Determination of f0 for fine magnetic particles

Journal of Magnetism and Magnetic Materials 125 (1993) 345-350 North-Holland

Determination of f0 for fine magnetic particles D . P . E . D i c k s o n a, N . M . K . R e i d a, C. H u n t a, H . D . W i l l i a m s b, M. E I - H i l o b a n d K. O ' G r a d y b a Department of Physics, University of Liverpool, Liverpool L69 3BX, UK b School of Electronic Engineering Science, University College of North Wales, Bangor, Gwynedd, LL57 1UT, UK Received 26 August 1992; in revised form 7 May 1993

In this paper we have determined a value for the pre-exponential factor in the N6el-Arhennius equation for the iron oxyhydroxideparticles found in the protein ferritin. The data were obtained using a combination of zero field magnetic and M6ssbauer spectroscopystudies yielding a value for f0 of (9.5 + 2.7) X 1011 Hz. This value is significantlydifferent to that of 2.8x 109 Hz commonlyused and in closer agreement to that of 10t3 Hz obtained for iron particles using an analogous technique. Using our experimental value for f0 givesa revised superparamagnetic criterion for DC magnetic measurements on a 100 second time-scale of KV < 32 kT and a M6ssbauer spectroscopycriterion for a measurement time-scale of 10 - 9 s of KV < 8 kT. Our results together with other published data would suggest that a more appropriate estimate for the value of f0 would lie in the range 10x2 to 1013 Hz.

but with a simplified expression for f0 which in zero field is given by

I. Introduction The relaxation time for the reversal of the direction of magnetisation of a small single domain magnetic particle is usually described by an Arrhenius type equation ,/.--1

=f0 exp(-AEa/kT )

(1)

first proposed by N6el [1]. Thus the relaxation time for this process, known as superparamagnetic relaxation, depends on the probability of thermal excitation over an anisotropy energy barrier A E a. The pre-exponential factor f0 plays a crucial role in determining the values of the relaxation time obtained from this equation. Brown [2] considered the p h e n o m e n o n of magnetic relaxation using a different approach to that of N6el and derived a similar expression to eq. (1)

Correspondence to: Dr. M. E1-Hilo, School of Electronic Engineering Science, University College of North Wales, Bangor, Gwynedd, LL57 1UT, UK.

fo = [ K V / a r k T ]I/2yoHK

(2)

where 3'o is the gyromagnetic ration, K is the anisotropy constant, V is the particle volume and H r is the anisotropy field. Using an approximate expression for f0 (----(Y0/2"rr)HK) given by Brown [3] in which f0 was equated to the gyromagnetic precession in the anisotropy field, Kneller [4] calculated a value of f0 = 2.8 × 10 9 Hz for a-iron metal particles. A similar value to this is often arbitrarily applied to many other materials. There are very few reports in the literature of experimental studies to determine the value of f0, despite its significance for theoretical predictions of A C and D C magnetic, magnetic resonance and M6ssbauer spectroscopy phenomena. A n experimental value of approximately 1013 n z has been obtained for Fe-(SiO 2) granular films by a method similar to that described in this paper [5], in which a combination of magnetic and M6ssbauer measurements were used essentially

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D.P.E. Dickson et al. / Determination off o for fine magneticparticles

to solve eq. (1) simultaneously for f0 via a knowledge of ~- and an average value of AEa for the system. However the value of AE~ used was determined from measurements of initial susceptibility and as such will be subject to the effects of dipolar interactions in the system [6,7]. Also it is important to note that in ref. [5] the two measurements are slightly different in that it is assumed that, at the temperature of the peak in the initial susceptibility, the irreversible component of magnetisation disappears, which is not the case [8,9]. The disappearance of blocked particle behaviour in the M6ssbauer spectra does imply the absence of an irreversible component. In any real system of particles there will be a distribution o f anisotropy energies which gives rise to a distribution of blocking temperatures. As the temperature is raised from zero kelvin a median blocking temperature (TB) can be found for a given time-scale of measurement. At T = T B half of the total magnetic volume will exhibit blocked behaviour (i.e. particles with T B < T B) in which the magnetic moment of the particle is essentially fixed in space over the time-scale of the measurement, however the other half of the total magnetic volume will give unblocked behaviour (i.e. particles with T B > T B) in which the moments fluctuate giving a net zero value. Thus measurements over a range of temperatures will show the blocked behaviour decreasing with increasing temperature and will allow the median blocking temperature for the particular measurement time-scale to be determined. The criterion for the median blocking temperature T B for a given time-scale is

where /3 = TB1//TB2 is the ratio of the median blocking temperatures corresponding to the measuring times t 1 and t 2. Eq. (4) can be simplified and then f0 is given by

f o = [(tl)Cl//(t2)] 1/(1-~)

However if the median anisotropy energies of the barrier for the two techniques are not equal, then f0 is given by

f0 = [ ( t l ) t l l ( t 2 ) a ] 1/(~-[3)

2. Measuring techniques

2.1. Magnetic measurement technique A suitable magnetic technique for investigating fine particle magnetic systems is to measure the temperature decay of remanence following the application of a saturating magnetic field. The remanence at any temperature is a measure of the fraction of the blocked particles, i.e. those whose moments have not relaxed to their thermal equilibrium magnetisation during the time of measurement (typically 100 s) when the field is switched off. This measurement time gives the Bean and Livingston [10] criterion for superparamagnetic behaviour which in zero field is (7)

(3)

where t is the measurement time and AE~ is the median anisotropy energy barrier of the system. Thus with two techniques which involve different measuring times tA and t2, the corresponding values o f TB1 and TB2 c a n be found and hence f0 can be determined. It is of course essential that both techniques are applied to precisely identical samples. In the simplest situation, where AE a is the same for both techniques, this leads to

ln(tl/t2) = [In(t1) + l n ( f 0 ) ] (1 - / 3 )

(6)

where a = AEal/AEa2 is the ratio of the median anisotropy energy barriers corresponding to two techniques with measuring times t I and t 2.

( A Ea)crit = In( tfo ) kT.

t-1 = f 0 exp(-- AEa/kTB)

(5)

(4)

For a system with a distribution of A E a, deriving from a distribution of either K, usually due to particle shape variation, or particle size, or even a combination of both the temperature variation of the remanence is then given by

where Ycrit= (AEa)crit//AEa = T/TB as defined from eqs. (3) and (7), is the reduced energy barrier and f ( y ) is the normalised distribution

D.P.E. Dick.son et al. / Determination o f f o for fine magnetic particles

function of reduced energy barriers. The formulation of eq. (8) in reduced parameters enables us to determine TB for a set of data measured at a particular time without the need to use a value for f0. Also irrespective of the form of f ( A E ) the median blocking temperature T B is that at which the remanent magnetisation has fallen to half its value at zero temperature, i.e It(T= T B ) = Ir(0)/2. From the form of eq. (8) it is then clear that the differential of the temperature decay of remanence is the energy barrier distribution f(AEa). It has been shown [6,7] that the form of A E a obtained is independent of the magnetic packing density in the system implying that the distribution obtained is unaffected by dipolar interactions.

2.2. M6ssbauer spectroscopy Another technique which can give information on the relaxation behaviour of iron-containing fine particle systems is 57Fe M6ssbauer spectroscopy. In a magnetic material, magnetic hyperfine structure is observed in the Mfssbauer spectrum if the atomic magnetic moments are fixed in space for a time of 5 × 10 -9 s or greater [11]. In fine particle systems relaxation of the magnetic moments occurs with a time-scale given by the N6el equation and if this relaxation is sufficiently fast the magnetic sextet structure in the M6ssbauer spectrum collapses. The M6ssbauer spectra exhibit a decrease in the blocked fraction of the sample with increasing temperature, which is analogous tO the temperature decay of remanence observed in the dc magnetic measurement. The median blocking temperature is defined as the temperature at which half of the total spectral intensity is associated with the magnetic sextet component of the spectrum.

3. Experimental The magnetic and M6ssbauer spectroscopy measurements were carried out on the same sample of freeze-dried horse spleen ferritin (Sigma Chemicals). The magnetic moment was measured with an

347

Oxford Instruments vibrating sample magnetometer. For the remanence measurements a fixed time of 100 s after reducing the applied field of 5 T to zero was used. The low temperatures were obtained using an Oxford Instruments CF1200 continuous flow cryostat, with temperature measurement via a gold-iron vs. chromel thermocoupie and temperature control with an Oxford Instruments ITC4 temperature controller. Elongated sample holders were used to avoid the need to correct for shape demagnetisation effects. The M6ssbauer spectroscopy measurements were made with a conventional constant acceleration spectrometer and a 100 mCi 57CoRh source, as previously described [12]. The variable temperatures were obtained with an Oxford Instruments CF500 continuous flow cryostat, with a carbonglass resistance thermometer to measure the temperature of the sample. The sample used in this study is the iron-storage protein ferritin. This material has a protein envelope which contains the iron as a microcrystal of the iron oxyhydroxide mineral ferrihydrite [13] with a diameter of up to 8 nm. The nature of the magnetic ordering in the ferritin core is generally considered to be antiferromagnetic, although each particle has a magnetic moment arising from uncompensated spins due to the finite particle size [14]. Ferritin was chosen because it is readily available, and in addition, the protein envelope keeps the magnetic particles well separated by a distance of at least 12 nm, thus greatly reducing the interaction effects which are a complicating feature of the relaxation behaviour in many small particle systems. The characteristic superparamagnetic relaxation behaviour is a well known feature of the M6ssbauer spectra of ferritin as a function of temperature [15,16]. For an antiferromagnetic material the values of T B obtained by the two techniques correspond to different values of A E a in that the magnetic measurement depends on the magnetic moment of each particle, whereas the M6ssbauer measurement depends on the number of iron atoms in each particle, and for antiferromagnetic particles these do not necessarily correspond, If there is either complete proportionality, or alternatively

D.P.E. Dickson et aL / Determination o f f o for free magnetic particles

348

no correlation at all, between the magnetic mom e n t of a particle and the n u m b e r of iron atoms in it, then the value of Aff~a will be the same for both techniques and a = 1 in eq. (6). T h e biggest deviation of a from unity would occur in the (unlikely) situation where the magnetic m o m e n t s of all particles are the same, irrespective of the n u m b e r of iron atoms in them. For this situation and for the extreme case of a fiat distribution of particle volumes, with a range of 3 in the volumes, a value for a of 0.9 is obtained.

70K

O

........

~

7

'"

7.5

O. . . .

50K

0

~

4.0

-.~

30K

Q ¢-

2.0 ,~ o..

20K

0

4. R e s u l t s 1.,

Figure 1 shows t e m p e r a t u r e decay of remanence for the sample of ferritin which we have studied. These data show classical behaviour similar to that which we have previously reported [6,7]. The solid line through the data was calculated from eq. (8). In the calculation a lognormal distribution of y ( = A E a / A / ~ a) has been used, i.e. f(y)

18 .

:D16

"t-~.~.

14 12

i ,° ~

8

o

2.5

I -10

I -5

I 0

I 5

L 10

V e l o c i t y / r n r n s -~

Fig. 2. The M6ssbauer spectra of the ferritin sample measured at different temperatures.

(1/2v~-~-~trry ) e x p ( - [ l n ( y ) ] Z / 2 t r y z )

=

as was found to be the case in our previous study [6,7]. F r o m this analysis a value of the median blocking t e m p e r a t u r e T B = (9.0 + 0.2) K for this

0 ,e-

4.2K

0

2

4

6

8

10

12

14

16

18

20

22

Temperature/K Fig. 1. The temperature decay of remanence curve for the ferritin sample examined.

system at t = 100 s is obtained, i.e. TB1 = (9.0 + 0.2) K. The M6ssbauer spectra of the same sample were obtained over a range of temperatures from 1.3 to 100K. A selection of these spectra is shown in fig. 2. These spectra consist of a superposition of sextet and doublet components, corresponding to the contribution of slowly relaxing (blocked) and rapidly relaxing (unblocked) particles. In order to account for the broadening in the spectral lines, which arises from heterogeneity of the iron sites as well as from the contribution of intermediate relaxation, these spectra were computer fitted to a distribution of hyperfine fields and the fitted lines are shown in the figure. Thus the percentage of the total spectral intensity associated with the sextet component as a function of t e m p e r a t u r e could be obtained and is shown in fig. 3. These data were fitted in the same way as the magnetic data to give a value for the median blocking t e m p e r a t u r e TB = (36 + 1) K for the same system at t = 5 × 1 0 -9 s, i.e. T B 2 = ( 3 6 + 1) K.

D.P.E. Dickson et aL / Determination o f f o for fme magnetic particles

349

error however is significantly less than that which arises if the calculated value of f0 is used when an error close to a factor of 103 results.

100

90 80

~ 7o co 6o

4. Conclusions

~ so c

~ 40 •

3o 2o 10 0

0

I0

1

I

I

I

I

I

,,I

20

30

40

50

60

70

Temperatu re / K

Fig. 3 ~ e temperature variation of the total spectral intensity associated with the sextet component for the sample examined.

T h e above values of the median blocking temperatures TB1 and TB2 for the different times of m e a s u r e m e n t t 1 = 100 s and t 2 = 5 × 10 - 9 S respectively yield a value o f / 3 = Tm/TB2= 0.25 + 0.01. For these data eq. (5) gives a value (5.4 + 2.4) × 1011 Hz for f0. If account is also taken of a possible difference between the median anisotropy energies corresponding to the two techniques, eq. (6) should be used with an appropriate value for a, the ratio of the median anisotropy energies. Putting a = 0.95 + 0.05, together with the above value for /3 = 0.25 + 0.01, gives a value of (9.5 _ 2.7) × 1011 I-'Iz for f0- Thus for those particles examined in this study, the superparamagnetic criterion for dc magnetic m e a s u r e m e n t s on a 100 s time-scale is K V < 32 kT and that for a M6ssbauer technique with a m e a s u r e m e n t time-scale of 5 × 10 -9 seconds is K V < 8 kT. The large error tolerance on the final value of f0 arises from the form of eq. (6) where the p a r a m e t e r s a and /3 are critical. F r o m our data we have measured /3, the ratio of the .average blocking t e m p e r a t u r e to 4%. The p a r a m e t e r a has to be estimated but the worst error is again only 5%. These errors combine in eq. (6) to yield a final error in f0 of around 30% which whilst being disappointing is inevitable from the form of this equation and thus beyond our control. This

In this p a p e r we have described a technique for the determination of the pre-exponential factor in the N6el equation for magnetic relaxation in particulate systems. The technique relies on studies of relaxation in zero field via two techniques with widely differing m e a s u r e m e n t timescales. These techniques have b e e n shown to be independent of interaction effects. The value obtained for our iron oxyhydroxide particles was f 0 ~ 1012 Hz, implying that the superparamagnetic criterion for a m e a s u r e m e n t time of 100 s is K V < 32 kT. Similar studies are in progress on other systems.

Acknowledgements The financial support of the Science and Engineering Research Council is gratefully acknowledged.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

L. N~el, Ann. Geophys. 5 (1949) 99. W.F. Brown Jr., J. Appl. Phys. 34 (1963) 1319. W.F. Brown Jr., J. Appl. Phys. 30 (1959) 130S. E. Kneller, Proceedings of the International Conference of Magnetism, 1964 (Physical Society, London, 1965) 174. Gang Xiao, S.H. Liou, A. Levy, J.N. Taylor and C.L. Chien, Phys. Rev. B. 34 (1986) 7573. R.W. Chantrell, M. EI-Hilo and K. O'Grady, IEEE Trans. Magn. 27 (1991) 3570. M. E1-Hilo, K. O'Grady and R.W. ChantreU, J. Magn. Magn. Mater. 117 (1992) 21. J. Souletie and B. Tissier, J. Magn. Magn. Mater. 15-18 (1980) 201. M. E1-Hilo and K. O'Grady, IEEE Trans. Magn. 26 (1990) 1807. C.P. Bean and J.D. Livingston, J. Appl. Phys. 30 (1959) 120. H.H. Wickman, in: M6ssbauer Effect Methodology, Vol. 2, ed. I.J. Gruverman (Plenum Press, New York, 1966) p. 39.

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D.P.E. Dickson et al. / Determination o f f o for fine magnetic panicles

[12] D.P.E. Dickson, N.M.K. Reid, S. Man, V.J. Wade, R.J. Ward and T.J. Peters, Biochim. Biophys. Acta 957 (1988) 81. [13] T.G. St. Pierre, J. Webb and S. Mann, in: Biomineralisation: Chemical and Biochemical Perspectives, eds. S. Mann, J. Webb and R.J.P. Williams (VCH Verlagsgesellschaft, Weinheim, 1989) 295.

[14] T.G. St. Pierre, D.H. Jones and D.P.E. Dickson, J. Magn. Magn. Mater. 69 (1987) 276. [15] J.F. Boas and B. Window, Aust. J. Phys. 19 (1966) 573. [16] S.H. Bell, M.P. Weir, D.P.E. Dickson, G.A. Sharp and T.J. Peters, Biochim. Biophys. Acta 787 (1984) 227.