N ELSEVIER Journal of Magnetism and Magnetic Materials 169 (1997) 199 206

Journal of magnetism and magnetic materials

Tensor magnetic hysteresis loops for a ferrite permanent magnet cube

Y. Yu, D.L. A t h e r t o n *

Department (?/'Physics, Queen's University, Kingston, Ont., Canada K7L 3N6

Received 27 August 1996; revised 25 September 1996

Abstract

Tensor hysteresis loops are measured for a 25.4 mm (1 in.) ferrite permanent magnet cube. The Atherton Beattie model is extended to fit the tensor hysteresis loops. Magnetization vector rotation processes are responsible for both low and high magnetization behaviours. Hysteresis comes from both the mean field interaction and the Stoner-Wohlfarth type irreversible magnetization vector rotation. The nonzero off-diagonal tensor elements result from the deviation of the sample easy axis from the cube edge direction. The relative magnitude and direction of the off-diagonal elements uniquely determine both the sample easy axis direction and the distribution of particle easy axis directions.

PACS: 75.30.Gw; 75.50.Ww; 75.60.Ej

Keywords. Hysteresis; Tensor; Mean field; Easy axis; Magnetization rotation

1. Introduction

When magnetic fields are applied to bulk fer- romagnetic materials, the magnetization processes occur through two fundamental mechanisms: do- main wall motion and/or magnetization rotation [1, 2]. Domain wall motion occurs at low magnet- ization while magnetization rotation dominates at high magnetization, which can be identified from initial magnetization curves or hysteresis loops.

*Corresponding author. Tel.: + 1 (613) 545-2701; fax: + 1 (613) 545-6463.

When a ferromagnet undergoes magnetization pro- cesses from low to high magnetization, there exists a transition field near which the primary magnetiz- ation mechanism switches between wall motion and magnetization rotation, as evidenced by tensor hysteresis loop measurements [3].

Typically, both wall motion and magnetization vector rotation are involved in the magnetization processes of bulk ferromagnetic materials. Because of the complexity of hysteresis phenomenon, no single model can satisfactorily describe every hysteresis phenomenon observed so far. The Stoner Wohlfarth (SW) model describes the mag- netization behaviour of non-interacting single do- main particles where coherent rotation is solely

0304-8853//97/'$17.00 (75 1997 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 6 ) 0 0 7 1 8 - 4

200 Y. Yu, D.L. Atherton / Journal of Magnetism and Magnetic Materials 169 (1997) 199 206

responsible for the magnetization processes [4] and hysteresis comes from irreversible rotation (switch- ing) due to the uniaxial anisotropy. The Ather- ton Beattie model (AB), which is an extension of SW model, takes into account the mean field inter- action which also contributes to hysteresis. The Jiles Atherton (JA) model applies to isotropic sys- tems with domain wall motion as the major mag- netization processes [5]. In the JA model, domain wall bending results in reversible processes and domain displacement due to unpinning and mean field interaction explain irreversible processes.

It has been shown that tensor hysteresis loops are useful for understanding magnetization pro- cesses as well as the hysteretic behaviour of ferromagnetic materials [3]. In order to study mag- netization vector rotation processes, we measured tensor hysteresis loops for a ferrite permanent mag- net sample in which magnetization rotation domin- ates throughout the entire magnetization processes. We extended the AB model to explain the tensor hysteresis loops.

1t

Sample essy /

~ artiele easy axis

-i,,i /,, ,,

- /

Y

Fig. 1. Definitions of angles for magnetic field applied in the z direction.

2. Experiment

A 25.4 mm (1 in.) Indiana lndox 5 ferrite perma- nent magnet cube cut previously [6] was used in this experiment. Small perpendicular grooves were milled across the centre on each of the six surfaces of the cube to contain 20 turn flux coils. The experi- mental apparatus and measurement technique have been described previously [3]. The hysteresis loop measurements start from high field in order to minimize the effect of the initial magnetization state of the sample.

Indox 5 and Ferrite 5 are similar permanent magnets whose nominal metallurgical composition is BaO- 6Fe203 [7]. They have a normal coercivity about 2200 Oe [7] and saturation magnetization of about 3.8 x 105 A/m [8].

3. Theory

The cube sample can be treated as an ensemble of interacting particles. Each grain is assumed to be a single domain uniaxial particle with its easy axis

distributed about the sample's easy axis. The inter- actions among particles are simplified as mean field interactions [6]. The AB model calculates the single hysteresis loop for the field applied in the magnetized direction of the permanent magnet un- der the assumption that the sample easy axis is in the cube edge direction. In this paper, we extend the AB model to the tensor hysteresis loop case where all nine hysteresis loops are calculated.

The energy density for such particles is given by

E = Ku sin2(0 - q) - yoMs(H + aM) cos 0, (1)

where K, is the uniaxial anisotropy constant, Ms is the saturation magnetization of the particle, H is the applied field, M is the total magnetization of the sample along the applied field direction, c~ is mean field interaction constant, r/ is the angle between applied field direction and particle easy axis, and 0 is the angle between applied field direction and the magnetization direction, as shown in Fig. 1.

For simplicity, Eq. (1) can be rewritten as

e = sin2(0 -- I'/) -- (h + am) cos 0, (2)

E Yu, D.L. Atherton /Journal of Magnetism and Magnetic Materials 169 (1997) 199-206 201

where e = E/Ku, h = H/H~, H , = K . /poM~, m = M/M~, and e = ~#oMZ/Ku . The equil ibrium posi- t ion 0o of the magnet iza t ion vector is given by

~e - - = sin 2(0 - r/) + {h + ~m) sin 0 = 0, (3) ~0

and

~2 e

~0 2 - 2 cos 2(0 - r/) + {h + xm) cos 0 > 0. (4)

Upon solving these equations, one obtains

0o = 0o(r/, h + ~m). (5)

Assuming that the applied field is in the z direc- tion, then for a single particle, the magnet iza t ion componen t s along the x, y, and z directions are given by

M~x = M~ sin 0o cos {b o, (6)

M z y = M~ sin 0o sin q~o, (7)

M=z = M~ cos 0o. (8)

In the symmetr ic case, the particle or ienta t ion dis- t r ibut ion is given byJ(O), where ~ is the angle that the particle easy axis at (r/, ~) makes with the sample easy axis at (7, ~), as shown in Fig. 1. The math- ematical expression for ~ is given by

where rn=x = M,x /M~, m=r = M~r/M~, m~z = M~z/M~. The mean field term ~rn~z is taken as the same for all componen t s because the diagonal ele- ment rn~z is typically at least an order of magni tude larger than the off-diagonal ones [3] and the total magnet iza t ion vector is a lmost parallel to the ap- plied field direction.

For field applied in the x or y direction, the same formulae can be used except that the sample easy axis direction has to be t ransformed accordingly. Specifically, if we denote the sample easy axis direc- t ion as (7=, 6=) when the magnet ic field is applied in the z direction, then the corresponding sample easy axis directions for field applied in the x and y direc- tions are given by, respectively

cos 7~ = sin 7: cos 6., (13)

sin 7z sin 6= cos 6~ = (14)

x/1 - sin27z COS2~/

and

cos 7}, = sin 7~ sin ~ , (15)

COS )~z 6y = (16) COS

x/1 -- sinZTz sin26/

The hysteresis loops calculated using Eqs. (10)-(12) can then be t ransformed as follows. For field ap- plied in the x direction,

cos ~9 = cos r/cos 7 + sin i /sin 7 cos(~ - 6). (9) m~z ~ mxx, mzx ~ mxr, m.r ~ mxz. (17)

For field applied in the z direction, the magnet iz- at ion componen t s are given by

t 2 ~ I ~/2 mzx = J( O ) sin Oo( r/, h + Emzz)

do do

x cos qSo sin r/dr/d~, (10)

i 2~ ( ~/2 m=r = riO) sin 0o(17, h +~_mzz)

do do

x sin {bo sin r/dr/d#, (11)

mzz = J(0) cos Oo(r/, h +~m=z) sin r/dr/d#, do do

(12)

For field applied in the y direction,

m z v ~ m y x , m = z ~ m y r , m z x ~ m y z . (18)

It is expected f rom the above analysis that a ran- domly oriented sample will have zero off-diagonal elements. On the other hand, if a sample is perfectly oriented or has a symmetr ic distr ibution abou t a preferred direction, the off-diagonal elements will also be zero when the applied field is parallel or perpendicular to the preferred orientat ion. Fo r all o ther cases, there exist nonzero off-diagonal ele- ments. Consequent ly, the nine componen t s of the tensor are correlated with each other through the sample easy axis and the or ienta t ion distr ibution of particle easy axes.

202 E Yu, D.L. Atherton /'Journal qf Magnetism and Magnetic Materials 169 (1997) 199 206

4. R e s u l t s

4.1. Measured data

Fig. 2 shows the measured tensor hysteresis loops for the ferrite permanent magnet. The H field values are not calibrated. The diagonal elements, xX and yY and zZ, are characteristic of magneti- zation rotation processes with the applied field di-

rection nearly parallel or perpendicular to the easy axis of the sample. The off-diagonal elements, al- though about 20 times smaller than the diagonal elements, are not negligible. Again these loops are typical of magnetization rotation. This justifies the use of a SW model in describing these data. Based on the SW model, one can tell the approximate easy axis direction of the cube sample by analysis of the shape, relative magnitude and sign of all nine

2.0 2.0

0.0

- 2 . (

-1000.0

' I '

J xX

I I I

0.0 1000.0

0.0

- 2 . 0 i

-1000.0

I °

xY

I ,

0.0 1000.0

2.0 ' I '

0 , 0

xZ

-2.0 I I -1000.0 0.0 1000.0

2.0 i I

0.0

y X

-2.0 I I J

-1000.0 0.0 1000.0

2.0

0.0

- 2 . £

-1000.0

' I '

/ yY

I I i

0.0 1000.0

2.0

0.0

' I '

yZ

-2.0 I I i

-1000.0 0.0 1000.0

2.0 ' I '

0.0

zX

-2.0 I I -1000.0 0.0

2.0 2.0

0.0

' I '

0.0

zY

-2.0 I I I -2.0

1000.0 -1000.0 0.0 1000.0 -1000.0

/ zZ

, I i

0.0 1000.0

Fig. 2. Measured tensor hysteresis loops for the Indox 5 cube sample. Figures xX, x Y, and x Z represent magnetic field in the x direction and magnetization components along the x, y, and z directions, respectively, and so on. In each ligure, the horizontal axis is the magnetic field H (kA/m) and the vertical axis the magnetic flux density B (T). For off-diagonal elements, B is multiplied by 20.

E Yu, D.L. Atherton ,/Journal Of Magnetism and Magnetic' Materials 169 (1997) 199 206 203

hysteres is loops . It can be d e d u c e d tha t the easy

axis of the cube s a m p l e is loca ted c lose to the z axis

in the s econd oc tan t . O u r ca l cu l a t ed d a t a in the

next sec t ion will give the d i r ec t ion of the s ample

easy axis.

A n o t h e r fea ture r evea led by Fig. 2 is tha t the

degree o f hysteres is is different f rom one t enso r

e l emen t to the o ther . T h e o b s e r v e d hysteres is for

the cube s ample d e p e n d s on the d i r ec t i on of the

app l i ed field a n d the m a g n e t i z a t i o n c o m p o n e n t

2.0 ' I ' 2.0

i J i

0.0 0.0

I 2.0 ' I '

0.0

xY

-2.0 -2.0 I J -2.0

-1000.0 0.0 I000.0 -1000.0 0.0 1000.0 -1000.0

xZ

i [ J

0.0 1000.0

2.0 , 2.0 2.0

0.0

I

0.0 0.0

] I

yX

-2.0 i I i -2.0 i t t -2.0

-1000.0 0.0 1000.0 -1000.0 0.0 1000.0 -1000.0

, [ ,

yZ

i I J

0.0 1000.0

2.0 2.0 2,0 ' I '

0.0 0.0 . r ~ 0.0

zX zY

- 2 . 0 t b I - 2 . 0 L I , - 2 . 0

-1000.0 0.0 1000.0 -1000.0 0.0 1000.0 -1000.0

' I '

0.0 1000.0

Fig. 3. Calculated tensor hysteresis loops for the Indox 5 cube sample. The solid and dashed curves are calculated from the SW model with and without mean field interactions, respectively. For field applied in the x direction, the magnetization components along the x, y, and z directions are represented by xX, xY, and xZ, respectively, and so on. In each figure, the horizontal axis is the magnetic field H (kA/m) and the vertical axis is the magnetic flux density B (T). For off-diagonal elements, B is multiplied by 20.

204 E Yu, D.L. Atherton / Journal o f Magnetism and Magnetic Materials 169 (1997) 199 206

measured. This phenomenon was also observed in steel cube samples [-3].

4.2. Calculated data from mean field Stone> Wohl farth model

Before calculating tensor hysteresis loops, we must first solve Eq. (3) to find the equilibrium posi- tion of the magnetization vector. The root of Eq. (3) is determined numerically by the bisection method [-9] and so is the root, re=z, of Eq. (12). With 00(q, h + _~m=z) and m=z solved, m=x and m~r can then be calculated from Eqs. (10) and (11).

The particle easy axis orientation distribution function used in our calculation is assumed to be symmetric about the sample easy axis, in the form of

f (~) = N e x p ( - ~/a), (19)

where ~ is defined by Eq. (9), a is the effective distribution width, and N a normalization con- stant. An exponential type of distribution function is chosen, due to its sharp drop at small angles, which describes very well a highly oriented sample, such as the Indox 5 sample studied here.

Fig. 3 shows the calculated tensor hysteresis loops using Eqs. (10)-(12). The solid curves in Fig. 3 reproduce the measured data very well. The param- eters used in the calculation are as follows: poM~=0.478T, K u = 1 .5x105J /m 3, : t=0.165, and a = 10 ° with sample easy axis located at (1.7 °, 150°). As a comparison, the tensor loops for the SW model without mean field interaction are cal- culated, as shown by dashed curves in Fig. 3. The parameters for the dashed curves are identical to the solid curves except ~ = 0. It is clear that the mean field SW model describes the measured data better than the generic SW model.

Fig. 4 shows the exponential distribution func- tion with a = 10 °, which indicates that the particle easy axes are distributed primarily within a conical region of about 60 ° from the sample easy axis. The small deviation ( < 2 °) of the sample easy axis from the cube edge results in significant off-diagonal elements. Thus tensor hysteresis loop measure- ments provide a sensitive method of determining the easy axis direction of a cube sample.

1.0 I I

. . . . . . o=5 O

o : 10 0 ',\ - - - - o= 1~0

i '-

> 0.5 •

0.0 - ~ . . . . . . 0.0 30.0 60.0 90.0

Angle u/(deg.)

Fig. 4. Exponential distribution function for different distribu- tion widths.

The mean field interaction constant, c~, used in the calculation is about two orders of magnitude larger than that used previously by Atherton and Beattie [-6]. The effect of :~ on zZ hysteresis loops is shown in Fig. 5 with all the other parameters being the same as those used in Fig. 3. The larger the

value, the narrower is the switching field distribu- tion. The experimental data in Fig. 2 indicate a nar- row switching field distribution, which necessitates the use of large ~ value. The mean field parameter, :~, originates primarily from the magnetic interac- tion of individual particles with the rest of the sample. Positive values of :~ indicate that mean field interaction favours a sample being magnetized while negative values of ~ indicate mean field intera- ction favours a sample being demagnetized. The effect of the particle distribution width, a, on hys- teresis loops is different from that of the mean field interaction constant, ~, in that the coercivity of the sample decreases with decreasing alignment of par- ticle easy axes (i.e., increasing a), while the coerci- vity increases with the increase of mean field interaction constant, :~, as shown in Figs. 6 and 5, respectively.

It is interesting to compare the coercivity of a single SW particle or a sample consisting of a

Y. Yu, D.L. Atherton ,/Journal of Magnetism and Magnetic Materials 169 (1997) 199 206 205

2.0

1.0

0.0

-1 .0

-2.C - 1 0 0 0 . 0

. . . . aff i 0 . 0 1 6 5

a = O, 1 6 5

. . . . . . . . . . . a = 1 6 5 7

I

/ ( ..

I . .J"

/ i

- 5 0 0 . 0 0 .0 5 0 0 . 0 1 0 0 0 . 0

H (kA/m)

Fig. 5. Effect of mean field constant, ~, on zZ hysteresis loops.

2.0

1.0

0.0

-1.0

-2.0 -1000.0

. . . . . . . o" = 5 0 / . j -

c = 1 0 °

. . . . . . . . . . . . . . o ' = 1 5 0

, I , I , I ,

-500.0 0.0 500.0 1000.0

H (kA/m)

Fig. 6. Effect of distribution width, ~, on zZ hysteresis loops.

collection of noninteracting SW particles with all their easy axes aligned with the applied field with our measured coercivity. The coercivity of the alig- ned SW particles is given by

_ 2K 2.0 x 1.5 x 10 s (J/m3)~ Hc #0Ms - 0.478 (T) = 630 kA/m.

(2o)

However, our calculated and measured coercivities are around 300 kA/m. This reduction is due to the distribution of particle easy axes since particles whose easy axes are not aligned with the applied field direction have a coercivity lower than 630 kA/m. The positive mean field interactions in- crease the coercivity of individual particles as well as that of the sample consisting of these particles, no matter in which direction the magnetic field is applied. Nevertheless, the increase of the sample coercivity due to positive mean field interactions is smaller than the coercivity reduction due to the distribution of particle easy axes, leading to the overall reduction in coercivity.

It should be noted that in our calculation we did not take into account the distribution of other parameters, such as anisotropy constant and satu- ration magnetization. It is assumed that the distri- butions of these parameters are rather narrow and can be ignored to a first order approximation.

5. Conclusions

The Atherton-Beattie model has been extended to calculate tensor hysteresis loops for a ferrite permanent magnet cube sample. The calculated data are in good agreement with the measured data. Tensor hysteresis loop measurement is a sen- sitive method of determining the easy axis of a cu- bic sample of a ferromagnet. The existence of off- diagonal loops is attributed to the small deviation of sample easy axis from the cube edge direction.

Acknowledgements

The authors would like to thank Pat Weyman and Derk Micke for their assistance during experi- ment and sample preparation. The research is sup- ported by Natural Sciences and Engineering Council of Canada, Pipetronix, and the Province of Ontario.

206 Y. Yu, D.L. Atherton / Journal of Magnetism and Magnetic Materials 169 (1997) 199 206

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[5] D.C. Jiles and D.L. Atherton. J. Magn. Magn. Mater. 61 (1986) 48 60.

[6] D.L. Atherton and J.R. Beattie, IEEE Trans. Magn. 26 (1990) 3059.

[7] L.R. Moskowitz, Permanent Magnet Design and Applica- tion Handbook (Cahners Books International, Boston. MA, 1996).

[8] B.D. Cullity, Introduction to Magnetic Materials (Addison- Wesley, New York, 1972).

[9] W.E. Grove. Brief Numerical Methods (Prentice-Hall. En- glewood Cliffs, NJ, 1966).