Determination of single particle coercivity from rotational hysteresis curves

ELSEVIER

~H ~1 i ~i

Journalof magnetism and magnetic materials

Journal of Magnetism and Magnetic Materials 196-197 11999) 794 795

Determination of single particle coercivity from rotational hysteresis curves K. Elk* Department qfPhy.s'ics. Technical Univer.siO'Dresden. D-01062 Dresden, Ge#v~anv

Abstract

The magnetization behaviour of polycrystalline samples is studied by applying rotating magnetic fields. Measurable values are the torque and the remanence angle_ If the field rotates perpendicularly to the easy direction of the sample, the magnetic remanence can be calculated analytically. This allows to determine the magnetic alignment of the sample and the single-particle coercivity without any model assumptions. 1999 Elsevier Science B.V. All rights reserved.

Kevwords: Permanent magnets: Measuring methods: Magnetic alignment: Switching field

To get information on the magnetization processes in permanent magnetic materials, investigations of polycrystalline samples can be used if the relation between the microscopic behaviour of the grains and the measurable macroscopic properties of the sample is known. As an alternative to c o m m o n studies of the dependence of the sarnple magnetization M on the strength H of an applied magnetic field H the dependence on the direction of H can be investigated. Thereby a field of constant amount H rotates with angle # around the sample, and the torque

D=MxH,

IDI = M ~ " t f ,

(1)

and the remanence angle 7 being defined by the angle between the projection o f M R in the rotational plane and the starting position of the rotating field H (corresponding to # = 0), i.e. ;' = [4 - tan

1 (MR ~lf'MR) " I1'

(2)

can be measured as functions of H and fi, In Eqs. (1) and (2) M~_rr and M~ eft mean the components of M and M ~, resp., perpendicular to H and the rotational axis of the field simultaneously, whereas M~ is the component of M R parallel to H [1]. * Fax: + 351-463-7060: e-n'mil: [email protected].

To carry out computer simnlations a combined model is used consisting of rotational processes with the singlegrain coercivity [2] HR(,g} = HA(COS2'3 ,9-} sin 2 ~ <9) 3e

(31

with H.x = 2 K l/Ms (K~ is tile anisotropy energy for tile assumed uniaxial anisotropy and Ms is the saturation magnetization of the grains, whereas 3 describes the angle between the applied field H a n d the crystal axis c of tile grain) and pinning processes described by [3]

HR(Ot = HC,?/Icos 0 I.

t4)

where H ° is an empirical hardness parameter. The resulting switching field of that model is the minimum value of Eqs. (3) and (4). If all fields and magnetizations are measured related to HA and Ms, resp., this model contains only one parameter h ° = H~,Ha. For polycrystalline samples a texture function is needed to describe the magnetic aligmnent of the grains, For simplicity a function with only one texture parameter n is used in the forin .ln(q~) = (2n -- 1)COS2" Y.

(5}

with c~ being the angle between the texture axis t of tile sample (easy directionl and the crystal axes c of the grains. (n = 0 in Eq. (5) corresponds to the isotropic case). Then M, M R and ~,,can be calculated in dependence on h~L

0304-8853/99/$ - see front matter ~ 1999 Elsevier Science B.V. All rights reserved. PIl: S 0 3 0 4 - 8 8 53(9 8 ) 0 0 9 3 2 - 9

K. Elk/Journal of Magnetism and Magnetic Materials 196-197 (1999) 794-795

/

(a) 270-

/

"~180 "0 p-.

0

0 ,o

~+o

O(de:gi

-

(b) 27O"

~180 ©

/,

p90

oo

Herein K is the opening angle of the remagnetization cone, defined by the single-particle coercivity field in the form of the solution of the equation MR(X) = H.

'

~,(/-/) = ~ - t a n - 1

S

(do i

-

'

2+0

'

'

360

Fig. 1. Remanence angle 7(fi) for textured polycrystals with n = 5 and hardness parameter H°/HA = 0.3 for 6 = 0 ° (a) and = 60 ° (b) for a rotating field H/HA = 0.51.

n and the direction of the rotational plane of the applied field H, given by the angle 6 between t and that plane. Some results are presented in Fig. 1. If the rotational plane is chosen prependicular to t the remanence M R and the remanence angle y can be calculated analytically. It results in

• (2n + 1)!!

M~ = Ms (~n + 2)!! sin2n+2 x,

(6)

m Reff = Ms 2 sin 2"+a K cos K,

(7)

7~

=/3 - t a n -

1 ~2 (2n + 2)11

[_~ (~n-n+ 1)!!-"cot x

1

.

(9)

If HR(,9) is a m o n o t o n o u s function, only one solution exists. For non-monotonous HR(,9), however, K means that solution with the maximum value of K. So for every function HR(O) the meaning of K is always well defined and K depends only on the strength H of the applied field H. Eq. (8) contains the macroscopic (texture parameter n) and the microscopic (HR(~9)) properties in the form of a product of two factors, each depending on one of these values only. Therefore, a separation is possible. For a known texture parameter n from Eq. (8) follows

/

360

795

(8)

- tan(/3 - 7) • (2n + 2)!!

(10)

Now, constructing the inverse function to K(H), the switching field HR(,9t is obtained without any model assumptions. Otherwise, if the texture parameter n is unknown, it can be determined by comparison with a sample of the same material with known texture. For instance, if an isotropic sample is available (usually this can be produced in a simple way) it follows tan(/3 - 7n) 1 (2n + 2)!! c~n - tan(/3 - 70) - 2 (2n + 1)!!'

(11)

In order to get the quotient c~, the remanence angles 7, and 7o have to be measured at the same applied field H. On the basis of this method measurements of the magnetic texture of bonded NdzFelgB-magnets were carfled out [-4]. Investigations of HR(,9) are in preparation.

References [1] K. Elk, J. Magn. Magn. Mater. 138 (1994) 339. [-2] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. Roy. Soc. A 240 (1948) 599. [-3] E.J. Kondorsky, Izv. Akad. Nauk SSSR, Ser. Fiz. 16 (1952) 398. [4] E.H. Klaus, C.P. Jambrich, H.R. Kirchmayr, K. Elk, Fifteenth International Workshop on Rare-Earth Magnets and their Applications, Dresden 1998, submitted.