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The complex transverse susceptibility
7 February 2000
Physics Letters A 265 Ž2000. 391–402 www.elsevier.nlrlocaterphysleta
The complex transverse susceptibility C. Papusoi, Jr.
)
Alexandru Ioan Cuza UniÕersity’’, Faculty of Physics, 6600 Iasi, Romania Received 6 September 1999; accepted 24 December 1999 Communicated by J. Flouquet
Abstract Previous attempts to explain the dependence of the transverse susceptibility ŽTS. on the DC field predict the occurence of a maximum at the particle anisotropy field Ž HK .. We demonstrate that the TS is also very sensitive to the particle volume distribution, which strongly influences the magnitude of the TS peak and the corresponding value of the DC field. The DC field has the same effect on the TS as the temperature on the AC susceptibility, giving rise to a complex TS with invaluable applications for recording media, like the determination of the particle volume and anisotropy field distributions. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 75.60.-d Keywords: Transverse susceptibility ŽTS.; Stoner-Wohlfarth model; Thermal relaxation; Easy axis distribution; Anisotropy field distribution; Particle volume distribution
1. Introduction The concept of transverse susceptibility ŽTS. was first introduced by Aharony et al. w1x, who calculated the reversible susceptibility tensor for a Stoner–Wohlfarth particle w2x. The experimental procedure for measuring the TS consists of the application of both a DC field H DC , which can be varied in a quite large range, and a small perturbing AC field HAC , usually having the frequency f in the kHz domain, oriented at right angle to the DC field. TS is usually measured as the amplitude ratio of the first harmonic of the induced reversible magnetization along the AC field direction and that of the AC field. As demonstrated in Ref. w1x and, later on, experimentally evidenced by Paretti and Turilli on BaFe12 O 19 thin films w3x, TS as a function of the DC field presents two maxima at H DC s "HK Žreferred hereafter the anisotropy peaks, HK s 2 KrMS being the particle anisotropy field. and a third one at H DC s yHC , Žthe coercivity peak. where HC is the coercivity in the H DC direction. Hoare et al. w4x have studied the effects of the particle easy axis distribution on the TS and demonstrated that the coercivity peak could be sharper than the anisotropy peaks for randomly oriented particle systems, but gradually disappears followed by a sharpening of
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0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 0 9 - 8
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the anisotropy peaks when the easy axis distribution narrows along the AC field direction. The peaks at H DC s "HK are always present, and they reflect the behaviour of the particles having the easy axis oriented perpendicular to the DC field. Thus, it was suggested that the detection of the anisotropy peaks could be used for measuring the particle anisotropy field and easy axis texture w5x. A refined method for measuring the anisotropy field and easy axis distribution was proposed by Chantrell et al. in Ref. w6x where the ‘non-linear susceptibility’ defined by the second harmonic of the magnetization along the DC field direction, was shown to give rise to anisotropy peaks sharper than those of the TS for systems with high dispersion of the easy axis. Sollis and Bissell w7x measured the volume of a commercial CrO 2 tape sample using a combined VSM-TS method relying on the shift of the anisotropy peak due to the sample demagnetizing field. TS measurements on samples consisting of randomly oriented barium ferrite and g-Fe 2 O 3 particles performed by Richter w8x shown that with increasing MSrHK Žwhich was considered to be a measure of the magnetostatic interactions. the field corresponding to the anisotropy peak of the TS decrease faster than the anisotropy field measured by the torque pendulum method. The author concluded that the TS method for measuring the magnetic anisotropy becomes inaccurate for particle systems with low anisotropy or strong interactions. These conclusions were theoretically confirmed by Chang and Yang w9x who shown, within a mean-field approach, that the anisotropy peak gradually converge with the coercivity peak with increasing mean-field parameter. The shift of the anisotropy peak was found to be linearly dependent on the mean-field parameter and independent on the easy axis texture which influences only the position of the coercivity peak w10x. In Ref. w11x the thermal relaxation effects on the TS of an identical Stoner–Wohlfarth particle system, subjected to the simultaneous action of a slowly varying field H Ž t . Ž f f 10y4 Hz. and of a perpendicular infinitesimal field hŽ t . Ž f f 10 Hz., were studied for different orientations of the particle easy axis. It was shown that the peaks occurring at H s "HK are temperature independent while the peak corresponding to H s yHC becomes smother and shifts towards lower values of H with increasing temperature. This result is obviously due to the simplifying assumption that the particle relaxation time is much higher than the reciprocal of the AC field frequency. The goal of this paper is to demonstrate that this hypothesis, which appears to be a common trend for all previous attempts for interpreting the TS w1–11x, is unjustified, the influence of the thermal relaxation on the anisotropy peaks being very important. When the DC field is decreased from HK down to zero, the particle relaxation time distribution shifts from zero to very high values. In the DC field range where the particle relaxation time becomes of the same order of magnitude as the reciprocal of the AC field frequency, the TS is strongly influenced by the particle volume distribution. This mechanism, similar to that giving rise to a complex AC susceptibility w12,13x Žwhere the role of the transverse field is played by the temperature. gives rise to a complex transÕerse susceptibility with invaluable applications for the physics of magnetic recording.
2. Theory Let us suppose that the DC field H DC acts along the Oz axis of a rectangular coordinates system and the AC 0 0 field HAC s HAC cos Ž v t . , Ž HAC < H DC ., perpendicular to the later, acts in the plane xOy, having the azimuth Ž . fAC Fig. 1 . For the beginning, we shall consider an identical particle system, having the volume V and uniaxial anisotropy of constant K, the easy axis being confined in the xOz plane, having the polar coordinate c . The particle magnetization coordinates are denoted by Ž u , f . . In the following, we shall refer only to the reversible behaviour of the TS with respect to the DC field Žfrom positive saturation down to zero. where the present theories predict only one maximum at H DC s HK , and not on the irreversible Žswitching. behaviour leading to the maximum at H DC s yHC .
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Fig. 1. Stoner–Wohlfarth particle, having the easy axis in the xOz plane, subjected to two, perpendicular fields, H DC and HAC s 0 HAC cos Ž v t . .
It is interesting to study the dependence of the TS on the orientation of the particle easy axis with respect to the DC field. According to the previous studies w4x, the most important contribution to the TS comes from the particles having the easy axis oriented perpendicular to the DC field Žfor which the two energy barriers are symmetric.. When the particle relaxation time becomes of the order of 1rv Žas a result of increasing the DC field. the transverse magnetization lags in phase with respect to the AC field, giving rise to a complex susceptibility. The simple mechanism responsible for this phenomenon is depicted in Fig. 2.
Fig. 2. The particle free energy ŽEq. Ž1.. for c s908, f s 08 and different values of h DC ; with increasing h DC the energy barriers separating the free energy minima decrease and so does the relaxation time which may become of the order of 1r v resulting in a strong influence of the thermal relaxation on the TS.
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Assuming the coherent rotation mode for the particle magnetization w2x, the particle free energy becomes: E s E0 Ž ycos 2a y 2 h DC cos u y 2 hAC cos b . ,
Ž 1.
where E0 s KV, hAC s HAC rHK , h DC s H DC rHK , and: cos a s cos u cos c q sin u cos f sin c ,
Ž 2.
cos b s sin u cos f cos fAC q sin u sin f sin fAC ,
Ž 3.
In the hypothesis that the AC field shifts the particle moment orientations corresponding to the extrema Ž i s 1,2,3. of Eq. Ž1. in the absence of hAC with the quantities Ž du i , df i . linear in hAC , a simple perturbation calculus gives:
Ž u i0 , fi0 .
½
du i s du i Ž c , u i0 . hAC
,
df i s df i Ž c , u i0 . hAC
i s 1,2,3
Ž 4.
Ž i s 1,2 correspond to the energy minima, 1 being closest to the AC field direction, and 3 to the maximum. where:
°d ~
0 u i Ž c ,u i . s " cos2
¢d
fi
Žc
, u i0
cos u i0 cos fAC
Ž u i0 . c . q h DC cos u i0
sin fAC
. s sin c cos
.
Ž 5.
Ž u i0 . c .
The pair Ž‘q’ before the fraction line, ‘y’ inside the fraction. corresponds to i s 1, while Žy,q . to i s 2,3. The master equation w14,15x describing the evolution of the occupation probability of the 1-st energy minimum becomes: dP1rdt s y Ž W120 q W210 . P q b Ž W120 g 1 q W210 g 2 . hAC P1 q W210 Ž 1 y b g 2 hAC . ,
Ž 6.
where b s E0r Ž k B T . . The transition rates in the absence of HAC are: Wi 0j s f 0 exp y
D Ei0j Ž h DC . k BT
,
i / j s 1,2 ,
Ž 7.
where f 0 f 10 10 Hz w13x, D Ei0j Ž h DC . being the energy barriers in the absence of HAC , and: g i s f Ž c , u 30 . y f Ž c , u i0 . , f Ž c , u i0 . s "cos fAC
i s 1,2 ,
sin2 Ž u i0 . c . cos u i0 y 2sin u i0 cos2 Ž u i0 . c . cos2 Ž u i0 . c . q h DC cos u i0
Ž 8. ,
i s 1,2 ,
Ž 9.
where ‘q’ before the fraction line in Eq. Ž9. corresponds to i s 1, while ‘y’ to i s 2. Explicitly, the parameters g i represent the perturbation of the energy barriers D Ei j introduced by the AC field, D Ei j s D Ei0j Ž h DC . q E0 g i hAC . 0 Solving Eq. Ž6. for hAC s hAC exp Ž i v t . and taking into account the expression of the particle magnetization along the AC field: m s MrMS s Ž sin u 10 q sin u 20 q cos u 10du 1 q cos u 20 du 2 . cos fAC q Ž sin u 10df 1 qsin u 20 df 2 . sin fAC P1 y sin u 20 cos fAC y cos u 20 cos fAC du 2 y sin u 20 sin fAC df 2 ;
Ž 10 .
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one easily finds the real and imaginary components of the TS, x ' mrhAC s xr q i x i :
°Ž
sin u 10 qsin u 20 . cos fAC b
~
xr s
¢ °Ž ~ ¢
W120 W210 Ž g 1 y g 2 .
v 2 q Ž W120 qW210 .
2
q w Ž cos u 10 du 1 qcos u 20 du 2 . cos fAC
cos u 10 cos fAC du 1 qsin u 10 sin fAC df 1 ,
xi s
h DC - h cr Ž c . ,
Ž 11 .
q Ž sin u 10 df 1 qsin u 20 df 2 . sin fAC x P10 Ž t . ycos u 20 cos fAC du 2 ysin u 20 sin fAC df 2 ,
sin u 10 qsin u 20 . cos fAC b
½
h DC 0 h cr Ž c . ,
W120 g 1 qW210 g 2
v
W210 Ž W120 qW210 .
v 2 q Ž W120 qW210 .
2
y P10 Ž t . q
v W210 g 2 v 2 q Ž W120 qW210 .
2
5
,
h DC - h cr Ž c . , h DC 0 h cr Ž c . ,
0,
Ž 12 . where h cr Ž c . s Ž sin P10
Ž t.
2r3
s P1eq q
c q cos
Ž
2r3
P1 i y P1eq
c.
y3 r2
and
. exp Ž ytrt .
Ž 13 .
is the occupation probability of the 1-st energy minimum under the action of the DC field. P1eq s y1 is the W210 r Ž W120 q W210 . is the equilibrium probability of the first energy minimum, t s Ž W120 q W210 . relaxation time and P1 i is the initial probability at t s 0. Like for the Field–Cooled magnetization process w14x, one may consider: P10 s
½
P1eq Ž h cr . ,
h DC - h cr ,
P1eq
h DC 0 h cr ,
Ž h DC . ,
Ž 14 .
where the critical field h cr is the solution of the blocking condition:
t s Ž W120 q W210 .
y1
st0 ;
Ž 15 .
t 0 f 100 s being a typical measurement time. The choice of t 0 is not critical, since the influence of the thermal relaxation on the TS occurs on a much lower time scale Žt 0 ( 1rv .. Usually, the ‘period’ of the DC field is much higher than that of the AC field, such that the influence of the thermal relaxation on the TS comes from the particles which are superparamagnetic on the time scale of a typical DC experiment. For illustrating the behaviour of the real and imaginary parts of the TS in relation to the easy axis orientation, in Figs. 3–6 are plotted the curves xr Ž h DC . , x i Ž h DC . , for different values of c . The parameter values used in the simulations are: K s 2 10 6 ergrcm3, D s 45 nm Ž V s p D 3r6., T s 300 K, f s 10 3 Hz, fAC s 0.
Fig. 3. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS as a function of h DC s H DC r HK for c s908.
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C. Papusoi, Jr.r Physics Letters A 265 (2000) 391–402
Fig. 4. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s89.98.
Fig. 5. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s89.88.
Fig. 6. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s808 Žcurve 4., 608 Žcurve 3., 408 Žcurve 2. and 208 Žcurve 1..
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According to Figs. 3–6 when c decreases from 908, both, the real and imaginary parts of the TS undergo a very abrupt decrease. In a sharp interval around c s 908, xr Ž h DC . presents two kinks, one of them corresponding to the critical field h cr Ž c . , and the other, occurring at a lower value of h DC , being due to the thermal relaxation on the time scale of the AC field period. These kinks rapidly vanish with decreasing c , and xr Ž h DC . starts to present a flat maximum shifting towards h DC s 0. For c s 08, the TS is zero because the first harmonic of the AC transverse magnetization vanishes. Thus, the TS is dominated by the behaviour of the particles having the easy axis perpendicular to the DC field Ž c s 908.. The particle easy axis distribution in the plane Ž H DC , HAC . could be found by rotating the sample in this plane and noticing the dependence of the TS peak magnitude on the sample rotation angle. Either of the peaks, occurring in the real and imaginary components of the TS, could be used for this purpose. Although the use of the imaginary peak would result in a higher resolution Žaccording to Figs. 3–6 the imaginary peak is due to the particles having the easy axis oriented in a very narrow angular interval around c s 908. it has the disadvantage of having a lower magnitude than the real peak Žabout 1:10 as it will be shown in the following, the same order of magnitude as for the classical AC susceptibility w12x.. The position of the TS peaks should not depend on the angle of rotation, unless the particle easy axis, volume and anisotropy field distributions are somehow connected, e.g. for ferrofluids frozen in a magnetic field. In the following, we shall restrict our attention to the case c s 908, the only orientation of the particle easy axis for which the thermal relaxation effects on the TS are important. In Figs. 7 and 8 and Figs. 9 and 10 are presented xr Ž h DC . , x i Ž h DC . , and also the phase lag w Žtg w s x irxr . of the first harmonic of the induced magnetization along the AC field with respect to the AC field Žwhich should be measured, together with the total susceptibility x s xr2 q x i2 in order to find the real and imaginary parts of the TS. for c s 908 and different values of the particle volume, anisotropy and AC field frequency. The parameter values used in the simulations presented in Figs. 7 and 8 are f s 10 3 Hz, T s 300 K, fAC s 08, and for Figs. 9 and 10, K s 2 10 6 ergrcm3. One notices that either an increase of the particle volume, anisotropy or AC field frequency Žby several orders of magnitude. sharpen the TS maxima and shift them to higher values of h DC . Thus, both xr Ž h DC . and x i Ž h DC . are sensitive to the particle volume, indicating that the TS could be used for measuring the particle size distribution for recording media. On the basis of the above considerations it becomes obvious that the mechanism leading to the occurrence of a complex TS is basically the same as that leading to the complex AC susceptibility w13x. The only difference is that in a transverse field experiment, the particle relaxation time distribution is shifted towards lower values with increasing H DC Žwhich shifts the energy barrier distribution towards lower values. while in a classical AC susceptibility experiment, the same shift is due to increasing temperature.
(
Fig. 7. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS as a function of h DC for two values of the particle diameter D: 40 nm Žcurves 1., 55 nm Žcurves 2. and two values of particle anisotropy K : 10 6 ergrcm3 Žcurves A. and 2 10 6 ergrcm3 Žcurves B..
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Fig. 8. The corresponding phase lag of the transverse magnetization with respect to the AC field Žtg w s x i r xr ..
As we have shown above, the TS is dominated by the behaviour of the particles having the easy axis oriented perpendicular to the DC field direction Ž c s 908.. For these particles, Eqs. Ž11. and Ž12. have the following expressions:
°8W b 2
x s~ r
1
1 y h2DC 4W 2 q v
cos 2fAC y 2
h2DC h2DC y 1
cos 2fAC q sin2fAC ,
2
¢h y 1 cos f , °y4Wvb 1 y h ~ xs ¢0 , 4W q v
h DC - 1 ,
Ž 16 . h DC 0 1 ,
AC
DC
2 DC
i
2
2
cos 2fAC ,
h DC - 1 ,
Ž 17 .
h DC 0 1 , 2
where W s f 0 exp yb Ž 1 y h DC . . In order to extend the calculus for a particle system having volume and
Fig. 9. The real Žsolid line. and imaginary Ždashed line. parts of the TS as a function of h DC for two values of the particle diameter D: 40 nm Žcurves 1. and 55 nm Žcurves 2. and two values of the AC field frequency f : 10 3 Hz Žcurves A. and 10 5 Hz Žcurves B..
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Fig. 10. The corresponding phase lag of the transverse magnetization with respect to the AC field.
anisotropy field distributions, the singularity occurring in Eq. Ž16. for h DC s 1 could be approximated by the Dirac function:
°8W b 1 y h cos f ~ xs 4W q v ¢dŽ h y 1. cos f , 2 DC
2
2
2
2
AC q sin
2
fAC ,
h DC - 1 ,
Ž 18 .
r
2
DC
h DC 0 1 .
AC
This approximation is not able to interpret the dependency w Ž h DC . , which is very sensitive to the second term in Eq. Ž16. for the case h DC - 1 Žespecially in the range of fields lower than the critical field for which vr Ž 2W . s 1.. However, as it will appear from the numerical results below, the influence of this approximation on the real part of the TS is very small comparatively to the superparamagnetic contribution given by the first term in Eq. Ž16. for h DC - 1. It is worthwhile mentioning that the real part of the TS could be written in a more simplified form using the concept of critical volume Vcr Ž h DC , v . s lnŽ2 f 0rv . k B Trw K Ž1 y h DC . 2 x, defined in the same manner as in Ref. w13x for the classical AC susceptibility. Eqs. Ž17. and Ž18. could easily be used for calculating the TS of a particle system with volume and anisotropy field distributions:
½
2
xr 0 s cos fAC h DC 0 G Ž h DC 0 . q q sin2fAC
x i0 s y
1 V
`
1 V
`
`
2
H0 H0 8W b
1 y Ž h DC 0rh K . 4W 2 q v 2
2
VF Ž V . dV G Ž h K . dh K
5
`
H0
Ž 1rh K . G Ž h K . dh K ,
`
H0 H0 4Wvb
1 y Ž h DC 0rh K . 4W 2 q v 2
Ž 19 . 2
VF Ž V . dV G Ž h K . dh K ,
Ž 20 .
where x 0 s mrh AC 0 , h AC 0 s HAC rHK 0 , h DC 0 s H DC rHK 0 , HK 0 being the most probable value of the anisotropy field distribution. G Ž h K . is the reduced anisotropy field distribution Ž H0` G Ž h K . dh K s 1. and F Ž V . is the particle volume distribution Ž H0`F Ž V . dV s 1.. In Figs. 11 and 12 we present the dependencies of the TS on the anisotropy field distribution ŽFig. 11. and on the particle volume distribution ŽFig. 12.. We have assumed a lognormal f Ž x . s 1rŽ'2p x s .exp ywln Ž x . y s 2 x 2rŽ2 s 2 .4 for both distributions, having the most probable values HK 0 , V0 and standard deviations sK , s V .
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Fig. 11. The real and imaginary parts of the TS for Ds 45 nm, K 0 s 2 10 6 ergrcm3 , MS s 300 emurcm3 , f s10 3 Hz, T s 300 K, fAC s 08, sK s 0.05 and s V s 0.01 ŽB., 0.3 Žv . and 0.6 Žq..
The flattening of the particle volume distribution gives rise to an increase of both, real and imaginary components of the TS, especially due to the larger volumes, at the queue of the particle volume distribution, which also shift both maxima towards the anisotropy field, as we have shown above. This effect could also explain the experimental results of Richter w8x, a decrease of the particle anisotropy giving rise to a decrease of the DC field corresponding to the anisotropy peak. As one can notice in Figs. 11 and 12, the anisotropy peak representing a singularity for an identical particle system, has a negligible effect on the TS of a particle system with volume and anisotropy field distributions, and becomes evident only for very narrow distributions, unlikely to those experimentally encountered w18,19x. Thus, the so-called anisotropy peak, is not directly related to the particle anisotropy field distribution, as previously stated on the basis of the Stoner–Wohlfarth theory w1x, w2–11x, being strongly influenced by the particle volume distribution. The real and imaginary components of the TS could be measured using a similar experimental setup to that used for the classical AC susceptibility w12x, which basically consists of two balanced signal coils and a lock-in detection scheme. In order to estimate the range of the induced voltage coming from the sample, let us consider a magnetic thin film having MS s 300 emurcm3 and K s 2 10 6 ergrcm3 , a surface of 1r1 cm and a thickness of 200 nm. If the anisotropy is in plane, a pair of rectangular detection coils, having the surface 1r0.5 cm, 1 cm 0 length and 1000 turns would be suitable. A very simple calculus shows that for HAC s 10 Oe, f s 10 4 Hz and a
Fig. 12. The real and imaginary parts of the TS for s V s 0.3 and sK s 0.01 ŽB., 0.05 Žv . and 0.1 Žq..
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perfect alignment of the particle easy axis, a reduced TS in the range x s 50 % 100 Žtypical values for the maximum of the TS. gives rise to a differential voltage in the range Ee f f s 1.3 % 2.5 mV. If the sample anisotropy is perpendicular, for the same sample shape and detection coils having the shape of rectangular plates having the surface 1r1 cm, 0.5 cm high and 1000 turns, Ee f f s 3.5 % 7 mV. Even if these values are expected to be attenuated by the dispersion of the easy axis orientation, always present within a real sample, they could still be measured using a lock-in detection scheme. For broad particle anisotropy and volume distributions, their deconvolution could be a difficult task unless complementary experiments are available. Different attempts to find the anisotropy field distribution for recording media, especially based on the remanence curves w16x, are proposed in the literature. Usually, the result is a convolution of the anisotropy and particle volume distributions, and a correction for the relaxation effects, requiring some knowledge about the particle volume distribution, is necessary. Among these attempts, the method proposed by Flanders and Shtrikman w20x and later on used by Pfeiffer et al. w17x- w19x for barium hexaferrite powders, has a remarkable resemblance to the TS. The later method relies on the fact that once the sample saturated in a certain direction, the application followed by the suppression of a progressively increasing sequence of applied fields along a direction performing a very small angle with respect to the previous saturation direction results in the switching of the particles having the easy axis oriented in the interval between the perpendiculars to the previous saturation and current applied field directions. The derivative of the remanence curve measured along the perpendicular to the previous saturation direction, corrected for the relaxation effects, gives the anisotropy field distribution. Just like the TS, the later method is capable to deconvolute the easy axis distribution Žby successively rotating the saturating field and measuring the remanence along the perpendicular to the previous saturation direction one can explore the whole angular interval of orientation of the particle easy axis.. Also, both methods involve the same particles, having the easy axis perpendicular to the applied field direction, for which the coherent rotation mode holds w20x. Still, the complex TS offers clear advantages with respect to the methods based on the remanence curves, like a well known time window for the relaxation effects Žt s 1rv . which could also be varied in a quite large range by varying the frequency of the AC field and a very simple procedure for measuring the particle easy axis distribution. 3. Conclusions During a TS experiment, the increasing DC field shifts the particle relaxation time towards lower values, making possible the occurrence of the thermal relaxation on the time scale of the AC field period, for frequencies of the later in the kHz range. For an identical particle system having the easy axis perpendicular to the DC field and in the domain of applied DC fields studied in this paper, the present theories predict the occurrence of only one peak at H DC s HK . We have demonstrated that the thermal relaxation gives rise to a secondary peak in the real part of the TS and to a peak in the imaginary part of it. For a particle system with volume and anisotropy field distributions, only one finite peak is present in the real part due, on one hand, to the anisotropy field distribution which tends to flatten the discontinuity occurring at H DC s HK for an identical particle system, and on the other, to the particle volume distribution which can induce a high peak via the contribution of the particles which become superparamagnetic at DC fields lower then HK . Usually, the latter peak is dominant in a TS experiment. The thermal relaxation influences the TS Õia the particles having the easy axis perpendicular to the DC field. The complex TS could be used as an alternative to the standard AC susceptibility measurements w12,13x. The disadvantages of the later method when applied to recording media are: Ø the negative effects of the temperature variation on the material structure of the sample Že.g. magnetic tapes.; Ø the implication of the usually unknown temperature variation laws of the particle anisotropy and spontaneous magnetization.
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The complex TS method has the following advantages: Ø uses a DC field for shifting the relaxation time distribution Žat constant temperature.; Ø only the particles having the easy axis oriented at 908 with respect to the DC field, which essentially behave in agreement to the coherent rotations model w20x Ženabling the calculus of the energy barriers and magnetization., contribute to the TS. Under these circumstances, the particle activation volume in a TS experiment is expected to be close to the physical volume, which makes this method very efficient for measuring the particle size distribution in recording media. For highly anisotropic media, particle interaction effects are not expected to interfere with the above mentioned effect for DC fields close to HK w7x, which gives the order of magnitude of the fields corresponding to the real and imaginary peaks of the TS.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
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Physics Letters A 265 Ž2000. 391–402 www.elsevier.nlrlocaterphysleta
The complex transverse susceptibility C. Papusoi, Jr.
)
Alexandru Ioan Cuza UniÕersity’’, Faculty of Physics, 6600 Iasi, Romania Received 6 September 1999; accepted 24 December 1999 Communicated by J. Flouquet
Abstract Previous attempts to explain the dependence of the transverse susceptibility ŽTS. on the DC field predict the occurence of a maximum at the particle anisotropy field Ž HK .. We demonstrate that the TS is also very sensitive to the particle volume distribution, which strongly influences the magnitude of the TS peak and the corresponding value of the DC field. The DC field has the same effect on the TS as the temperature on the AC susceptibility, giving rise to a complex TS with invaluable applications for recording media, like the determination of the particle volume and anisotropy field distributions. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 75.60.-d Keywords: Transverse susceptibility ŽTS.; Stoner-Wohlfarth model; Thermal relaxation; Easy axis distribution; Anisotropy field distribution; Particle volume distribution
1. Introduction The concept of transverse susceptibility ŽTS. was first introduced by Aharony et al. w1x, who calculated the reversible susceptibility tensor for a Stoner–Wohlfarth particle w2x. The experimental procedure for measuring the TS consists of the application of both a DC field H DC , which can be varied in a quite large range, and a small perturbing AC field HAC , usually having the frequency f in the kHz domain, oriented at right angle to the DC field. TS is usually measured as the amplitude ratio of the first harmonic of the induced reversible magnetization along the AC field direction and that of the AC field. As demonstrated in Ref. w1x and, later on, experimentally evidenced by Paretti and Turilli on BaFe12 O 19 thin films w3x, TS as a function of the DC field presents two maxima at H DC s "HK Žreferred hereafter the anisotropy peaks, HK s 2 KrMS being the particle anisotropy field. and a third one at H DC s yHC , Žthe coercivity peak. where HC is the coercivity in the H DC direction. Hoare et al. w4x have studied the effects of the particle easy axis distribution on the TS and demonstrated that the coercivity peak could be sharper than the anisotropy peaks for randomly oriented particle systems, but gradually disappears followed by a sharpening of
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0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 0 9 - 8
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the anisotropy peaks when the easy axis distribution narrows along the AC field direction. The peaks at H DC s "HK are always present, and they reflect the behaviour of the particles having the easy axis oriented perpendicular to the DC field. Thus, it was suggested that the detection of the anisotropy peaks could be used for measuring the particle anisotropy field and easy axis texture w5x. A refined method for measuring the anisotropy field and easy axis distribution was proposed by Chantrell et al. in Ref. w6x where the ‘non-linear susceptibility’ defined by the second harmonic of the magnetization along the DC field direction, was shown to give rise to anisotropy peaks sharper than those of the TS for systems with high dispersion of the easy axis. Sollis and Bissell w7x measured the volume of a commercial CrO 2 tape sample using a combined VSM-TS method relying on the shift of the anisotropy peak due to the sample demagnetizing field. TS measurements on samples consisting of randomly oriented barium ferrite and g-Fe 2 O 3 particles performed by Richter w8x shown that with increasing MSrHK Žwhich was considered to be a measure of the magnetostatic interactions. the field corresponding to the anisotropy peak of the TS decrease faster than the anisotropy field measured by the torque pendulum method. The author concluded that the TS method for measuring the magnetic anisotropy becomes inaccurate for particle systems with low anisotropy or strong interactions. These conclusions were theoretically confirmed by Chang and Yang w9x who shown, within a mean-field approach, that the anisotropy peak gradually converge with the coercivity peak with increasing mean-field parameter. The shift of the anisotropy peak was found to be linearly dependent on the mean-field parameter and independent on the easy axis texture which influences only the position of the coercivity peak w10x. In Ref. w11x the thermal relaxation effects on the TS of an identical Stoner–Wohlfarth particle system, subjected to the simultaneous action of a slowly varying field H Ž t . Ž f f 10y4 Hz. and of a perpendicular infinitesimal field hŽ t . Ž f f 10 Hz., were studied for different orientations of the particle easy axis. It was shown that the peaks occurring at H s "HK are temperature independent while the peak corresponding to H s yHC becomes smother and shifts towards lower values of H with increasing temperature. This result is obviously due to the simplifying assumption that the particle relaxation time is much higher than the reciprocal of the AC field frequency. The goal of this paper is to demonstrate that this hypothesis, which appears to be a common trend for all previous attempts for interpreting the TS w1–11x, is unjustified, the influence of the thermal relaxation on the anisotropy peaks being very important. When the DC field is decreased from HK down to zero, the particle relaxation time distribution shifts from zero to very high values. In the DC field range where the particle relaxation time becomes of the same order of magnitude as the reciprocal of the AC field frequency, the TS is strongly influenced by the particle volume distribution. This mechanism, similar to that giving rise to a complex AC susceptibility w12,13x Žwhere the role of the transverse field is played by the temperature. gives rise to a complex transÕerse susceptibility with invaluable applications for the physics of magnetic recording.
2. Theory Let us suppose that the DC field H DC acts along the Oz axis of a rectangular coordinates system and the AC 0 0 field HAC s HAC cos Ž v t . , Ž HAC < H DC ., perpendicular to the later, acts in the plane xOy, having the azimuth Ž . fAC Fig. 1 . For the beginning, we shall consider an identical particle system, having the volume V and uniaxial anisotropy of constant K, the easy axis being confined in the xOz plane, having the polar coordinate c . The particle magnetization coordinates are denoted by Ž u , f . . In the following, we shall refer only to the reversible behaviour of the TS with respect to the DC field Žfrom positive saturation down to zero. where the present theories predict only one maximum at H DC s HK , and not on the irreversible Žswitching. behaviour leading to the maximum at H DC s yHC .
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Fig. 1. Stoner–Wohlfarth particle, having the easy axis in the xOz plane, subjected to two, perpendicular fields, H DC and HAC s 0 HAC cos Ž v t . .
It is interesting to study the dependence of the TS on the orientation of the particle easy axis with respect to the DC field. According to the previous studies w4x, the most important contribution to the TS comes from the particles having the easy axis oriented perpendicular to the DC field Žfor which the two energy barriers are symmetric.. When the particle relaxation time becomes of the order of 1rv Žas a result of increasing the DC field. the transverse magnetization lags in phase with respect to the AC field, giving rise to a complex susceptibility. The simple mechanism responsible for this phenomenon is depicted in Fig. 2.
Fig. 2. The particle free energy ŽEq. Ž1.. for c s908, f s 08 and different values of h DC ; with increasing h DC the energy barriers separating the free energy minima decrease and so does the relaxation time which may become of the order of 1r v resulting in a strong influence of the thermal relaxation on the TS.
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Assuming the coherent rotation mode for the particle magnetization w2x, the particle free energy becomes: E s E0 Ž ycos 2a y 2 h DC cos u y 2 hAC cos b . ,
Ž 1.
where E0 s KV, hAC s HAC rHK , h DC s H DC rHK , and: cos a s cos u cos c q sin u cos f sin c ,
Ž 2.
cos b s sin u cos f cos fAC q sin u sin f sin fAC ,
Ž 3.
In the hypothesis that the AC field shifts the particle moment orientations corresponding to the extrema Ž i s 1,2,3. of Eq. Ž1. in the absence of hAC with the quantities Ž du i , df i . linear in hAC , a simple perturbation calculus gives:
Ž u i0 , fi0 .
½
du i s du i Ž c , u i0 . hAC
,
df i s df i Ž c , u i0 . hAC
i s 1,2,3
Ž 4.
Ž i s 1,2 correspond to the energy minima, 1 being closest to the AC field direction, and 3 to the maximum. where:
°d ~
0 u i Ž c ,u i . s " cos2
¢d
fi
Žc
, u i0
cos u i0 cos fAC
Ž u i0 . c . q h DC cos u i0
sin fAC
. s sin c cos
.
Ž 5.
Ž u i0 . c .
The pair Ž‘q’ before the fraction line, ‘y’ inside the fraction. corresponds to i s 1, while Žy,q . to i s 2,3. The master equation w14,15x describing the evolution of the occupation probability of the 1-st energy minimum becomes: dP1rdt s y Ž W120 q W210 . P q b Ž W120 g 1 q W210 g 2 . hAC P1 q W210 Ž 1 y b g 2 hAC . ,
Ž 6.
where b s E0r Ž k B T . . The transition rates in the absence of HAC are: Wi 0j s f 0 exp y
D Ei0j Ž h DC . k BT
,
i / j s 1,2 ,
Ž 7.
where f 0 f 10 10 Hz w13x, D Ei0j Ž h DC . being the energy barriers in the absence of HAC , and: g i s f Ž c , u 30 . y f Ž c , u i0 . , f Ž c , u i0 . s "cos fAC
i s 1,2 ,
sin2 Ž u i0 . c . cos u i0 y 2sin u i0 cos2 Ž u i0 . c . cos2 Ž u i0 . c . q h DC cos u i0
Ž 8. ,
i s 1,2 ,
Ž 9.
where ‘q’ before the fraction line in Eq. Ž9. corresponds to i s 1, while ‘y’ to i s 2. Explicitly, the parameters g i represent the perturbation of the energy barriers D Ei j introduced by the AC field, D Ei j s D Ei0j Ž h DC . q E0 g i hAC . 0 Solving Eq. Ž6. for hAC s hAC exp Ž i v t . and taking into account the expression of the particle magnetization along the AC field: m s MrMS s Ž sin u 10 q sin u 20 q cos u 10du 1 q cos u 20 du 2 . cos fAC q Ž sin u 10df 1 qsin u 20 df 2 . sin fAC P1 y sin u 20 cos fAC y cos u 20 cos fAC du 2 y sin u 20 sin fAC df 2 ;
Ž 10 .
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one easily finds the real and imaginary components of the TS, x ' mrhAC s xr q i x i :
°Ž
sin u 10 qsin u 20 . cos fAC b
~
xr s
¢ °Ž ~ ¢
W120 W210 Ž g 1 y g 2 .
v 2 q Ž W120 qW210 .
2
q w Ž cos u 10 du 1 qcos u 20 du 2 . cos fAC
cos u 10 cos fAC du 1 qsin u 10 sin fAC df 1 ,
xi s
h DC - h cr Ž c . ,
Ž 11 .
q Ž sin u 10 df 1 qsin u 20 df 2 . sin fAC x P10 Ž t . ycos u 20 cos fAC du 2 ysin u 20 sin fAC df 2 ,
sin u 10 qsin u 20 . cos fAC b
½
h DC 0 h cr Ž c . ,
W120 g 1 qW210 g 2
v
W210 Ž W120 qW210 .
v 2 q Ž W120 qW210 .
2
y P10 Ž t . q
v W210 g 2 v 2 q Ž W120 qW210 .
2
5
,
h DC - h cr Ž c . , h DC 0 h cr Ž c . ,
0,
Ž 12 . where h cr Ž c . s Ž sin P10
Ž t.
2r3
s P1eq q
c q cos
Ž
2r3
P1 i y P1eq
c.
y3 r2
and
. exp Ž ytrt .
Ž 13 .
is the occupation probability of the 1-st energy minimum under the action of the DC field. P1eq s y1 is the W210 r Ž W120 q W210 . is the equilibrium probability of the first energy minimum, t s Ž W120 q W210 . relaxation time and P1 i is the initial probability at t s 0. Like for the Field–Cooled magnetization process w14x, one may consider: P10 s
½
P1eq Ž h cr . ,
h DC - h cr ,
P1eq
h DC 0 h cr ,
Ž h DC . ,
Ž 14 .
where the critical field h cr is the solution of the blocking condition:
t s Ž W120 q W210 .
y1
st0 ;
Ž 15 .
t 0 f 100 s being a typical measurement time. The choice of t 0 is not critical, since the influence of the thermal relaxation on the TS occurs on a much lower time scale Žt 0 ( 1rv .. Usually, the ‘period’ of the DC field is much higher than that of the AC field, such that the influence of the thermal relaxation on the TS comes from the particles which are superparamagnetic on the time scale of a typical DC experiment. For illustrating the behaviour of the real and imaginary parts of the TS in relation to the easy axis orientation, in Figs. 3–6 are plotted the curves xr Ž h DC . , x i Ž h DC . , for different values of c . The parameter values used in the simulations are: K s 2 10 6 ergrcm3, D s 45 nm Ž V s p D 3r6., T s 300 K, f s 10 3 Hz, fAC s 0.
Fig. 3. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS as a function of h DC s H DC r HK for c s908.
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Fig. 4. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s89.98.
Fig. 5. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s89.88.
Fig. 6. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS for c s808 Žcurve 4., 608 Žcurve 3., 408 Žcurve 2. and 208 Žcurve 1..
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According to Figs. 3–6 when c decreases from 908, both, the real and imaginary parts of the TS undergo a very abrupt decrease. In a sharp interval around c s 908, xr Ž h DC . presents two kinks, one of them corresponding to the critical field h cr Ž c . , and the other, occurring at a lower value of h DC , being due to the thermal relaxation on the time scale of the AC field period. These kinks rapidly vanish with decreasing c , and xr Ž h DC . starts to present a flat maximum shifting towards h DC s 0. For c s 08, the TS is zero because the first harmonic of the AC transverse magnetization vanishes. Thus, the TS is dominated by the behaviour of the particles having the easy axis perpendicular to the DC field Ž c s 908.. The particle easy axis distribution in the plane Ž H DC , HAC . could be found by rotating the sample in this plane and noticing the dependence of the TS peak magnitude on the sample rotation angle. Either of the peaks, occurring in the real and imaginary components of the TS, could be used for this purpose. Although the use of the imaginary peak would result in a higher resolution Žaccording to Figs. 3–6 the imaginary peak is due to the particles having the easy axis oriented in a very narrow angular interval around c s 908. it has the disadvantage of having a lower magnitude than the real peak Žabout 1:10 as it will be shown in the following, the same order of magnitude as for the classical AC susceptibility w12x.. The position of the TS peaks should not depend on the angle of rotation, unless the particle easy axis, volume and anisotropy field distributions are somehow connected, e.g. for ferrofluids frozen in a magnetic field. In the following, we shall restrict our attention to the case c s 908, the only orientation of the particle easy axis for which the thermal relaxation effects on the TS are important. In Figs. 7 and 8 and Figs. 9 and 10 are presented xr Ž h DC . , x i Ž h DC . , and also the phase lag w Žtg w s x irxr . of the first harmonic of the induced magnetization along the AC field with respect to the AC field Žwhich should be measured, together with the total susceptibility x s xr2 q x i2 in order to find the real and imaginary parts of the TS. for c s 908 and different values of the particle volume, anisotropy and AC field frequency. The parameter values used in the simulations presented in Figs. 7 and 8 are f s 10 3 Hz, T s 300 K, fAC s 08, and for Figs. 9 and 10, K s 2 10 6 ergrcm3. One notices that either an increase of the particle volume, anisotropy or AC field frequency Žby several orders of magnitude. sharpen the TS maxima and shift them to higher values of h DC . Thus, both xr Ž h DC . and x i Ž h DC . are sensitive to the particle volume, indicating that the TS could be used for measuring the particle size distribution for recording media. On the basis of the above considerations it becomes obvious that the mechanism leading to the occurrence of a complex TS is basically the same as that leading to the complex AC susceptibility w13x. The only difference is that in a transverse field experiment, the particle relaxation time distribution is shifted towards lower values with increasing H DC Žwhich shifts the energy barrier distribution towards lower values. while in a classical AC susceptibility experiment, the same shift is due to increasing temperature.
(
Fig. 7. The real xr Žsolid line. and imaginary x i Ždashed line. parts of the TS as a function of h DC for two values of the particle diameter D: 40 nm Žcurves 1., 55 nm Žcurves 2. and two values of particle anisotropy K : 10 6 ergrcm3 Žcurves A. and 2 10 6 ergrcm3 Žcurves B..
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Fig. 8. The corresponding phase lag of the transverse magnetization with respect to the AC field Žtg w s x i r xr ..
As we have shown above, the TS is dominated by the behaviour of the particles having the easy axis oriented perpendicular to the DC field direction Ž c s 908.. For these particles, Eqs. Ž11. and Ž12. have the following expressions:
°8W b 2
x s~ r
1
1 y h2DC 4W 2 q v
cos 2fAC y 2
h2DC h2DC y 1
cos 2fAC q sin2fAC ,
2
¢h y 1 cos f , °y4Wvb 1 y h ~ xs ¢0 , 4W q v
h DC - 1 ,
Ž 16 . h DC 0 1 ,
AC
DC
2 DC
i
2
2
cos 2fAC ,
h DC - 1 ,
Ž 17 .
h DC 0 1 , 2
where W s f 0 exp yb Ž 1 y h DC . . In order to extend the calculus for a particle system having volume and
Fig. 9. The real Žsolid line. and imaginary Ždashed line. parts of the TS as a function of h DC for two values of the particle diameter D: 40 nm Žcurves 1. and 55 nm Žcurves 2. and two values of the AC field frequency f : 10 3 Hz Žcurves A. and 10 5 Hz Žcurves B..
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Fig. 10. The corresponding phase lag of the transverse magnetization with respect to the AC field.
anisotropy field distributions, the singularity occurring in Eq. Ž16. for h DC s 1 could be approximated by the Dirac function:
°8W b 1 y h cos f ~ xs 4W q v ¢dŽ h y 1. cos f , 2 DC
2
2
2
2
AC q sin
2
fAC ,
h DC - 1 ,
Ž 18 .
r
2
DC
h DC 0 1 .
AC
This approximation is not able to interpret the dependency w Ž h DC . , which is very sensitive to the second term in Eq. Ž16. for the case h DC - 1 Žespecially in the range of fields lower than the critical field for which vr Ž 2W . s 1.. However, as it will appear from the numerical results below, the influence of this approximation on the real part of the TS is very small comparatively to the superparamagnetic contribution given by the first term in Eq. Ž16. for h DC - 1. It is worthwhile mentioning that the real part of the TS could be written in a more simplified form using the concept of critical volume Vcr Ž h DC , v . s lnŽ2 f 0rv . k B Trw K Ž1 y h DC . 2 x, defined in the same manner as in Ref. w13x for the classical AC susceptibility. Eqs. Ž17. and Ž18. could easily be used for calculating the TS of a particle system with volume and anisotropy field distributions:
½
2
xr 0 s cos fAC h DC 0 G Ž h DC 0 . q q sin2fAC
x i0 s y
1 V
`
1 V
`
`
2
H0 H0 8W b
1 y Ž h DC 0rh K . 4W 2 q v 2
2
VF Ž V . dV G Ž h K . dh K
5
`
H0
Ž 1rh K . G Ž h K . dh K ,
`
H0 H0 4Wvb
1 y Ž h DC 0rh K . 4W 2 q v 2
Ž 19 . 2
VF Ž V . dV G Ž h K . dh K ,
Ž 20 .
where x 0 s mrh AC 0 , h AC 0 s HAC rHK 0 , h DC 0 s H DC rHK 0 , HK 0 being the most probable value of the anisotropy field distribution. G Ž h K . is the reduced anisotropy field distribution Ž H0` G Ž h K . dh K s 1. and F Ž V . is the particle volume distribution Ž H0`F Ž V . dV s 1.. In Figs. 11 and 12 we present the dependencies of the TS on the anisotropy field distribution ŽFig. 11. and on the particle volume distribution ŽFig. 12.. We have assumed a lognormal f Ž x . s 1rŽ'2p x s .exp ywln Ž x . y s 2 x 2rŽ2 s 2 .4 for both distributions, having the most probable values HK 0 , V0 and standard deviations sK , s V .
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Fig. 11. The real and imaginary parts of the TS for Ds 45 nm, K 0 s 2 10 6 ergrcm3 , MS s 300 emurcm3 , f s10 3 Hz, T s 300 K, fAC s 08, sK s 0.05 and s V s 0.01 ŽB., 0.3 Žv . and 0.6 Žq..
The flattening of the particle volume distribution gives rise to an increase of both, real and imaginary components of the TS, especially due to the larger volumes, at the queue of the particle volume distribution, which also shift both maxima towards the anisotropy field, as we have shown above. This effect could also explain the experimental results of Richter w8x, a decrease of the particle anisotropy giving rise to a decrease of the DC field corresponding to the anisotropy peak. As one can notice in Figs. 11 and 12, the anisotropy peak representing a singularity for an identical particle system, has a negligible effect on the TS of a particle system with volume and anisotropy field distributions, and becomes evident only for very narrow distributions, unlikely to those experimentally encountered w18,19x. Thus, the so-called anisotropy peak, is not directly related to the particle anisotropy field distribution, as previously stated on the basis of the Stoner–Wohlfarth theory w1x, w2–11x, being strongly influenced by the particle volume distribution. The real and imaginary components of the TS could be measured using a similar experimental setup to that used for the classical AC susceptibility w12x, which basically consists of two balanced signal coils and a lock-in detection scheme. In order to estimate the range of the induced voltage coming from the sample, let us consider a magnetic thin film having MS s 300 emurcm3 and K s 2 10 6 ergrcm3 , a surface of 1r1 cm and a thickness of 200 nm. If the anisotropy is in plane, a pair of rectangular detection coils, having the surface 1r0.5 cm, 1 cm 0 length and 1000 turns would be suitable. A very simple calculus shows that for HAC s 10 Oe, f s 10 4 Hz and a
Fig. 12. The real and imaginary parts of the TS for s V s 0.3 and sK s 0.01 ŽB., 0.05 Žv . and 0.1 Žq..
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perfect alignment of the particle easy axis, a reduced TS in the range x s 50 % 100 Žtypical values for the maximum of the TS. gives rise to a differential voltage in the range Ee f f s 1.3 % 2.5 mV. If the sample anisotropy is perpendicular, for the same sample shape and detection coils having the shape of rectangular plates having the surface 1r1 cm, 0.5 cm high and 1000 turns, Ee f f s 3.5 % 7 mV. Even if these values are expected to be attenuated by the dispersion of the easy axis orientation, always present within a real sample, they could still be measured using a lock-in detection scheme. For broad particle anisotropy and volume distributions, their deconvolution could be a difficult task unless complementary experiments are available. Different attempts to find the anisotropy field distribution for recording media, especially based on the remanence curves w16x, are proposed in the literature. Usually, the result is a convolution of the anisotropy and particle volume distributions, and a correction for the relaxation effects, requiring some knowledge about the particle volume distribution, is necessary. Among these attempts, the method proposed by Flanders and Shtrikman w20x and later on used by Pfeiffer et al. w17x- w19x for barium hexaferrite powders, has a remarkable resemblance to the TS. The later method relies on the fact that once the sample saturated in a certain direction, the application followed by the suppression of a progressively increasing sequence of applied fields along a direction performing a very small angle with respect to the previous saturation direction results in the switching of the particles having the easy axis oriented in the interval between the perpendiculars to the previous saturation and current applied field directions. The derivative of the remanence curve measured along the perpendicular to the previous saturation direction, corrected for the relaxation effects, gives the anisotropy field distribution. Just like the TS, the later method is capable to deconvolute the easy axis distribution Žby successively rotating the saturating field and measuring the remanence along the perpendicular to the previous saturation direction one can explore the whole angular interval of orientation of the particle easy axis.. Also, both methods involve the same particles, having the easy axis perpendicular to the applied field direction, for which the coherent rotation mode holds w20x. Still, the complex TS offers clear advantages with respect to the methods based on the remanence curves, like a well known time window for the relaxation effects Žt s 1rv . which could also be varied in a quite large range by varying the frequency of the AC field and a very simple procedure for measuring the particle easy axis distribution. 3. Conclusions During a TS experiment, the increasing DC field shifts the particle relaxation time towards lower values, making possible the occurrence of the thermal relaxation on the time scale of the AC field period, for frequencies of the later in the kHz range. For an identical particle system having the easy axis perpendicular to the DC field and in the domain of applied DC fields studied in this paper, the present theories predict the occurrence of only one peak at H DC s HK . We have demonstrated that the thermal relaxation gives rise to a secondary peak in the real part of the TS and to a peak in the imaginary part of it. For a particle system with volume and anisotropy field distributions, only one finite peak is present in the real part due, on one hand, to the anisotropy field distribution which tends to flatten the discontinuity occurring at H DC s HK for an identical particle system, and on the other, to the particle volume distribution which can induce a high peak via the contribution of the particles which become superparamagnetic at DC fields lower then HK . Usually, the latter peak is dominant in a TS experiment. The thermal relaxation influences the TS Õia the particles having the easy axis perpendicular to the DC field. The complex TS could be used as an alternative to the standard AC susceptibility measurements w12,13x. The disadvantages of the later method when applied to recording media are: Ø the negative effects of the temperature variation on the material structure of the sample Že.g. magnetic tapes.; Ø the implication of the usually unknown temperature variation laws of the particle anisotropy and spontaneous magnetization.
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The complex TS method has the following advantages: Ø uses a DC field for shifting the relaxation time distribution Žat constant temperature.; Ø only the particles having the easy axis oriented at 908 with respect to the DC field, which essentially behave in agreement to the coherent rotations model w20x Ženabling the calculus of the energy barriers and magnetization., contribute to the TS. Under these circumstances, the particle activation volume in a TS experiment is expected to be close to the physical volume, which makes this method very efficient for measuring the particle size distribution in recording media. For highly anisotropic media, particle interaction effects are not expected to interfere with the above mentioned effect for DC fields close to HK w7x, which gives the order of magnitude of the fields corresponding to the real and imaginary peaks of the TS.
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