic regime was also very small (less than 1 K) confirming a very

weak strength of the interparticle interactions if they actually exist.

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020 4060 80 100120 140

Temperature (K)

0

0.2

0.4

0.6

0.8

1

Moment (arbitrary units)

Figure 2. Temperature dependence of the magnetization. Field cooling (upper curve)

and zero field cooling (lower curve) measured under an applied magnetic field of 50

Oe. Both curves are indistinguishable for T > Tirr≈ 40 K.

4. Results and discussion

4.1. Core anisotropy

M¨ ossbauer spectra in Ref. [26] showed that the particles undergo a Verwey transition

at a certain temperature within 40 and 16 K, in accordance with previous results of

other authors [31] for magnetite particles of similar size. The reduced value of the

Verwey temperature with respect to that of bulk magnetite may be due to finite size

effects and/or the non–perfect stoichiometry of the particles. Below the temperature

at which the Verwey transition takes place, magnetite is in a monoclinic configuration

with uniaxial magnetic anisotropy. The effective unixial anisotropy along the [111] easy

direction for stoichiometric magnetite [32, 33] is Ku≈ 2.1 × 105erg cm−3and almost

temperature independent below 40 K. This value of the uniaxial anisotropy is also a good

approximation for that of partially oxidized magnetite [32] with the Verwey transition

occurring at a tempertaure above about 20 K, as in the case of the sample studied in

this work. This is the effective bulk anisotropy that should be expected to contribute

to the energy barriers blocking the particle magnetization in thermoremanent and ac

magnetization experiments below about 20 K. It is worth noting that ZFC-FC curves

(Fig. 2) did not show any anomaly associated with the Verwey transition, likely due to

smearing effects related to particle size distribution.

Taking into account that the nanoparticles are not perfect spheres we carried out

a statistical evaluation of the shape anisotropy of about 100 particles from the TEM

images. We assumed the particles to be ellipsoids with equal minor axes. The average

value was Ksh = 5.3 × 104erg cm−3, which is one order of magnitude less than Ku.

Consequently, shape anisotropy was neglected and a value Kv ≈ Ku≈ 2.1 × 105erg

cm−3was expected.

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Figure 3. Scaling of the relaxation curves measured at several temperatures with

an attempt time of τ0= (5 ± 4) × 10−10s. The temperature in K corresponding to

each relaxation curve is indicated beside it. Right hand side Y-axis: Derivative of the

master curve with respect to the scaling variable.

4.2. Energy barrier distribution from magnetic relaxation

In order to obtain the effective distribution of energy barriers which block the switching

of the net magnetization of the particles, we measured the magnetic relaxation

of the sample at constant temperature (thermoremanent magnetization) towards a

demagnetized state in zero applied field after a previous cooling in the presence of 50

Oe (FC process). The obtained relaxation curves corresponding to several temperatures

were plotted as a function of the scaling variable T ln(t/τ0), selecting an attempt time

τ0 = (5 ± 4) × 10−10s that brought all the curves onto a single master curve [34].

Due to inaccuracy in the determination of the initial value of the magnetization at

each temperature (the value of M at t = 0), it was also necessary to normalize the

experimental data dividing them by an arbitrary reference magnetization value M0

which was very close to MFC(T). The opposite of the derivative of the master curve

with respect to T ln(t/τ0) gave the distribution of energy barriers of the system [35].

Figure 3 shows the master curve for the relaxation data and its derivative as a function

of the scaling variable. When magnetic interactions among particles are very weak if

present, as in the case of the sample studied in this work, the distribution of energy

barriers directly represents the distribution of the effective anisotropy of the particles.

The contribution of the surface anisotropy to the effective anisotropy of the particle

can be evaluated by simply comparing the distribution of energy barriers in Fig. 3 with

the distribution of anisotropies corresponding to the particle cores. To do that, the

volume distribution was first calculated from the size distribution by transforming the

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particle diameter into volume assuming that the particles have spherical shape. The

obtained histogram was properly renormalized by dividing the height of each bin by

its width and assigning the new value of the height to the volume corresponding to the

center of the bin. In the next section, we discuss a method to obtain the effective density

of surface and core anisotropy energies by transforming the particle volume distribution

into the total energy barrier distribution.

4.3. Effective surface and volume anisotropies

According to Eq. 1, the total anisotropy energy of a single domain nanoparticle can

be described in a simple model as the sum of two contributions, one proportional to

its volume and another proportional to its surface area, in the form E = KvV + KsS.

Assuming that the nanoparticle has spherical shape, one can rewrite the right hand of

this equation in terms of the volume only, resulting in

3√36πKsV2/3.

E = KvV +

(2)

In a statistical set of spherical, non-interacting nanoparticles the mean values of

both sides of Eq. 2 have to coincide provided one uses an energy barrier distribution

f(E) for the left hand side and a volume distribution g(V ) for the right hand side.

Namely

?∞

0

0

3√36πKs?V2/3?.

Calculating the derivative of Eq. 3 with respect to V , and making use of Eq. 2 to

calculate dE/dV and for substituting E in the resulting expression, we obtain

3√36πKsV2/3?

that relates the distribution of the effective energy barriers to the particle volume

distribution and enables the transformation between them. In fact, Eq. 5 depends only

on one of the anisotropy constants, provided that Eq. 4 is used to find a relationship

between Ksand Kv. From Eq. 4 we obtain

Ks=?E? − Kv?V ?

Ef(E)dE =

?∞

?

KvV +

3√36πKsV2/3?

g(V )dV.(3)

?E? = Kv?V ? +

(4)

f

?

KvV +=

?

Kv+ 2

3?

4π/(3V )Ks

?−1

g(V ),(5)

3√36π?V2/3?. (6)

In the previous discussion there is no need to make any assumption about the

specific form of the distributions f(E) and g(V ). In order to evaluate the averages in

Eq. 6 it is sufficient to have a conveniently large set of experimental data and carry

out a numerical calculation. In our case, we fitted the experimental f(E) and g(V )

distributions to Poisson–like distribution functions. Then, the averages in Eq. 6 were

calculated by numerical integration using the corresponding fitted functions and leading

to

Ks= 4.9 × 10−2− 8.8 × 10−8Kv,(7)

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0.0

0.5

1.0

1.5

2.0

2.5

3.0

Probability density x10-3

Kv

SSWR

02004006008001000

-1.0

-0.5

0.0

0.5

1.0

Residuals x10-3

E/kB (K)

0123456

1

10

100

Figure 4. Upper panel: Core anisotropy distribution g(V ) (?) transformed using Eq.

5 superimposed to the energy barrier distribution f(E) (•) with Kv= 2.3 × 105erg

cm−3and Ks= 2.9×10−2erg cm−2. Solid line corresponds to a Poisson–like function

simultaneously fitted to both sets of experimental data. Lower panel: Residuals when

fitting Eq. 5 (?), imposing either Ks= 0 (?) or Kv= 0 (△). Inset: Sum of squared

weighted residuals of the transformed g(V ) from f(E) using Eq. 5

where Ksand Kvwere in erg cm−2and erg cm−3, respectively.

A weighted least-squares fitting of the Poisson–like distribution function,

corresponding to f(E), to Eqs.5 and 7 yielded the determination of an optimum

value of Kvthat allowed the superimposition of the two distributions, f(E) and g(V )

(see Fig. 4). Weighting of the data in the fitting procedure was done by dividing the

residuals by f(E). The fitted value was Kv= (2.3 ± 0.7) × 105erg cm−3from which

Ks = (2.9 ± 0.6) × 10−2erg cm−2was estimated by using Eq. 7. The univocity of

the fitting is demonstrated in the inset in Fig. 4 where the sum of the squared weighted

residuals shows a well-defined minimum at the fitted value of Kv. It is also worth noting

that the quality of the fitting got signifincantly worse when trying to fit f(E) imposing

either Kv= 0 or Ks= 0 in equation 2 (see lower panel in Fig 4 where the deviations from

f(E) are compared for the three cases). Moreover, the fitted values Ks= (25±2)×10−2

erg cm−2and Kv= (4.7 ± 0.1) × 105erg cm−3obtained when imposing either Kv= 0

or Ks= 0 in equation 2, respectively, were completely unphysical.

The fitted value of Kvwas in good agreement with the value expected for the core

anisotropy of magnetite nanoparticles at a temperature below the Verwey transition

(see Section 4.1).Besides, the obtained Ks lay within the range from 2×10−2to

6×10−2erg cm−2which was reported in previous experimental results for iron oxide

nanoparticles [36, 7, 37, 8, 9].

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020040060080010001200

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

Probability density

E/kB (K)

Volume plus surface

energy distribution

Volume only energy distribution

Figure 5.

nanoparticles studied in this work. They correspond to 0% of Ks(volume only) and

100% of Ks(volume plus surface). Dashed lines correspond to the transformed g(V )

distribution accordingly to Eq. 5 for 20, 40, 60, and 80 percent of Ks= 2.9×10−2erg

cm−2with a fixed value of Kv= 2.3 × 105erg cm−3.

Poisson–like fittings to g(V ) and f(E) (solid lines) for the magnetic

In the preceeding calculations, it was implicitly assumed that Ks is a size

independent constant.This assumption seems to be supported by the good

superimposition of both distributions, f(E) and g(V ), achieved when g(V ) is

transformed accordingly to Eq. 5. Besides, these results also confirm the applicability

of Eq. 1 to describe, at least in a first approximation, the effective anisotropy of the

spheroidal magnetite particles studied in this work.

Interestingly, a relatively small value of Ksstrongly modified g(V ) giving rise to a

broader energy barrier distribution centered at a much higher value of the energy than

that corresponding to the g(V ) function (core–anisotropy contribution). The strong

effect of Ks on g(V ) is emphasized in Fig. 5 where transformed g(V ) distributions

using Eq. 5 for gradually increasing values of Kswithin 0 and 2.9 × 10−2erg cm−2and

Kv = 2.3 × 105erg cm−3are shown. If no surface anisotropy is present, the energy

barrier distribution is identical to g(V ), while increasing values of Ksgradually shift

energy barriers to higher temperatures (energies) producing a significant broadening

effect.

4.4. Effective anisotropy from ac susceptibility

The study of the blocking temperature close to the superparamagnetic regime as

a function of the observational time window in ac susceptibility measurements is a

conventional method to evaluate the mean value of the energy barrier which blocks

switching processes of the particle magnetization. In particular, the real part of the ac

susceptibility χ′

roughly equal to the mean value of the energy barriers become blocked [38]. Therefore,

at Tmax, Arrhenius’ law τ = τ0exp[?E?/(kBTmax)] is accomplished with an attempt time

acpeaks at a temperture Tmaxfor which the particles having an energy

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20406080100120140

Temperature (K)

χ' (arbitrary units)

0.05 0.055 0.06 0.065 0.07 0.075 0.08

1/Tmax

-6

-4

-2

0

2

ln(1/f)

Figure 6. Real part of the ac susceptibility measured at frequencies (in Hz) 0.1

(upper curve), 1, 5, 10, 33, 60, 120, 240, 481, 919, and 1320 (lower curve). Inset shows

ln(1/f) as a function of 1/Tmax. Solid line corresponds to the linear regression of the

experimental points.

τ equal to the reciprocal of the frequency f of the applied ac field. Then, substituting

τ by 1/f, it can be obtained

ln(1/f) = lnτ0+ ?E?/(kBTmax).

Consequently, ?E? can be determined by linear regression of the experimental data for

ln(1/f) plotted as a function of 1/Tmax.

In Fig. 6, χ′

0.1 and 1320 Hz are shown. The inset in Fig. 6 shows ln(1/f) data as a function of 1/Tmax

and the corresponding linear regression, the slope of which is ?E?/kB= 300 K. This value

is in good agreement with that obtained by averaging the energy barrier distribution

f(E) from the relaxation data which for the nanoparticles studied was 263 K. This fact

confirms the reliability of the method to obtain the energy barrier distribution associated

with the effective anisotropy of the particles from magnetic relaxation analysis.

Besides, the values of the attempt time τ0 estimated from the ac susceptibility

[(2±1)×10−10s] and magnetic relaxation [(5±4)×10−10s] are also in agreement taking

into account the error intervals. However, it is worth noting the fact that, indeed, τ0

for fine particles is not a constant and shows an approximately square root dependence

on the temperature [39]. Bearing this in mind, it should be considered that the peak

shifting of the real part of the ac susceptibility extended to the temperature interval from

12 to 20 K, while magnetic relaxation data scaled onto the master curve corresponded

to a larger interval within 3 and 40 K. Therefore, magnetic relaxation data took into

account additional switching processes at lower and higher temperatures, associated

with, respectively, shorter and longer attempt times. Then, both experimental values

of τ0 should be understood as a result of appropriate averages within the particular

temperature interval where each experiment was carried out.

(8)

ac(T) curves for magnetite nanoparticles measured at frequencies within

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5. Conclusions

We have proposed a method to determine the volume and surface contributions

to the effective anisotropy energy in magnetic fine particles which is based on the

superimposition of the distributions corresponding to volumes and energy barriers,

after transformation of the volume distribution assuming a specific expression for the

dependence of the energy barrier on the particle size, E(D).

this method to a colloidal suspension of non–interacting magnetite nanoparticles of

spheroidal shape has shown that the widely used expression given by Eq. 1 may be a

good approximation for E(D) in this case. It is stated In Ref. [20] that surface anisotropy

in spherical particles can never produce solely an effective uniaxial anisotropy which

scales with 1/D. However, in our case, the significant uniaxial anisotropy observed

may be associated with the slight deviations from the spherical shape observed in

TEM micrographs and it is very plausible that the complex disorder taking place

at the surface of real particles (including local modifications of the crystallographic

structure, composition gradients, vacancies, dislocations and other defects) plays a key

role in determining the character of their effective anisotropy since disorder may break

spherical symmetry giving rise to a non–vanishing first–order uniaxial contribution to

the surface anisotropy which could be dominant. Besides, magnetic frustration yielding

non–collinear arrangements of the spins, modifications of the exchange interactions or

the occurrence of a dead magnetic layer are phenomena having a strong potencial effect

on the surface anisotropy that, in principle, cannot be excluded at the outermost shell of

the particle. Moreover, the hypothetical existence of a disordered layer of finite thickness

at the particles’ surface would imply a reduction of the volume contributing to the core

anisotropy. Thus Eq. 1 should be rewritten in terms of the core volume and the volume

of the surface shell, giving raise to a slight reduction in Ksand an increase in Kv.

The good matching between the volume and energy barrier distributions after

transformation using Eq. 1 also suggests that the effective density of surface anisotropy

can be considered as a size independent constant in magnetite nanoparticles, at least

as a first approximation. This is contrary to the hypothesis used in Ref. [8] to obtain

surface anisotropy from ac suceptibility data for magnetite nanoparticles, where the ad

hoc equation Ks(D) = K0

It is worth noting that the actual nature of the anisotropy energy in nanoparticles

may be much more complex than the Eq. 1 implies, for instance, due to correlation

effects between surface and core of the nanoparticles, and temperature dependence of

the anisotropy constants.

However, our results suggest that Eq. 1 can be used to build an effective description

of the surface anisotropy contribution from the distribution of energy barriers blocking

the switching of the particle magnetization, and the obtained values of the anisotropy

constants should be understood as averaged over the range of temperatures in the

experiments.

Finally, we would like to emphasize the strong broadening effect produced by surface

The application of

stanh(D/λ) was introduced without further justification.

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Surface anisotropy broadening...

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anisotropy on the energy barrier distribution of magnetic fine particles even when the

size distribution is quite narrow, as in the case of the sample studied in this work. This

effect is an obvious consequence of the different functional dependence on the particle

diameter of the energy contributions due to the core and surface anisotropy, which makes

their relative importance to change dramatically as the size of the particle is reduced.

This energy broadening may be relevant to give a proper interpretation of the dynamical

response in systems of magnetic particles.

Acknowledgements

The financial support of the Spanish MEC through the projects NAN2004–08805–

C04–02, NAN2004–08805–C04–01, MAT2006–03999, MAT2005–02454 and Consolider–

Ingenio 2010 CSD2006–00012 is largely recognized. The Catalan DURSI (2005 SGR

00969) is also acknowledged.

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properties. J. Phys. D, 35(6):R15–R42, 2002.

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Energy barrier distributions in magnetic

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