Magnetic heating effect of nanoparticles with different sizes and size distributions

Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

Contents lists available at SciVerse ScienceDirect

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

Magnetic heating effect of nanoparticles with different sizes and size distributions a ¨ R. Muller , S. Dutz a, A. Neeb b, A.C.B. Cato b, M. Zeisberger c,n a

Department of Nano Biophotonics, Institute of Photonic Technology, Jena, Germany Institute of Toxicology and Genetics, Karlsruhe Institute of Technology, Karlsruhe, Germany c Department of Spectroscopy and Imaging, Institute of Photonic Technology, Jena, Germany b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 May 2012 Received in revised form 3 September 2012 Available online 5 October 2012

We present a comparative study of dynamic and quasistatic magnetic properties of iron oxide nanoparticles. The samples are prepared by different wet chemical precipitation methods resulting in different sizes and size distributions. The structural characterization was performed by X-ray diffraction and transmission electron microscopy. The heating effect in an ac field in the range 0-30 kA/m at 210 kHz was measured calorimetrically. In addition, a vibrating sample magnetometer was used for hysteresis and remanence curve measurements. & 2012 Elsevier B.V. All rights reserved.

Keywords: Magnetic nanoparticle Hysteresis Ne´el relaxation Hyperthermia Size distribution

1. Introduction Magnetic iron oxide nanoparticles (MNP) are promising tools for medical applications like hyperthermia [1] which relies on the fact that the particle system absorbs energy when it is subjected to an alternating magnetic field. The relevant property of the magnetic material is the specific heating power (SHP) that strongly depends on the size D, the size distribution fðDÞ, and the microstructure of the particles as well as on the ac magnetic field amplitude Hac and the frequency f. The amount of energy W dissipated during one field cycle is given by the area of the hysteresis loop of the M(H) curve. For single domain particles the reason for the hysteresis is the energy barrier KV (K anisotropy constant, V particle volume) between states of opposite magnetization. The reversal of the magnetization can be achieved by two means. Firstly, by an external field H which causes an additional energy contribution in the order of m0 M s H. This mechanism is the basis of the Stoner–Wohlfarth model (SWM) [2], which was developed for randomly oriented uniaxial particles but can be extended to particles of other symmetry (e.g. [3]). Secondly, the barrier can be overcome by thermal activation (Ne´el relaxation) with the relaxation time

t ¼ t0 exp

KV kB T

ð1Þ

where kB is the Boltzmann constant and T  300 K the temperature. This mechanism is the basis of the linear response model (LRM) as used in Ref. [4]. In real particles both mechanisms are present and the models mentioned above can be regarded as the limiting cases for t b1=f (SWM) or tu1=f and Hac 5HK (LRM). Here HK ¼ 2K=m0 Ms is the anisotropy field. A general treatment that covers both effects is given in Ref. [5]. We will compare our experimental data with the SWM and LRM as they allow a relatively simple interpretation. Moreover, in real samples both limiting cases are present because of the size distribution of the particles. In the SWM the specific hysteresis losses WðHac Þ depend on the ratio between the field amplitude and the anisotropy field. For Hac oHK =2 the losses are zero. For larger fields the specific hysteresis losses (SHL) per cycle increase and reach their limit at ^ s . Here M ^ s ¼ Ms =r is the specific Hac ¼ HK which is W  m0 HK M saturation magnetization, Ms is the saturation magnetization, and r the density of the magnetic material. The specific heating power for this quasistatic hysteresis process is P ¼ fW. In real samples the size distribution of the particles causes a distribution of HK which modifies the onset of WðHac Þ. In the framework of the LRM the specific heating power can be expressed by [6] P¼

m0 pf w00 H2ac r

ð2Þ

where w00 is the imaginary part of the susceptibility n

Corresponding author. Tel.: þ49 3641 206 109. E-mail address: [email protected] (M. Zeisberger).

0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2012.09.064

w00 ðDÞ ¼

m0 M2s VðDÞ

2pf tðDÞ

3kB T

1 þ ð2pf tðDÞÞ2

ð3Þ

R. M¨ uller et al. / Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

This model also allows to include the size distribution fðDÞ of the particles by a volume weighted average value /w S ¼ 00

R1 0

fðDÞw00 ðDÞD3 dD R1 3 0 fðDÞD dD

ð4Þ

In this paper we present investigations on four different types of MNP in the size range 10–20 nm. The heating effect at 210 kHz is measured calorimetrically for field amplitudes up to 30 kA/m. In addition, quasistatic magnetization measurements using a vibrating sample magnetometer (VSM) are performed. The calorimetric data are directly related to the heating applications whereas the VSM measurement provides additional information on the samples. In particular, it allows to obtain magnetic data at higher fields. In previous investigations we have shown the size dependence of the static hysteresis and the SHP (at one fixed field amplitude) [7,8] of particles in this size range. In this paper we include field dependent SHP measurements which could not be found in the literature. The most investigations in the literature were done on superparamagnetic MNP or cover field amplitudes that are not sufficient for ferrimagnetic MNP. In addition to the loss mechanisms discussed above, the movement of domain walls in multidomain particles and the mechanical rotation of particles suspended in a fluid (Brown relaxation) can cause a heating effect. These effects will not be regarded here as they are not relevant for medical hyperthermia. The first effect appears only for very large particles (D 4 100 nm) [4], the second does not appear when the particles are located in biological tissue [9].

2. Preparation All samples were prepared by wet-chemical methods. Two samples (SD1, SD2) of single domain particles were prepared by a precipitation method similar to [10]. A KOH solution was added to a FeCl2/FeCl3 solution up to pH  9. After that, the solution was boiled for 5 min. The magnetic particles were then washed with water and freeze-dried. The samples SD1 and SD2 differ in the addition rate of the KOH-solution (SD1: 1.8 ml/min; SD2: 1.2 ml/min). Sample MCNP (multi-core nanoparticles) was prepared by a precipitation method according to [11]. NaHCO3 solution was slowly added to FeCl2/FeCl3 solution up to pH  8. After that, the solution was boiled to form an almost black precipitate. The magnetic particles were then washed with water and freezedried. Sample LSDP (large single domain particles) was prepared by an oxidation process in aqueous solution similar to [12]. Into an aqueous NaOH solution, which was sufficiently deaerated by N2 gas, sodium nitrate (NaNO3) as an oxidant was added. Deaerated ferrous chloride aqueous solution was added to the alkaline solution and the mixture was kept at 25 1C for 24 h. MNPs of all samples were separated from the solution with a magnet and washed three times with pure water. Finally, the particles were coated with the biocompatible coating material carboxymethyldextran (CMD; initial material: CMD sodium salt from Fluka, Buchs, Switzerland) as follows. After adjusting the pH with diluted HCl to 3–4, the suspension was warmed up to 45 1C. The suspension was homogenized by ultrasonic treatment for 1 min (Sonopuls GM200, Bandelin electronic, Berlin, Germany) and an aqueous solution of CMD (CMD/iron oxide weight ratio 41=3) was added. The suspension was stirred for further 60 min at 45 1C and the coated nanoparticles were separated magnetically and washed once with water. The suspensions of MCNP and LSDP are not longterm stable. The particles sedimentate within some hours (LSDP) or several days (MCNP), respectively.

81

3. Experimental/methodology The structural characterization of our samples was performed by X-ray diffraction (XRD) measurement using an X’PERT MPD Pro (PANalytical, Almelo, NL) system. In addition, TEM investigations using JEM 3010 System (Jeol, Japan) system were done with three of the samples in order to obtain size distribution data. The quasistatic magnetic properties were investigated by a vibrating sample magnetometer (VSM). We use a MicroMagTM3900 (Princeton Measurements Corp., USA) system. The measurements include hysteresis loops in a wide range of maximum fields up to the saturation state and remanence curve measurements. For superparamagnetic samples there is a well established method to measure the size distribution [13]. However, our samples consist of a mixture of superparamagnetic and stable ferrimagnetic particles, and the measurement of the remanence curve Mr(H) allows to obtain data which are related to the size distribution of the particles [14]. In particular, the saturation remanence ratio mrs ¼ M rs =M s (with Mrs the maximum remanence) corresponds to the ratio of ferrimagnetic (hysteretic) and superparamagnetic particles. Moreover, the switching field distribution (SFD) which can be calculated from the remanence data [15] by SðHÞ ¼

1 dM r ðHÞ M rs dH

ð5Þ

provides more detailed information on the hysteretic fraction. S(H) can be regarded as the distribution of the amount of particles that switch their magnetization irreversibly at the field H. It should be noted here that there is no sharp distinction between superparamagnetic and ferrimagnetic particles. The criterion is the relation between the relaxation time tðDÞ and the typical time tloop for measuring a magnetization loop. Besides superparamagnetic (t 5 tloop ) and ferrimagnetic (t b tloop ) there is an intermediate fraction (t  tloop ). For the calorimetric determination of SHP the particle suspensions (usually 1 ml) were thermally isolated in PUR foam and placed into a coil that provides the alternating magnetic field (frequency 210 kHz, amplitudes up to 30 kA/m). In order to measure temperature changes over time we used a fibreoptical sensor (OPTOcon, Dresden, Germany). There is no effect of the magnetic field on the temperature sensor. The field generator is based on a HTG-10000 device by Linn HighTherm, Eschenfelden, Germany. The SHP was calculated by P¼

c ms dT mp dt

ð6Þ

where c is the specific heat capacity and ms the total mass of the fluid sample. mp is the mass of the iron oxide particles in the sample, and dT=dt is the heating rate determined in the linear range (the first  20 s) after switching on the magnetic ac field. The specific heat capacity of water was used for the calculation as the particle concentration in the suspension was in the order of a few percent. The latter was determined by measuring the specific saturation magnetization using the VSM and assuming the specific magnetization values which were obtained from the powder samples (Table 2).

4. Results and discussion 4.1. X-ray data The measured XRD data are shown in Fig. 1. From the width of the [440] peak we calculated the mean particle diameter DX according to the Scherrer formula whereas the position of the peak provides information on the composition. As magnetite

82

R. M¨ uller et al. / Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

intensity (a.u.)

SD1 SD2 MCNP LSDP

61°

62°

63°

64°

65°

2θ

Fig. 2. TEM image of sample SD1.

Fig. 1. XRD pattern (y2y-scan near the [440] peak).

Table 1 Data from XRD and TEM investigations: DX mean diameter due to the width of the [440] peak, 2y value and corresponding magnetite percentage x as well a related anisotropy constant K; D0 and sD are the parameters of the log-normal size distribution. DX (nm)

2y (deg)

x (%)

K (104 J=m3 )

D0 (nm)

sD

SD1 SD2 MCNP LSDP

10.9 12.6 15.2 20.5

62.765 62.790 62.850 62.670

53 47 32 76

1.76 1.85 2.05 1.44

11.2 – – 25.4

0.37 – – 0.51

tends to oxidize to maghemite the particles usually consist of an intermediate phase which can be characterized by the peak position that is 62.9811 for maghemite and 62.5721 for magnetite according to the JCPDS database (39-1346 and 19-0629). From the actual peak positions and these two limits we calculated effective percentages x of magnetite which are shown in Table 1, assuming a linear shift of the peak position with the mass fraction of the phases and a formation of nanoparticles with a homogeneous stoichiometry. This might not be the real case since an oxidation ¨ of Fe3O4 starting from the surface (see [16]: Mossbauer-investigations on iron oxide but different preparation method) or distortion of the lattice could occur but it was assumed for simplicity reasons. The table also shows an effective anisotropy constant that is obtained from averaging the values of magnetite (1:1  104 J=m3 [17]) and maghemite (2:5  104 J=m3 [17]) according to the composition x. The data show that the largest particles (LSDP) have a composition close to Fe3O4 whereas the other particles are closer to g-Fe2 O3 . In the literature the influence of a surface anisotropy of nanoparticles ‘with a few atomic layers’ [18] is discussed. The effect was shown on maghemite particles with about 4 nm [19] or 5 nm [20] in size but only a weak effect was shown in 12 nmparticles [19]. According to the relations given in [20] a small contribution of surface anisotropy cannot be excluded in sample SD1 but is weak in the other samples.

Fig. 3. TEM image of sample LSDP.

20

SD1 lognormal fit

18 16 14 number

Sample

12 10 8 6 4 2 0

0

5

10

15

20

25

30

35

D [nm] Fig. 4. Particles size distribution of sample SD1: Histogram obtained from the TEM data and log-normal fit with D0 ¼ 11:2 nm and sD ¼ 0:37.

fðDÞ ¼ LðD0 , sD ,DÞ with " # 2 1 ln ðx=x0 Þ Lðx0 , sx ; xÞ ¼ pffiffiffiffiffiffi exp  , 2s2x 2psx x

xZ0

ð7Þ

4.2. TEM investigations Figs. 2 and 3 show typical TEM images of the samples SD1 and LSDP. Several images were used obtain statistical size data. In the literature (e.g. Ref. [13]) a log-normal size distribution

is often mentioned as suitable for nanoparticles. The histograms of our data (Figs. 4 and 5) show that a log-normal distribution can also be used for our particles. For sample SD2 we have no TEM data. But we expect a similar distribution as for SD1 because of

R. M¨ uller et al. / Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

30

Table 2 Magnetic parameters of dried powders at room temperature (HC coercivity, mrs ^ s : specific magnetization, HS: mean value of the fitted switchremanence ratio, M

LSDP lognormal fit

25

ing field distribution S(H), sS width of the fitted S(H).

20 number

83

15

Sample

Hc (kA/m)

mrs

^ s (Am2/kg) M

HS (kA/m)

sS

SD1 SD2 MCNP LSDP

0.83 1.2 4.0 11.2

0.018 0.033 0.084 0.122

64.9 66.8 59.5 79.9

11.8 10.5 14.5 27.0

0.767 0.726 0.590 0.618

10

100

0

0

10

20

30

40

50

60

70

80

90

D [nm]

103

SD1 SD2 MCNP LSDP

102

Fig. 5. Particles size distribution of sample LSDP: Histogram obtained from the TEM data and log-normal fit with D0 ¼ 25:4 nm and sD ¼ 0:51.

80

W[J/kg]

10-1

101

10-2

100

10-3

10-1

spec. magnetization[Am2/kg]

60 40

10-4

LSDP SD 2 SD 1 MCNP

20

10-2 1

0

P [W/g] (210 kHz)

5

10

100

1000

H[kA/m]

-20

Fig. 7. Specific hysteresis losses per cycle W from VSM measurements and corresponding specific heating power P ¼ fW at 210 kHz.

-40 -60 -80 -1000

-500

0 H[kA/m]

500

1000

Fig. 6. Magnetization curves of the samples.

the similar preparation procedure. The investigation of the size distribution of sample MCNP is more complicated as these particles consist of clusters of smaller grains [11]. Our TEM investigation revealed the distribution parameters of D0 ¼ 71:0 nm and s ¼ 0:17 for the clusters. However, the magnetic properties are mainly related to the size of the grains inside the clusters which are much smaller (see Table 1). As these grains are hard to distinguish in the TEM images we could not obtain a size statistics of them. A relation between the log-normal distribution parameters and the apparent diameter in XRD is given in [23] DX ¼ D0 expðs2D Þ

ð8Þ

Using this equation for our data results in DX ¼ 9.8 nm for SD1 and 19.6 nm for LSDP which is close to the values measured by XRD. 4.3. Quasistatic magnetic properties Fig. 6 shows the magnetization curves of the samples. The main data from the VSM measurements are given in Table 2. The specific magnetization of SD1 and SD2, and LSDP is typical for particles with a mixed g-Fe2 O3 2Fe3 O4 stoichiometry though it should be noted that the value of the LSDP is much closer to that of Fe3O4. The MCNP sample shows a magnetization a little lower than that of g-Fe2 O3 . Beside the influence of the composition on Ms there exists a dependence on the particle size. Ref. [21] gives

an empirical formula containing a decrease of Ms of g-Fe2 O3 particles, that depends linearly on the specific surface. A decrease of Ms with decreasing particle size is revealed as well for Fe3O4 in [22]. Taking this size dependence (from [21]) into account the magnetization of MCNP fits quite well with g-Fe2 O3 data, whereas the Ms values of SD1 and SD2 are higher than expected for pure g-Fe2 O3 . The value of LSDP is close to that of Fe3O4. Despite there is no significance by XRD the comparatively low magnetisation of MCNP might be caused as well by a small fraction of nonmagnetic iron oxide resulting from a stronger oxidation of Fe2 þ in the initial solutions due to the long precipitation time. These magnetic data confirm the tendency of the magnetite contents taken from XRD. The coercivity increases from SD1 to LSDP indicating that the fraction of superparamagnetic particles decreases with increasing mean particle size. A similar trend can be seen in the specific hysteresis losses which we investigated for fields from 0.8 to 800 kA/m (Fig. 7). SHL were investigated because they are related to the heat generation in bigger particles (t b tloop ) and can be determined in a wide field range up to the saturation. Remanence data provide information on the statistical distribution of the particle properties. A low remanence ratio indicates a large fraction of superparamagnetic particles. The remanence curve itself and the SFD which is derived from it provide information on the hysteretic fraction in the sample and correlate with their particle sizes. Fig. 8 shows an increasing mean value from sample SD1 to LSDP, i.e. with increasing mean particle size. The SFD was fitted [14] by a Log-normal distribution SðHÞ ¼ LðHS , sS ; HÞ. The fits are shown in Fig. 8. The SFD is not uniquely related to the size distribution but it depends on the distribution of HK and is additionally widened because of the statistical orientation of the particles.

R. M¨ uller et al. / Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

The comparably weak correlation of the coercivity with the losses at low fields (Hmax o5 kA=m) is remarkable. Moreover, the small mean crystallite size DX and the low remanence even for LSDP suggest the presence of a significant fraction of small particles. The MCNP and LSDP samples show a much higher hysteresis compared to the small particles (SD1, SD2) which can be an advantage in hyperthermia applications. However, this advantage can only be used if the hyperthermia is performed at relatively high field amplitudes of ]15 kA=m.

cal SWM LRM

100 P[W/g]

84

10 4.4. Dynamic magnetic properties Figs. 9–12 show the calorimetrically determined SHP values which were calculated from the measured heating rates dT=dt. The latter is usually obtained with a relative error of 7 5% by a linear fit to the heating curve T(t). In addition, despite the PUR foam isolation, our setup shows a temperature drift of a few mK/s due to the temperature difference between the sample and the environment. This effect becomes important when the heating rate due to the magnetic field is low. For this effect we estimate an absolute error of the specific heating power in the order of 1 W/g. The data of the MCNP at the highest fields ( Z27 kA=m) are less accurate ( 715%) as the sample was heated very quickly allowing only a few data points between room temperature and

3

5

10

20

30

H [kA/m] Fig. 10. Specific heating power of sample SD2: calorimetrically measured values (squares), values calculated from quasistatic hysteresis data (circles), and data calculated for the linear response model (line).

1000

cal SWM 100

0.07

P[W/g]

SD1 SD2 MCNP LSDP

0.06

S(H) [m/kA]

0.05 0.04

10

1

0.03 0.02

0.1

0.01

3

5

10

20

30

H [kA/m]

0 -0.01

Fig. 11. Specific heating power of sample MCNP: calorimetrically measured values (squares), and values calculated from quasistatic hysteresis data (circles).

0

5

10

15

20

25

30

35

40

H [kA/m] Fig. 8. Switching field distribution calculated from the remanence data (symbols) and log-normal fit (lines).

P[W/g]

cal SWM LRM

100

100

cal SWM

10

P[W/g]

10 1

1

10

20

30

H [kA/m] 0.1 1

2

3

5

10

20

30

H [kA/m] Fig. 9. Specific heating power of sample SD1: calorimetrically measured values (squares), values calculated from quasistatic hysteresis data (circles), and data calculated for the linear response model (line).

Fig. 12. Specific heating power of sample LSDP: calorimetrically measured values (squares), and values calculated from quasistatic hysteresis data (circles).

50 1C. We stopped the heating at 50 1C in order to avoid degeneration of the particle coating. The samples show maximum values (at 30 kA/m and 210 kHz) of the specific heating power between 216 W/g (SD1) and 702 W/g

R. M¨ uller et al. / Journal of Magnetism and Magnetic Materials 328 (2013) 80–85

(MCNP). For the samples SD1 and SD2 the slopes in the log–log plots (Figs. 9 and 10) are approximately 2 with a slight variation, i.e. PpHn with n  2. The ‘LRM’ labeled lines in Figs. 9 and 10 are calculated by the LRM and show an approximate agreement with the measured data. For sample SD1 we used Eq. (4), and for SD2, where we have no distribution data we used Eq. (3). For the larger particles we found a steeper field dependence of n ¼ 4:2 for sample MCNP and n ¼ 3:7 for sample LSDP. For comparison Figs. 9–12 show SHP values calculated by the SWM, i.e. from the extrapolating the static hysteresis losses to the frequency of 210 kHz. For the samples SD1 and SD2 these SWM values are about one order of magnitude lower than the calorimetrically measured data which results from the small fraction of SWM like particles in these samples. For sample MCNP and LSDP the SWM data are closer to the calorimetric data but the SWM data show a significantly lower slope compared with the calorimetric data.

5. Conclusions We have investigated four different samples of iron oxide nanoparticles which have been prepared by wet chemical precipitation. The samples SD1 and SD2 with small particles (mean diameters of 10.9 nm and 12.6 nm) show a mainly superparamagnetic behavior with very low static hysteresis losses and a specific heating power that is in approximate agreement with the linear response theory. The samples MCNP and LSPD which consist of larger (mean diameter 15.2 nm and 20.5 nm) particles show the behavior of a mixture of superparamagnetic and stable ferrimagnetic particles. Their static hysteresis losses are significantly higher than for SD1/SD2 and the heating effect shows a steeper field dependence. For medical hyperthermia the MCNP are particularly attractive because of the high specific heating power. As there is a trade-off between frequency and field in this application the steep P vs H curve suggests to operate at higher fields rather than higher frequencies.

Acknowledgments The work was supported by the EU (Project 214137 Nano3T) and the DFG (ZE 825/1-1). The authors thank Dr. A. Undisz

85

(Institute of Materials Science and Technology, FSU Jena) for preparing the TEM images.

References [1] I. Hilger, R. Hiergeist, R. Hergt, K. Winnefeld, H. Schubert, W.A. Kaiser, Investigative Radiology 37 (10) (2002) 580–586. [2] E.C. Stoner, E.P. Wohlfarth, Philosophical Transactions of the Royal Society A 240 (1948) 599–642. [3] C.E. Johnson, W.F. Brown, Journal of Applied Physics 32 (suppl.) (1961) 243S. [4] R. Hergt, W. Andr¨a, C.G. d’Ambly, I. Hilger, W.A. Kaiser, U. Richter, H.G. Schmidt, IEEE Transactions on Magnetics 34 (5) (1998) 3745–3754. [5] J. Carrey, B. Mehdaoui, M. Respaud, Journal of Applied Physics 109 (2011) 083921. [6] R. Hergt, R. Hiergeist, I. Hilger, W.A. Kaiser, Y. Lapatnikov, S. Margel, U. Richter, Journal of Magnetism and Magnetic Materials 270 (2004) 345–357. ¨ [7] R. Muller, S. Dutz, T. Habisreuther, M. Zeisberger, Journal of Magnetism and Magnetic Materials 323 (2011) 1223–1227. ¨ [8] R. Hergt, S. Dutz, R. Muller, M. Zeisberger, Journal of Physics: Condensed Matter 18 (2006) S2919–2934. ¨ [9] S. Dutz, M. Kettering, I. Hilger, R. Muller, M. Zeisberger, Nanotechnology 22 (2011) 265102. [10] R. Massart, IEEE Transactions on Magnetics 17 (1981) 1247–1248. ¨ R. Hergt, R. Muller, ¨ ¨ [11] S. Dutz, W. Andra, C. Oestreich, C. Schmidt, J. Topfer, M. Zeisberger, M.E. Bellemann, Journal of Magnetism and Magnetic Materials 311 (2007) 52–54. [12] K. Nishio, M. Ikeda, N. Gokon, S. Tsubouchi, H. Narimatsu, Y. Mochizuki, S. Sakamoto, A. Sandhu, M. Abe, H. Handa, Journal of Magnetism and Magnetic Materials 310 (2007) 2408–2410. [13] R.W. Chantrell, J. Popplewell, S.W. Charles, IEEE Transactions on Magnetics 14 (5) (1978) 975–977. ¨ [14] M. Zeisberger, S. Dutz, J. Lehnert, R. Muller, Journal of Physics: Conference Series 149 (2009) 012115. [15] H. Pfeiffer, Physica Status Solidi A 118 (1990) 295. ¨ [16] R. Muller, R. Hergt, S. Dutz, M. Zeisberger, W. Gawalek, Journal of Physics: Condensed Matter 18 (2006) S2527–2542. ¨ [17] Landoldt–Bornstein, Zahlenwerte und Funktionen, Springer, Berlin, 1967. [18] X. Batlle, A. Labarta, Journal of Physics D: Applied Physics 35 (2002) R15–42. [19] M.P. Morales, M. Andres-Verges, S. Veintemillas-Verdaguer, M.I. Montero, C.J. Serna, Journal of Magnetism and Magnetic Materials 203 (1999) 146–148. [20] N. Perez, P. Guardia, A.G. Roca, M.P. Morales, C.J. Serna, O. Iglesias, F. Bartolome, L.M. Garcia, X. Batlle, A. Labarta, Nanotechnology 19 (2008) 475704. [21] D.H. Han, J.P. Wang, H.L. Luo, Journal of Magnetism and Magnetic Materials 136 (1994) 176–182. [22] G.F. Goya, T.S. Berquo, F.C. Fonseca, M.P. Morales, Journal of Applied Physics 94 (2003) 3520–3528. [23] F. Sanchez-Bajo, A.L. Ortiz, F.L. Cumbrera, Acta Materialia 54 (2006) 1–10.