FC curves in the case of a magnetic anisotropy energy distribution

Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Magnetic susceptibility curves of a nanoparticle assembly II. Simulation and analysis of ZFC/FC curves in the case of a magnetic anisotropy energy distribution F. Tournus a,b,, A. Tamion a,b a b

Universite´ de Lyon, F-69000, France Univ. Lyon 1, Laboratoire PMCN; CNRS, UMR 5586; F69622 Villeurbanne Cedex, France

a r t i c l e i n f o

a b s t r a c t

Available online 20 November 2010

Starting from the theoretical results established in Tournus and Bonet (2010 [1]) to describe ZFC/FC (zerofield cooled/field cooled) susceptibility curves, we examine the limitations of the widely used two states model (or abrupt transition model) where the magnetic particles are supposed to be either fully blocked or fully superparamagnetic. This crude model appears to be an excellent approximation in most practical cases, i.e. for particle assembly with broad enough size distributions. We improve the usual model by taking into account the temperature sweep existing in experimental measurements. We also discuss a common error made in the use of the two states model. We then investigate the relation between the ZFC peak temperature and the particle anisotropy constant, and underline the strong impact of the size dispersion. Other useful properties of ZFC/FC curves are discussed, such as invariance properties, the reversibility of the FC curve and the link between the susceptibility curves and the magnetic anisotropy distribution. All these considerations lay solid bases for an accurate analysis of experimental magnetic measurements. & 2010 Elsevier B.V. All rights reserved.

Keywords: Magnetic nanoparticle Magnetic anisotropy Magnetic susceptibility Superparamagnetism ZFC/FC curve

1. Introduction It has been shown (see preceding article [1]) that a convenient and physically appealing analytical expression can be used to describe the magnetic moment evolution of an assembly of randomly oriented magnetic nanoparticles (macrospins) having the same magnetic anisotropy energy (MAE), during a zero field-cooled (ZFC) measurement. The magnetic moment at temperature T is in this case well approximated by the so-called ‘‘zeroth order’’ approximation: MZFC ðTÞ ¼ Mb endt þMeq ð1endt Þ where n is the macrospin relaxation frequency at temperature T and dt is an effective measurement time (or waiting time) which is related to the dynamical temperature variation (sweeping) met in experimental acquisitions. This expression makes very clear the crossover between the blocked regime where MZFC ¼Mb, at low temperature, and the equilibrium (i.e. superparamagnetic) one where MZFC ¼Meq, at high temperature. This theoretical description which we call progressive crossover model (PCM) goes beyond the usual ‘‘two states’’ model or ATM  Corresponding author at: Univ. Lyon 1, Laboratoire PMCN; CNRS, UMR 5586; F69622 Villeurbanne Cedex, France E-mail address: fl[email protected] (F. Tournus).

0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.11.057

(abrupt transition model) where the system undergoes an abrupt blocked to superparamagnetic ‘‘transition’’ at a given temperature [2]. As discussed before [1], strictly speaking, the crossover temperature is different from the blocking temperature TB and, even for a single MAE, the ZFC susceptibility peak does not coincide with TB. Things get much more complicated when going from a single MAE to an assembly of particles with a distribution of MAE, reflecting for instance the size distribution. What is then the physical meaning of the ZFC peak temperature Tmax and how is it related to a blocking temperature? This question has already been addressed in the literature and the critical influence of the size distribution has been discussed (showing how a bad assumption on the size distribution can lead to a harmful misinterpretation of the Tmax value). The analysis has been however performed only in the framework of the two states model [3–7], and unfortunately using a wrong expression for the ZFC curve [3,4,6] (this question is addressed further). This model (again, with often a wrong expression of the curves) is also the only one used, up to now, when a quantitative analysis is performed, using a fit of the ZFC curve and sometimes additionally of the FC curve [8,3,4,9–13]. One can then wonder if the improved ZFC/FC curves description we have just established [1] can be used to perform a more accurate analysis of low field susceptibility measurement on real samples. While nanoparticles samples with better and better mono-dispersity are now available, the size dispersion may be low enough so that

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

the progressive crossover from a blocked to a superparamagnetic regime could be detected: the validity of the crude two states model should then be examined. In this article, we will first give an overview on the influence of various parameters on the ZFC/FC curves. The efficiency of the two states model will then be examined and we will compare our improved two states model to the usual one. A frequent error made in the ZFC/FC formulation will also be discussed. The question of an improved FC curve simulation will then be addressed, together with some other considerations related to ZFC/FC experimental curves analysis.

2. Influence of the parameters on ZFC/FC curves 2.1. ZFC/FC curves shape for a single MAE: a survey of the influence of V and Keff As it is often assumed, we make the hypothesis that both the magnetic moment m and the MAE K of a particle are simply proportional to its volume V:

m ¼ MS V and K ¼ Keff V The saturation magnetization MS and the magnetic anisotropy constant Keff are then supposed to be identical for each particle and, for the sake of simplicity, they are supposed to be constant in the temperature range considered. The following relations (which are not rigorous, but approximate [1]) control the general aspect of the ZFC/FC susceptibility curves and are also summarized in Table 1. They will be of great help to rationalize the influence of the two key parameters, Keff and V:

 The initial moment MZFC(0) of the ZFC curve is proportional to V and inversely proportional to Keff.

 The blocking temperature TB, the crossover temperature TX, as    

well as the temperature Tmax of the ZFC peak, are proportional to Keff and V. We have Keff V C 25kB TB C 23:6kB Tmax . The width of the ZFC peak increases linearly with Tmax (and TB) and verifies DT=Tmax C 11%. DT is thus proportional to Keff and V. The amplitude of the ZFC peak verifies Mmax C 23:6MZFC ð0Þ. Mmax is then proportional to V and inversely proportional to Keff. The final moment of the FC curve verifies MFC ð0Þ C25:5MZFC ð0Þ. It is then proportional to V and inversely proportional to Keff. The amplitude of the signal in the superparamagnetic regime (for T higher enough than TB) is proportional to V2.

From these relations, it is clear that the shape of ZFC/FC curves for a realistic particle assembly will strongly depend on the size distribution, and in a complicated way. This is why experimental curves have to be theoretically adjusted in order to perform a quantitative analysis of the particles magnetic properties.

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2.2. Non-uniqueness of a parameter set corresponding to a curve We recall that, in the model we use, seven parameters fully determine the ZFC/FC curves in the case of a single MAE particle assembly. These are the number of particles Ntot, the applied field H, the saturation magnetization MS , the particle volume V, the magnetic anisotropy constant Keff, the ‘‘attempt’’ frequency n0 and the temperature sweeping rate vT. Among these parameters, the first three have no impact on the curve shape, they just change the amplitude (we indeed restrict the discussion to the frame of linear response): therefore, any reduction in one parameter can be compensated by an increase in another parameter. For the parameters pair V and Keff, the only quantity that can modify the curves shape is the product KeffV, i.e. the MAE. Thus, once again, if the particle volume is doubled while the anisotropy constant is divided by two, there is no consequence on the curves. More precisely, the consequence is only a change in amplitude which can be ‘‘absorbed’’ in a modification of Ntot for instance. Indeed, some parameters are supposed to be known (except in some particular cases): H, MS , vT; whereas some others usually need to be determined from the magnetic measurements: V (here, we are dealing with the magnetic volume, which may be different from the observed ‘‘geometric’’ volume), Keff and Ntot. The case of the frequency n0 is particular: it is not really known but a fixed value is often assumed, typically 109 or 1010 Hz. However, n0 has only an influence through a logarithm, which somehow justifies that the poor knowledge we have on this quantity is not a major problem. Nevertheless, it should be noted that for instance, if n0 is lowered while Keff also decreases it is possible to keep the ZFC/FC curves almost unchanged. In fact, there will be a slight modification of the blocked-superparamagnetic crossover width, and an increase in the initial value of MZFC : these changes may be in the error bars and may be completely undetectable on a real experimental curve (see Fig. 1). These considerations on single MAE curves are indeed applicable to ZFC/FC curves on assemblies of particles with a size distribution. This shows that there is necessarily an indetermination when ZFC/FC curves (or worse, if only a ZFC curve) are adjusted. In particular, it should be kept in mind that it is the product KeffV which controls the curves shape, so that a rescaling of the size distribution, with a jointly modified anisotropy constant, can also reproduce the curves. In addition, many couples of n0 and Keff values can provide a good fit: the lack of information on n0 thus results in an uncertainty on Keff. The indetermination existing for ZFC/FC curves can be removed, at least partly, by adjusting simultaneously different magnetization curves sharing common parameters (as for instance the particle volume and the number of particles). This approach has been shown recently to be very successful, in the frame of the ‘‘triple fit’’ where a high temperature magnetization loop is adjusted simultaneously with the ZFC/FC curves [14–16]. Following this idea, a gain in accuracy can be expected when additional experimental curves (low temperature magnetization loops, or isothermal remanence magnetization curves for instance) are also simultaneously theoretically adjusted.

3. Improved two states model Table 1 Influence of the anisotropy constant Keff and the particle volume V on the various features of ZFC/FC curves for an assembly with a single magnetic anisotropy energy K¼ KeffV.

MZFC(0), MFC(0) TB, TX, Tmax Absolute width of the ZFC peak Amplitude of the ZFC peak Superparamagnetic signal

Keff

V

Inversely proportional Proportional Proportional Inversely proportional –

Proportional Proportional Proportional Proportional Proportional to V2

3.1. Efficiency of the improved two states model In the so-called ‘‘two states model’’ or ATM, an assembly of particles having a given MAE (K¼KeffV) is considered to be either completely blocked or perfectly superparamagnetic, which means at thermodynamic equilibrium. Thus, the crossover between the two regimes is supposed to be infinitely abrupt and occurs at a transition temperature directly related to K. We have shown previously the expression of the crossover temperature TX which is the best choice: this temperature is neither the blocking temperature TB, nor simply

M (arb. unit)

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

M (arb. unit)

1120

9

ν0 = 10 Hz Keff = 100 kJ/m

3

9

ν0 = 10 Hz 3

Keff = 100 kJ/m

6

6

ν0 = 10 Hz Keff = 72.5 kJ/m

0

5

10

ν0 = 10 Hz 3

15

3

Keff = 72.5 kJ/m

20

0

5

T (K)

10

15

20

T (K)

Fig. 1. Calculated ZFC/FC curves for two different values of the n0 and Keff parameters, in the case of a single MAE (single particle size Dm) (a) and for a gaussian size distribution (b) with s=Dm ¼ 0:05. In each case, the calculations are performed with Dm ¼ 4 nm and vT ¼ 2 K/min.

w = 0.05

σ / Dm = 5 %

σ / Dm = 2.5 %

M (arb. unit)

M (arb. unit)

w = 0.17

w = 0.1

0

σ / Dm = 7.5 %

0 Analytical expression MZFC

0 Analytical expression MZFC

Improved two states model

Improved two states model

3

6 T (K)

9

5

10

15 T (K)

Fig. 2. Comparison between the improved two states model and the progressive blocked-superparamagnetic crossover model for a lognormal size distribution (a) and for a gaussian size distribution (b) of various relative dispersions. The calculations are performed with n0 ¼ 1010 Hz, vT ¼ 2 K/min, Keff ¼100 kJ/m3, and a median diameter of respectively Dm ¼2.3 and 4 nm for the lognormal and the gaussian size distributions. The relative dispersion is indicated by the ratio s=Dm (s being the diameter standard deviation) for the gaussian distribution and by the dimensionless dispersion parameter w for the lognormal distribution.

proportional to K following the Teff ¼ K=ðaeff kB Þ relation (note that by setting aeff ¼ 25, we recover the familiar K ¼ 25kB Teff equality). The use of TX in the well-known two states model will be referred in the following as the ‘‘improved two states model’’. In the two states model (usual or improved), we neglect the progressive nature of the blocked-superparamagnetic crossover which corresponds to a natural width of the ZFC susceptibility peak: this relative width is almost constant and of the order of 10% [1]. Therefore, it is expected that this approximation is not too crude for assemblies of magnetic nanoparticles with a wide size distribution: since the ZFC peak position varies essentially linearly with V, the size dispersion will ‘‘rub out’’ the details of the blockedsuperparamagnetic crossover. On the other hand, in the limit of a single size, which means a single MAE, this crossover is an essential feature of the ZFC/FC curves. How sharp has the size distribution to be, for the progressive crossover to be needed? To answer this question, we have computed ZFC/FC curves with the improved two states model (with transition at TX) and with the more realistic model where the system evolves continuously between the blocked and the superparamagnetic regime: this is the progressive crossover model (PCM) corresponding to the analytic expressions established in the preceding article [1]. We have considered the case of a normal or a lognormal size distribution

rðDÞ, which are certainly the two most experimentally relevant functions. As it can be seen in Fig. 2, the difference between PCM and ATM are detectable only for a relative size dispersion below 5% (or a dispersion w less than 0.05 for the lognormal). Note that for the progressive crossover model, the ZFC curves have been calculated 0 using the zeroth order approximation MZFC , and that the FC curves are not displayed since the differences between the two models are even more minor. The improved two states model is then applicable, with a high degree of precision, in most experimental cases: size distributions with such a narrowness are not widely met! Moreover, when coming to extremely sharp size distributions, we can expect the K¼KeffV relation with a single anisotropy constant to be no more valid: other sources of anisotropy dispersion may then surpass the effect of size dispersion [17,16]. Nevertheless, the limitations of the model should be kept in mind and using the PCM is always preferable: the additional computational cost is negligible, at least for the ZFC curve where the M0ZFC analytic expression is very handy. From a simulation point of view, the use of M0ZFC, which naturally smoothes the curves, also allows to increase the particle size discretization step (for the numerical integration on the size distribution) without having ‘‘sawtooth artefacts’’ as is can be the case for the two states model.

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

αeff = 27

αeff = 27

Transition at TX

αeff = 26

αeff = 26

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Transition at TB Transition at Teff

M (arb. unit)

M (arb. unit)

αeff = 25

αeff = 25

Transition at TX Transition at TB Transition at Teff

5

10 T (K)

15

20

0

10

5

15

T (K)

Fig. 3. Simulated ZFC (a) and FC (b) curves with the two states model for a lognormal size distribution (median diameter Dm ¼ 2.3 nm and dispersion parameter w¼ 0.33) with different choices for the transition temperature. The improved two states model corresponds to a transition at TX. The calculations are performed with n0 ¼ 1010 Hz, vT ¼2 K/min and Keff ¼ 100 kJ/m3.

28

ρ (D) (arb. unit)

α = KeffV / (kBTX)

26

24

22

20 0

1

2

3

4

5

6

D (nm) Fig. 4. Size distribution and evolution of a ¼ Keff V=ðkB TX Þ with the particle size corresponding to the calculations of Fig. 3.

3.2. Comparison with the usual two states model In the improved two states model, the transition between the blocked regime and the superparamagnetic one occurs at a temperature TX given by   K V n0 Keff V TX ¼ eff with a C 0:9283ln 3:69 kB vT akB The V dependence of a means that TX is not strictly proportional to V. On the contrary, in the usual two states model the transition occurs at a temperature Teff ¼

Keff V

aeff kB

where the parameter aeff is a constant, of the order of 25, related to the ‘‘attempt’’ frequency n0 and a ‘‘measurement time’’. Note that the use of TX clearly constitutes an improvement over the usual two states model, in the sense that it takes into account the experimental temperature sweeping rate. Within this model, it is easy to compute a ZFC/FC susceptibility curve: for each temperature T, and for each particle volume V, T is compared to TX(V). If T Z TX then the magnetic moment corresponding to particles of volume V is Meq(V), while if T oTX it is Mb(V) for the ZFC and M0(V) for the FC curve. As noted before [1], the definition

itself of TX ensures that we have the correct magnetic moment M0 for the blocked regime in the FC protocol. We can then write M0 ¼ Meq(TX): there is no discontinuity in the FC curve. Using Teff instead of TX, as it is the case in the literature [2–4, 8–13], would result either in a discontinuity, or a wrong low temperature limit for the FC curve. In Fig. 3 we compare the impact of different choices for the transition temperature on calculated ZFC/FC curves for a lognormal size distribution. We also show, for the sake of completeness, the calculated curves with the transition at TB (which is somehow equivalent to TX, but with another V dependent a coefficient). It can be seen that the curves obtained within the improved two states model (transition at TX) are significantly different from those obtained by the usual two states model, even if we try to find the best aeff value. In particular, the temperature Tmax of the maximum of the ZFC curve (ZFC peak) can slightly vary. In Fig. 4 we can see how the a coefficient evolves with the particle size, which explains these differences. It is then clear, especially because of the long tail of the lognormal distribution, that taking a single value for a will not be very accurate. The problem is less important for a gaussian size distribution which generally goes with a lower dispersion that in lognormal samples (especially, it drops faster to zero at large sizes).

4. ZFC curve for a particle assembly with a size distribution 4.1. Analytic expression In the improved two states model, a particle of volume V will be blocked as long as T is lower than TX(V) and at thermodynamic equilibrium for higher temperatures. Therefore, for a size distribution rðVÞ and at a given temperature T, all the particles with a volume larger than Vlim will be blocked while the smaller particles will be superparamagnetic. This threshold volume, which corresponds to the blocked-superparamagnetic limit, depends on T and is given by the equality TX ðVlim Þ ¼ T

equivalent to Vlim ¼

akB T Keff

Unfortunately, because the a coefficient depends itself on Vlim, this reads as an implicit equation over Vlim which does not allow a direct analytical resolution. Note that if we neglect the variation of a, which corresponds to the usual two states model, Vlim is then directly proportional to the temperature. We will see later, that it is possible to get an approximate analytical expression for Vlim(T). Knowing the expression of the blocked and equilibrium magnetic moment for a population of particles having a volume V,

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F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

we then have

which are, respectively [1] Mb ¼

2 0 HMS V

m

3Keff

and

Meq ¼

2 2 0 HMS V

m

wtot ¼

3kB T

we can express the total magnetic moment of the particle assembly as a sum of two integral terms (here, and for the following, we drop the total number of particles Ntot which would appear as a simple prefactor): Z Z m HM2S 0 2 m HM2S 1 V rðVÞ dV þ 0 V rðVÞ dV ð1Þ MZFC ¼ 0 3kB T 3Keff Vlim Vlim The first term corresponds to the contribution of superparamagnetic particles, while the second term corresponds to the blocked ones. Note that the first term involves V 2 rðVÞ whereas the second term involves V rðVÞ.

Mtot =Vtot 1 ¼ Vtot H

Z

wV V rðVÞ dV

The missing V multiplicative term in the wrong formula has a quite dramatic consequence on the simulated ZFC (and FC) curves: the contribution of large particles is erroneously minimized. This means in particular that, by fitting experimental data with the wrong formula, a deceptive particle size distribution and an overestimated anisotropy constant will be obtained. As can be seen in Fig. 5 the difference between the correct and the wrong formula is highly significant: in the example of a lognormal size distribution, the use of the wrong formula will lead to a severe Keff overestimation!

4.2. Frequent error met in the literature

4.3. Evolution of Vlim with T

The above expression has already been used by several authors [10,11,7,12,18]. A similar formula is also met in the literature [3,4,9,6,19,13], following the article of Chantrell et al. [8], where the two integral terms involve on one hand V rðVÞ for the superparamagnetic contribution and on the other hand rðVÞ for the blocked contribution (a reduced variable, based on the particle volume or particle blocking temperature, is sometimes used). This expression is in fact erroneous and the missing V contribution comes from a mistake in writing down the susceptibility of an assembly with a size distribution. The susceptibility, which is a dimensionless quantity is indeed defined as the ratio between the magnetization and the applied field H. For an assembly of particles having the same volume V, the magnetic susceptibility wV is then given by

In the above expression of MZFC, the difference brought by the improved two states model, with respect to previously published results, is in the variation of Vlim with T. It is clear (see the expression of TX) that TX is completely determined by n0 , vT and K ¼ Keff V. Since TX(K) is a monotonous function, K is then conversely fixed by the parameters TX, n0 and vT. By introducing the dimensionless quantity zX ¼ n0 TX =vT , we can write without any loss of generality: KðTX Þ ¼ kB TX gðzX Þ where the function g is unknown. A large number of simulations with various zX values allows us to obtain an excellent approximate expression (see Fig. 6) for g : gðzX Þ C 0:9609lnzX 1:629. We can then write KðTX ¼ TÞ ¼ Keff Vlim ðTÞ, which finally gives   gkB T n0 T with g C 0:9609ln 1:629 ð2Þ Vlim ðTÞ ¼ Keff vT

wV ¼

MV =V H

where MV is the magnetic moment of the assembly. One can verify that both MV/V and H have the same dimension: their value can both be written in A/m unit. When considering an assembly with different particle volumes, the additive quantity is the magnetic moment and not the susceptibility. The total magnetic moment Mtot, which is the experimentally accessible quantity, then verifies (as before, we drop the total number of particle Ntot which is simply a prefactor): Z Z MV rðVÞ dV ¼ wV HV rðVÞ dV Mtot ¼ While H can be taken out the integral, this is not the case of the V term multiplied by wV . If we want to express the total susceptibility,

This expression is similar to the previous equation on Vlim involving a. In fact, the two quantities a and g correspond to the ratio K/(kBTX), but they depend on a different variable: this is why it is more convenient to adopt two distinct notations. We now have an explicit expression that allows us to directly determine Vlim at any temperature. As previously noticed, Vlim is not exactly proportional to T and this non-linearity depends on the temperature sweeping rate. To give an order of magnitude, the g relative variation is about 10% when the temperature goes from 10 to 100 K. In order to obtain the most accurate results, one should then use the above expression for Vlim(T) in the integral formulation of MZFC: this is again the improved two states model, the impact of the temperature sweep appearing here on Vlim(T) instead of TX(V).

Correct formula Wrong formula

M (arb. unit)

M (arb. unit)

Correct formula Wrong formula

0

10

20

30 T (K)

40

50

0

20

40 T (K)

60

80

Fig. 5. Comparison between the correct formulation [with, respectively, V 2 rðV Þ and V rðV Þ contributions for the superparamagnetic and blocked term] and the wrong one [with, respectively, V rðVÞ and rðVÞ contributions for the superparamagnetic and blocked term] met in the literature (originally in [8,3]), for a lognormal (a) and gaussian (b) size distribution. The calculations are performed with n0 ¼ 1010 Hz, vT ¼2 K/min, Keff ¼ 100 kJ/m3 and, respectively, Dm ¼ 2.3 nm and w¼0.33 for the lognormal distribution, and Dm ¼4 nm and s=Dm ¼ 20% for the gaussian distribution.

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

1123

28 30

Numerical calculations Fit : α = 0.9283 ln(y) - 3.69

γ = K / (kB TX)

28 α = K / (kB TX)

Numerical calculations Fit : γ = 0.9609 ln(zX) - 1.629

26

26 24 22

24 22 20

20

18

18 24

26

28 30 32 34 ln(y) = ln[ν0K / (kB vT)]

20

36

22

24 26 28 ln(zX) = ln(ν0TX / vT)

30

Fig. 6. Variation of the ratio K/(kBTX) as a function of the dimensionless variable y ¼ n0 K=ðkB vT Þ (a) and zX ¼ n0 TX =vT (b). The results of numerical calculations can be adjusted with an analytical function corresponding to aðKÞ (a) and to gðTX Þ (b).

10

3.0 Numerical calculations Fit

Numerical calculations Fit

2.5 Vmax/ Vm

Vmax/ V

8 6 4

2.0 1.5

2 0.0

0.1

0.2

0.3 w

0.4

0.5

0.6

1.0 0.00

0.05

0.10

0.15

0.20

0.25

σ/ Dm

Fig. 7. Evolution of Vmax =V with the dispersion parameter w for a lognormal size distribution (a) and of Vmax/Vm with the relative size dispersion s=Dm for a gaussian size distribution (b). The value of Vmax relates the ZFC peak temperature Tmax to the anisotropy constant Keff (see text). The equations of the analytical fits shown on the figures are given in the text.

leading to the following relation by setting it to zero for T¼Tmax:

4.4. Link between Tmax and Keff A widespread strategy, when analyzing ZFC susceptibility curves, is to obtain information on the magnetic particles just from a single point on the curve: namely the temperature of the ZFC maximum, Tmax. It is clear that Tmax is related to the anisotropy constant Keff, but as it has already been pointed out [3–6], it strongly depends on the particle size distribution. Tmax is by no way a blocking temperature: a blocking temperature is defined only for a given magnetic anisotropy energy. Of course, one can always define a certain volume, the blocking temperature of which would correspond to Tmax. However, such a volume has no simple link with the size distribution: in particular, it is not the mean or median volume. We then insist on the fact, already pointed out, that Tmax should not be considered as a mean blocking temperature, or the blocking temperature of median size particles [20]. Nevertheless knowing the size distribution rðVÞ, we can find a link between Tmax and Keff. For this purpose, we introduce the volume Vmax defined by Vmax ¼Vlim(Tmax) which corresponds to Vmax ¼

gm kB Tmax Keff

with gm ¼ 0:9609ln



n0 Tmax vT

 1:629

To a good approximation, we may then write Vlim ðTÞ C

gm kB T=Keff . Thus, we have removed the non-linearity of Vlim with T, keeping track of the temperature sweep through the value of gm . This simplification allows an analytical calculation of dMZFC =dT,

3 ¼ Vmax

gm

1 gm 1 rðVmax Þ

Z

Vmax

V 2 rðVÞ dV

ð3Þ

0

This implicit equation can be use to compute quite easily the value of Vmax for any size distribution. Note that a similar, but wrong, equation can be obtain from the wrong formulation discussed earlier [4]. Therefore, the impact of the size dispersion has not been correctly investigated yet: in particular, in the analysis of Jiang et al. [5] the contribution of the largest particles has been severely minimized due to the missing V term (see previous discussion), leading to a underestimation of b (i.e. the ratio between Tmax and the median transition temperature). Once the above equation has been solved, the anisotropy constant is directly deduced from the equality Keff ¼

gm kB Tmax Vmax

ð4Þ

We have determined numerically the value of Vmax in the case of a lognormal and a gaussian size distribution. Note that since Vmax depends slightly on the value of gm and consequently on the ZFC peak position, no universal formula can be provided. We have taken gm ¼ 28 as a reference value for our calculations: this corresponds to vT ¼0.01 K/s, n0 ¼ 1010 Hz and Tmax ¼25 K. In the lognormal case, the ratio between Vmax and the mean particle volume V can be adjusted with an excellent precision

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F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

(see Fig. 7) by the following analytical expression:

will then be either M0(V) if T oTX or Meq(V,T) if T ZTX , where

Vmax ¼ 1 þ 3:2w þ 0:1expð7:14wÞ V

Meq ðV,TÞ ¼

Note that the coefficients of this fit may be slightly different if gm has another value, and that the precision is expected to be poorer for large w since it becomes too crude an approximation to neglect the variation of g with T. We recall that a lognormal size distribution on the particle diameter corresponds to (  2 ) 1 1 lnðD=Dm Þ pffiffiffiffiffiffi exp  rðDÞ ¼ 2 w Dw 2p with w the dimensionless dispersion parameter and Dm the median diameter. The mean volume is then given by  2 p 9w V ¼ D3m exp 6 2 Interestingly, this result shows that the anisotropy constant can be severely overestimated if the mean volume is used for the direct determination of Keff from Tmax. Even if such an erroneous procedure is not used, this emphasizes the huge impact of a bad estimation or uncertainty on w on the calculated Keff value. Note also that there is no simple recipe relating Vmax to some particular volumes (as for instance taking the ratio V 2 =V ) [21]. In the case of a gaussian size distribution rðDÞ, we can also provide an analytical fit (see Fig. 7) relating the ratio Vmax/Vm to the relative size dispersion s=Dm , where Dm and Vm ¼ ðp=6ÞD3m are, respectively, the median diameter and volume and s is the diameter standard deviation:  2  3 Vmax s s s ¼ 1 þ 5:8 5:5 þ 45:9 Vm Dm Dm Dm Once again, this has been calculated with the reference value of gm ¼ 28. The range of relative size dispersion was limited to a maximum value of 25% since gaussian size distributions are usually used to describe only particles with a well-defined size (as for instance size-selected clusters made by a physical route [16], or chemically synthesized particles). This is why the Vmax variation is less important than for the lognormal distribution: the error on Vmax, and consequently on Keff, coming from a bad estimation of the relative size dispersion should be less dramatic in this case because this parameter is known, in principle, with a few percent precision. Note however that in both cases (lognormal and normal size distribution) the relative error on Keff is greater than the one on the relative size dispersion. This shows how it is crucial to have a precise determination of the magnetic size distribution if an accurate determination of the magnetic anisotropy constant is wanted.

5. Simulation of a FC curve for a particle assembly with a size distribution 5.1. Improved two states model for the FC curve We have already discussed the excellent precision of the simplified two states model as compared to the more realistic progressive blocked-superparamagnetic crossover model. However, this only holds if the crossover temperature for each particle volume V is taken to be TX(V), and not TB(V) or a temperature Teff(V) purely proportional to V. TX is the only choice of transition temperature for which the blocked moment of the FC curve, M0, is equal to the equilibrium moment at the transition temperature, Meq. This is what we call the improved two states model. The contribution of particles of volume V to the FC curve of an assembly

m0 HM2S V 2 3kB T

and

M0 ðVÞ ¼ Meq ðV,TX Þ ¼

am0 HM2S V 3Keff

The coefficient a is the same as before: because it depends on V, the blocked magnetic moment of the FC is not exactly proportional to V. As for the ZFC curve, we can introduce for each temperature T a volume Vlim defined as TX (Vlim)¼T that allows us to express the FC magnetic moment as a sum of two integral terms corresponding to the respective contribution of superparamagnetic and blocked particles (as before, we drop the Ntot factor): Z Z m HM2S Vlim 2 m HM2S 1 V rðVÞ dV þ 0 aV rðVÞ dV ð5Þ MFC ¼ 0 3kB T 3Keff 0 Vlim Note that the a coefficient cannot be taken out of the integral since it depends on V: there was no such problem for the ZFC curve. Moreover, we recall that Vlim ¼ gkB T=Keff , meaning that it is not strictly proportional to T. It is still possible to write an approximate expression dropping, on one hand the variation of g, and on the other hand the variation of a: we can use the simplification g C gm and, with V the mean volume, ! n K V a C am ¼ 0:9283ln 0 eff 3:69 kB vT With these approximations, we are brought back to a usual two states model expression [22], but the dynamic temperature sweep still plays a role through the value of am and gm . However, as shown earlier (see Fig. 3) neglecting the non-proportionality of TX with respect to V (and of Vlim with respect to T) goes with a small error on the FC curve, especially for the low temperature limit: the larger the size dispersion is, the worse the approximation is. 5.2. Irreversibility of the FC curve: can we use the equation of a ZFC curve to describe the FC curve? As discussed in the case of a single MAE particle assembly [1], the FC curve is irreversible. It means that the curve measured on cooling will not be the same as that measured on heating the sample from the final FC point at low temperature (0 K in the ideal case). The later curve can be described with the same equation as for the ZFC curve, except that it has a different starting point: once the initial magnetic moment M0 is known, it is very convenient to simulate this curve (FC, then measurement on heating the sample) using a zeroth order expression similar to that of the ZFC curve, with a progressive crossover between the blocked and superparamagnetic regime: 0 MFC ¼ M0 endt þMeq ð1endt Þ

Experimentally, we have never detected any irreversibility in the case of a real sample with a size distribution. By simulating FC curves (on cooling and heating the sample) for realistic assemblies of particles, and with the accurate analytical formulas seen before [1], we understand that this apparent reversibility comes from a smoothing effect due to the MAE dispersion. Samples with an extremely sharp MAE distribution are needed to detect the FC irreversibility. Note that an irreversibility may be observed because of other (spurious) physical reasons, as for instance inter-particle magnetic interactions or electronics issues due to the experimental acquisition. Therefore, since the two curves appear to be identical in almost any case, we can adopt the ZFC-like analytical expression to describe a FC curve, which can be convenient from a computational point of view. Nevertheless, contrary to the ZFC case where the blocked magnetization is the same for each particle (the blocked magnetic moment Mb is simply proportional to V), the blocked moment for a given size, M0(V) does not vary strictly linearly with V. This complicates a little the calculations and for more simplicity we

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

can assume that the blocked magnetization M0(V)/V is constant: we can either consider a ‘‘mean’’ am coefficient, or even adjust its value from the experimental data [14,15].

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variation of gm and am ). Note that this invariance involves the ZFC curve, since the normalization is made with respect to the ZFC peak which does not correspond to any particular point on the FC curve. 6.2. Width of the ZFC peak

6. Other considerations on FC and ZFC curves 6.1. Invariance of normalized ZFC/FC curves We can observe, while simulating ZFC/FC curves for different parameters (size distribution and anisotropy constant), that the curves show an invariance, in some particular cases, when they are normalized with respect to the ZFC peak at Tmax: we then plot MFC(T)/ MZFC (Tmax) and MZFC(T)/MZFC(Tmax) as a function of T/Tmax. With this representation, changing Keff while keeping the other parameters constant has no effect on the curves shape. The same invariance is observed when modifying only the scale of the particle size (i.e. the median diameter Dm is changed, without changing the shape of the distribution, which means keeping the same w for a lognormal and the same s=Dm for a normal distribution). To our knowledge, no theoretical explanation to this feature has been given in the literature. These properties of invariance can however be demonstrated using the two states model for the ZFC/FC curves. We start by defining reduced variables y ¼ T=Tmax and v¼ V/Vmax. We recall that we have, by definition Vlim(Tmax)¼ Vmax. The ZFC curve in the two states approximation then reads Z vlim Z 1 m HM2S 2 m HM2S MZFC ðyÞ ¼ 0 Vmax v2 rðvÞ dv þ 0 Vmax vrðvÞ dv 3kB yTmax 3Keff 0 vlim where vlim ¼ Vlim/Vmax can be approximated (this is an approximation since it neglects the variation of the g parameter between T and Tmax) by T/Tmax. Noting that Keff Vmax ¼ gm kB Tmax , we finally end up with MZFC ðyÞ ¼

m0 HM2S Vmax 3Keff

gðyÞ

where gðyÞ ¼

gm y

Z

y

v2 rðvÞ dv þ

Z

0

1

vrðvÞ dv

ð6Þ

y

The function gðyÞ, which entirely control the shape of the ZFC curve, does almost not depend on the value of Keff or Vmax. In fact, there is a slight dependence through the parameter gm which is determined by the value of Tmax. Letting aside this subtle consideration, we can see that changing Keff alone has no effect on the normalized curve, as well as changing only the scale of the particle size (which does not affect the reduced parameter v). The expression above is also useful to see how the amplitude of the ZFC peak, which corresponds to Mmax ¼ MZFC ðy ¼ 1Þ, is affected by the various parameters: Mmax is proportional to Vmax (and thus proportional to the mean particle volume if the shape of the size distribution is unchanged), and inversely proportional to Keff. On the other hand, for the superparamagnetic part of the curve the magnetic moment is proportional to the mean quadratic volume. The same approach can be applied for the FC curve, in the framework of the two states model. This leads, letting aside the fact that a and g are non-constant, to the following equation: MFC ðyÞ ¼

m0 HM2S Vmax 3Keff

hðyÞ

with hðyÞ ¼

gm y

Z 0

y

v2 rðvÞ dvþ am

Z

1

vrðvÞ dv

ð7Þ

y

Once again, the curve shape is entirely governed by the function hðyÞ which is almost independent of Keff and Vmax (by neglecting the

The width of the ZFC peak, which is the range of temperature values where MFC aMZFC , is also related to the anisotropy constant and to the particle size distribution. The FC and ZFC curves remain separated at a temperature T0 as long as there exist in the sample particles of volume V0 such that TX(V0)¼T0, which corresponds to V0 ¼

gðT0 ÞkB T0 Keff

This criterion is not very useful, since the size distributions of real samples are not strictly null above a given volume V0: the decrease of rðVÞ is gradual. Therefore, it is in fact impossible to define a definite merging point of the ZFC and FC curves. Moreover, an experimentally determined merging temperature necessarily depends on the data precision. Nevertheless, Keff being fixed, it is clear that the more particles of large size are present in a sample, the higher the merging temperature will be: this implies that the ZFC peak width increases with the size dispersion (when the mean size is fixed), and with the mean size (when the relative dispersion is fixed). Besides, for a given V0, TX(V0) varies almost linearly with Keff which means that for a fixed particle size distribution, the ZFC peak width must increase with Keff. As a general remark, it should be kept in mind that large particles (even in small quantity) have a great impact on the susceptibility curves: we indeed recall that for a given volume V the ZFC peak amplitude is nearly proportional to V. This analysis was on the absolute width of the ZFC peak. As far as its relative width is concerned, we have just shown that the ZFC/FC curves display some invariance properties. For instance, increasing the anisotropy constant without changing the size distribution will not affect the ZFC peak relative width (i.e. the width of the normalized curve), while the peak position and its absolute width will change. These considerations may seem unconnected to real measurements, but they can be helpful for instance to check if any change in the particle size has occurred in a sample (after annealing), to verify that only the anisotropy constant has evolved, or to compare different samples. 6.3. Use of DM ¼ MFC MZFC Since analytical formulas, within the two states model framework, are available both for the FC and the ZFC curve, we can also obtain a simple expression of the difference DM ¼ MFC MZFC . The contribution of superparamagnetic particles (i.e. particles at thermodynamic equilibrium) is the same for the two curves, and is cancelled. We then have Z 1 m HM2S DM ¼ 0 ðam 1Þ V rðVÞ dV ð8Þ 3Keff Vlim This equation can be used to adjust the difference between experimental FC and ZFC curves. DM is also interesting because it is related to the MAE distribution, or equivalently to the distribution of crossover temperatures TX in a sample. By using the fact that rðVÞ dV ¼ rðKÞ dK (the MAE distribution is supposed to come directly from the size distribution), and introducing Klim ¼KeffVlim, we can write Z 1 m HM2S K DM ¼ 0 ðam 1Þ rðKÞ dK 3Keff Klim Keff The derivative of DM with respect to T, is then  2 dDM 1am gm kB ¼ m0 HM2S T rðK ¼ gm kB TÞ dT 3 Keff

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Here, we have neglected the variation of g with T and considered that Klim is simply proportional to T. This finally shows how DM is related to the MAE distribution:

rðTX ¼ TÞp

1 dDM rðK ¼ gm kB TÞp T dT

ð9Þ

Note also that the crossover temperature distribution rðTX Þ can be related in the same way to DM, neglecting the variations of a and g (we have indeed kB TX ¼ K=a or equivalently K ¼ gkB TX ):

rðTX ¼ TÞp

temperature. He indeed establish, by neglecting the contribution of the blocked particles:

1 dDM T dT

These relations, which are not strictly exact, can be very useful to have an idea of the MAE distribution in a sample (or the crossover temperature distribution, which is almost equivalent to the blocking temperature distribution), directly using the experimental curves, without any theoretical fit. A limitation comes however from the fact that the experimental noise will be amplified first by the derivation and second by the 1/T multiplication. This 1/T factor should by the way not be omitted, as it sometimes happens in the literature [23–25]. As already noticed by some authors [10,26] a plot of dDM=dT provides the crossover temperature distribution, weighted with the magnetic moment m of each particle (thus, it is proportional to T rðTÞ assuming that both K and m are proportional to V). The omission of the 1/T can be ascribed to the use of the wrong expression for the ZFC/FC curves: as discussed before, a V multiplicative term is missing in the two integral contributions (superparamagnetic and blocked particles) when the susceptibilities (i.e. the magnetizations) are carelessly added instead of the magnetic moments. In the case of a single MAE (i.e. a single volume) and for the improved two states model, the DM curve is just a step function, constant before TX and going to zero at T¼TX. Therefore, the derivative will correspond to a dirac peak at TX(V) for each particle volume in an assembly (or equivalently, for each MAE), the amplitude of this dirac scaling almost linearly with TX. These considerations provide another way to understand the above formulas relating rðKÞ or rðTX Þ to the DM curve. However, since the true DM curve for a single MAE is not a step function but varies continuously (because the blocked-superparamagnetic crossover is progressive), the derivative will not be a dirac peak but a peak having a finite width (see Fig. 8a). As long as this ‘‘natural’’ width is smaller than the dispersion of TX values due to size distribution, the above relations are good approximations. As a final remark, we can examine the direct method proposed by Wohlfarth [2] to obtain the distribution of blocking

dðTM ZFC Þ dT

This relation seems even more simple than the one we have established above. Are they equivalent? We have computed rðTÞ with three different formulas: the correct one, by calculating ð1=TÞ dDM=dT, the erroneous one where the (1/T) term is omitted, and the one proposed by Wohlfarth. As can be seen in Fig. 8, the three approaches are almost identical as far as a single particle size (i.e. single MAE and crossover temperature) is concerned. From Fig. 8a, we can also see that the crossover temperature TX lies more in the middle of the peak (which is asymmetrical) than the blocking temperature TB. On the other hand, when we consider a particle assembly with a size distribution, the three approaches give completely different curves: only the use of ð1=TÞ dDM=dT provides the correct crossover temperature distribution rðTX ¼ TÞ. The reason of this discrepancy is in fact easy to understand. The omission of the (1/T) term corresponds to the omission of a V multiplication in each contribution (superparamagnetic and blocked), while the equation proposed by Wohlfarth omits a V2 multiplication in the contribution of superparamagnetic particles to the total magnetic moment. The contribution of large particles is then severely minimized in both cases (the Wohlfarth formula is even worse): when extracted from experimental susceptibility curves, the deduced rðTÞ are consequently artificially displaced towards higher crossover temperature. Looking at the original paper of Wohlfarth [2], we can see that he has considered the case of a TB distribution but with particles having the same volume! In fact, he never envisages that the MAE distribution would be related to a volume distribution. This unfortunately makes quite useless his proposed method for the direct determination of rðTÞ. Moreover, while the proposition of Wohlfarth to neglect the contribution of blocked particles is justified for a single volume, this is no more a good approximation for samples with a size distribution. Neglecting the contribution of blocked particles, as it is sometimes done [25], is in fact not a sound choice.

7. Conclusion We have examined the limitations of the widely used two states model initially proposed by Wohlfarth [2], where the magnetic particles are supposed to be either fully blocked or fully

TX TB (-1 / T) dΔM / dT dΔM / dT d(T MZFC) / dT

ρ (T) (arb. unit)

ρ (T) (arb. unit)

(-1 / T) dΔM / dT dΔM / dT d(TMZFC) / dT

7

8

9

10 T (K)

11

12

0

10

20

30

40

T (K)

Fig. 8. Crossover temperature distribution rðTÞ calculated in three different ways, for a single particle size (i.e. a single MAE and a single crossover temperature) (a) and for a gaussian size distribution (b). Only the correct method, using rðTÞpð1=TÞdDM=dT provides the true distribution in the case of a size distribution. The calculations are performed using n0 ¼ 1010 Hz, vT ¼ 2 K/min, Keff ¼ 100 kJ/m3 and Dm ¼ 4 nm (with a relative dispersion of s=Dm ¼ 20% in the gaussian case). In the case of a single particle volume, the crossover temperature TX is indicated on the figure, together with the blocking temperature TB.

F. Tournus, A. Tamion / Journal of Magnetism and Magnetic Materials 323 (2011) 1118–1127

superparamagnetic (i.e. at thermodynamic equilibrium). This crude model, the abrupt transition model (ATM), appears to be an excellent approximation in most practical cases: an extremely sharp size distribution (less than a 5% relative diameter dispersion) would be necessary to detect the details of the progressive crossover from the blocked regime to the superparamagnetic one. Nevertheless, it is still preferable to use an improved two states model which takes into account the experimental temperature sweeping rate, contrary to the usual model. Such a model allows a convenient analysis of experimental susceptibility curves in the case of a particle assembly with a size distribution rðVÞ. A particular attention must be paid to the fact that the contribution of the superparamagnetic particles, respectively blocked particles, involves a V 2 rðVÞ term, respectively a V rðVÞ term: an erroneous expression is still often used in the literature. We have established that the ZFC peak temperature Tmax can be related to the anisotropy constant Keff, provided that the particle magnetic size distribution is precisely known. We have shown that the particle volume corresponding to a blocked-superparamagnetic crossover situated at Tmax is not linked in a simple way to the size distribution parameter: therefore, Tmax is in particular not equal to the mean, or median, transition temperature (or blocking temperature). In addition, it should be kept in mind that, since Tmax increases rapidly with the relative size dispersion, a quite dramatic error on the Keff determination can be made by considering that Tmax reflects the behavior of particles having the mean volume V : the oversimplified and erroneous relation Keff V ¼ 25kB Tmax should definitely not be used. The expression of ZFC/FC curves involving two distinct contributions (from blocked and superparamagnetic particles) also allows to demonstrate some invariance properties of the curves: they can be useful to analyze experimental results, for instance to ensure that no particle coalescence has occurred. For experimental curves fit, simulating ZFC/FC curves with the established analytical formulas, where the blocked-superparamagnetic crossover is progressive (PCM: progressive crossover model), can be advantageous without any significant additional computational cost. A very convenient expression similar to the one of the ZFC curve can then be used for the FC curve, because the irreversibility that exists for a single particle size is smoothed out in the case of a size distribution. Such an approach, within the PCM, has been successfully used recently on samples made of magnetic nanoparticles diluted in matrices and it has enabled an accurate determination of their magnetic anisotropy [14–16]. We have restricted here our discussion to the commonly admitted model where both the magnetic moment and the anisotropy energy of a particle are directly proportional to its volume: this may not be true for some particles, and in particular there exist other sources of anisotropy dispersion that can play a role [17,16]. Nevertheless, the described approach can easily englobe other models. Finally, we emphasize that all the theory we have developed relies on strong hypotheses (uniaxial macrospin, no inter-particle interaction, linear response to the applied field): they may easily be wrong assumptions as long as real samples are concerned and a special care must then be taken to verify the validity of these simplifying hypotheses. The use of a wrong theoretical framework will certainly correspond to a magnetic characterization with

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significant errors. The present theory may however be extended in the future to more complex cases.

Acknowledgements The authors acknowledge S. Rohart and E. Bonet for fruitful discussions. This work has been partially funded by the ‘‘Agence Nationale de la Recherche’’ (ANR DYSC).

References [1] F. Tournus, E. Bonet, J. Magn. Magn. Mater. (2010), this issue, doi:10.1016/j. jmmm.2010.11.056. [2] E.P. Wohlfarth, Phys. Lett. 70A (1979) 489. [3] M. El-Hilo, K. O’Grady, R.W. Chantrell, J. Magn. Magn. Mater. 114 (1992) 295. [4] M. El-Hilo, K. O’Grady, R.W. Chantrell, J. Magn. Magn. Mater. 117 (1992) 21. [5] J.Z. Jiang, S. Mørup, Nanostruct. Mater. 9 (1997) 375. [6] M.F. Hansen, S. Mørup, J. Magn. Magn. Mater. 203 (1999) 214. [7] H. Kachkachi, W.T. Coffey, D.S.F. Crothers, A. Ezzir, E.C. Kennedy, M. Nogue s, E. Tronc, J. Phys. Condens. Matter 12 (2000) 3077. [8] R.W. Chantrell, M. El-Hilo, K. O’Grady, IEEE. Trans. Magn. 27 (1991) 3570. [9] H. Pfeiffer, R.W. Chantrell, J. Magn. Magn. Mater. 120 (1993) 203. [10] R. Sappey, E. Vincent, N. Hadacek, F. Chaput, J.P. Boilot, D. Zins, Phys. Rev. B 56 (1997) 14551. [11] M. Respaud, J.M. Broto, H. Rakoto, A.R. Fert, L. Thomas, B. Barbara, M. Verelst, E. Snoeck, P. Lecante, A. Mosset, J. Osuna, T. Ould Ely, C. Amiens, B. Chaudret, Phys. Rev. B 57 (1998) 2925. [12] C. Antoniak, J. Lindner, M. Farle, Europhys. Lett. 70 (2005) 250. [13] H.T. Yang, D. Hasegawa, M. Takahashi, T. Ogawa, Appl. Phys. Lett. 94 (2009) 013103. [14] A. Tamion, M. Hillenkamp, F. Tournus, E. Bonet, V. Dupuis, Appl. Phys. Lett. 95 (2009) 062503. [15] A. Tamion, C. Raufast, M. Hillenkamp, E. Bonet, J. Jouanguy, B. Canut, E. Bernstein, O. Boisron, W. Wernsdorfer, V. Dupuis, Phys. Rev. B 81 (2010) 144403. [16] F. Tournus, N. Blanc, A. Tamion, M. Hillenkamp, V. Dupuis, Phys. Rev. B 81 (2010) 220405(R). [17] F. Tournus, S. Rohart, V. Dupuis, IEEE Trans. Magn. 44 (2008) 3201. [18] O. Margeat, M. Tran, M. Spasova, M. Farle, Phys. Rev. B 75 (2007) 134410. [19] F. Luis, J.M. Torres, L.M. Garcı´a, J. Bartolome´, J. Stankiewicz, F. Petroff, F. Fettar, J.-L. Maurice, A. Vaure s, Phys. Rev. B 65 (2002) 094409. [20] Some authors [8,3,4] have used the notation Tg ¼ b/TB S where Tg stands for the ZFC peak temperature (i.e. Tmax with our notation). This is however confusing since they use in fact the median volume to express /TB S, and they make the erroneous statement that /TB S is ‘‘the temperature at which half the magnetic volume will be superparamagnetic’’: this is not true, for the mean blocking temperature as well as for the median one. In the end, their /TB S is the median blocking temperature and corresponds to the temperature at which half the magnetic particles will be superparamagnetic. Other authors [5,6] have used Tpeak ¼ bTBm where TBm is the median blocking temperature: however, they also make the wrong statement that the median volume Vm ‘‘is defined such that the sum of the volumes of the particles with V 4 Vm contributes to 50% of the total particle volume in the sample’’. Let us indeed remind the reader that the correct definition of the median volume Vm is given by the condition that 50% of the particles have a higher volume than Vm and 50% a lower volume. [21] J.L. Dormann, F. D’Orazio, F. Lucari, E. Tronc, P. Prene´, J.P. Jolivet, D. Fiorani, R. Cherkaoui, M. Nogue s, Phys. Rev. B 53 (1996) 14291. [22] The same error as for the ZFC curve (having carelessly dropped a V term in each integral contribution) must also be avoided. [23] J.C. Denardin, A.L. Brandl, M. Knobel, P. Panissod, A.B. Paknomov, H. Liu, X.X. Zhang, Phys. Rev. B 65 (2002) 064422. [24] W.C. Nunes, L.M. Socolovsky, J.C. Denardin, F. Cebollada, A.L. Brandl, M. Knobel, Phys. Rev. B 72 (2005) 212413. [25] Y. Shiratsuchi, M. Yamamoto, Phys. Rev. B 76 (2007) 144432. [26] H. Mamiya, M. Ohnuma, I. Nakatani, T. Furubayashim, IEEE Trans. Magn. 41 (2005) 3394.