Journal of Magnetism and Magnetic Materials 92 (1991) 417-423 417 North-Holland

Magnetic viscosity in Ba-ferrite

Franc i sco Carmona , Aure l io Mar t in Blanco a n d Carlos A l e m a n y

Instttuto de Ciencia de Materiales, Serrano 144, 28006 Madrid, Spare

Received 24 April 1990

Results are presented on magnetic viscosity measurements performed at room temperature on Ba-fernte pressed powder samples of different particle sizes. Maximum values of viscosity coefficients (S) occur around coercive fields, except for a sample with an intrinsic coercive force of 4639 Oe, in which the maximum is at a much higher field (5706 Oe). Activation volumes are of the order of 10-17 cm 3 and decrease along with parttcle size (except, again, for the high coercivaty sample) Experimental values of coercive helds and viscosity parameters (kT/q) , used m connection with Gaunt's theory, suggest that the viscosity mechanism in Ba-ferrite, for particle sizes from 7 to 0.7 I~m is the strong pinning of domain walls.

1. Introduction

Magnetic viscosity is a phenomenon which dis- plays the following characteristics: when the ap- plied magnetic field experiences some change, the magnetization of a ferromagnetic (or ferri- magnetic) material is not attained instantly, but is approached asymptotically.

The first rigorous experimental study on mag- netic viscosity was made by Street and Woolley [1], who showed how the magnetization of a sam- ple, after introducing some variation in the mag- netic field, varies with time following a logarith- mic law

M = S log t + constant,

where t is the time that has elapsed since the variation of the magnetic field took place. The constant S (viscosity coefficient) measures the number of irreversible processes contributing ' o the ,n.'...:^~ ~,~,~,~,,,,, of magnetization r-,, tin, and its value is maximum around the coercive field (where the number of irreversible processes involved is greater).

The subject has raised renewed interest due to the recent development of magnetically hard materials [2.3]; but there has been a number of years in which it was almost forgotten, and some

materials seem to have been overlooked. Namely, references to barium ferrite are scarce, if any. In this paper we report on the study of magnetic viscosity in Ba-ferrite. A number of samples were available, all from the same starting material, but subject, in each case, to a different milling time. This implied different size distributions and, pre- sumably, differences in the value of S that seemed worthwhile to be studied.

2. Experimental method

All samples except one came from a single batch of barium ferrite, obtained at 1350°C by standard ceramic techniques [4]. Different por- tions received milling treatments of different dura- tion, as shown in table 1 together with mean particle sizes. Also included for comparison is sample SII, prepared ~Ath Ba-ferrite obtained b~' chemical precipitation techniques, whose magneuc properties have been previously reported [51.

The samples, in the form of pressed pox~der cylinders, were measured at room temperature ,vith a vibrating sample magnetometer, linked to a computerized data acquisition device with which magnetic field and time were also registered. The magnetometer was calibrated with a nickel cylin-

0304-8853/91/$03.50 ~; 1991 - Elsevier Science Pubhshers B V (North-Holland)

418 F. Carmona et aL / Magnetw vtscostty in Ba-fernte

Table 1 Ba-ferrite samples. Mdhng times, parttcle sizes, maxima of viscosity coefficients, magnettc fields at which maxima occur, and intrinsic

coercive forces

Sample # Mdhng ttme Parttcle size Smax Hm~x He, (rain) (l~m) (emu/cm 3) (Oe) (Oe)

Hd 10 7.1 0.376 961 725 Ht 30 4.9 0.516 1091 948 HI 60 3.4 0.631 1327 1260 H3 180 2.3 0.745 1589 1694 HX 600 1.7 0.797 2280 2268 HT 1800 1.45 0.806 2645 2533 HN 5400 1.40 0.669 2792 2602 S|l 0.6~ 0.613 5706 4639

der of the same shape as the samples. Magnetic fields were measured with a calibrated Hall-effect probe. And times were read from a computer time-card to within 1 ms.

Several types of measurements were made: - After "saturat ing" the sample in a field of - 2 5 500 Oe, the field was brought to zero, and increased in the opposite direction at a rate of 5 Oe / s . At regular intervals the field was made to remain constant for 7 rain. Data acquisition took place uninterruptedly throughout. - D e m a g n e t i z i n g curves without interruptions were also traced for each sample.

In fig. 1 we have represented, together, a curve with 7 min interruptions every 150 Oe and the

curve without interruptions for the same sample. As can be seen, they superpose very well. There- fore, every time a stop is made so that viscosity can show off, one starts from the same curve; and when the field is allowed to resume variation one returns to the same curve.

In fig. 2 we have the variation of magnetization with time during one of the stops. We have also represented the logarithmic law that best fits the experimental data. This law was found by search- ing for the value of t o, within 0.1 s, that gave the best fit to

M = S log( t - t o) + constant .

We would thus have overcome the inaccuracy,

xXX

t _ .

O o o I x~

H (Oer sfecl) -29L00

M(au)

o,o, 9 oO° !

ox°×x xx~ t oox

o o i "9 °' o o , ,

0 0 o

O°xxX~ ~o~ 025

000

-025

-050

J i

-3003 -2CO0 -2"/00 -26~00 -25~00 -24'00

Ftg. 1. Two M versus H measurements on the same sample. Circles, slow vanaiton of f~eld wtthout interruptions. Crosses, the same but stopping for 7 rain every 150 Oe.

F. Carmona et al /Magnettc ctscostty m Ba-ferrtte 419

iM(a u } 0 075 [o

0050

0025

O00C

-0 025

= - .

l(s) 29'o0 2~so 3o'oo 3d50 3,oo 3,'~o 32'00

Fig. 2. Variation of magnetic moment during one of the stops of fig. 1. The numerical equatton is a least squares fit to a loganthnuc law.

noted by other authors [2], regarding the precise m instant in which the field stabilizes, but taking for granted that the law of variation were logarithmic. ,~

This is the assumption that we made all along, and, although it does not completely solve the

.3 above mentioned inaccuracy, it is coherent with our method of measurement, and appropriate for the determination and comparison of values of S 2 for different samples and different values of the applied field. .1

In fig. 3 magnetic viscosity is shown to be due to irreversible processes: after maintaining the field stable for 7 rain (fig. 3a), so that aftereffects may .4 take place, variation of the field is allowed to continue, and afterwards inverted in order to re- turn and remain stable for another 7 min. The 3 recoil curve is almost a straight line, and the second stop reveals no aftereffect, showing that the .2

variation observed in the first s t o p was irreversi- ble.

Variations of magnetizations that occur when the field is made to recoil do not depend on having made a stop or not (compare figs. 3a and 3b). Therefore, in order to evaluate the reversible susceptibility, normal demagnetization curves were traced, without interruptions but describing minor

emu)

(a)

0 0

0 0

0

0 0

0 0 J

o oooooO o 8 ® oo oo oO oOoOoOO° °~ ~ o

(b) 0

0 0

0 0

0 0

0 0

0

O .~ 9 0 0 0 0 O0 O0 O0 O0 0 0 0 0 0 ~ 00000

-1200 -1150 -1100 H ( O e ) -~o5o

Fig. 3. (a) Af t e r a stop at - 1 0 5 0 Oe. variation of field ts

resumed, and at - 1 2 0 0 Oe he ld ~s made to recod the - 1 0 5 0

Oe and stop again. N o variation of magne,~c moment ts

observed at thts second stop. (b) Same as (a) but wtthout the first stop at - 1050 Oe

420 F. Carmona et al. /Magnettc vascosity in Ba-femte

0

0

0

0

p o

o ] ° o 0 ° ° ~ !

o o

AH d

O

o

o

0

AM

Fig, 4, Total susceptibdlty is defined, for an interval of 5:50 Oe around point P, as AM/AH. Reversible susceptibility is

assimilated to the slope of the recoil curve.

loops of + 150 Oe every 300 Oe as shown in fig. 4. These loops are very thin and their slopes (that change, though very little, along the demagnetiz- ing curve) were taken as the reversible susceptibil- ity corresponding to their starting points (P in fig. 4). In turn, the total susceptibility was assimilated to the ratio

XTOT = A M / A H ,

where AM is tile variation of magnetization in an interval of AH = + 50 Oe around point P.

3. Results and discuss ion

15

10

S (a u) H o "-1260 Oe

o o

x X X X X

5 o x

x o 9 x x

X

X

O X

X

o O

O

I0'0o ' ' 40'o0 H (Oe.) 2000 3000

Fig. 5. Variation of viscosity coefficient S versus applied field for sample HI. Circles: stops every 300 Oc. Crosses: stops every 150 Oe. The maximum of S is next to the intrinsic

coercive force (1260 Oe).

Coercive force values, maxima of S, and fields corresponding to these maxima are all given in table 1. Although S is sample-shape dependent, this does not happen with the viscosity parameter: k T / q = So/X~ RR (where S O and XI RR correspond to the case of zero demagnetizing factor of the sample), and, according to Street et al. [2]

k r / q = S(1 - DXREv)/(XToT -- XREV), (1)

where D is the demagnetizing factor of the sample and S, XTOT and XRt:.V are measured quantities.

In fig. 5 the variation of S as a function of the applied field is shown for one of the samples. The maximum of S is very near to the intrinsic coer- cive field. This happens for all samples except SII, for which the maximum is at a substantially higher field. This sample consists of particles of opt imum size, and has an intrinsic coercive force that is 32% snort of the theoretically achievable value of 6830 Oe, or, if the coercive force were defined as the field for which XtRR is a maximum, as Givord et al. [3] do, 5709 Oe, only 16% short of the above mentioned.

In fig. 6 we have the central portions of the curves of the seven samples of the series, showing how their maxima increase in magnitude along with the coercive force.

S (a.u.)

3h

.6

4 ~ 3 0 m

lOre 2

I I I 1ooo ?ooo 3000 H(Oe)

Fig 6. Variatlon of S versus applied field for all samples of the ~enes. The mdhng time is noted next to each curve. Only the

central porttons of curves are represented.

F. Carmona et al. / Magnetw wscoslt) m Ba-femte 421

Now, q is the rate of activation energy with H, and the corresponding volume of activation v (de- fined by q = vM~) is given by

v = ( q / k T ) ( k T / M s ) . (2)

Making use of (1) and (2) we have thus been able to trace k T / q versus H and v versus H for our samples. In fig. 7 the family of curves v versus H is represented. They were obtained by fitting quadratic expressions to the experimental points; and they are concave upwards, progressively flatten for decreasing particle sizes, and finally become concave downwards. The separation be- tween the "upward" and "downward" regions would be around the optimum size, or around the most uniform distribution of sizes (or both). The fact is that the curve for sample SII (not shown in the figure), made from particles of opt imum size and narrow distribution of sizes, is almost a hori- zontal line, with

v = ! . 1 X 1 0 -17c m 3 for H = 3600 Oe and

v = l . 0 × 1 0 - l ~ c m 3 for H = 6300 Oe.

The order of magnitude of the activation volumes,

V

2~

20

15

10

05

O0

× 10 '~ (cm 3)

©0

[] A

Fig. 7. Activation volumes versus applied field for all samples of the series. Ctrcles, sample Hd. Squares, Ht. Triangles, HI. Stars, H3. Plus signs, HX. Crosses, HT. *, HN, Continuous

lines are least squares fits to quadratic equauons.

1 0 - n cm-3, would correspond, assuming spheric- ity, to a diameter of 2.7 × 10-6 c m , a value that compares well with an estimate of 1.7 × 10 - 6 c m

for the width of a 180" domain wall in Ba-ferrite (using 8 = 4 ( A . , / K ) 1/2 with A., = 6 x 10 -7 erg cm - t and K ~ 3.3 × l0 t' erg cm-3).

It is surprising that v values for SII are not different from those of conventional samples. According to Wohlfarth [6] v is the particle volume for an isolated single domain particle, or the effec- tive volume covered by a single jump between pinning centers for domain-wall motion processes. And, according to Gaunt [7], v should be, for single domain particles, a non-negligible fraction of the particle volume. Of course, sample SII is not an isolated single domain particle, but some of its properties seem to be largely due to single domain behaviour. One would thus expect v to be somewhat different from the v's of samples in which domain walls are prevailing. But this is not the case, and one is driven to the conclusion that, except for minor differences, all the samples be- have in a very similar way as far as magnetic viscosity is concerned.

It is possible to use o , r experimental data m connection with Gaunt 's r.lodel [7,8]. according to which, if we assume tha, the prevailing mechanism is domain wall pinning, the distmction may be made between "weak" and "strong" pruning.

For weak pinning [7]

k T / q -'- ( H o - Hc ) /25 ,

where H 0 is the field required to unpin the wall if there were no thermal activation: and

H c / H o -" 1 - 25kT /31yb 2,

where ), is the domain-wall energy and 4b the domain-wall width. For strong pinning

k T / q = v~H~/~-( H1o/~-- H2 ,'2 )

and

where f is the maximum restoring force per pro. (The approximation consists in assuming that the activation energies to be considered are ~ 25kT).

422 F. Carmona et aL / Magnettc oiscostty in Ba-femte

Table 2 Ba-ferrite samples. Viscosity parameters (kT/q) for H = He, characteristic constants for weak (3Ivb 2) and strong (fb) pinmng mechamsms, and - assuming strong pruning - force per pin ( f ) and density of pinning sites (p)

Sample = (kT/q)nc 31yb 2 ]b f p (Oe) ( x 10-12 erg) ( x 10- n erg) ( x 10- s dyn) ( x 10 Is cm- s)

Hd 5.10 6.92 1.95 4.5 4.1 Ht 6.35 7.22 2.08 4.8 4.6 HI 7.52 7.97 2.43 5.7 4.3 H3 8.19 9.60 3.24 7.5 3.2 HX 10.14 10.29 3.61 8.4 3.4 HT 11.66 10.03 3.47 8.1 4.1 HN 14.41 8.51 2.69 6.3 7.1 SII 10.10 20.05 9.99 23.2 0.83

These expressions allow us to calculate 31-tb 2, and fb from our experimental values k T / q and H e, as shown in table 2. But if 4b is the wall thickness, b- -4 .3 x 10-Tcm, and if 7 is taken to be 7 =5.6 erg cm -2 ( 7 = 4 ( A . , K ) 1/2 with the above mentioned values for Ax and K), then 317b 2 should be -- 3.2 × 10 - u erg. Furthermore, considering that the wall thickness is the minimum value of 4b [8] we should rather write

317b 2 > 3.2 × 10-11 erg

in contradiction with the values listed in table 2. On the other hand, the condition for strong pin- ning [8]

3f/87r~b > 1

ts sausfied if we use for 7b 2 the above mentioned estimate. Values of the ratio range from 2.3 for

log kT]q

15' 0

10

2'8 2'9 3'0 3~1 3'2 3'3 3'4 3L5 3'6 3'7 tog Hc Fig. 8 Segment of Barbler's plot and log(~T/q) versus log(H~)

of our samples

sample Hd to 4.2 for sample HX (and 11.7 for sample SII). We would thus conclude that, accord- ing to Gaunt 's model, the viscosity mechanism in Ba-ferrite would be mainly governed by the strong pinning of domain walls. Also in table 2 we have f , calculated from fb and our estimate of b, and p, density of pinning sites, given by [81

O = 4"rrTM~Ho/3f 2.

It is interesting to observe that, although there are no great changes in f and p with particle size, the number of particles per unit volume change appre- ciably, and so does the number of pins per par- ticle, that can be estimated at 7.7 x 105 for sample Hd, 8.7 × 103 for sample HX, and 1.3 x 102 for sample SII.

Let us finally point out that, as we she , in fig. 8, the values we have obtained for k T / q fit quite neatly into Barbier's plot [9] which correlates k T / q and H c for a number of very different samples, with H c ranging from 1 0 - 1 0 e to 103 Oe. Sample SII seems not to fit so well, but it does not separate from the straight line more than other materials which were considered in order to trace it [9]. According to Gaunt [7] there is no obvious reason why this relation should work. We agree with that, but let us just note down that it also works for Ba-ferrite.

References

[1] R. Street and J C. Woolley, Proc. Roy. A 62 (1949) 562. [2] R. Street, R K. Day and J.B. Dunlop, J ivlagn. Magn. Mat.

69 (1987) 106.

F Carmona et al. / Magnetac vtscostt.v m Ba-femte 423

[3] D. Givord, P. Tenaud, T. Viadieu and G. Hadjipanayls, J. Appl. Phys. 61 (1987) 3454.

[4] J.G. Zato and A. Martin, Advances in Cerarmcs, vol 15, ed. F.F.Y. Wang (American Ceramic Soc., New York, 1985) p. 513.

[5] A. Martin and F. Carmona, IEEE Trans, Magn. MAG-4 (1968) 259.

[6] E.P. Wohlfarth, J Phys. F 14 (1984) L155. [7] P. Gaunt, J. Appl. Phys. 59 (1986) 4129 [8] P. Gaunt, Phil. Mag. B 48 (1983) 261 [9] J.C. Barbier, Ann. de Phys. 9 (1954) 84.