Journal of Magnetism and Magnetic Materials 215}216 (2000) 280}283

Experimental test of NeH el's theory of the Rayleigh rule usinggradually devitri"ed Co-based glass

H.K. Lachowicz*

Institute of Physics, Polish Academy of Sciences and College of Science, Al.Lotniko&w 32/46, 02-668 Warszawa, Poland

This work is dedicated to Professor Louis NeH el

Abstract

It is shown that gradually devitri"ed Co-based nonmagnetostrictive metallic glass is an excellent model material toverify Louis NeH el's theory of the Rayleigh rule. In the course of the calculations, NeH el showed that the parameterp"bH

#/a (where H

#is the coercivity, a and b are the coe$cients of a quadratic polynomial expressing the Rayleigh rule)

is expected to range between 0.6 (hard magnets) and 1.6 (soft). However, the experimental values of this parameter,reported in the literature for a number of mono- and poly-crystalline magnets, are much greater than those expected fromthe theory presented by NeH el (in some cases even by two orders of magnitude). The measurements, performed for a seriesof Co-based metallic glass samples annealed at gradually increasing temperature to produce nanocrystalline structureswith di!erentiated density and size of the crystallites, have shown that the calculated values of the parameter p fall withinthe range expected from NeH el's theory. ( 2000 Elsevier Science B.V. All rights reserved.

PACS: 75.50.Kj; 75.60.Ej

Keywords: Rayleigh rule; Metallic glasses; Nanocrystalline magnets; Coercivity; Magnetization processes

1. Introduction

In 1887, Lord Rayleigh [1] discovered an empiricalrelation which describes the virgin magnetization curvemeasured in the range of weak applied "elds (much lowerthan the coercivity of a given material), now commonlyknown as the Rayleigh rule. This relation is given asa quadratic polynomial of the form

M"aH#bH2, (1)

where M is the magnetization at the applied "eld H, anda, b are the coe$cients characteristic for a given materialexpressing reversible and irreversible changes of magnet-ization, respectively.

In 1938, Kersten [2] had proved Eq. (1). For thispurpose he introduced the formalism of the so-called

*Tel.: #48-22-843-5232; fax: #48-22-843-0926.E-mail address: lacho@ifpan.edu.p1 (H.K. Lachowicz).

`model of potential function <(x)a. This model assumesthat the interaction of a Bloch wall may be described bya conservative potential energy which is a random func-tion <(x) of the coordinate x lying along the direction ofthe wall motion. Later on, NeH el [3,4], using the samemodel and assuming that the distribution of the forcespinning the moving walls follows a Gaussian probabilitylaw, had improved the calculations made by Kersten.Under this assumption, NeH el, combining the calculatedvalues of the coe$cients a, b and of the coercivity, H

#,

had derived a dimensionless parameter p"bH#/a, the

value of which is expected to range between 0.6 (hardmagnetic materials) and 1.6 (soft materials) [5]. Com-parison of the experimental and calculated values of theparameter p is usually considered as a practical numer-ical test for the validity of the model used by NeH el.

However, to the best of our knowledge, the only valueof the parameter p ("1.4) which was in agreement withthe theoretical expectation, was obtained in the experi-ment of Porteseil and Geo!roy on rapidly quenched

0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 1 3 4 - 7

Fig. 1. Annealing temperature dependencies of the coercivityand the parameter p (hatched area shows theoretical limitationof the parameter p).

Co-based amorphous ribbon specimens [6] (even in itsas-quenched state very tiny crystallites of iron are presentin the material which act as the pinning centers).

Considering the work cited above, it may be expectedthat the parameter p, calculated from the data obtainedfor a series of Co-based amorphous ribbon specimensgradually devitri"ed in a controlled way to produce in-tentionally nanocrystalline structure with di!erentiateddensity of these nanograins, should verify NeH el's theoryin a more satisfactory way.

2. Experimental, results and discussion

For the experiment, a rapidly quenched amorphousribbon of nominal composition Co

71Fe

1Mo

1Mn

4Si

14B9

(Vitrovac 6030 produced by Vacuumschmelze,GmbH) was used as the parent material. Extremelylow magnetostriction which this material exhibits((0.2]10~6) was the main reason for the choice. Forsamples showing non-negligible magnetostriction, thecoercivity will be mainly determined by magnetoelasticenergy created by long-range correlated stress "eld.Therefore, reliable determination of the in#uence of thenanocrystallites generated would not have been possible.

The specimens used were 25lm thick, 5mm wide and80mm long. All the samples were annealed for 1 h attemperatures ranging from 693 to 873K. The virginmagnetization curves as well as the hysteresis loops atsaturation have been measured along the sample axis bymeans of a conventional inductive technique in a com-puterized high-sensitivity set-up with a built-in furnace,operating at a frequency 25Hz of the sawtooth magnetiz-ing "eld.

The microstructure of the annealed specimens wasstudied by means of transmission electron microscopy(TEM) (Philips EM-300, operating at 100keV).

Fig. 1 shows the dependence of the coercivity of thesamples as a function of the annealing temperature. In

the same "gure, the dependence of the parameterp"bH

#/a on the temperature is also presented. The

values of this parameter have been computed using thevalues of the coercivity determined from the hysteresisloops and the values of the coe$cients a and b obtainedfrom the best "t (least-squares method) to Eq. (1) of thevirgin magnetization curve. The error in calculations isestimated to be not greater than 20%. As shown in Fig. 1,the calculated values of the parameter p fall within therange predicted by NeH el's theory (0.6(p(1.6). Fig.1 shows that the parameter p varies with the annealingtemperature in a rather erratic fashion. According toNeH el's theory, this parameter should evolve monotoni-cally, smoothly decreasing with the increase of coercivity.However, at higher annealing temperatures the volumedensity of the crystallites formed can be high enough sothat the assumption on non-correlated random energylandscape becomes invalid.

In order to test the accuracy of the "t to Eq. (1), themeasured virgin magnetization curves have also beenapproximated by a cubic polynomial by adding to Eq. (1)a third term cH3. The "t has shown that the coe$cientc is at least three orders of magnitude smaller than thecoe$cients a and b. Since the cubic term re#ects thedeviation from the Rayleigh rule, the above relationbetween the coe$cients a, b and c shows that the mea-sured virgin magnetization curves can be described withsu$cient accuracy by a quadratic polynomial, indicatingthat the Rayleigh rule is indeed satis"ed.

As shown in Fig. 1, an increase in the annealing tem-perature causes a rapid rise in coercivity. In the range ofthese temperatures from 693 to 873K, the coercivityincreases by more than three orders of magnitude. Thise!ect is, however, quite obvious since the higher thetemperature of annealing, the larger is the density of thecrystallites formed, leading to an increase of the pinningforces which retard the wall motion. As seen in Fig. 2,which shows TEM-micrographs obtained for the sam-ples annealed at the temperatures in the range wherea rapid increase of the coercivity is observed, the densityand the mean size of the crystallites increase with thetemperature of annealing, as expected.

An expression for the coercivity in a non-magnetostric-tive, amorphous medium containing randomly distrib-uted, non-interacting nanocrystallites has been derivedby Porteseil and Geo!roy [6] and is given in the form

H#"n1@2d~3@2Kv/12J

4, (2)

where n is the volume density of nanocrystallites, d thewall width, K the anisotropy constant of nanocrystallites,v the mean volume, and J

4the magnetic polarization of

the sample.It can be easily noticed from the above expression that

the coercivity should rise with an increase in the densityand size of the crystallites, assuming that the wall width is

H.K. Lachowicz / Journal of Magnetism and Magnetic Materials 215}216 (2000) 280}283 281

Fig. 2. TEM micrographs obtained for samples: annealed at693K (a), at 708K (b) and at 753K (c). The magni"cation is thesame for all the micrographs.

Fig. 3. Dependencies of the coercivity and the parameter p asa function of the temperature of measurement.

mainly determined by the properties of the amorphousmatrix and therefore, does not change signi"cantly. How-ever, it should be noted that annealing the samples athigher and higher temperatures leads to an increase inthe volume density of the crystallites as well as an in-crease in their mean size. As a consequence, the composi-tion of the matrix undergoes some changes. Similarly, thecomposition of the crystallites can vary since they aremost probably composed of Co-based solid solution (seeRef. [7]). Both these lead to uncontrolled changes in themagnetic properties of both phases. Since NeH el's theoryhas been formulated for a material of well-de"ned prop-erties, it is not surprising that the dependence of theparameter p, obtained for a series of samples graduallydevitri"ed, does not follow precisely the behavior oneexpects theoretically.

The value of the parameter p as well as its changes canbe observed for a nanocrystalline sample for which thevirgin magnetization curve and its coercivity are mea-sured as a function of temperature. Such a procedureshould allow one to avoid eventual changes of the com-position of both phases, amorphous and crystalline, ifonly the temperature of measurement is lower than thetemperature of crystallization. Fig. 3 presents themeasuring temperature dependencies of the coercivity

and the parameter p for the sample annealed at 723K. Asshown in this "gure, the coercivity monotonically de-crease with an increase in the temperature of measure-ment from 303 up to 473K, whereas, the parameterp increases in this temperature range. The coercivityscales with the quotient of anisotropy and magnetization.The anisotropy usually decreases with temperature muchfaster than magnetization, in particular, in the temper-ature range much lower than the Curie point and this factcan be taken into account to interpret the observed dropof coercivity. Fig. 3 also shows that the value of theparameter p increases with a decrease in the coercivity.Such a dependence is, in principle, in agreement withNeH el's theory which anticipates an increase of p witha decrease of coercivity. However, the two-fold variationof this parameter, while the change of coercivity is only25%, is not consistent with the theory. The values ofp give, however, the right order of magnitude.

The experimental results shown above allow one toconsider the series of nanocrystalline magnets as a suit-able model material to verify NeH el's theory of theRayleigh rule. The basic property of the model material isthat the main obstacle for the moving walls is the nanoc-rystallites which in#uence this movement through theiranisotropy energies and that these energies can be treatedas statistically independent random variables. It is notsurprising since the formed nanocrystallites grow ran-domly so that their crystallographic directions are alsorandomly distributed. Owing to the exchange coupling atthe crystal}matrix interfaces, the magnetization of thecrystallites is forced to be aligned along that inside thesurrounding matrix. If there is no direct magnetic inter-action between the crystallites, the orientations of theeasy axes of magnetization are independent and conse-quently, the anisotropy energies are independent vari-ables thus ful"lling the assumptions made by NeH el in histheory.

282 H.K. Lachowicz / Journal of Magnetism and Magnetic Materials 215}216 (2000) 280}283

Acknowledgements

The work was supported in part by the State Commit-tee for Scienti"c Research (Poland) under Grant No.8T11B 048 10.

References

[1] L. Rayleigh, Philos. Mag. 23 (1887) 225.

[2] M. Kersten, Phys.Z. 9 (1938) 860.[3] L. NeH el, Cah. Phys. 12 (1942) 1.[4] L. NeH el, Cah. Phys. 13 (1943) 18.[5] J.L. Porteseil, Phys. Stat. Sol. A 51 (1979) 107.[6] J.L. Porteseil, O. Geo!roy, J. Magn. Magn. Mater. 140}144

(1995) 1855.[7] H.K. Lachowicz, T. Kulik, R. Z0 uberek, L. Mal1 inH ski,

M KuzHminH ski, A. SD lawska-Waniewska, J.S. Mun8 oz, J.Magn. Magn. Mater. 190 (1998) 267.

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